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Tel-Aviv University Faculty of Exact Sciences Department of Statistics and Operations Research. CHARACTERIZING UNCERTAINTY FOR MODELING RESPONSE TO TREATMENT. David M. Steinberg. UCM 2012 Sheffield, UK. July 2012. Based on Joint Work With. Mirit Kagarlitsky, TAU Zvia Agur, IMBM - PowerPoint PPT Presentation
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CHARACTERIZING UNCERTAINTY FOR MODELING
RESPONSE TO TREATMENT
Tel-Aviv UniversityFaculty of Exact Sciences
Department of Statistics and Operations Research
David M. Steinberg
July 2012UCM 2012
Sheffield, UK
Based on Joint Work With
Mirit Kagarlitsky, TAU Zvia Agur, IMBM Yuri Kogan, IMBM
Institute for Medical Bio Mathematics
Overview
Goals Mathematical models for immunotherapy Data Patient and population models NLME models for separating sources of
variance Protocol assessment Summary and Conclusions
Goals
Use mathematical models and data to predict outcomes from new treatment protocols in a patient population.
Characterize the variation in response to treatment in a patient population.
Exploit existing trial data to describe the population.
Patient level – use the model to personalize treatment.
Goals
Patients observed under Protocol A.
How would they respond to Protocol B?
Treated patients.
Math Models for Cancer
Biomathematics is a science that studies biomedical systems by mathematically analyzing their most crucial relationships. Incorporating biological, pharmacological and medical data within mathematical models of complex physiological and pathological processes, the model can coherently interpret large amounts of diverse information in terms of its clinical consequences.
Agur – 2010, Future Medicine
Math Models for Cancer
We work with models for immunotherapy treatment of cancer.
The models reflect the natural growth of the cancer, the response of the immune system to chemotherapeutic agents, and the consequent effect on the cancer.
The models involve compartments and rate constants that govern growth, growth suppression and flows between compartments.
Create an updated training data set adding the recent individual data
Model validation assessmentNo
Yes
No Yes
Construct a personalized model using the current data set
Construct a mathematical model and
a validation criterion
Predict treatment outcome and suggest improved regimens
Monitoring model accuracy
Preparation
Personalization
Prediction
Compare the current model predictions to those of previous modelsC
oll
ect
mo
re d
ata
Kogan et al., Cancer Research, 2012 72(9), pp.2218-2227
Math Models for Cancer
Kogan et al. proposed a “success of validation” criterion for the model.
The criterion compares data thresholds and asks when sufficient data have been collected to enable accurate prediction of future results.
The criterion requires agreement in predictions following three successive observations.
The SOV is used to determine a “learning” data set for each subject, from which a personalized treatment regime can be determined.
Math Models for Cancer
Our model uses a system of ODE’s to describe vaccination therapy for prostate cancer in terms of interactions of tumor cells, immune cells and vaccine.
Assumptions: Vaccine injection stimulates maturation of dendritic cells. These become mature antigen-presenting DCs. Some DCs migrate into lymph nodes. DCs are exhausted at a given rate and give rise to
regulatory DCs. Antigen-presenting DCs stimulate T-helper cells and
activate cytotoxic T lymphocyte (CTL) cells. Some of these cells die or are inactivated by regulatory cells.
Cancer cells grow exponentially at a rate r but are destroyed, with a given efficiency, by CTLs.
Lymph node
Skin
Tumor
DC DR
V
Dm
P
Phh
CParPP
RDaR
CRkCDaC
DDkD
DkDkD
DkVpVkD
VnkV
P
PP
RRR
RCCC
RDCCRR
CCRmmlC
mmim
vi
)(
Immunostimulation Immunoinhibition
C R
Math Models for CancerKogan et al., Cancer Research, 2012 72(9), pp.2218-2227
Prediction from the Model
The model tracks tumor size over time. Expected tumor size can be computed
by solving the system of differential equations.
The solution depends on the parameter values and the treatment protocol.
Alternative protocols can be compared for a patient or a population by running the model.
Data
Various data sources are available. Observation of patients. Direct study of rate constants.
The observational data is not sufficient to estimate all model parameters.
Relevant literature may provide estimates or distributions for some parameters. These may involve “generic” research, not specifically on prostate cancer.
Data
We have data on 38 patients. The data tracks a biomarker Y over
time. The marker should reflect tumor size.
Calibrating the marker to tumor size is subject-specific.
DataBiomarker data for two typical patients, with
fitted curves. Time is relative to the start of treatment.
Data
Residuals for 16 patients. Plot shows observed/predicted.
Patient Models
The general model for a particular patient:
Here 1 includes “common” parameters, 2 includes four subject-specific parameters, and is a random error term.
The subject-specific parameters are the tumor growth rate, the CTL killing efficacy and two linear calibration terms.
The treatment protocol is specified by P.
1 2 ,( ) ( ; , , )i ij ij i i i jY t g t P
Patient Models
Distribution of the calibration parameters from nonlinear least squares fits for 40 patients.
Patient Models
Distribution of the calibration parameters from nonlinear least squares fits for 40 patients.
20
Patient ModelsStatistical distributions of the parameter estimates.
21
Patient ModelsStatistical distributions of the parameter estimates; confidence ellipses for first two parameters.
Patient Models Substantial variation in parameter
values across patients. High correlations among the
parameter values. The variation could reflect:
– Statistical (estimation) uncertainty.– True population heterogeneity.
Population Model
Treat the individual parameters 2 as random effects.
Their distribution describes the heterogeneity of the population.
This generates a nonlinear mixed effects (NLME) model.
NLME Models
Common to assume normal distributions.
But is this plausible for our application?
If not, is there any hope to estimate a more general multivariate density?
1 2 ,( ) ( ; , , )i ij ij i i i jY t g t P
Fi ~,2
NLME Models
The covariance matrix for our model is too rich to estimate: 4 variances and 6 covariances.
The empirical subject-specific parameter estimates are correlated.
NLME Models
Our suggestion: replace the original parameters with the empirical principal components.
Assume the new parameters are independent.
NLME Models
Model estimation is challenging.
Many convergence problems.
Work still in progress.
Protocol Assessment
Algorithm 1
1. Sample patients by generating patient-specific parameter vectors.
2. For each patient, run the model to assess the expected outcome for this patient under different protocols of interest.
3. Characterize population behavior for each protocol.
4. Use paired data to compare protocols or make a factorial analysis.
Protocol Assessment
Paired outcomes are used to compare protocols – how do particular patients succeed on a new protocol versus an old protocol.
Marginal outcomes are important to present an overall population picture of protocol success.
Protocol Assessment
Algorithm 2
Like Algorithm 1, but in summarizing each patient-protocol pair:
1. Average over a sample of values of the common parameters, reflecting their distribution.
2. For each sampled value of the common parameters, re-analyze the data to estimate the conditional (on the common parameters) distribution of the patient parameters.
Summary & Conclusions Bio-Mathematical models provide a stronger
basis for prediction than empirical models. They enable us to assess potential treatment
protocols that have not been tested in vivo. It may be difficult to estimate the needed
population descriptions. It is essential to distinguish estimation
uncertainty from population heterogeneity.