Development and evaluation of a tumor growth inhibition ...enrigri/Public/PremiGNB/Morten/Tosca_Tesi.pdf · Chapter 1 Introduction The recent advances in biomedical science have raised

Universita degli Studi di Pavia

DIPARTIMENTO DI MATEMATICA

Corso di Laurea in Matematica

Tesi di laurea Magistrale

Development and evaluation of a tumor growthinhibition (TGI) model integrating dynamic

energy budget (DEB) theory

Laureanda:

Elena Maria ToscaRelatore:

Chiar.mo Prof. Paolo Magni

Correlatore:

Chiar.ma Prof.ssa Raffaella Guglielmann

Anno Accademico 2014-2015

Abstract (Italian)

Questa tesi si colloca nell’ambito della modellistica matematica, una com-ponente integrante del processo di ricerca e sviluppo di nuovi farmaci. Ilfocus di questo lavoro consiste nello sviluppo e nella valutazione di un nuovomodello matematico per lo studio dell’efficacia e della tossicita di trattamentiantitumorali in topi xenograft durante la fase preclinica oncologica.

I modelli matematici usualmente utilizzati per descrivere la crescita tu-morale in animali sono a volte criticati in quanto non tengono in consider-azione l’organismo ospitante e la sua interazione con il tumore. Per colmarequesta mancanza e stato proposto un nuovo modello piu complesso e mec-canicistico che consideri esplicitamente le relazioni energetiche tra tumore eorganismo [40]. Tale modello, fondendo i concetti cardine del modello TGISimeoni [37] con quelli della Dynamic Energy Budget (DEB) theory [13],e in grado di spiegare i dati sperimentali relativi alla crescita del tumore edell’organismo ospitante, nel nostro caso topi xenograft, includendo anche iltrattamento farmacologico. In particolare, la struttura del modello consentedi separare l’effetto del farmaco sul tumore da quello sul peso del topo, evi-tando un’errata interpretazione dell’azione specifica del farmaco sul tumore.

La prima parte del lavoro di tesi consiste nell’implementazione, identi-ficazione e validazione di questo nuovo modello realizzata con l’ausilio delsoftware Monolix v.4.3.3. E’ stata elaborata un’opportuna strategia perl’identificazione del modello in numerosi dataset relativi ad esperimenti contopi xenograft provenienti da studi condotti negli anni passati durante il pro-cesso di sviluppo di un farmaco antitumorale orale. Nel dettaglio, sono statianalizzati i dati provenienti da nove esperimenti che coinvolgono tre diverselinee tumorali e tredici differenti farmaci antitumorali, alcuni dei quali daanni in commercio altri ancora in sviluppo, la cui somministrazione variasia per l’entita della dose sia per il protocollo di somministrazione seguito.In particolare le scelte adottate per il fitting hanno portato alla risoluzionedi alcuni problemi di identificabilita insorti, dimostrando le ottime capacitadescrittive del modello.

Nella seconda parte del lavoro e stato realizzato un confronto tra il nuovo

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modello DEB-TGI e quello proposto da Simeoni e coautori. In particolaredall’analisi condotta si evincono alcune affinita che, da un lato, contribuis-cono a garantire la validita del modello in fase di sviluppo, dall’altro, for-niscono una possibile interpretazione biologica alle ipotesi su cui si basa ilpiu empirico modello Simeoni.

Abstract (English)

Within the model-based drug development paradigm, this thesis focuses onthe development and the evaluation of a new mathematical model to assessthe safety and the efficacy of anticancer drug administration in xenograftmice during oncology preclinical studies.

Mathematical models for describing the tumor growth in animals aresometimes criticized because they absolutely neglects the relationship be-tween tumor and host organism. To overcome this limitation, a more complexand mechanistic model, based on an energetic rate balance between tumorand host, was developed [40]. This model combines the key concepts of theDynamic Energy Budget (DEB) theory [13] and with these of the Simeonitumor growth inhibition (TGI) model [37] in order to describe the dynam-ics of the tumor-host interaction and the effect of anticancer treatments. Inparticular, this new model allows to distinguish the drug effect on the tu-mor growth from that on the body weight, thus, avoiding a confoundinginterpretation of the specific drug action on the tumor growth.

In the first part of this thesis the new DEB-TGI model was implementedin Monolix v.4.3.3 and tested upon several datasets relative to nine experi-ments conducted on xenograft mice some years ago during the developmentof a new oral anticancer drug. In particular, we analyzed data relative tothree different cell tumor lines and thirteen anticancer drugs, administeredwith different doses and scheduling, some of which already on the marketother still in development. Even if some assumptions had to be made forsolving some identifiability issues, the model was able to describe standardxenograft experiments.

In the second part of this work a comparison was made between theDEB-TGI model and the widely used Simeoni TGI model. In particular,from this analysis some similarities between the two models were deduced.These affinities not only contribute to ensure the validity of the model un-der development but also provide a possible biological interpretation of theassumptions underlying the more empirical Simeoni model.

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Contents

1 Introduction 31.1 Drug Discovery and Development . . . . . . . . . . . . . . . . 41.2 Model-based drug development (MBDD) . . . . . . . . . . . . 61.3 Model-based drug development applied to oncology . . . . . . 7

1.3.1 In vivo preclinical tumor experiment: xenograft models 81.3.2 Simeoni tumor growth inhibition (TGI) model . . . . . 9

2 A new tumor-in-host DEB-based model 132.1 Tumor free individual: DEB theory . . . . . . . . . . . . . . . 132.2 Tumor-bearing individual: van Leeuwen model . . . . . . . . . 172.3 Tumor-bearing individual under anticancer treatments: DEB-

TGI model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 The drug effect on the host body weight . . . . . . . . 222.3.2 The tumor growth saturation . . . . . . . . . . . . . . 24

3 Monolix: a tool for the model identification and validation 283.1 Monolix projects . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.1 The data and the model . . . . . . . . . . . . . . . . . 313.1.2 The initialization frame . . . . . . . . . . . . . . . . . 33

3.2 Executing tasks . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 The popolation parameters estimation: SEAM algorithm 333.2.2 The estimation of the Fisher information matrix and

standard errors . . . . . . . . . . . . . . . . . . . . . . 353.2.3 Graphics and results . . . . . . . . . . . . . . . . . . . 36

4 Model identification and validation 374.1 Dataset presentation . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.1 Tumor-free individual . . . . . . . . . . . . . . . . . . . 374.1.2 Tumor-bearing individual and drug treatments . . . . . 37

4.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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5 TGI model and DEB-TGI model 535.1 Exponential and linear phase of the tumor growth . . . . . . . 545.2 A numerical comparison of λ0 and λ0 . . . . . . . . . . . . . . 575.3 Analysis of the Simeoni switch from the exponential to the

linear phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Conclusions 616.1 Advantages of a mechanistic approach . . . . . . . . . . . . . 626.2 A quantitative measurement of the drug toxicity . . . . . . . . 626.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Appendix A 66

Appendix B 70

Appendix C 90

Bibliography 96

Acknowledgements 97

Acknowledgements 99

List of Figures

1.1 Drug discovery and development process . . . . . . . . . . . . 41.2 Central focus of Model-based drug discovery . . . . . . . . . . 71.3 Schematic representation of the evaluation of TGI and TGD . 91.4 Simeoni TGI model . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Energy fluxes in an individual organism, according to the DEBtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Tumor-bearing individual: energy-allocation rule . . . . . . . . 182.3 Inclusion of the anticancer effect into the tumor-in-host model:

a schematic representation in terms of differential equations . 232.4 Comparison of experimental data between untreated and treated

mice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Model identification and validation steps . . . . . . . . . . . . 283.2 Monolix graphical user interface . . . . . . . . . . . . . . . . . 303.3 Loading data in Monolix . . . . . . . . . . . . . . . . . . . . . 313.4 Monolix window to check initial fixed effects . . . . . . . . . . 333.5 Diagnostics graphics made by Monolix . . . . . . . . . . . . . 36

4.1 Growth chart of HSD athymic nude mice . . . . . . . . . . . . 384.2 Body weight prediction together with experimental data . . . 424.3 Simulation of reserve energy, structural biomass and total body

weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.4 Problems arisen in the identification of the tumor model against

the control groups . . . . . . . . . . . . . . . . . . . . . . . . . 444.5 Controls groups:fitting of the body weight and the tumor growth 464.6 Simulated concentration profile for Experimental 1, Drug A . . 474.7 Experiment 1, Drug A: fitting of body weight and tumor growth 484.8 Simulated concentration profile for Experiment 6, Drug O . . 494.9 Experiment 6, Drug O: fitting of body weight and tumor growth 494.10 Simulated concentration profile for Experiment 9, Drug I . . . 50

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4.11 Experiment 9, Drug I: fitting of body weight and tumor growth 514.12 Experiment 9, Drug I: fitting of body weight and tumor growth

without Group 4 . . . . . . . . . . . . . . . . . . . . . . . . . 524.13 Observed and predicted curves obtained for the Experiment

9, Drug I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1 Simeoni TGI and DEB-TGI tumor growth . . . . . . . . . . . 545.2 Switch point of the DEB-TGI model . . . . . . . . . . . . . . 555.3 Identification of the Simeoni model upon data simulated by

DEB-TGI model . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Switch points of the Simenoni model and of the DEB-TGI model 59

6.1 Sensivity analysis of the DEB-TGI model for parameter µu . . 636.2 Drug action modulated by an effect compartment . . . . . . . 64

List of Tables

2.1 Parameters of the tumor-free individual model . . . . . . . . . 172.2 Tumor growth parameters . . . . . . . . . . . . . . . . . . . . 212.3 Tumor-in-host DEB-based model parameters . . . . . . . . . . 27

4.1 Dataset information: tumor cell line, sex, age . . . . . . . . . 394.2 Dataset information: drug . . . . . . . . . . . . . . . . . . . . 394.3 Dose and scheduling information of Experiment 1, Drug A . . 404.4 Dose and scheduling information of Experiment 6, Drug O . . 404.5 Dose and scheduling information of Experiment 9, Drug I . . . 414.6 PK parameters of anticancer drugs A, I, O . . . . . . . . . . . 414.7 Physiological parameters estimates of the tumor-free model . . 424.8 Physiological parameters of the tumor-free model . . . . . . . 434.9 PD model parameters estimated for the control groups . . . . 454.10 PD model parameter estimates for Experiment 1 Drug A . . . 474.11 PD model parameter estimated for Experiment 6 Drug O . . . 484.12 PD model parameter estimated for Experiment 9 Drug I . . . 504.13 PD model parameter estimated for Experiment 9 Drug I with-

out Group 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1 Estimates of the parameter λ0 and the corresponding λ0 . . . 58

6.1 Estimates of the toxicity index for Experiments 1, 6 and 9 . . 64

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Chapter 1

Introduction

The recent advances in biomedical science have raised new hope for the pre-vention, treatment and cure of serious illnesses. However, there is growingconcern that the current drug development path is becoming increasinglychallenging, inefficient and costly. This inefficiency becomes even more acutewhen one considers the number of compounds that undergo attrition in clin-ical and preclinical research: more than 90% of new chemical entities failsin clinical trials, and many more in the preclinical stages of development forpharmacodynamic (PD) (e.g., lack of efficacy and safety) or pharmacokinetic(PK) reasons1. Therefore, the industry has being forced to focus on attritionrates to balance the costs of drug development, to explore cost containmentmeasures while still investing significantly in drug research and development.The resulting demand of scientific and technological innovations that affectefficacy and safety has led to a growing interest in the model-based drug de-velopment (MBDD) paradigm, as the development and application of mathe-matical and statistical models to better characterize, understand and predictthe drug behavior in terms of PK, PD and efficacy and toxicity biomarkers.In this context, my thesis deals with the development and the evaluation ofa new mathematical model, based on PK/PD concepts, able to describe thetumor growth inhibition in xenograft mice and the behavior of anticancercandidates in several experimental settings.

1Pharmacokinetics is the relationship between drug inflow and drug concentration atvarious body sites, notably the so-called biophases, of drug action, and for which subpro-cesses for drug absorption, distribution, metabolism, and elimination determine the rela-tionship; pharmacodynamics, is the relationship between drug concentrations and phar-macological effects (called bioresponses), and the relationship, in turn, of these responsesto clinical outcomes [36].

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1 Introduction 4

1.1 Drug Discovery and Development

The development of a new drug is a long, complicated and very expensiveprocess, that starts with the discovery of a new medicine and ends when itis available for treating patients, process that is characterized by an averageduration of about 13 years. The average cost to research and develop eachsuccessful drug is estimated to be $800 million to $2 billion, numbers thatinclude the costs of thousands of failures. Therefore, pharmaceutical compa-nies are continuously involved in the optimization of this process, predictingin advance compounds with high probability of failure, while making the de-velopment of the most promising candidate drugs faster and more effective.

The research activities leading to the discovery of a new drug can bedivided in different phases (Fig.1.1) [32].

Figure 1.1: Drug discovery and development process.

• Pre-discovery process: before any potential new medicine can be dis-covered, scientists work to understand the disease to be treated as wellas possible, and to unravel the underlying causes of the pathologicalconditions.

• Discovery process: the discovery process includes all early research toidentify a new drug and testing it in the laboratory. It comprises severalstages:

1. Target identification and validation: once understood the underly-ing causes of a disease, pharmaceutical researchers select a targetfor a potential new medicine, that is a single molecule which isinvolved in the disease to be treated, and show that can be actedupon it by a drug.

2. Hit identification: molecules showing affinity for the target, calledhits, are searched within a huge number of compounds, which arecollected in commercially available chemical libraries or synthe-sized by the drug companies themselves with the aim of finding

1 Introduction 5

a promising molecule, a lead compound, that may act on the se-lected target to alter the disease progression and could become adrug.

3. Hit to lead: lead compounds go through a series of tests to pro-vide an early safety assessment. Scientists test Absorption, Dis-tribution, Metabolism, Excretion and Toxicological (ADME/Tox)properties of each lead with studies performed in living cells, inanimals and via computational models.

4. Lead optimization: lead compounds that survive the initial screen-ing are then optimized or altered to make them more effective andsafe.

• Preclinical Phase or Phase 0: once having obtained one or more op-timized compounds, researchers turn their attention to testing themextensively to determine if they should proceed to test in humans. Thecompounds are administered to a small number of animals (usually ro-dents and/or other animal species, like dogs and rabbits) to assess thePK, pharmacology and safety of the compound in vivo and to identifythe conditions (in terms of exposure and duration of the exposure) thatachieve the best compromise between pharmacological and toxicologi-cal effects. The main objective of this research phase is to evaluate andintegrate all the generated available data in order to predict the actionof the drug in man.

• Clinical phase: the candidate drug is tested in clinical setting in threephases of trials:

1. Phase I : during this phase the candidate drug is administeredin humans for the first time. These studies are usually carriedout with about 20 to 100 healthy volunteers to test the safetyin humans. Researchers look at the pharmacokinetics of a drug,mode of absorption, metabolism and elimination from the body,but also they study the drug pharmacodynamics, with a particularinterest for the dose-response or exposure-response relationshipsin human and, therefore, for the safe dose range.

2. Phase II : in Phase II researchers evaluate the candidate drug ef-fectiveness in about 100 to 500 patients suffering from the diseaseunder study, and examine the possible short-term side effects andrisks associated with the drug. Researchers also analyze optimaldose strength and schedules for using the drug.

1 Introduction 6

3. Phase III : in the last step of the clinical phase, the drug candidateis studied in a larger number (about 1000-5000) of patients togenerate statistically significant data about safety, efficacy and theoverall benefit-risk relationship of the drug. This phase of researchis key in determining whether the drug is safe and effective.

• New drug approval and marketing or Phase IV : once all three phases ofthe clinical trials are complete, all the results are tested to determine ifthe drug can be approved for patients to use. Even after the approvaland the launch on the market, the research on a new medicine continuesand the entire population assuming it is monitored to evaluate the long-term safety and the effectiveness on a specific subgroup of patients.

1.2 Model-based drug development (MBDD)

High development costs and low success rates in bringing new medicines tothe market demand a better set of prognostic tools to improve the efficiencyin developing safe and efficacious drugs. Model-based drug development,MDBB, has been identified in 2004 by the FDA critical path document [41]and well-recognized by many others [1, 25, 27], as a key tool to supportthe optimization of the drug development process2. This mathematical andstatistical approach constructs, validates and utilizes disease models, drugexposure-response models and pharmacometric models from preclinical andclinical data to improve drug development knowledge and decision making[15, 47].

These mathematical models help answering two basic questions: whichcompound should be selected for development and how it should be dosed?The process of applying model-based approach to answer these questions canbe summarized into three steps: knowledge gathering, model constructionand simulation (Fig.1.2 as in [47]). The first one is the collection of all thepossible information: assumptions, prior information and experimental data[10]. Starting from the available knowledge, a model is built and validatedwith the aim to capture the casual relationship between disease state, prog-nostic factors, drug characteristics, safety and efficacy outcomes. Finally,models developed can be used to simulate outcomes helping to refine dose

2The mission of the FDA, Food and Drug Administration, is, in part, to protect thepublic health by assuring the safety, efficacy, and security of human and veterinary drugs,biological products, and medical devices. The FDA is also responsible for advancing thepublic health by helping to speed innovations that make medicines more effective, safer,and more affordable; and helping the public get the accurate, science-based informationthey need to use medicines to improve their health [41].

1 Introduction 7

selection and study design and to represent dose-response and time-responsebehavior [1, 27, 47].

Figure 1.2: Constructing and utilizing disease models and drug exposure-response mod-els in model-based drug development contain three steps: knowledge gathering, modelconstruction, and simulation.

MBDD covers the whole spectrum of the drug development process. In-deed, PK/PDmodeling concepts can be applied in all stages of preclinical andclinical drug development, and their benefits are manifold. At the preclinicalstage, potential applications might include the evaluation of in vivo potency,the identification of bio-markers, as well as dosage form and regimen selectionand optimization. At the clinical stage, PK/PD applications include charac-terization of the relationship between dose, concentration, effect and toxicity,evaluation of food, age and gender effects, but also drug-drug interactions,tolerance development and inter/intra-individual variability in response [25].

For all these reasons model-base drug development, and in particularPK/PD concepts, are believed to play a pivotal role in optimizing and stream-lining the drug development process of the future.

1.3 Model-based drug development applied

to oncology

Over the past decade, a large number of novel anticancer drugs have beendeveloped and many are now used into routine clinical practice. However,the development of new anticancer drugs remains an expensive and inefficientprocess. In anticancer drug development, attrition rate is the major factorthat reflects the level of loss of new candidate drugs during their preclinicaland clinical development. Less than 5% of drugs that reach Phase I gaina marketing authorisation [11]. Numerous solutions have been proposed totackle the issue of attrition in anticancer drug development by many authors[8, 28, 30, 35, 39, 44]; one of these consists in the implementation of in vivo

preclinical models which can act as predictors of success in clinical trials [26].

1 Introduction 8

1.3.1 In vivo preclinical tumor experiment: xenograftmodels

Numerous murine models have been developed to study human cancer al-ready starting from 1940s [45]. These models are used to investigate thefactors involved in malignant transformation, invasion and metastasis, andto examine response to therapy as well. The most preclinical models, usedfor evaluating the anticancer activity of new compounds under developmentin the oncology therapeutic area, are xenografts of human tumors grown inimmunodeficient mice. Despite discussions about their ability to generatemeaningful data for the translation from animal to humans, they still have amajor role in cancer drug development due to more robust approaches thatcombine high ability in predicting clinical efficacy with simplicity and lowcost in their implementation [34, 38].

During these experiments, fragments of about 20-30mg of tumor hu-man cell lines are implanted subcutaneously in immunodeficient mice suchas athymic (nude) or severe combined immunodeficient mice. When theanimals bearing a palpable tumor (approximately 100-200 mm3), they areselected, randomized and divided in two or more groups (in general eachincluding several animals). After the randomization the experiment can be-gin: some groups are treated with a vehicle (control groups), others with ananticancer compound (treated groups) following prescribed protocols. Thecontrols and treated groups are clinically evaluated daily and the tumor di-mensions (length and width (mm)) are measured, typically from once a dayto every 4 days. The tumor mass (mg) is, then, calculated as:

weight = ρtrlength ∗ width2

2(1.1)

approximating the tumor shape with the ellipsoid generated by the rotationof a semi-ellipse around its larger axis (length) and assuming that the tumordensity is ρtr = 1 mg/mm3.

Drug administrations can differ for the following aspects: dosages, du-ration of the treatment, schedule (number of administrations and times ofadministration), way of administration (intravenous or intra-peritoneal) andadministration profile (bolus or infusion).

To compare the ability to inhibit tumor growth of different compounds orof different dosages/schedules of the same compound the distances betweenthe different tumor growth curves is measured, either at specific weights(TGD: Tumor Growth Delay) or times (TGI: Tumor Growth Inhibition), seeFig.1.3. Unfortunately, most of these tumor growth inhibition metrics arenot invariant with respect to the experimental conditions. For this reason,

1 Introduction 9

a certain number of experiments have to be performed to obtain a valuableestimate of the drug activity and, in addition, it is difficult to extrapolatethese results to human patient. Only mathematical models that are ableto describe tumor growth by dissecting the system-specific properties canprovide compound specific and experiment-independent model parameters.

Figure 1.3: Schematic representation of the evaluation of Tumor Growth Inhibition andTumor Growth Delay between tumor weight curves in treated (red line) and untreated(black line) groups.

For these reasons, in the past 40 years several mathematical models havebeen introduced to describe the relationship between drug administrationand the dynamics of tumor growth (PK-PD models) in xenograft models.These mathematical models are generally based on some biological and phys-iological grounds, so that their parameters can have a biological meaning.Furthermore, these models may be used as predictive tools for anticipatingthe outcome of new dosing regimens, for the optimization of the preclinicalexperimental design and for the transfer from preclinical to clinical setting.Among these models, the Simeoni Tumor Growth Inhibition (TGI) model[22, 37] is one of the most popular and, often, acts as a reference.

1.3.2 Simeoni tumor growth inhibition (TGI) model

Simeoni TGI model is a simple and effective semi-empirical PK/PD model,linking the plasma concentration of anticancer drugs on tumor growth inxenograft mice, able to describe successfully the inhibition of tumor growthobserved at different dose levels and schedules, independently of the mech-anism of action and the therapeutic indications of the compounds. Themain features and the formulas of the Simeoni TGI model are summarizedin Fig.1.4.

1 Introduction 10

Figure 1.4: Scheme and equations of the Simeoni TGI model [22, 37].

The model is based on the observation that in vivo tumor growth inxenograft models seems to follow exponential growth, at least in its earlyphases of development. Subsequently, the tumor shows a slowdown in itsgrowth and follows a linear growth, reaching eventually a plateau. Because aplateau was never observed in the experimental datasets, the model focuseson the exponential and linear phases. Therefore, it is assumed that thereis a threshold tumor mass wth, at which the tumor growth switches fromexponential to linear. The model for untreated animals (Fig.1.4, left panel)is characterized by three parameters: w0, that represents the tumor weight atthe inoculation time t0, λ0 and λ1, that are the parameters characterizing therate of exponential and linear growth, respectively. Imposing the continuityof the derivatives of the model, the value of the threshold wth is

wth =λ1λ0. (1.2)

In treated animals it is supposed that the anticancer treatment makessome cells non-proliferating eventually bringing them to death (Fig.1.4, right

1 Introduction 11

panel). The model assumes that the drug elicits its effect decreasing thetumor growth rate by a factor proportional to the drug concentration c(t)and to the portion of proliferating cells x1(t) through the constant parameterk2. Because the death of tumor cells is delayed with respect to the drugtreatment, a transit compartment model is used for describing this feature:it was assumed that cells affected by drug action stop proliferating and passthrough 3 different stages (named x2, x3 and x4), characterized by progressivelevels of damage before they die. The dynamics by which the cells proceedthrough progressive degrees of damage is modulated via a rate constant k1that can be interpreted in terms of kinetics of cell death. Then, the systemof differential equations involves now the three parameters to describe thegrowth of the proliferating cells in the control animals (w0, λ0, λ1) and twofor the drug action: k1, the micro-rate constant describing the kinetics ofnon-proliferating cells, and k2, the proportionality factor linking the plasmaconcentration to the effect. In this context, k2 is the parameter describingthe anti-tumor potency of the compound. The model involves, also, a set ofequations depending on the PK model for the drug kinetics description.

The Simeoni TGI model has demonstrated several times to have excel-lent in vivo predictive capabilities although relying only on a few identifiableand biologically relevant parameters, whose estimation requires only the datatypically available in the preclinical setting: the pharmacokinetics of the anti-cancer agents and the tumor growth curves in vivo. Obviously these featuresoffer various advantages such as generalizability and a simple description ofprocesses associated with tumor growth, however the model presents all thelimits of an empirical model approach. In particular, the transition fromthe exponential phase to the linear phase, that so-good describes the tumorgrowth, is not supported by a real biological justification.

An other limiting aspect of the Simeoni TGI model is that, as almost allmathematical tumor models, it absolutely neglects the relationship betweentumor and host organism with two relevant consequences. First, not everydata available in the preclinical experiments are properly exploited, indeedfree information, such as mice weights, fell by the wayside. Second, the modeldoes take into account neither the influence of tumor on host organism, northe anticancer drug side effects.

To fill these lacks, starting from a tumor-in-host model, based on theDynamic Energy Budget (DEB) theory [13] and able to describe the in vivo

tumor and body weight growth, a new model has been proposed in whichthe pharmacological treatment is included as well [40]. In particular, forthis purpose the concept of the mortality chain coming from the SimeoniTGI model has been embedded. This new TGI DEB-based model is able todescribe the tumor growth inhibition effect of an anticancer drug, also taking

1 Introduction 12

into account the side effects on the host body weight.

Chapter 2

A new tumor-in-hostDEB-based model

The inoculation of human cancer cells in immunodeficient rodents (xenograftmodels) constitutes one of the major preclinical screen for the developmentof novel cancer therapeutics. The main assumption underlying xenograftmodels is that human cancers xenografted into immunocompromised animalsclosely reflect the human condition. Despite this evidence, the interactionsbetween tumor and host are often neglected into the mathematical modelsused to describe data on tumors growing in vivo.

To supply this lack, van Leeuwen et al.[43] proposed a tumor-in-hostmodel exploiting the Dynamic Energy Budget (DEB) theory as a mathemat-ical framework that describes the host physiology. Starting from the assump-tion that the host features can influence deeply the tumor behavior and viceversa, their aim was to develop a model to explore the energetic interactionsbetween the tumor growth and the physiology of the host organism.

Starting from the van Leuuwen model and the Simeoni TGI model, anew tumor-in-host DEB-based model has been defined and developed to alsoinclude the pharmacological effect of anticancer treatments in xenograft mice[40]. The result is a model able to describe both the pharmacological effectof a compound on the treated tumor and the side effects on the host on thebasis of the energetic interactions.

2.1 Tumor free individual: DEB theory

To model the interaction between tumor and host, it is necessary to introducea general framework describing the physiology of the host organism. Such aframework is provided by the DEB theory which starts with a set of rules

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2 A new tumor-in-host DEB-based model 14

Figure 2.1: Energy fluxes in an individual organism, according to the DEB theory. Foodis conceived as material that bears energy. Part of this energy is taken up via blood anddelivered to the reserves. Energy required to carry out the various physiological processesis obtained from these reserves.

to characterize an individual organism, based on fundamental mechanismsin common to all organisms. From these rules, quantitative expressions forseveral physiological processes are derived. In this section, only the aspectsof the theory mandatory to understand the model for tumor growth are ex-plained. A more complete and exhaustive formulation of the theory can befound in [12, 13].

The basic structure of the DEB theory is depicted in Fig.2.1. It as-sumes that the body is divided in two components: the structural biomassand the reserve compounds. Both the components have by assumption aconstant, but not necessarily identical, chemical composition. Structuralbiomass, V (t), can be conceived as volume, while the second pool representsthe stored energy, E(t), essential to carry out the physiological processes.The energy reserve is made partially committed for the physiological pro-cesses, in particular the utilized energy is spent on somatic processes (growthand maintenance) and on reproductive processes (development and reproduc-tion).

The authors assume that the assimilation efficiency is independent of thefood ingestion rate, [42]. So, if the animal receives at time t a fixed fractionρ of food consumption, the assimilation rate is then given by

A(t) = ρAm (2.1)

where Am denotes the maximum assimilation rate. Then, if we define the


surface-specific maximum assimilation rate as: {Am} = AmV−2/31∞ with V1∞

being the ad libitum asymptotic maximum structural volume, the assimila-tion rate can be rewritten as

A(t) = ρ{Am}V 2/31∞ . (2.2)

As it can be seen in Fig.2.1, the relationship between the structuralbiomass and the amount of reserves is represented by the utilization rateC(t), that is the rate at which the energy mobilized from the reserve is madeavailable for the physiological processes. According to the DEB theory, theutilization rate is given by

C(t) =E(t)

V (t)

(

νV (t)2/3 − dV (t)

dt

)

(2.3)

where ν is the energy conductance defined as

ν ={Am}[Em]

=AmV

−2/31∞

[Em](2.4)

with [Em] the maximum reserve density for unit of volume.Given the expression of the utilization rate C(t) and the assimilation

rate A(t), the change in the amount of reserves over time is then given bytheir difference: dE(t)/dt = A(t)−C(t) (Fig.2.1). Replacing the expressions(2.3) and (2.1) for C(t) and A(t) and defining the scaled energy densitye(t) = E(t)/[Em]V (t), we obtain

de(t)

dt=

ν

V 1/3

(

ρV

2/31∞

V (t)2/3− e(t)

)

(2.5)

with initial condition e(t0) = e0.The DEB theory also assumes that somatic processes (growth and main-

tenance) and reproductive processes (development and maintenance) takeplace in parallel, so, in accordance to the so-called k -rule, an individualspends only a fixed fraction k of the available energy on the first. Therefore,if we denote the costs of growth and maintenance for time unit with G(t)and M(t) respectively, we can obtain the following energy rate balance

G(t) = kC(t)−M(t) . (2.6)

This balance equation says that an animal has to give maintenance priorityover growth to stay alive; consequently increase in size ceases when all energy


available for maintenance and growth is spent on maintenance only, thus,the maintenance is key determing the ultimate size an organism can reach.Assuming that the costs of growth and maintenance per unit of structuralvolume are constant, the costs of growth per time unit G(t) turn out tobe proportional to the increase in structural volume, whereas the costs ofmaintenance per time unit M(t) to the structural volume:

G(t) = [G]dV (t)

dt

M(t) = [M ]V (t) .

(2.7)

Given (2.7), the change of structural volume is obtained from (2.6) as

dV (t)

dt=kC(t)− [M ]V (t)

[G]=νe(t)V (t)2/3 − gmV (t)

g + e(t)(2.8)

where g = [G]/k[Em] is the energy-investment ratio and m = [M ]/[G] themaintenance-rate coefficient. The initial condition is V (t0) = V0. Fromequations (2.5) and (2.8), it can be shown that V (t) tends to an asymptoticmaximum value, Vρ∞ = ρV1∞.

We have said that the body has two components, so the total body weightis the sum of the weight of both structure and reserve: W (t) = WV (t) +WE(t) = dV V (t) + dEE(t)/rE where the coefficient dV and dE represent thevolume specific weight of structural biomass and reserve respectively and rEis defined as amount reserves

volume reserves. As reported in [42], the expression above can be

rewritten asW (t) = dV (1 + ξe(t))V (t) , (2.9)

where ξ is a dimensionless compound parameter representing the scaled re-serve specific weight given by

ξ =dE[Em]

dV rE. (2.10)

Overall, the change in size of a tumor-free organism is characterized by thefollowing system of differential equations whose parameters are reported inTab. 2.1.

de(t)

dt= ν

(

ρV2/31∞

V (t)− e(t)

V (t)1/3

)

dV (t)

dt=νe(t)V (t)2/3 − gmV (t)

g + e(t)

W (t) = dV (1 + ξe(t))V (t)

(2.11)


with e(t0) = e0 and V (t0) = V0.

Parameter Dimension Interpretationν LT−1 Energy conductanceρ - Food-supply coefficientV1∞ L3 Maximum structural volumeg - Growth energy-investment ratiom T−1 Maintenance-rate coefficientξ - Scaled reserve specific weight

Table 2.1: Parameters of the tumor-free individual model.

2.2 Tumor-bearing individual: van Leeuwen

model

In the previous section the tumor-free individual model derived by DEBtheory was presented; now we are going to introduce the extension of thismodel proposed by van Leeuwen et al. [43], to describe tumor growing in

vivo. Some considerations about the changes in the energetic framework isrequired. First, since tumor tissue is generally less differentiated than othertissues, tumor growth and maintenance costs per volume unit may be lower,allowing tumor cells to proliferate faster than normal cells. Furthermore,because a tumor is a part of the body out of control, a second energeticaspect may also changes: tumor cells may become gluttonous, taking all theavailable energy at the expense of the normal cells. To model tumor growthdynamics, we need to introduce some additional variables and parameters. Inaddition to the body size V (t), we consider the tumor size Vu(t). Obviously,to survive and proliferate, the tumor has to obtain nutrients from the host.The gluttony of the tumor is characterized by a coefficient µu; if µu = 1,each tumor cell demands the same amount of energy as an average normalcell, if µu > 1 a tumor cell takes more than an average normal cell: in otherwords, the gluttony coefficient µu can be conceived as a measurement of theaggressiveness of the tumor.

In the new framework, the k -rule has been modified on the assumptionthat the tumor appropriates a fraction ku(t) of the energy available for thesomatic processes (Fig.2.2). This assumption implies that the tumor haspriority for the resources over the host, which is supported by experimentalevidence [2].

Experimental observations substain that the tumor energy demand in-creases with tumor size. Therefore, as reported in [43], the coefficient of


Figure 2.2: Tumor-bearing individual: energy-allocation rule within DEB framework.

energy-allocation among tumor and host is defined as

ku(t) =µuVu(t)

V (t) + µuVu(t), (2.12)

with ku(t) that takes values between 0 and 1 as k. Eq.(2.12) implies thatat small tumor size, the fraction of the resources appropriated by the tumoris approximately proportional to tumor size. As the tumor becomes larger,the fraction still increases but at a diminishing rate. With the generalizationof the k -rule, the equation of the energy rate balance in host (2.6) is thenmodified as

G(t) = k(1− ku(t))C(t)−M(t) . (2.13)

Consequently, the change of structural volume in a tumor-bearing individualis described by

dV (t)

dt=

(1− ku(t))νe(t)V (t)2/3 − gmV (t)

g + (1− ku(t))e(t). (2.14)

Then, if the growth and maintenance costs for tumor are denoted with Gu(t)and Mu(t) respectively, the tumor energy rate balance can be written as

Gu(t) = kku(t)C(t)−Mu(t) . (2.15)

Also the tumor growth costs are proportional to the increase in structuralvolume, whereas the maintenance costs are proportional to structural volume,so the equation (2.15) leads to

dVu(t)

dt=

(νV (t)2/3 +mV (t))gku(t)e(t)

ggu + (1− ku(t))gue(t)−muVu(t) (2.16)

with energy-investment ratio gu and maintenance-rate coefficient mu definedas in (2.17) and (2.18), respectively

gu =[Gu]

k[Em](2.17)


mu =[Mu]

[Gu]. (2.18)

Since tumor growth and maintenance costs seem to be less than host growthcosts, as previously introduced, it is expected that gu ≤ g and mu ≤ m.Moreover, if the condition mugu = µumg holds, the tumor grows accordingto a S-shaped saturating growth pattern1 [43]. As a rule, it is shown thatmugu << µumg.

As the tumor appropriates reserves originally destined to be spent onphysiological processes such as body growth, the host is no longer able toreach its maximum size Vρ∞ and, in particular, it has been adjusted as in(2.19) to take into account the dependence on the tumor size Vu(t) for atumor-bearing individual:

Vρ∞(t) = ρV (t)

V (t) + Vu(t)V1∞ . (2.19)

To account for this, the scaled reserve density equation (2.5) has been gen-eralized as follows:

de(t)

dt=

ν

V (t)1/3

(

ρ( V1∞Vu(t) + V (t)

)2/3

− e(t)

)

. (2.20)

The equations (2.20), (2.14) and (2.16) specify the change in size for bothtumor and host as long as the organism disposes the energy to carry outnormal physiological processes and, in particular, for growth (dV/dt ≥ 0).Indeed, the tumor exploits the resources of the host, so the latter disposesof less energy; as maintenance has always priority over growth, the energyspending-cut initially results in a decrease of the host growth rate. If in aspecific time instant t1, it decreases to zero and tumor size still increases,the host organism has to start degrading structural biomass to survive whilesatisfying the tumor energy demand. Although the generalized k -rule equa-tion (2.14) allows body-weight loss, there are two reasons why it would beinappropriate to use these equations to describe tissue degradation. Firstly,if these equations were used, all energy originally invested in “building” aunit biomass would be regained, which is thermodynamically impossible.Secondly, equations (2.14) imply that the host exploits all energy releasedfrom tissue degradation to pay its own maintenance costs. This contradictsaccepted knowledge, indicating that both host and tumor benefit from the

1This condition marks the bifurcation between tumors growing (mugu < µumg) ordying of (mugu > µumg). The adopted approach does not assume a priori the existenceof an asymptotic maximum size (see [43] for more details).


released resources. Therefore, van Leeuwen et al., in [43], rewrote the equa-tions of energy rate balance (2.13) and (2.15), as

G(t) = (1− ku(t))(kC(t) + S(t))−M(t) (2.21)

Gu(t) = ku(t)(kC(t) + S(t))−Mu(t) , (2.22)

where S(t), the rate at which energy is regained from the degradation ofstructural biomass, is given by

S(t) = −ω[G]dV (t)

dt. (2.23)

This means that the amount of energy that becomes available per time unitis proportional to the tissue degradation rate2. The parameter ω is an effi-ciency coefficient whose the thermodynamic upper limit ω = 1 means 100%efficiency and, however, can never be achieved. In the realistic case thatω < 1, part of the degraded structural biomass is actually wasted.

Therefore, from (2.21), (2.22) and (2.23), for t > t1 the loss of struc-tural biomass and the increase in tumor size are described by the followingequations, respectively:

dV (t)

dt=

(1− ku(t))νe(t)V (t)2/3 − gmV (t)

(1− ku(t))(e(t) + ωg)(2.24)

dVu(t)

dt=gmku(t)V (t)

gu(1− ku(t))−muVu(t) (2.25)

with initial conditions V (t0) = V0 and Vu(t0) = Vu0. Replacing the expression(2.12) for ku(t) in (2.25), we can rewrite the derivative dVu(t)/dt as

dVu(t)

dt=(mgµu

gu−mu

)

Vu(t) . (2.26)

Then, a tumor-bearing individual is described by the two systems of differ-ential equations (2.27) and (2.28), characterized by the parameters reportedin Tab.2.1 and the new reported in Tab.2.2.

2Notice that the host loses structural volume, dV (t)/dt is negative and, consequently,S(t) is positive.


Parameter Dimension Interpretationµu - Coefficient of gluttonygu - Tumor growth energy-investment ratiomu T−1 Tumor maintenance coefficientω - Thermodynamic efficiency coefficient

Table 2.2: Tumor growth parameters.

• Case dV (t)dt

≥ 0

de(t)

dt=

ν

V (t)1/3

(

ρ( V1∞Vu(t) + V (t)

)2/3

− e(t)

)

dV (t)

dt=

(1− ku(t))νe(t)V (t)2/3 − gmV (t)

(1− ku(t))(e(t) + ωg)

dVu(t)

dt=


ggu + (1− ku(t))gue(t)−muVu(t)

W (t) = dV (1 + ξe(t))V (t)

Wu(t) = dVuVu(t)

(2.27)

with e(t0) = e0, V (t0) = V0 and Vu(t0) = Vu0.

• Case dV (t)dt

< 0

de(t)

dt=

ν

V (t)1/3

(

ρ( V1∞Vu(t) + V (t)

)2/3

− e(t)

)

dV (t)

dt=

(1− ku(t))νe(t)V (t)2/3 − gmV (t)

(1− ku(t))(e(t) + ωg)

dVu(t)

dt=(mgµu

gu−mu

)

Vu(t)

W (t) = dV (1 + ξe(t))V (t)

Wu(t) = dVuVu(t)

(2.28)

with e(t1) = es, V (t1) = V1 and Vu(t1) = Vu1.

In both cases the coefficient of energy-allocation among tumor and host isdefined as in (2.12).


2.3 Tumor-bearing individual under anticancer

treatments: DEB-TGI model

The tumor-in-host model, introduced in the last section and described bythe systems (2.27) and (2.28), is able to describe the experimental data ofboth tumor and host organism in absence of a pharmacological treatment.In order to take into account the effect of an administered anticancer drug,a new model has been developed and implemented merging the concepts ofthe Simeoni TGI model with the DEB theory [40].

On the basis of the Simeoni TGI model, the inhibition effect of an an-ticancer drug on the tumor has been modeled by assuming that a portionof proliferating cells becomes non-proliferating due to the anticancer treat-ment and goes to death through three different stages of damage. Also themodel assumes that the drug elicits its effect decreasing the tumor growthrate by a factor proportional to the drug concentration and to the portionof proliferating cells. Therefore, the mortality chain, representing the non-proliferating cells decay path, as well as the inhibition drug effect on theproliferating tumor cells have been included. So, the original tumor-in-hostmodel has been modified both adjusting the tumor growth rate in order toinclude the inhibition due to the drug and annexing the equations relativeto the non-proliferating cells (Fig.2.3).

Differently from the Simeoni TGI model, in which the equations refer tothe changes of tumor masses (proliferating and non-proliferating), now thechanges of different tumor compartments (Vu1, Vu2, Vu3 and Vu4) are expressedin volume terms. Consequently, the total tumor weight is obtained as

Wu(t) = dVu(Vu1(t) + Vu2(t) + Vu3(t) + Vu4(t)) (2.29)

where dVuis the tumor density considered equal to 1, as in the Simeoni TGI

model.As it can be seen in Fig.2.3, the model assumes, also, that only the

proliferating cells are able to exploit the host resources. Therefore, in placeof considering the whole tumor volume Vu as in the van Leeuwen model, onlythe volume of the proliferating cells Vu1 appears in all the energetic balances.

However, the new model not just integrates the concepts given by theDEB theory with those ones present in the Simeoni TGI model, but alsosome substantial modifications have been introduced.

2.3.1 The drug effect on the host body weight

The first change introduced has the purpose to include in the model the effectof the drug treatments on the host organism. Indeed, a qualitative analysis


Figure 2.3: System on the right represents the new TGI DEB-based model in termsof differential equations: the anticancer drug effect is included into the tumor-bearingindividual model (system in the top-left corner) by adding the drug effect expression andthe equations relative to the mortality chain of the Simeoni TGI model (system in thebottom-left corner).

showed that the body weight data relating to treated subjects are charac-terized by a further decreasing in addition to the normal decreasing due tothe tumor growth and its energy demand. In particular, it was observed thatthese weight losses occur in conjunction with the days of the anticancer treat-ment and that subsequently the body weight retakes to grow. The idea isthat in these days the organism is affected by the drug side effects like weak-ening and lack of appetite or limited assimilation and, as a consequence, hasa temporally structural biomass decreasing. Therefore, the food-supply co-efficient ρ, that in the previous model was considered a fixed fraction of foodconsumption, has been modified to consider the reduced ability in introduc-ing energy by food intake during treatment and include it into the model.

Obviously, the assumption of a time-variant of the food-supply coefficientimplies an effect on the tumor growth dynamic as well. Indeed, van Leeuwenet al. [43], studying the changes in the growth capacity of the tumor underdifferent caloric regimes, showed that a tumor grows slower in caloricallyrestricted animals than in ad libitum fed ones 3.

To model the temporary caloric restriction linked to the drug side effects,

3The authors considered three different feeding regimes: ad libitum (ρ = 1), 25% caloricrestriction (ρ = 0.75), and 55% caloric restriction (ρ = 0.45).


0

10

20

30

40

10 20 30Time(day)

Bod

y w

eigh

t (g)

0.00

0.05

0.10

0.15

0 10 20 30 40 50Time(day)

Con

cent

ratio

n(m

g/m

l)Figure 2.4: A comparison of the experimental data between untreated and treated mice(left panel): the red data are relative to body-weight of an untreated host, the black dataof a treated host. Simulated concentration profile of the drug treatment (right panel). Thebody-weight loss is observed in correspondence of the treatment of the anticancer drug.

the parameter ρ has been replaced by a time variable function given by thefollowing expression:

ρ(t) = 1− c(t)

C50 + c(t). (2.30)

Note the ρ(t) takes values between 0 and 1 and, in particular, the param-eter C50 represents the concentration producing 50% of the maximum sideeffect of the drug on the food intake process, that is the drug concentrationcorresponding to 50% caloric restriction (ρ = 0.5). The Eq.(2.30) implicityassumes that the ad libitum food consumption is still equal to 1 when thetumor is not treated.

2.3.2 The tumor growth saturation

The second change introduced affects the host ability to degrade struc-tural biomass. Indeed, during the structural biomass degradation (casedV (t)/dt < 0), the previous model assumes that the tumor continues growingundisturbed and exponentially by exploiting the host resources and the en-ergy regained from the degradation. Then, the host dies before tumor growthsaturation. However, on the basis of experimental observations, partly em-bodied in the Simeoni TGI model, it is expected an exponential phase oftumor growth followed by a slowdown in growth (linear phase) and, then,a saturation for high tumor size values (even if, in general, these are notmeasured for ethical reasons). It is reasonable to assume that the degrada-tion rate can not reach an infinite value, so a maximum threshold VdegMax


for degradation rate has to be introduced into the model. In absence ofanticancer treatment, the resulting model is the following:

• CasedV (t)

dt≥ 0

de(t)

dt=

ν

V (t)1/3

(

ρ(t)( V1∞Vu(t) + V (t)

)2/3

− e(t)

)

dV (t)

dt=

(1− ku(t))νe(t)V (t)2/3 − gmV (t)

(1− ku(t))(e(t) + ωg)

dVu1(t)

dt=


ggu + (1− ku(t))gue(t)−muVu(t)

ρ(t) = 1

W (t) = dV (1 + ξe(t))V (t)

Wu(t) = dVuVu(t)

(2.31)

• Case −VdegMax ≤ dV (t)

dt< 0

From the system (2.31), only the equations relative to dV (t)dt

and dVu1(t)dt

are modified as follows:

dV (t)

dt=

(1− ku(t))νe(t)V (t)2/3 − gmV (t)

(1− ku(t))(e(t) + ωg)

dVu1(t)

dt=(mgµu

gu−mu

)

Vu1(t)

(2.32)

• CasedV (t)

dt< −VdegMax

The new equations introduced from (2.32):

dV (t)

dt= −VdegMax

dVu(t)

dt=ku(t)

gu

(

e(t)νV (t)2/3 + VdegMaxe(t) + VdegMaxωg)−muVu(t)

(2.33)

If the host organism is subjected to an anticancer treatment, the modelis given by the equations:


• CasedV (t)

dt≥ 0

de(t)

dt=

ν

V (t)1/3

(

ρ(t)( V1∞Vu1(t) + V (t)

)2/3

− e(t)

)

dV (t)

dt=

(1− ku(t))νe(t)V (t)2/3 − gmV (t)

(1− ku(t))(e(t) + ωg)

dVu1(t)

dt=


ggu + (1− ku(t))gue(t)−muVu1(t)− k2c(t)Vu1(t)

dVu2(t)

dt= k2c(t)Vu1(t)− k1Vu2(t)

dVu3(t)


dVu4(t)


ρ(t) =c(t)

C50 + c(t)

W (t) = dV (1 + ξe(t))V (t)

Wu(t) = dVu(Vu1(t) + Vu2(t) + Vu3(t) + Vu4(t))

(2.34)

• Case −VdegMax ≤ dV (t)

dt< 0

From the system (2.34), only the equations relative to dV (t)dt

and dVu1(t)dt

are modified as follows:

dV (t)

dt=

(1− ku(t))νe(t)V (t)2/3 − gmV (t)

(1− ku(t))(e(t) + ωg)

dVu1(t)

dt=(mgµu

gu−mu

)

Vu1(t)− k2c(t)Vu1(t)

(2.35)

• CasedV (t)

dt< −VdegMax

The new equations introduced from (2.35):


Parameter Dimension Interpretationν LT−1 Energy conductanceV1∞ L3 Maximum structural volumeg - Growth energy-investment ratiom T−1 Maintenance-rate coefficientξ - Scaled reserve specific weightVdegMax - maximum degradation rateµu - Coefficient of gluttonygu - Tumor growth energy-investment ratiomu T−1 Tumor maintenance coefficientω - Thermodynamic efficiency coefficientk1 T−1 First-order rate constant of transitk2 (ConcT )−1 Drug potencyC50 Conc Half maximal inhibitory concentration

Table 2.3: Tumor-in-host DEB-based model parameters.

dV (t)

dt= −VdegMax

dVu1(t)

dt=ku(t)

gu

(

e(t)νV (t)2/3 + VdegMaxe(t) + VdegMaxωg)−muVu1(t)− k2c(t)Vu1(t)

(2.36)

In all the cases the coefficient of energy-allocation among tumor and host isdefined as in (2.12). Furthermore, the model including the anticancer drugeffect collapses into the untreated model in the absence of treatment.

The new tumor-in-host DEB-based model (2.34), (2.35) and (2.36) ischaracterized by the physiological parameters reported in Tab.2.3.

Chapter 3

Monolix: a tool for the modelidentification and validation

Model identification and validation are fundamental steps in the process ofbuilding a new model (Fig.3.1). Identification phase consists substantiallyin the parameter estimation: given a set of experimental data, the goal is tofind the best parameters for the model according to some optimality criteria.Instead, the aim of the validation step is to assess the predictive capabilityof the model in a specific context. This assessment is made by comparingthe predictive results of the model with validation experiments.

Figure 3.1: Model identification and validation.

To identify the parameters of the new tumor-in-host model (see Section2.3) and test its ability to describe data coming from mice xenograft experi-ments, we have chosen the software Monolix as a support tool1.

1In particular the model has been implemented in MONOLIX v.4.3.3.

28

3 Monolix: a tool for the model identification and validation 29

Monolix (MOdeles NOn LIeaires a effets miXtes) is a software created byINRIA (Institut National de la Recherche en Informatique et Automatique)and now developed and commercialized by Lixsoft, which has become a plat-form of reference for model-based drug development. Indeed, combining themost advanced algorithms with really ease of use, it represents a fast andpowerful tool for parameter estimation in nonlinear mixed effect models.

Briefly, it is possible to built a model with two different approaches:

• Individual approach: let consider data, y = (y1 . . . , yn), coming for asingle subject (as in our case), and model these measurements as

y = f(ψ) + ǫ (3.1)

where f represents the model prediction and ǫ is the residual error.

• Population approach: let consider data coming from N different sub-jects, yi = (yij, 1 ≤ j ≤ ni), so it is possible to model the measurementsfor the i-th individual, yi as

y = f(ψi) + ǫi (3.2)

where ψi are the individual parameters for subject i (unknown) and ǫiis the residual error. Moreover we can write ψi as

ψi = g(θ) (3.3)

with θ the population parameters of the model, unknown too.

In particular, the nonlinear mixed effect model decomposes ψi into fixed andrandom effects, that is

ψi = g(θ, ai, ηi) (3.4)

where θ is the unknown vector of fixed effects (population parameters), aiare the individual covariates and ηi is the unknown vector of random effectsof the individual parameters2. Obviously, the individual model can be seenas a particular case of the population model in which the set of subjects isreduced to a single individual.

In this context, the objective of Monolix is to perform:

1. Parameter estimation:

2See [3] cap.7 for a full discussion of population models.


• computing the maximum likelihood estimator of the populationparameters using the Stochastic Approximation Expectation Max-imization (SAEM) algorithm;

• computing standard errors for the maximum likelihood estimator;

• computing the conditional modes, the means and the standarddeviations of the individual parameters, using the Metropolis-Hastings algorithm.

2. Model selection: comparing several models using some information cri-teria (AIC, BIC).

3. Easy description of pharmacometrix models (PK, PK-PD) with theMlxtran language.

4. Goodness of fit plots.

5. Data simulation.

3.1 Monolix projects

Monolix offers a graphical user interface (GUI) to create projects (Fig.3.2).

Figure 3.2: Monolix graphical user interface.


A Monolix project contains information about the dataset to use, thestructural model, the statistical model for random effects, the tasks to run,their settings and initial values. All this information is stored in a projecttext file that can be edited [19]. Let us illustrate some basic features of thesoftware exploited in this work.

3.1.1 The data and the model

The dataset is an ASCII file containing all the data necessary for the studyarranged in a matrix (Fig.3.3). The columns of this matrix contain (in ourcase):

• ID to identify the subjects;

• TIME, the sampling time of the measurements;

• the observations Y, i.e. the tumor and the mice weights;

• YTYPE to specify the type of observations when there are several typesof observations (1 for mice weight, 2 for tumor weight);

• AMT or DOSE, for the entity of the dose;

• EVID, for dose events.

Nevertheless, further information can been included in the dataset file.

Figure 3.3: Loading data in Monolix.

Once having loaded the data, the user has to specify the model for the ob-servations that consists of the structural model and the residual error model.


The first is the parametric model that defines the computation of the pre-dictions; in Monolix it is possible to select one from a library included in thesoftware or to implement your own model using Mlxtran [19].

Mlxtran is a declarative, human-readable language describing simple andcomplex structural models. A Mlxtran script for Monolix is composed ofseveral blocks, an example is reported in Appendix A:

• DESCRIPTION consists in the description of the script content.

• INPUT declares some variables that are defined outside of the currentscript in the project or in the data. This block also declares the typesof these variables: the keyword parameter declares the input individualparameters.

• EQUATION contains the mathematical equations defining the inter-mediate variable, the PK elements of a optional prediction sub-model,the derivatives of an ODE system, the initial time and the initial values.

• OUTPUT declares variables whose values are exported for the varioustasks.

The observation model section shows, for each observation, the variablesname and their type (continuous or discrete) and their residual error model.Monolix considers the general form y = f + he where f is the model predic-tion, e is their residual error i.e. a sequence of independent random variablesnormally distributed with mean 0 and variance 1, and h specifies the errormodel. Here are some examples of error models available in Monolix:

• constant error model (const): y = f + ae;

• proportional error model (prop): y = f + bfe;

• combined error model (comb1): y = f + (a+ bf)e;

• combined error model (comb2): y = f + ae1 + bfe2;

• proportional error model + power (propC): y = f + bf ce

• combined error model + power (comb1C): y = f + (a+ bf c)e;

• combined error model + power (comb2C): y = f + ae1 + bf ce2.

It is also possible to specify the distribution of individual parameters (theavailable distributions are Normal, logNormal, logitNormal, probitNormaland powerNormal) and the covariance model (Fig.3.2).


3.1.2 The initialization frame

Initial values are specified for the fixed effects, for the variances of the ran-dom effects and for the residual error parameters. For the fixed effects wehave three options: the default choice is “Estimate”, in which a maximumlikelihood estimation is made, the second possibility is “Fixed” that meansthe parameter must be kept to its initial value and so, it is not estimatedand the last one is “Prior” in which can be specified a prior distribution fora parameter. Only the two options “Estimate” and “Fixed” are available forthe variances and residual error model parameters.

Moreover, the fittings obtained with the initial fixed effects are displayedfor continuous observations (Fig.3.4). It can be very useful in case of complexmodels (as in our case) in order to find some good initial values.

Figure 3.4: Monolix window to check initial fixed effects.

3.2 Executing tasks

Monolix includes several estimation algorithms: the estimation of the pop-ulation parameters, the Fisher information matrix and standard errors, theindividual parameters and the log-likelihood. Also, different types of resultsare available in the form of graphics and tables. In this section a brief pre-sentation of some of these algorithms is included.

3.2.1 The popolation parameters estimation: SEAMalgorithm

Given the observed data yi, our goal is to find the “best” parameters θ forthe model. A traditional framework to solve this kind of problem is called


maximum likelihood estimation (MLE). The likelihood is defined as

L(θ) = p(y|θ) =∫

p(y, ψ|θ)dψ (3.5)

i.e., the conditional joint density function of the observations y given theparameters θ, but looked at as if the data are known and the parameters not.The maximum likelihood estimation of the parameters consists of maximizingwith respect to θ the likelihood function L(θ):

θ = argmaxθ

L(y|θ) = argmaxθ

∫

p(y, ψ|θ)dψ . (3.6)

This maximization problem does not usually have an analytical solutionfor nonlinear models, so an optimization procedure needs to be used. Thestochastic approximation expectation-maximization (SAEM) algorithm asimplemented in Monolix has been shown to be extremely efficient for a widevariety of complex models [5, 14, 20, 29, 33, 46].

SAEM is a generalization of the expectation-maximization (EM) method.The EM algorithm is an iterative procedure that starts from a initial valueθ0 and then, at the k-th iteration, updates θEM

k−1 to θEMk with the following

two steps:

• E-step: Evaluate the quantity

QEMk (θ) = E

[

log p(y, ψ|θ)∣

∣

∣y, θEM

k−1

]

. (3.7)

• M-step: Update the estimation of θ

θEMk = argmax

θQEM

k (θ). (3.8)

Unfortunately, in the framework of nonlinear mixed effects models, there is noexplicit expression for the E-step since the relationship between observationsy and individual parameters ψ is nonlinear. The SAEM algorithm replacesthe E-step by a stochastic approximation based on a single simulation of ψ.

Then, the k-th iteration of SAEM consists of three steps:

• Simulation step: For i = 1 . . . , N draw ψki from the conditional distri-

bution p(ψi|yi, θk−1).

• Stochastic approximation: Update Qk−1(θ) according to

Qk(θ) = Qk−1(θ) + γk

(

log p(y, ψk|θ)−Qk−1(θ))

, (3.9)

where (γk) is a decreasing sequence of positive numbers with γ1 = 1.


• Maximization step: Update θk−1 according to

θk = argmaxθ

Qk(θ). (3.10)

It is shown that SAEM converges to a maximum (local or global) of thelikelihood of the observations under very general conditions [4].

Note that initial estimates must be provided by the user. Even thoughSAEM is relatively insensitive to initial parameter values, it is always prefer-able to provide as good initial values as possible to minimize the number of it-erations required and also increase the probability of converging to the globalmaximum of the likelihood. Moreover, SAEM as implemented in Monolix hastwo phases. The goal of the first is to get in a neighborhood of the solutionin only a few iterations. A simulated annealing version of SAEM acceleratesthis process when the initial value is far from the actual solution. The secondphase consists of convergence to the located maximum with behavior thatbecomes increasingly deterministic, like a gradient algorithm. See [17] forfurther information about SAEM algorithm.

3.2.2 The estimation of the Fisher information matrixand standard errors

The variance of the maximum likelihood estimate θ and, thus, its confidenceintervals can be obtained from the observed Fisher information matrix (FIM)itself, derived from the observed likelihood

Iy(θ) = −∂2 logL(θ)

∂θ2. (3.11)

The variance-covariance matrix of θ can then be approximated by the inverseof the observed FIM. Standard errors (s.e.) for each component of θ are com-puted as the square root of the diagonal elements of the variance-covariancematrix. Monolix displays for each estimated parameter its estimated relativestandard error (r.s.e.), i.e., the estimated standard error divided by the valueof the estimated parameter. In particular, a stochastic approximation algo-rithm using a Markov Chain Monte Carlo (MCMC) algorithm is implementedin the software to estimate the FIM. This method is extremely general andcan be used for many data and model types (continuous, categorical, time-to-event, mixtures, etc.). In the case of continuous data, it is also possible touse a method based on model linearization. This method is generally muchfaster than stochastic approximation and also gives good estimates of theFIM. See [17] for more information.


3.2.3 Graphics and results

From the main interface toolbar, it is possible to represent the results withseveral graphics and tables which can help to evaluate the model identifica-tion:

• spaghetti plot : displays the original data as a spaghetti plot;

• individual fits : displays the individual fits. It is also possible to displaythe median and a confidence interval estimated with a Monte Carloprocedure;

• obs. vs pred.: displays observations versus the predictions computedusing the population parameters;

• convergence SAEM : displays the sequence of the estimated parameters,

and further informative graphics (Fig.3.5). Also all the outputs of the algo-rithm are available in a results folder, see Appendix A for an example.

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Figure 3.5: Diagnostics graphics made by Monolix: spaghetti plot and fitting plot in theupper left and right corner respectively; obs. vs pred. and convergence SAEM below onthe left and on the right respectively.

Chapter 4

Model identification andvalidation

In the present section the strategy adopted to identify the new DEB-TGImodel is presented. In particular, the parameters relative to the host or-ganism growth (Tab.2.1) have been estimated with model (2.11) and thenfixed in the successive identification phases of the complete model (2.36).Such a choice has proved to be necessary because of the large number of theparameters (Tab.2.3) that could have compromised the performance of theestimation algorithm.

4.1 Dataset presentation

4.1.1 Tumor-free individual

Different growth charts relative to athymic nude mice, published by HarlanSprague Dawley, Inc. (HSD), have been analyzed to characterize the hostphysiology model (Sec.2.1). The choice of the growth chart is important sincethe model parameter estimates significantly change using a dataset ratherthan another one. Among the available growth charts, that differ for year,strain and geographic origin, the one with the most similar characteristicsto the available experimental data was chosen. In Fig.4.1 the body weightdata of the selected growth chart are shown for both male and female miceof aging 3-13 weeks.

4.1.2 Tumor-bearing individual and drug treatments

To identify the complete tumor-in-host model (2.36), we analyzed differentdatasets relative to xenograft experiments conducted on athymic nude mice.

37

4 Model identification and validation 38

Harlan Laboratories B.V., Horst

Barrier HNL-ISO

Date: March 2014

Cage type RTT (780 cm2)

15 animals per cage

continuing data of 30 animals per week

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r)Hsd:Athymic Nude-Foxn1nu

Teklad Global Rodent Diet 2918

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Growth data must be used as a guideline only.

Data can be subject to differences in maintenance of rats.

Growth chart incl. mean +/- 2 SD's for 95% confidence interval.

3 4 5 6 7 8 9 10 11 12 13

Male 14,6 22,9 25,5 28,0 30,0 30,2 31,4 32,5 33,2 33,4 34,3

Female 13,4 19,1 20,3 21,1 23,1 23,1 23,6 24,3 25,2 25,4 25,9

0

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Figure 4.1: Growth chart of HSD mice selected for tumor-free individual model identifi-cation [9].

All the available data refer to the net body weight and the tumor weight forexperiments relative to anticancer drugs, synthesized by Nerviano MedicalSciences, Nerviano, Italy. Therefore, each measure is the average of a groupof different mice.

More precisely, the datasets refer to 9 experiments conducted on athymicnude mice (male or female) with 5 or 6 weeks of age, bearing human cancercells of three different tumor lines (A2780, A375, HTC116). Each experimentis composed of a control group, in which the growth was monitored withoutany drug administration, and one or more treated groups that can differ forscheduling, dose or administered drug. Mainly, the data refer to 13 drugs(denoted below with the capitol letters A-O) for a total number of 33 differenttreated groups. A summary of the dataset information is reported in Tab.4.1and Tab.4.2.

Because of the impossibility of reporting the analysis for all the availabledatasets, below we will only describe the results obtained for three of them.In particular, we will focus on Experiment 1, Experiment 6 and Experiment9 with drug I, including in this way data relating to both male and femalemice and to three tumor cell lines. The information about doses and drugadministration schedules are reported in Tab.4.3, Tab.4.4 and Tab.4.5. In allthe three Experiments the Group 1 is the control group.


Experiment Tumor cell line Sex AgeExp 1 A2780 female 5 weeksExp 2 A2780 female 6 weeksExp 3 A2780 female 5 weeksExp 4 A2780 female 5 weeksExp 5 A2780 female 6 weeksExp 6 HTC116 male 5 weeksExp 7 A375 male 5 weeksExp 8 A375 male 5 weeksExp 9 A375 male 5 weeks

Table 4.1: Dataset information: tumor cell line, sex, age.

Exp. Treated A B C D E F G H I L M N Ogroups

Exp 1 1 1 - - - - - - - - - - - -Exp 2 1 - 1 - - - - - - - - - - -Exp 3 4 - - 2 2 - - - - - - - - -Exp 4 6 - - 2 - 2 2 - - - - - - -Exp 5 8 - - - - - - 6 2 - - - - -Exp 6 1 - - - - - - - - - - - - 1Exp 7 6 - - - - - - 2 - 3 - - 1 -Exp 8 2 - - - - - - - - 1 - 1 - -Exp 9 4 - - - - - - - - 3 1 - - -

Table 4.2: Dataset information: drug. Drugs A-F and Drug O are already on the market:A=Taxolo, B=Doxorubicina, C=Vincristina, D=Vinblastina, E=Gemzar, F=Ciplastin,O=5FU

4.2 Data analysis

PK and PD models were implemented in Monolix v.4.3.3, see Sec.3 for somefurther details about this tool and Appendix A for an example of the modelscript. The concentration profiles required by the new model were simulatedfor each experiment using the PK parameters estimated in single drug studiesand reported in Tab.4.6.

Experimental data were analyzed adopting the following strategy. First,the physiological parameters (Tab.2.1) were estimated by fitting the tumor-free model (2.11) against the male and the female subjects growth chart data(Fig.4.1). In particular, the parameters ν, g, V1∞ were estimated. Indeed, theparameter m was derived by the algebraic relationship (4.1) and the fractionof food consumption ρ was fixed to 1 according to the indications for an ad

libitum food feeding reported in [43].

m =ν

V1/31∞ g

. (4.1)


Drug Groups Dose Route Days of treatmentDrug A Group 2 30 mg/Kg i.v. 8,12,16

Table 4.3: Information about doses and schedules relative to Experiment 1 (Drug A).

Drug Groups Dose Route Days of treatmentDrug O Group 2 50 mg/Kg i.v. 8,15,23,29

Table 4.4: Information about doses and schedules relative to Experiment 6 (Drug O).

The density of structural biomass dV was fixed to 1 g cm−3. Nevertheless,performing the identification of the tumor-free model without fixing any otherparameter, high r.s.e. and high correlation between parameters estimatesarise (e.g., between ξ, V1∞ and e0). For these reasons, it have been decidedto exploit biological information gathered from literature.

In particular, the mice birth has been chosen as the initial time from whichto start; the initial weightW0 was fixed to 1 g in according with experimentaldata reported in [7]. As regards the energy initial value e0, many studiesabout body composition of mice [6, 16, 23, 24] show that the percentage ofbody fat is extremely low in early life stages of mice; so, approximating theenergy reserve with the body fat, we have fixed e0 = 0.

Another choice consisting in fixing the parameter ξ. As reported in (2.10),ξ is given by the expression

ξ =dE[Em]

dV rE

where dV and dE represent the volume specific weight of structural biomassand reserve respectively, [Em] is the maximum reserve density for unit ofvolume and rE is defined as amount reserves

volume reserves. So, always approximating the

stored energy with the body fat, it can be supposed that the fraction [Em]/rErepresents the body fat percentage and, thus,

ξ = fat density × body fat percentage . (4.2)

Using the values of the body fat percentage and fat density reported in[31, 21], the parameter ξ was fixed to 0.2116 for male mice and to 0.184 forfemale mice.

Finally, initial volume V0 can be derived from

V0 =W0

1 + e0ξ. (4.3)

Once identified the tumor-free individual model, we have used the es-timated values of the parameters to simulate the profile of the structural


Drug Groups Dose Route Days of treatmentDrug I Group 2 30 mg/Kg i.v. 10,11,12,14,15,16 (bid)Drug I Group 3 60 mg/Kg i.v. 10,11,12,14,15,16,30,31,32,34,35,36 (bid)Drug I Group 4 60 mg/Kg i.v. 10,11,12,13,14,15 (bid)

Table 4.5: Information about doses and schedules relative to Experiment 9 (Drug I).

Drug V1 k10 K12 K21

[ml/kg] [1/day] [1/day] [1/day]Drug A 813.1812 20.82801 0.1451209 2.010171Drug I 362.6153 134.4873 31.34683 16.21491Drug O 713.8977 151.2248 5.619443 2.312445

Table 4.6: PK parameters of anticancer drugs A, I, O.

biomass, the energy reserve and the total body weight using Simulx1. Thisstep has allowed us to identify a good value for the energy reserve at thebeginning of the experiment. So, unlike the approach followed by Leuweenet al., the parameter e0 was fixed to the value predicted by the simulationfor the energy reserve in mice of 5/6-weeks of age2.

Once we fixed the tumor-free model parameters and the initial valuee0 to the estimated values, the fitting of the tumor-bearing model againsteach experiment was performed. In particular, with the datasets reportedin Tab.4.1 and Tab.4.2 we have two different analysis. First, we consideredonly the data relative to the control groups identifying the tumor-relatedparameters (µu, gu, mu), VdegMax and the initial conditions Vu10 and W0

3.This step has allowed us some considerations upon the parameters relativeto the tumor growth and also has brought out some problems of the model.Once solved these issues, the complete model was able to describe both theuntreated and the treated data: so, for each experiment, we fitted control andtreated arms at the same time identifying simultaneously the tumor-relatedparameters and the drug-related parameters (Tab.2.3). In every case, thethermodynamic efficiency coefficient ω was fixed to 0.75 as in literature4.

For the identification of both models, we had to specify the residual stan-dard error model: the proportional standard error for the tumor-free modeland the proportionalC standard error for the tumor-in-host model were the

1Simulx is an R function for easily computing predictions and simulating data fromMlxtran models.

2In [42] the parameter e0 has been fixed always to 1 justifying this choice by the adlibitum feeding of the animals.

3Now the parametersW0 represents the mice weight at the beginning of the experiment.4The results reported in [42] was obtained fixing the parameter ω to 0.75 as well as 0.5.


best models among all those we have examined5.

4.3 Results

Considering the growth data of male and female HSD mice (Fig.4.1), thephysiological parameters of the tumor-free model were estimated. In partic-ular, as described in Sec.4.2, the estimation was limited to g, ν and V1∞ sinceof identification problems. The obtained parameter values together with theirr.s.e. are reported in Tab.4.7. The body weight predictions together withthe experimental data are shown in Fig.4.2.

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Figure 4.2: Experimental data and body weight prediction of male (graphic on the left)and female (graphic on the right) HSD mice obtained from tumor-free model identification.

Parameter g ν V1∞

- cm/week cm3

Values 15 8.82 31.2r.s.e(%) 6% 4% 3%

Parameter g ν V1∞

- cm/week cm3

Values 12.2 8.57 22.6r.s.e(%) 8% 4% 4%

Table 4.7: Physiological parameters estimates of the tumor-free model identified on male(left table) and female (right table) HSD mice.

The obtained physiological parameters were used to perform a simulationof the state variables, V (t), e(t) and W (t), identifying reasonable value forthe energy reserve in mice of 5/6-weeks of age. The profiles obtained for thestructural biomass, the energy reserve and the total body weight are reportedin Fig.4.3. From the simulation, the parameter e0 that represents the energyreserve value at the beginning of the experiment, was fixed to 1.4592 for malemice of 5 weeks of age, to 1.3 and to 1.2164 for female mice of 5 and 6 weeksof age, respectively.

So, to perform the DEB-TGI model identification against each exper-iment, the physiological parameters were fixed to the values reported inTab.4.8 (after unit conversion).

5For the tumor-in host model the residual standard error was modeled as y = f+b√fe,

see Sec.3.1.1 for more information.


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Parameter g ν V1∞ ξ m e0 ρ- cm/day cm3 - 1/day - -

Exp 1-3-4 12.2 1.2243 22.6 0.184 0.0355 1.3 1Exp 2-5 12.2 1.2243 22.6 0.184 0.0355 1.2164 1Exp 6-7-8-9 15 1.26 31.2 0.2116 0.0267 1.4592 1

Table 4.8: Physiological parameters of the tumor-free model fixed in the next steps of themodel identification.

In some control group experiments, the model ((2.31),(2.32) and (2.33))was able to describe correctly the data for both the mice and the tumorgrowth (Fig.4.4 panels on the top). In other experiments two different issuesemerged: in some cases the model showed an initial underestimation of themice weight (Fig.4.4, central panels), in others the fitting was good, but thetumor-relative parameters gu and mu do not respect the conditions reportedin Sec.2.2 (Fig.4.4 panels on the bottom).

Analyzing these problematic cases and, in particular, the Experiment6, we saw that the inadequacy of the fitting is often accompanied by animportant mice weight loss despite a slow growth of the tumor. So, wesupposed that this decrease in the body weight is justified not only by thereduction of the available energy due to the tumor, but there are some otherhidden causes. For these reasons, we guessed that in these experiments theanimals were afflicted by a reduced appetency or, more in general, by a low


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Figure 4.4: Problems arisen in the identification of the tumor model against the controlgroups. On the top the fittings relative to Experimental 1: the model is able to correctlydescribe the data. In the center, the results of the Experiment 6: an initial underestimationof the mice weight is shown. On the bottom, the graphics relative to Experiment 2:although the fitness of the fitting, the values gu = 13.2 and mu = 0.0521 violate therestrictions gu ≤ g and mu ≤ m.

assimilation capacity due to experimental conditions or to the tumor. Thus,to include the new considerations in the model, we decided to estimate alsothe parameter ρ representing the feeding condition. Not fixing any morethe ρ parameter, the model described adequately the data coming from allthe control groups and also the biological conditions upon the parameters guand mu (gu ≤ g and mu ≤ m) were always satisfied. Furthermore, in theseexperiments that did not showed problems, the parameter ρ was estimatedequal or very close to 1, so in these cases it was possible to fix it. Theobtained parameter estimates and their r.s.e. are reported in Tab.4.9. In Fig4.5 the resulting body weight and the tumor weight predictions together withthe experimental data are reported: note that there is no more the initialunderestimation of the mice weights.

Two important considerations arising from the results obtained by theidentifications of the control groups are: first, as can be seen from Tab.4.9,the estimated values of relative-tumor parameters µu, gu and mu are verysimilar for the experiments involving the same tumor cell line. Althoughthe number of cases is not high enough to draw a final conclusion, the data


Exp Tumor line µu gu mu VdegMax W0 Vu10 ρ- - 1/day - cm3 g -

Exp1 A2760 13.8 11.8 0.0193 0.179 20.8 0.00192 113% 15% 115% 17% 6% 27% 10%

Exp2 A2780 13 11.1 0.0289 0.0721 23.5 0.00555 0.74342% 233% 130% 80% 56% 392% 234%

Exp3 A2780 13.6 11.5 0.0304 0.102 23.2 0.00211 114% 24% 170% 63% 14% 50% 25%

Exp4 A2780 12.3 11.6 0.018 0.182 24.6 0.00529 0.757135% 267% 983% 43% 81% 511% 203%

Exp5 A2780 13.4 12 0.0169 0.195 27.4 0.00377 1174% 223% 103% 35% 29% 127% 60%

Exp6 HCT115 8.5 11 0.0126 0.397 34.7 0.0231 0.439255% 328% 103% 14% 30% 231% 175%

Exp7 A375 5.32 9.9 0.00887 1.47e-17 29.7 0.016 0.98244% 396% 103% 1013% 16% 45% 27%

Exp8 A375 4.58 11.3 0.0104 1.96e-17 30.5 0.0265 0.863514% 826% 103% 1015% 12% 66% 27%

Exp9 A375 4.51 12.4 0.014 0.0734 31.5 0.0183 1139% 266% 103% 76% 7% 37% 10%

Table 4.9: PD model parameters estimated for the control groups of the Experiments 1-9.

analyzed suggest that these parameters are tumor cell line dependent.Furthermore, estimating the parameter ρ resulted in a small change in the

function ρ(t) defined in the Sec.2.3.1. Indeed, the expression of ρ(t) reportedin (2.30) assumes that the coefficient of food consumption ρ is equal to 1 whenthe tumor is not treated. Then, to include the reduced assimilation capacitynot due to the drug administration but to the experimental condition, wehave modified the function ρ(t) as

ρ(t) = ρb

(

1− c(t)

C50 + c(t)

)

. (4.4)

The coefficient ρb represents the basal food consumption and takes valuebetween 0 and 1: it will be close to 1 if, in absence of treatments, thereis an ad libitum feeding condition, otherwise it will be less than 1 if thehost organism shows a reduction of the assimilation capacity. During theidentification of the complete model ((2.34),(2.35) and (2.36)) we use thenew expression for ρ(t), fixing ρb = 1 if in the fitting of the control groupthe parameter ρ, are estimated close to 1; otherwise ρb is estimated from thedata.


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(g)

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ght (

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Figure 4.5: Body weight (on the left) and tumor growth (on the right) profiles togetherwith experimental data for all the control groups. From the top to the bottom the graphicsare relative to Experiments 1-9.


The complete model ((2.34), (2.35) and (2.36)) was identified on theexperimental data relative to Drug A in the Experiment 1. The model wasable to fitting simultaneously the untreated and the treated arms; besides,the estimates of the tumor-relative parameters are comparable with thoseestimated only on the control group data and they satisfy the biologicalconditions too. In Fig.4.6 the simulated concentration profile is reported;the body weight and the tumor growth profiles with experimental data areshown in Fig.4.7; the obtained parameter estimates are reported in Tab.4.10together with their r.s.e..

µu gu mu VdegMax Vu10 W0 ρb k1 k2 C50

- - 1/day - cm3 g - 1/day ml/mgday mg/ml13.6 11.6 0.022 0.182 0.00186 21.1 - 0.401 686 4.64e-0747% 50% 404% 10% 75% 2% - 90% 24% 49%

Table 4.10: PD model parameters estimates for Experiment 1 Drug A.

0.00

0.01

0.02

0.03

0 10 20 30 40 50Time(day)

Con

cent

ratio

n(m

g/m

l)

Figure 4.6: Simulated concentration profile of the drug A given i.v. at 30mg/kg q4dx3 tothe treated group of Experimental 1. Administration protocols are reported in Tab.4.3.

In this case, according to the control fitting, the parameter ρb was fixedto 1. The decrease in body weight correspondence of the treatment is welldescribed by the model prediction (Fig.4.7, left panels) thanks to the caloricrestriction modeled through ρ(t) (4.4). Besides, the model correctly describesthe tumor growth inhibition as shown in Fig.4.7, right panels.

An analogous evaluation was performed on the data relative to Drug O inthe Experiment 6. Relying on the analysis of the control group, in this casethe parameter ρb has been estimated. With this measurement, the DEB-TGI model fits simultaneously the mice body weight and the tumor growthof tumors in both untreated and treated arms as in the last experiment. Theobtained tumor-related parameter and drug-related parameter estimates are


8 9 10 11 12 13 14 15 16 170

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ght (

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ght (

g)

Figure 4.7: Experiment 1, Drug A: in the left panels the obtained body weight profilestogether with experimental data are reported for the controls (on the top) and for thetreated group (on the bottom) of Experiment 1. In the right panels the correspondingtumor growth profiles together with experimental data are shown.

reported in Tab.4.11: note that in this case the estimates are affected byhigh uncertainty. The simulated concentration profile of the administereddrug and the fitting of the host body weight and the tumor growth data areshown in Fig.4.8 and Fig.4.9 respectively.


- - 1/day - cm3 g - 1/day ml/mgday mg/ml8.59 9.97 0.00431 0.239 0.00183 32.6 0.478 0.546 1320 8.95e-07131% 171% 103% 11% 106% 14% 86% 80% 41% 333%

Table 4.11: PD model parameters estimated for Experiment 6 Drug O.


0.00

0.02

0.04

0.06

0 10 20 30 40 50Time(day)

Con

cent

ratio

n(m

g/m

l)

Figure 4.8: Simulated concentration profile of Drug O given i.v. at 50mg/kg q7dx4 to thetreated group of Experiment 6. Drug administration protocols are reported in Tab.4.4.

8 10 12 14 16 18 20 220

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ght (

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10 15 20 25 30 350

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ght (

g)

Figure 4.9: Experiment 6, Drug O: in the left panels the obtained body weight profilestogether with experimental data are reported for the control group (above) and for thetreated group (below). In the right panels the corresponding tumor growth profiles to-gether with experimental data are shown.


Finally, we report the results regarding the model identification with thedata relative to Experiment 9, Drug I. In this case the DEB-TGI based modelfits simultaneously not only the control arm and a single treated arm, butalso the untreated and all the three treated groups together. As for theExperiment 1, the parameter ρb was fixed to 1. The obtained tumor-relatedparameter and drug-related parameter estimates are reported in Tab.4.12;the simulated concentration profiles for all the single arms and the fitting ofthe host body growth and the tumor growth data are shown in in Fig.4.10and Fig.4.11, respectively.

0.00

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cent

ratio

n(m

g/m

l)

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cent

ratio

n(m

g/m

l)

0.00

0.05

0.10

0.15

0 10 20 30 40 50Time(day)

Con

cent

ratio

n(m

g/m

l)

Figure 4.10: Simulated concentration profile of the Drug I for each single agent arm ofthe Experiment 9: concentration for drug I given i.v. at 30mg/kg 1-3bid for 2 cycles inGroup 2 (left panel), concentration for Drug I given i.v. at 60mg/kg 1-3bid for 4 cyclesin Group 3 (central panel), concentration for Drug I given i.v. 60mg/kg qodx64 in Group4 (right panel). Drug administration protocols are reported in Tab.4.5.


- - 1/day - cm3 g - 1/day ml/mgday mg/ml4.16 12.1 0.000171 0.138 0.0185 31.8 - 29.6 60.5 0.00010124% 23% 103% 28% 22% 1% - 103% 11% 35%

Table 4.12: PD model parameters estimated for Experiment 9 Drug I.

It is important to notice that the fitting of the control and the treatmentarms was simultaneous, which is indicative of consistency of the drug-relatedparameters k1, k2 and C50, across different dose levels. This features of themodel is confirmed by the results reported in Appendix B: in the experimentsreported here the model was able to fit data coming from control and sevendifferent treated groups with only a single set of parameters. Also in thesecases the goodness of the simultaneous fitting confirms the independence ofthe drug-related parameters from the administration protocol, supportingthe use of the model for prediction purpose.

To validate the within-experiment predictivity of the model, the treat-ment arm relative to Group 4 has been intentionally omitted from the dataset


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ght (

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Figure 4.11: Experiment 9, Drug I: on the left the obtained body weight profiles togetherwith experimental data are reported for the control group (above) and for the three treatedgroups (below). On the right the corresponding tumor growth profiles together withexperimental data are shown.

of the Experiment 9, Drug I, and the model was fitted simultaneously withthe data of the control arm and with only two of the active treatments (Group2 and Group 3). The results of the PK-PD modeling were good (Fig.4.12):the time courses of the mice body weight and of the tumor weight were welldescribed (Fig.4.12), and the model parameter values were estimated closeto the obtained from the complete dataset (Tab.4.12 and Tab.4.13).

Using these parameter values and the appropriate pharmacokinetic pro-files (Tab.4.5), the tumor weight curves for the remaining treatment was sim-ulated with Simulx. As expected, predicted mice body weights and tumorgrowth rates resulted in excellent agreement with the observations (Fig.4.13).This example demonstrates the across-experiments predictive power of themodel using different administration scheduling.


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Wei

ght (

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Figure 4.12: Experiment 9, Drug I without Group 4: on the top the obtained body weightand the tumor weight profiles together with experimental data are reported for the controlgroup; below the corresponding profiles together with experimental data are shown for thetreated Groups 2 and 3.


- - 1/day - cm3 g - 1/day ml/mgday mg/ml4.11 12.1 0.000258 0.131 0.0202 31.7 - 55.5 68.3 0.00012728% 26% 103% 48% 30% 2% - 103% 14% 55%

Table 4.13: PD model parameters estimated for Experiment 9 Drug I without Group 4.

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ght(

g)

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ght(

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Figure 4.13: Observed data and model predicted curves of the mice body weight (on theright) and the tumor growth (on the left) relative to the Group 4 of Experimental 9 withDrug I.

Chapter 5

TGI model and DEB-TGImodel

The van Leeuwen model [43] does not require a priori assumptions on theshape of the tumor growth curve but it depends on the relative values of thetumor and the host parameters. In particular, if the relation mugu = µumg isverified, the tumor grows according to a S-shaped saturating growth pattern;this condition marks the bifurcation between a nonsaturing tumor growing(mugu < µumg) and a tumor dying off (mugu > µumg).

With the introduction of the maximum degradation rate VdegMax (Sec.2.3.2),the new DEB-TGI model predicts a S-shaped saturating pattern of the tumorfor any value of the parameters for which it is not expected tumor death. So,as in the Simeoni TGI paradigm and in according to the data evidence, thetumor follows an exponential growth in the early phase of its development,then it shows a slowdown in its growth until reaching a plateau (Fig.5.1).Due to the similarities between the tumor growth curve of the DEB-basedmodel and of the Simeoni TGI model, a comparison of the two models hasbeen made. In particular, we were interested to answer these two questions:

1. Is there any correlation between the Simeoni tumor-related parameters(λ0 and λ1) and those of the DEB-TGI model?

2. Is the switch from exponential to linear phase of the Simeoni modelrelated to the degradation process predicted by the new model? Inwhich stage of the DEB-based model (dV (t)/dt > 0, dV (t)/dt = 0,dV (t)/dt < 0) falls this switch?

53

5 TGI model and DEB-TGI model 54

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or w

eigh

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−6

−4

−2

0

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log(

dVu(

t)/d

t)

−6

−4

−2

0

0 10 20 30 40Time(day)

log(

dVu(

t)/d

t)

Figure 5.1: On the top the simulated profiles of the tumor growth are reported for theSimeoni model (left panel) and for DEB TGI model (right panel). On the bottom therespective time derivatives in semilogarithmic scale are shown.

5.1 Exponential and linear phase of

the tumor growth

To perform a comparison of the tumor growth predicted by the Simeoni TGImodel and the DEB-TGI model we have focused on the untreated case. Soin the next sections we will refer to the model formulation reported in Fig.1.4on the left panel for the Simeoni model and to the systems (2.31), (2.32) and(2.33) for the DEB-based model.

The new tumor-in-host model consists of three different phases:

• dV (t)/dt ≥ 0 or Phase 0: there is not degradation of structural biomassand the model is described by the equations (2.31);

• 0 < dV (t)/dt ≤ −VdegMax or Phase 1: it ranges from the beginning ofstructural biomass degradation dV (t)/dt = 0 to the instant in whichthe degradation rate reaches its maximum value (system (2.32));

• dV (t)/dt > −VdegMax or Phase 2: the structural biomass is degradedwith the maximum constant rate VdegMax and the model is given by(2.33).

The simulation of the model has showed that the transition through thesethree phases follows a precise temporal trend, as suggested by biological con-


siderations. In the early phases of the tumor development, the host organismis able to satisfy both its energy demands and those of the tumor, so it keepsgrowing (Phase 0, dV (t)/dt ≥ 0). When the tumor grows, its energy demandincreases, consequently the host growth becomes slower until an instant t1in which it stops (dV (t1)/dt = 0) and the organism is forced to degradestructural biomass to fulfill the increased energy requirements (Phase 1).Finally, if the tumor growth is not hindered by drug treatment, the hostcontinues to consuming biomass with a increasing rate until an instant t2, inwhich the degradation rate reaches its maximum threshold VdegMax (Phase 2,dV (t2)/dt = −VdegMax). The Fig.5.2 shows the simulation profiles for V (t),e(t) and Vu(t) and for the respective time derivatives: the two marked pointsindicate the switch from Phase 0 to Phase 1 and from Phase 1 to Phase 2,respectively.

16.5

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uctu

ral b

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ass,

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t

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0 5 10 15 20Time(day)

de/d

t

0.0

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dVu1

/dt

Figure 5.2: Simulation profiles for structural biomass, energy and tumor mass performedwith the parameters identified against Experiment 1 are reported on the bottom. Belowthe respective time derivatives are shown. In all the graphics the first marked pointindicates the switch from Phase 0 to Phase 1, the second the transition from Phase 1 toPhase 2.

Imposing the continuity of the system, it is possible to characterize thepoints of switch between the three stages of the model; in particular at theinstant time t1 the relation

Vu(t1) =νe(t1)V (t1)

2/3

gmµu

− V (t1)

µu

(5.1)


marks the transition from Phase 0 to Phase 1, while at the instant t2 thefollowing relation between Vu(t), V (t) and e(t) holds

Vu(t2) =(νV (t2)

2/3 + VdegMax)e(t2) + VdegMaxωg

gmµu

− V (t2)

µu

. (5.2)

In according with these considerations, in order to analyze the tumorgrowth in the early stages of its development, we have focused on Phase 0considering the model (2.31). In particular, imposing the condition Vu = 0,the follow equilibrium point

V = Vρ∞ = ρV1∞ , e = ρ1/3 , Vu = 0 (5.3)

has been calculated. We have linearized the model in a neighborhood of thesteady state (5.3) computing the Jacobian matrix and evaluating it in thepoint (5.3):

− ν

ρ1/3V1/31∞

−2

3

ν

ρV4/31∞

−2

3

ν

ρV4/31∞

νρ2/3V2/31∞

g + ρ1/3− ν

3V1/31∞ (g + ρ1/3

− µuν

V1/31∞ (g + ρ1/3)

0 0mgµu

gu−mu

. (5.4)

From the linearized system analysis, two considerations could be made. First,when the tumor mass is close to 0, the tumor growth is independent fromthe structural biomass V and the energy reserve e. Second, considering thesubmatrix of (5.4)

− ν

ρ1/3V1/31∞

−2

3

ν

ρV4/31∞

νρ2/3V2/31∞

g + ρ1/3− ν

3V1/31∞ (g + ρ1/3

, (5.5)

we note that its trace is always negative for biologically relevant values ofthe parameters (ρ > 0, ν > 0, g > 0 and V1∞ > 0). For this reason, thestability of the equilibrium point (5.3) is determined only by the sign of theeigenvalue (mgµu/gu) − mu. In particular, the steady state is unstable ifmgµu > gumu and it is asymptotically stable if mgµu < gumu. From this


observation it follows that the relation mgµu = gumu marks the bifurcationbetween tumor growing (mugu < µumg) or tumor dying off (mugu > µumg)as in the van Leeuwen model [43].

Since the inoculated tumor mass is negligible at the beginning of theexperiment, the previous remarks allow to approximate the derivative dVu/dtwith its linearization in a neighborhood of Vu = 0, that is

dVu(t)

dt≈(

mgµu

gu−mu

)

Vu(t) . (5.6)

In conclusion, in the early phases of the tumor development, it is possibleto approximate the tumor growth with an exponential growth in which therate is given by the parameter combination (mgµu/gu)−mu.

Let’s consider now the system (2.32) that describe the host and the tumorgrowth during Phase 1. As we have seen in Sec. 2.2, in this case the sub-stitution of the expression (2.12) for ku(t) has allowed to rewrite derivativedVu/dt as

dVu(t)

dt=

(

mgµu

gu−mu

)

Vu(t) . (5.7)

So, also in the second phase the tumor growth is independent from the struc-tural biomass V and the energy reserve e and, moreover, the tumor growsexponentially with the same rate (mgµu/gu) − mu that characterized theearly stages of Phase 0.

The previous remarks suggest that the tumor growth is approximable withan exponential growth during all the Phase 0 and the Phase 1. Furthermore,the exponential rate is the same for the two stages, that is

λ0 =mgµu

gu−mu . (5.8)

For this reason, we have hypothesized a relationship between the parameterλ0 of the Simeoni TGI model and the parameter combination (5.8).

5.2 A numerical comparison of λ0 and λ0

To check the correspondence between λ0 and λ0 hypothesized in the previoussection, a numerical analysis has been carried out. In particular, because wewere interested to put in comparison the two models for the control group


abstracting from the estimation errors, we adopted the following strategy.First, we generated several datasets with the tumor weights simulated withthe DEB-TGI model using the parameters reported in Tab.4.9. Then, thesedata were used to identify the Simeoni TGI model and in particular to esti-mate the parameter λ0.

In Fig.5.3 the tumor weight data simulated by Simulx with the param-eters related to the control group of the Experiment 1 (Tab.4.9) and thecorresponding fitting with the Simeoni TGI model are reported. We have re-peated these steps for the control group of all the experiments; the values ofthe parameter λ0 estimated upon the simulated data are reported in Tab.5.1together with the corresponding λ0.

0

2

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6

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Tum

or w

eigh

t (g)

0 2 4 6 8 10 12 14 16 180

1

2

3

4

5

6

Time (day)

Wei

ght (

g)

Figure 5.3: Tumor weight data simulated with the DEB-TGI model and the parametersrelated to Experiment 1 (Tab.4.9) (left panel) and the corresponding fitting with theSimeoni TGI model (right panel).

Experiment Tumor line λ0 r.s.e. λ0

Exp1 A2780 0.497 2% 0.487Exp2 A2780 0.354 2% 0.478Exp3 A2780 0.464 2% 0.481Exp4 A2780 0.373 2% 0.441Exp5 A2780 0.432 2% 0.467Exp6 HTC116 0.216 2% 0.27Exp7 A375 0.221 1% 0.21Exp8 A375 0.148 1% 0.152Exp9 A375 0.139 1% 0.134

Table 5.1: Estimates of the parameter λ0 and their r.s.e. obtained with the data simulatedwith the DEB-TGI model together with the corresponding λ0.

From the results in Tab 5.1, we can conclude that the parameter λ0 andthe secondary parameter λ0 assume quantitative comparable values provingso a further evidence that the parameter combination in (5.8) has the samerule of λ0 in the DEB-TGI model.


5.3 Analysis of the Simeoni switch from the

exponential to the linear phase

Finally, we were interested to characterize the switch point from the exponen-tial to the linear phase of the Simeoni model in terms of the three DEB-TGImodel stages. For this purpose, first we simulated the tumor weight curvespredicted by the DEB-based model with the parameters reported in Tab.4.9.On these profiles we located the tumor weights at the times t1 and t2 and theweight threshold, Wth, that marks the transition to the exponential to thelinear growth in the Simeoni model1. In Fig.5.4 the results of the simulationare shown.

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Tum

or w

eigh

t (g)

0

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4

6

0 10 20 30 40Time(day)

Tum

or w

eigh

t (g)

Figure 5.4: The tumor weight profiles predicted by DEB-TGI model with the parametersin Tab.4.9 are reported for the Experiment 1-9 from the top left corner to the bottom rightcorner. In each panel the black marked points are the tumor weights in the two switchpoints of the DEB-based model (in the last two panels the points are overlapping due tothe numerical approximation); the red marked point is the Simeoni weight threshold.

1The value of the weight threshold, wth = λ1/λ0, was obtained from the parametersestimated on the datasets simulated by DEB-TGI model.


This analysis, even if it does not provide a rigorous proof, shows thatthe transition point from the exponential to the linear phase of the Simeonimodel falls in the Phase 2 of the DEB-TGI model for all the experiments.This means that the change in the tumor growth predicted by the TGI modelhappens always once the structural biomass degradation reached its maxi-mum rate VdegMax and then it is due to the limited availability of energy fortumor growth.

From the remarks reported in this section we can conclude that also theDEB-based model predicts an exponential growth of the tumor in the earlyphases of its development. The relationship between the parameter λ0 ofthe Simeoni model and the parameters of the DEB-TGI model is given by(5.8). Furthermore, because the DEB-based model parameters have a bio-logical meaning, the relation (5.8) shows that the growth rate of the tumor isinfluenced not only by the tumor characteristics, but also by the condition ofthe host organism. After the instant time t2, the DEB-TGI model predictsa slowdown in the tumor growth. Indeed, if during Phase 1 the increasingenergy demands of the tumor are offset by an increasing structural biomassdegradation rate, in Phase 2 the organism degrades biomass with a constantrate and so the greater and greater energy demands of the tumor remainsunfulfilled. This slower growth is approximated and described by the Sime-oni model with the linear phase in according with its minimalist approach.The Simeoni switch from the exponential to the linear phase has often beencriticized because not supported by a specific biological justification; in thiscontext, showing that the switch point falls always in the Phase 2 of theDEB-TGI model is of considerable relevance. Indeed, we showed that theenergy balances underlying the new mechanistic model lead to the same re-sults of the simpler Simeoni model justifying its empirical assumptions withthe mutated interaction between tumor and host organism.

Chapter 6

Conclusions

One important challenge for oncology model-based drug development is theinclusion of interactions between tumor and host organism into mathematicalmodels used to describe data coming from in vivo experiments for testing thetumor growth inhibition effect of anticancer drugs. To this aim Leeuwen etal.[42] proposed a tumor-in-host model, based on the DEB theory. Thismodel is able to describe the energetic interactions between tumor and host,but without considering the possibilities of an anticancer drug treatment.Therefore, starting from the van Leeuwen model, a new tumor-in-host DEB-based model has been developed. In particular, the new model is able todescribe the pharmacological effect of anticancer treatments in xenograftmice by including the mortality chain of the Simeoni TGI model.

The proposed model has been validated on nine experiments involvingthirteen different drugs (reported in Sec.4 and in Appendix B), showing agood capability in describing both tumor growth and host body growth evenwhen an anticancer treatment is administered. In particular, the model hasproven to be reliable enough to describe the effect of different dosing regimensin different cell tumor lines. Also, the example concerning Experiment 9,Drug I suggested that, given the control and some treatment arms, the modelcould be able to simulate other treatment arms with good accuracy showingthe predictive power of the model.

Obviously, because of its complexity, the model suffers from some iden-tifiability problems, as pointed out by the high uncertainty that affects theparameter estimates. On the other hand, thanks to the mechanistic andphysiology-based hypotheses on which it is based, the model is characterizedby biologically meaningful parameters. In this context, the new DEB-basedmodel provides a further validation of the empirical assumptions of the Sime-oni model, as reported in Sec.5. Moreover, a physiological meaning has beengiven to the transition from the exponential to the linear phase for a specific

61

6 Conclusions 62

weight threshold of tumor cells, giving more importance to the Simeoni TGIfunction.

The new TGI DEB-based model also provides a quantitative measure-ment of the side effects on the organism due to the anticancer treatment.Indeed, if the tumor growth inhibition effect of the drug is quantified by theparameter k2, the parameter C50 can be conceived as a negative potency ofthe drug linked to its toxicity.

6.1 Advantages of a mechanistic approach

As all functional models, the DEB-TGI model is based on mechanistic andphysiological hypotheses. In particular, the tumor growth and the host bodygrowth predicted by the model are the result of energy balances underlyingthe tumor-host interactions. Then, all the parameters of the model are char-acterized by a precise biological meanings. The tumor-related parameters(Tab.2.2) seem to be especially significant from a biological point of view.Indeed, as we have seen in Sec.4.3, they show a strong dependency on thecell tumor line and so they could provide a valuable support to describe thetumor characteristics.

For these reasons, a sensitivity analysis of the model to the tumor-relatedparameters has been performed. In particular, the tumor gluttony coefficientµu seems to be especially relevant for the tumor growth. In Fig.6.1 the modelpredictions for the tumor and the host body weight are reported for differentvalues of µu in absence of anticancer treatment1.

The Fig.6.1 shows that for high values of µu the tumor reaches greaterweights faster. This observation is confirmed by the results reported in Sec.4:the three cell tumor lines are characterized by different values of µu (Tab.4.9);in particular, the estimated value of the parameter is higher for the experi-ments in which the tumor reaches a higher weight. From these observationswe can conclude that the parameter µu can be considered as an index oftumor aggressiveness.

6.2 A quantitative measurement of the drug

toxicity

Testing the safety of the drug compounds is a very important issue in in

vivo preclinical experiments. The new tumor-in-host model could provide

1The results of the sensitivity analysis of the model for the parameters gu and mu arereported in Appendix C.

6 Conclusions 63

Figure 6.1: Sensivity analysis of the DEB-TGI model for parameter µu: the host bodyweight (left panel) and the tumor (right panel) predicted by the DEB-based model usingthe parameters identified upon the control group of Experiment 1 varying µu from 1 to20.

a valuable tool for evaluating the toxicity of an anticancer treatment. In-deed, since the model includes the action of the drug treatment on the hostorganism (Sec.2.3.1), it takes into account the possible drug side effects onthe organism when an anticancer compound is administered. In particular,a quantitative measurement of the negative potency of the drug is providedby the parameter C50.

Furthermore, we have attempted to define an index of the drug toxicity.To this aim, the area under the curve described by

c(t)

C50 + c(t)(6.1)

was estimated for the treated groups of all the experiments. It representsthe non-assimilation due to the side effects of the drug administration. Inparticular, we estimated the AUC on the time interval between the beginningof the experiments (t = 0) and the time instant in which the concentrationbecome not detectable, divided by the number of the days. The resultsobtained for the experiments discussed in Sec.4 are reported in Tab.6.1 (thecomplete table of the estimated values are included in Appendix B).

To evaluate the validity of the drug toxicity index (DTI) proposed, itmight be interesting to cross the results obtained for DTI with the infor-mation available on the toxicity of the anticancer compounds tested in theexperiments.

The previous remarks underline the importance of an adequate descrip-tion of the drug effects on the host organism. For this reason, starting from

6 Conclusions 64

Experiment Tumor Drug Dose Schedulig C50 DTIline mg/ml

Exp 1 A2780 Drug A 30 mg/kg q4dx3 i.v. 4.64e-7 0.388Exp 6 HTC116 Drug O 50 mg/kg q7dx2 i.v. 8.95e-7 0.275Exp 9, Group 2 A375 Drug I 30 mg/kg 1-3die x 2 i.v. 1.01e-4 0.361Exp 9, Group 3 A375 Drug I 60 mg/kg 1-3die x 4 i.v. 1.01e-4 0.516Exp 9, Group 4 A375 Drug I 60 mg/kg qodx6bid i.v. 1.01e-4 0.422

Table 6.1: Estimates of the proposed toxicity index for Experiments 1, Drug A, Experi-ments 6, Drug O and Experiments 9, Drug I.

the available data, we considered two different possibilities for taking intoaccount the action of the anticancer treatment on the assimilation capabil-ity.

In some experiments, the decrease in the mice body weight of the treatedgroups is delayed compared to the days of the drug administration. There-fore, in these cases we suggested a delayed side effect of the anticancer treat-ment upon the mice assimilation capacity. To take into account this belatedaction of the drug, an effect compartment was included in the model as shownin Fig.6.2.

Figure 6.2: Schematic representation of the drug action upon the tumor and the hostorganism after the inclusion of an effect compartment in the model.

In particular, we supposed that the effect compartment modulates thedrug action only upon the host organism and not upon the tumor. Theresults obtained by this modification of the model for the Experiments 1 and6 are reported in Appendix B.

As a further possibility to describe the side effects of the anticancer treat-ment, we hypothesized that the reduced assimilation capacity there is not di-rectly connected with the drug concentration c(t), but it is linked to the massof the non-proliferatig tumor cells. In particular, we rewrote the function ρ(t)as

ρ(t) = ρb

(

1− Vu4(t)

VM + Vu4(t)

)

, (6.2)

6 Conclusions 65

relating the toxicity effect to the tumor cells at the last stage of the apoptosisprocess. Also in this case, the results obtained by the model identificationare reported in Appendix B.

6.3 Discussion

The new tumor-in-host DEB-based model proved to be able to describe tumorand host body growth in xenograft mice with anticancer drug administra-tion. In particular, the developed model wants to be an answer to the needof considering the interactions between tumor and host into mathematicalmodels used to describe in vivo tumor growth data. An interesting objectiveof the model is the opportunity to evaluate the possible side effect of the an-ticancer treatment in term of a decreasing in the intake of energy trough foodproving, so, a quantitative measurements of the drug toxicity. Moreover, thetumor growth profiles predicted by the more mechanistic model are similar tothose obtained with the Simeoni TGI model, deepening this affinity a furthervalidity to its empirical growing function has been provided.

Finally, the proposed model can be object of a further analysis, which maylead to several interesting results. In particular, the mechanistic functiondescribing tumor growth could be adequately modified to take into accountthe effect of an antiangiogenetic treatment. Moreover, even if the model hasbeen tested and identified by using data of xenograft mice, from a theoreticalpoint of view the translation from a specie to another one should be not sodifficult. Indeed, it would be sufficient estimating the growth parameters ofthe specific specie and applying the identification strategy described in Sec.4:in this direction some attempts has been made with data of xenograft rats.An interesting open point is how this model could be applied and used as anefficient tool in translation from animal to human studies of anticancer drugactivity.

Appendix A

In this section some examples of Mlxtran scripts for the new tumor-in-hostmodel and the outputs summary are reported.

DESCRIPTION:

DEB-TGI model - Experimental 5. Drug H - female mice, 6 weeks

INPUT:

parameter={mu_u, gu, mu, Vdeg_Max, Vu10, W0, k1,k2,C50}

EQUATION:

ni=1.2242

g=12.2

xi=0.184

V1inf=22.6

dV=1

dVu=1

omega=0.75

m=ni/(V1inf^(1/3)*g)

t0=0

e_0=1.2164

Z_0=W0/(1+e_0*xi)

Vu1_0=Vu10

Vu2_0=0

Vu3_0=0

Vu4_0=0

V=1108.397315

k=2.004733

k12=1.529543333

k21=0.225656

Cc=pkmodel(V,k,k12,k21)

rho=1-Cc/(C50+Cc)

ku=(mu_u*Vu1)/(Z+mu_u*Vu1)

66

Appendix A 67

ddt_e=(ni/Z^(1/3))*(rho*(V1inf/(Vu1+Z))^(2/3)-e)

devV=((1-ku)*ni*e*Z^(2/3)-g*m*Z)/(g+(1-ku)*e)

if (devV>=0)

int_Vu1=((ni*Z^(2/3)+m*Z)*g*ku*e)/(g*gu+(1-ku)*gu*e)-mu*Vu1-k2*Vu1*Cc

int_Z=((1-ku)*ni*e*Z^(2/3)-g*m*Z)/(g+(1-ku)*e)

else

devV2=((1-ku)*ni*e*Z^(2/3)-g*m*Z)/((1-ku)*(e+omega*g))

if (devV2<0)&(devV2>-Vdeg_Max)

int_Z=((1-ku)*ni*e*Z^(2/3)-g*m*Z)/((1-ku)*(e+omega*g))

int_Vu1=(g*m*ku*Z)/(gu*(1-ku))-mu*Vu1-k2*Cc*Vu1

else

int_Z=-Vdeg_Max

int_Vu1=(ku/gu)*(e*ni*Z^(2/3)+Vdeg_Max*e+Vdeg_Max*omega*g)-mu*Vu1-k2*Cc*Vu1

end

end

ddt_Z=int_Z

ddt_Vu1=int_Vu1

ddt_Vu2=k2*Cc*Vu1-k1*Vu2

ddt_Vu3=k1*Vu2-k1*Vu3

ddt_Vu4=k1*Vu3-k1*Vu4

Wu=dVu*(Vu1+Vu2+Vu3+Vu4)

W=dV*(1+xi*e)*Z

odeType=stiff

OUTPUT:

output={W,Wu}

Appendix A 68

******************************************************************

* exp5_DrugH.mlxtran

* June 07, 2015 at 15:56:44

* Monolix version: 4.3.3

******************************************************************

Estimation of the population parameters

parameter s.e. (lin) r.s.e.(%)

mu_u_pop : 13.4 12 89

gu_pop : 11.5 11 97

mu_pop : 0.0125 0.16 1.29e+003

Vdeg_Max_pop : 0.162 0.057 35

Vu10_pop : 0.00226 0.0026 114

W0_pop : 25.9 0.69 3

k1_pop : 0.43 1.2 276

k2_pop : 15.6 30 192

C50_pop : 0.00148 0.0011 77

omega_mu_u : 0 - -

omega_gu : 0 - -

omega_mu : 0 - -

omega_Vdeg_Max: 0 - -

omega_Vu10 : 0 - -

omega_W0 : 0 - -

omega_k1 : 0 - -

omega_k2 : 0 - -

omega_C50 : 0 - -

b_1 : 0.241 0.035 14

c_1 : 0.5 - -

b_2 : 0.166 0.024 14

c_2 : 0.5 - -

______________________________________________

correlation matrix of the estimates(linearization)

mu_u_pop 1

gu_pop 0.98 1

mu_pop -0.91 -0.97 1

Vdeg_Max_pop-0.24 -0.26 0.36 1

Vu10_pop -0.84 -0.73 0.62 0.33 1

W0_pop 0.52 0.55 -0.52 0.23 -0.3 1

k1_pop -0.19 -0.24 0.24 -0.26 -0.05 -0.08 1

k2_pop 0.49 0.53 -0.4 0.34 -0.22 0.22 -0.85 1

C50_pop -0.51 -0.55 0.64 0.64 0.44 -0.45 -0.24 0.27 1

Eigenvalues (min, max, max/min): 3.5e-005 4.6 1.3e+005

Appendix A 69

b_1 1

b_2 0 1

Eigenvalues (min, max, max/min): 1 1 1

Population parameters and Fisher Information Matrix estimation...

Elapsed time is 611 seconds.

CPU time is 1.4e+003 seconds.

______________________________________________________________

Log-likelihood Estimation by linearization

-2 x log-likelihood: 53.19

Akaike Information Criteria (AIC): 75.19

Bayesian Information Criteria (BIC): 65.28

______________________________________________________________

Appendix B

Experiment 2, Drug B


- - 1/day - cm3 g - 1/day ml/mgday mg/ml13 11.1 0.0289 0.0721 0.00555 23.5 0.743 0.127 21.7 7.89e-842% 233% 130% 80% 392% 56% 234% 105% 104% 26%

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Experiment 2, Drug B: on the top mice body weight and tumor weight profiles with datafor the control group, on the bottom the respective profiles for the treated group.

70

Appendix B 71

Experiment 3, Drug C



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Simulated concentration profiles of the Drug C for each single agent arm of the Experiment3: concentration for Drug C given i.v. at 0.8mg/kg in day 8 (left panel), concentrationfor Drug C given i.v. at 0.8mg/kg in days 8 and 12 (right panel).

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Experiment 3, Drug C: on the top mice body weight and tumor weight profiles with datafor control, below the respective profiles for the two treated groups.

Appendix B 72

Experiment 3, Drug D



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Simulated concentration profiles of the Drug D for each single agent arm of the Experiment3: concentration for Drug D given i.v. at 3mg/kg in day 8 (left panel), concentration forDrug D given i.v. at 3mg/kg in days 8 and 12 (right panel).

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Appendix B 73

Experiment 4, Drug C



0.00000

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Appendix B 74

Experiment 4, Drug E



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Appendix B 75

Experiment 4, Drug F


- - 1/day - cm3 g - 1/day ml/mgday mg/ml12.6 11.7 0.0209 0.138 0.00478 23 0.785 0.399 7480 1.7e-0967% 144% 297% 44% 416% 70% 161% 192% 65% 103%

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Experiment 4, Drug F: on the top mice body weight and tumor growth profiles with datafor the control group, on the bottom the respective profiles for the two treated groups.

Appendix B 76

Experiment 5, Drug G


- - 1/day - cm3 g - 1/day ml/mgday mg/ml13.2 11.4 0.0239 0.157 0.00263 24.9 - 0.0366 3.04e3 3.55e-648% 48% 354% 18% 86% 2% - 103% 25% 52%

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Simulated concentration profiles of the Drug G for each single agent arm of the Experiment5. From the top on the left to the bottom on the right: concentration for Drug G given i.v.at 20mg/kg 1-4die, concentration for Drug G given i.v. at 20mg/kg qodx5, concentrationfor drug G given i.v. at 20mg/kg q3dx4, concentration for drug G i.v. given 40mg/kgqodx4, concentration for Drug G given i.v. at 40mg/kg q3dx4, concentration for Drug Ggiven i.v. at 50mg/kg qodx4.

Appendix B 77

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Experiment 5, Drug G: in the left panels the obtained body weight profiles together withexperimental data for the control and the treated Groups 2-7 (from top to bottom re-spectively). In the right panels the corresponding tumor growth profiles together withexperimental data.

Appendix B 78

Experiment 5, Drug H



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Simulated concentration profiles of the Drug H for each single agent arm of the Experiment5. On the left concentration for Drug H given i.v. at 50mg/kg qodx4 and on the rightconcentration for Drug H given i.v. at 50mg/kg q3dx4.

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Experiment 5, Drug H: on the top mice body weight and tumor growth profiles with datafor the control group, on the bottom the respective profiles for the two treated groups.

Appendix B 79

Experiment 7, Drug I


- - 1/day cm3 cm3 g - 1/day ml/mgday mg/ml5.3 11.2 0.00575 0.0058 0.0255 29 0.959 27 8.16 8.66e-10113% 147% 103% 60% 88% 8% 14% 103% 805% 396%

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cent

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n(m

g/m

l)Simulated concentration profiles of the Drug I, Experiment 7. On the left, concentrationfor Drug I given i.v. at 45mg/kg q3dx4bid, in the central panel, concentration for DrugI given i.v. at 50mg/kg q3dx4bid, concentration on the right for Drug I given i.v. at60mg/kg q3dx4bid.

Appendix B 80

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Experiment 7, Drug I: on the top mice body weight and tumor growth profiles with datafor the control group, below the respective profiles for the three treated groups.

Appendix B 81

Experiment 7, Drug G


- - 1/day - cm3 g - 1/day ml/mgday mg/ml5.07 9.45 0.00596 0.0627 0.0115 28.2 0.956 631 466 2.59e-784% 91% 103% 48% 60% 8% 11% 103% 68% 125%

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Simulated concentration profiles of the Drug G for each single agent arm of the Experiment7. On the left concentration for Drug G given i.v. at 30mg/kg q3dx4die and on the rightconcentration for Drug G given i.v. at 45mg/kg q3dx5die.

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Experiment 7, Drug G: on the top mice body weight and tumor growth profiles with datafor the control group and the respective profiles for the two treated groups.

Appendix B 82

Experiment 7, Drug N



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Experiment 7, Drug N: on the top the mice body weight and tumor growth profiles withdata for the control group, on the bottom the respective profiles for the treated group.

Appendix B 83

Experiment 8, Drug I


- - 1/day - cm3 g - 1/day ml/mgday mg/ml4.62 10.9 0.0139 5.9e-5 0.0219 30.7 0.887 41.2 36.5 3.22e-694% 154% 906% 104% 37% 7% 9% 103% 119% 370%

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cent

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Simulated concentration profile of the Drug I given i.v. at 60mg/kg 1-3bid for two cyclesin Experiment 8.

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Experiment 8, Drug I: on the top mice body weight and tumor growth profiles with datafor control group, on the bottom the respective profiles for the treated group.

Appendix B 84

Experiment 8, Drug M


- - 1/day - cm3 g - 1/day ml/mgday mg/ml4.62 10.9 0.00209 1.68e-12 0.018 30.8 0.857 240 39.5 0.000288205% 330% 104% 109% 46% 7% 11% 104% 323% 139%

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Simulated concentration profile of the Drug M given o.s. at 40mg/kg qodx10 in Experi-ment 8.

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Experiment 8, Drug M: on the top mice body weight and tumor growth profiles with datafor control group, on the bottom the respective profiles for the treated group.

Appendix B 85

Experiment 9, Drug L


- - 1/day cm3 cm3 g - 1/day ml/mgday mg/ml4.07 12 0.00628 0.0259 0.0207 31.7 - 206 263 0.00049362% 117% 103% 201% 22% 2% - 103% 34% 54%

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Simulated concentration profile of the Drug L given i.v. at 50mg/kg q0dx6 in Experiment9.

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Experiment 9, Drug L: on the top mice body weight and tumor growth profiles with datafor the control group, on the bottom the respective profiles for the treated group.

Appendix B 86

Drug toxicity index

Experiment Tumor Drug Dose Schedulig C50 DTIline mg/ml

Exp 2 A2780 Drug B 12 mg/kg 1 bolus i.v. 7.89e-8 0.557Exp 3, Group 2 A2780 Drug C 0.8 mg/kg 1 bolus i.v. 2.3e-10 0.247Exp 3, Group 3 A2780 Drug C 0.8 mg/kg q4dx2 i.v. 2.3e-10 0.393Exp 3, Group 4 A2780 Drug D 3 mg/kg 1 bolus i.v. 2.22e-7 0.163Exp 3, Group 5 A2780 Drug D 3 mg/kg q4dx2 i.v. 2.22e-7 0.232Exp 4, Group 2 A2780 Drug C 3 mg/kg 1 bolus i.v. 1.05e-7 0.161Exp 4, Group 3 A2780 Drug C 3 mg/kg q4dx2 i.v. 1.05e-7 0.237Exp 4, Group 4 A2780 Drug E 120 mg/kg 1 bolus i.v. 1.89e-10 0.111Exp 4, Group 5 A2780 Drug E 120 mg/kg q4dx2 i.v. 1.89e-10 0.142Exp 4, Group 6 A2780 Drug F 4 mg/kg 1 bolus i.v. 1.07e-9 0.1Exp 4, Group 7 A2780 Drug F 4 mg/kg q4dx2 i.v. 1.07e-9 0.154Exp 5, Group 2 A2780 Drug G 20 mg/kg 1-4die i.v. 3.55e-6 0.257Exp 5, Group 3 A2780 Drug G 20 mg/kg qodx5 i.v. 3.55e-6 0.5Exp 5, Group 4 A2780 Drug G 20 mg/kg q3dx4 i.v. 3.55e-6 0.419Exp 5, Group 5 A2780 Drug G 40 mg/kg qodx5 i.v. 3.55e-6 0.5Exp 5, Group 6 A2780 Drug G 40 mg/kg q3dx4 i.v. 3.55e-6 0.445Exp 5, Group 7 A2780 Drug G 50 mg/kg q6dx2 i.v. 3.55e-6 0.19Exp 5, Group 8 A2780 Drug H 50 mg/kg qodx4 i.v. 1.62e-5 0.411Exp 5, Group 9 A2780 Drug H 50 mg/kg q3dx4 i.v. 1.62e-5 0.376Exp 7, Group 2 A375 Drug I 45 mg/kg q3dx4bid i.v. 4.93e-4 0.191Exp 7, Group 3 A375 Drug I 50 mg/kg q3dx4bid i.v. 4.93e-4 0.192Exp 7, Group 4 A375 Drug I 60 mg/kg q3dx4bid i.v. 4.93e-4 0.192Exp 7, Group 5 A375 Drug G 30 mg/kg q3dx4die i.v. 2.59e-7 0.468Exp 7, Group 6 A375 Drug G 45 mg/kg q3dx5die i.v. 2.59e-7 0.531Exp 7, Group 7 A375 Drug N 15 mg/kg q3dx5die i.v. 4.06e-7 0.509Exp 8, Group 2 A375 Drug I 60 mg/kg 1-3bid x 2 i.v. 3.22e-6 0.364Exp 8, Group 4 A375 Drug M 40 mg/kg qodx10 o.s. 2.88e-4 0.023Exp 9, Group 5 A375 Drug L 50 mg/kg qodx6die i.v. 8.95e-7 0.250

Estimates of the proposed toxicity index (see Sec.6.2).

Appendix B 87

Drug effect upon host organism

We tested the two variants for the modeling of the drug effect on the hostorganism introduced in Sec.6.2 against the datasets relative to Experiment1 and Experiment 6.

The results relative to the introduction of the effect compartment arereported below.

µu gu mu VdegMax Vu10 W0 ρb k1 k2 C50 keff- - 1

day- cm3 g - 1

dayml

mgdaymgml

1

day

13.7 11.7 0.0222 0.187 0.00214 21.2 - 0.371 765 6.15e-07 53.742% 47% 370% 9% 69% 2% - 93% 25% 340% 103%

PD model parameter estimates for Experiment 1, Drug A after the introduction of theeffect compartment for the drug concentration.

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Experiment 1, Drug A: the obtained body weight (left panel) and tumor weight (rightpanel) profiles together with experimental data are reported for the control (on the top)and for the treated group (on the bottom) of Experiment 1. An effect compartment forthe drug concentration action upon host organism is included in the model.

µu gu mu VdegMax Vu10 W0 ρb k1 k2 C50 keff- - 1

day- cm3 g - 1

dayml

mgdaymgml

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day

7.58 9.9 0.013 0.255 0.00259 32.5 0.521 0.568 1260 3.63e-06 5.68122% 103% 19% 9% 193% 30% 148% 87% 51% 103% 103%

PD model parameter estimates for Experiment 6, Drug O after the introduction of theeffect compartment for the drug concentration.

Appendix B 88

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Experiment 6, Drug O: the obtained body weight (left panel) and tumor weight (rightpanel) profiles together with experimental data are reported for the control (on the top)and for the treated group (on the bottom) of Experiment 6. An effect compartment forthe drug concentration action upon host organism is included in the model.

The fittings resulting from the introduction of the function (6.2) are re-ported below for Experiments 1 and 6.

µu gu mu VdegMax Vu10 W0 ρb k1 k2 VM

- - 1/day - cm3 g - 1/day ml/mgday cm3

13.7 12.2 0.0127 0.178 0.00234 20.8 - 1.53 608 0.047950% 58% 818% 8% 78% 2% - 13% 11% 25%

PD model parameter estimates for Experiment 1, Drug A with the drug side effect linkedto the non-proliferating cells Vu4.

µu gu mu VdegMax Vu10 W0 ρb k1 k2 VM

- - 1/day - cm3 g - 1/day ml/mgday cm3

7.27 10.1 0.00464 0.239 0.0245 31.3 0.567 0.601 1670 0.886150% 222% 103% 18% 85% 14% 43% 59% 38% 171%

PD model parameter estimates for Experiment 6, Drug O with the drug side effect linkedto the non-proliferating cells Vu4.

Appendix B 89

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Experiment 1, Drug A: the obtained body weight (left panel) and tumor weight (rightpanel) profiles together with experimental data are reported for the control (on the top)and for the treated group (on the bottom) of Experiment 1. The drug side effect uponassimilation capacity is modeled by (6.2).

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Experiment 6, Drug O: the obtained body weight (left panel) and tumor weight (rightpanel) profiles together with experimental data are reported for the control (on the top)and for the treated group (on the bottom) of Experiment 6. The drug side effect uponassimilation capacity is modeled by (6.2).

Appendix C

The model shows a high sensivity for the tumor-related parameters µu, guand mu. In particular, from the graphics below we can see that the tumorgrows more and faster with decreasing of gu and mu values. Conversely thehost organism shows a more pronounced decrease of the body weight forhigher values of the two parameters.

Sensivity analysis of the DEB-TGI model for parameter gu

Sensivity analysis of the DEB-TGI model for parameter gu: the host body weight (leftpanel) and the tumor weight (right panel) predicted by the DEB-based model using theparameters identified upon the control group of Experiment 1 varying gu from 5 to 12.2.

90

Appendix C 91

Sensivity analysis of the DEB-TGI model for parameter mu

Sensivity analysis of the DEB-TGI model for parameter mu: the host body weight (leftpanel) and the tumor weight (right panel) predicted by the DEB-based model using theparameters identified upon the control group of Experiment 1 varying mu from 0.0001 to0.035.

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Acknowledgements

Dopo avervi annoiato con un’ infinita di pagine colme di grafici ed equazioni,queste ultime righe sono riservate a ringraziare tutte le persone che, piu omeno a lungo, mi hanno accompagnata in questi cinque anni di Universita.

Un enorme grazie va al Prof.re Paolo Magni, per avermi guidato in tuttii mesi di lavoro che la stesura di questa tesi ha richiesto. Quando tempofa mi “sono trasferita” al dipartimento di Bioingegneria, con un po la tipicapresunzione che caratterizza i matematici, non immaginavo di trovare unacosı immensa competenza, preparazione e professionalita. Ma cio per cuisento maggiormente di rivolgerLe il mio grazie e la forte passione con cuil’ho vista dedicarsi ogni giorno all’attivita di ricerca, alle tante lezioni, agliesami, alle mille riunioni trovando sempre tempo da riservare a studenti edottorandi per un confronto, un consiglio o una semplice battuta.

Un doveroso ringraziamento anche al Dr. Maurizio Rocchetti per l’interessedimostrato e il prezioso contributo che la sua grande esperienza ha portato.Un grazie anche alla Prof.ssa Raffaella Guglielmann che con grande gentilezzaha supervisionato questo lavoro di tesi.

Un pensiero speciale al gruppo di Modelli e a tutti i membri del Labo-ratorio di Bioinformatica e Biologia Sintetica che mi hanno accolto tra loroin questi mesi. Per quanto la stanzetta in cui vi rinchiudono sia poco ac-cogliente, sentiro la mancanza di quello che ormai sento un po’ come il mioangolo di scrivania: in voi ho visto un gruppo affiatato e appassionato, sem-pre pronto ad aiutare chi di volta in volta si trova in difficolta, anche se ilbisognoso di turno e una disgraziata tesista come me. Un immenso grazie adElisa che ha collaborato e revisionato questo lavoro: brucia le mie bozze enon svelare a nessuno le castronerie del mio inglese. Insomma concludo comefarebbe Maiara: sento gia saudade e spero di rivedervi a Settembre.

Cari familiari tocca ora a voi. Un grazie infinito ai miei genitori che mihanno sostenuto, senza mai farmi mancare nulla, in questi cinque lunghi annidi Universita. Grazie per aver creduto in me piu di quanto io abbia fatto, per

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aver condiviso la fatica, le ansie, le notti insonni e i risvegli all’alba per i viaggiin treno, per aver sopportato i nervosismi e gli sfoghi pre-esame ma ancheper aver gioito ad ogni mio piccolo successo. Grazie mamma per i rientri acasa del venerdı con l’immancabile tappa all’Esselunga, per le chiamate dellapausa pranzo, per sopportare (con molte lamentele) il disordine che portoin casa e in cucina ogni fine settimana: preparati che nei prossimi mesi ilcaos sara all’ordine del giorno. Grazie papa per il letto rifatto la domenica,per tutte le casse d’acqua con cui ti ho costretto a farti i muscoli, per tuttele domande che mi hai fatto sulla tesi e per aver sopportato la mia tipicarisposta: “Non farmi perder tempo che tanto non capisci!!!”

Grazie al Big Brother nonche quasi avvocato Alessandro, per i suoi “Chetorta e?!?” o “Caffe?!?!” dei fine settimana che tradotto significa “Elena faiil caffe!!”: e inutile che continui a celarti dietro a risposte monosillabilicheai miei messaggi tanto lo so che, in fondo, in fondo, sei un fratello tenerone.Grazie al Little Brother Francesco, per i passaggi per Pavia strappati a liti-gate, mi raccomando preparati all’idea di trovare vestiti appesi alla manigliadel bagno o le bottiglie di acqua perennemente vuote nel frigorifero percheI’m back!

Un ringraziamento speciale a nonno Piero, per il suo immancabile “Al-lora li hai sbaragliati tutti?!?” dopo ogni esame superato, per i racconti diquando ha sfilato davanti al Presidente della Repubblica e per i suoi rimedidomenicali ai mali dell’Italia. Grazie poi a nonna Piera, per i suoi “24 annilaureata come il papa!”, per credere ancora che diventero Professoressa diMatematica e per l’infinitamente ripetuta domanda “Quando hai il prossimoesame?”, finalmente posso risponderti “Mai piu!!”.

Un pensiero va poi a tutti gli amici piacentini o pavesi che ho incontratoin questi anni. A Claudia, fedele compagna di studi, dall’asilo fino al liceo:dobbiamo tornare a salutare la nostra amica toscana e la sua campanellaper la cena e poi, ora che mi sono laureata, non ho piu scuse, Crossfit Paviaaspettami. A Martina, super Dottoressa di cui ho seguito le orme. A Leuriche mi ha inizializzato al magico mondo dei concerti. A Valeria, per i viaggiin treno del lunedı mattina in cui ha sopportato i resoconti delle mie disgrazie.A Laura, che mi ha anticipato nel prendere il titolo di Dottoressa. A Luca,sempre presente in questi anni. A Giulia, a quando la prossima vacanzainsieme? Ed a tutti i compagni del corso di Matematica: Francesca, Irene,Nicoletta, Cecilia, Edoardo, Vittoria, Francesco, Simona e Andrea.

Infine un grazie infinito a Luca, in casa noto come Scarpa o il genio, nonpotevo trovare compagno migliore per questi anni di Universita. Ammetto,ho messo alla prova la nostra amicizia in tutti i modi possibili ed immagin-abili: ti ho inflitto le piu temibili torture cinesi durante le ore lezione, ti ho

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costretto a ripetere il programma di ogni esame almeno dieci volte nonos-tante a te ne sia sempre bastata mezza, ti ho convinto a ripassare AnalisiFunzionale a Capodanno, ti ho anche fatto indossare uno scolapasta a mo dicappello (quella foto e memorabile), ti ho inviato messaggi alle sei di mattinocon qualche dubbio da risolvere (anche se, diciamolo, la risposta non e maiarrivata prima di mezzogiorno, ora abituale del tuo risveglio), ti ho stressatocon il rituale “Mi boccia, non so nulla!”, ho condiviso con te ogni tipo diansia e preoccupazione, ho salutato mezza Pavia sporgendomi dal finestrinodalle tua macchina e, ricordiamolo, ti ho umiliato correndo alla Vernavola.Insomma, per descrivere questi anni passati insieme non c’e modo miglioreche riportare i messaggi con cui abbiamo appena commentato la tua immi-nente partenza per Londra:E: “Chissa chi risolvera i miei casini dall’anno prossimo, dovro trovare unnuovo insegnante di sostegno!”L: “Posso assisterti a distanza via Skype.”E: “Ecco, prima devi insegnarmi ad usarlo!”

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