Intratumor heterogeneity alters most effective drugs indesigned combinationsBoyang Zhaoa,b, Michael T. Hemannb,c, and Douglas A. Lauffenburgerb,c,d,1

aComputational and Systems Biology Program, bThe David H. Koch Institute for Integrative Cancer Research, and Departments of cBiology and dBiologicalEngineering, Massachusetts Institute of Technology, Cambridge, MA 02139

Edited by Gordon B. Mills, The University of Texas, M. D. Anderson Cancer Center, Houston, TX, and accepted by the Editorial Board June 13, 2014 (receivedfor review December 26, 2013)

The substantial spatial and temporal heterogeneity observed inpatient tumors poses considerable challenges for the design ofeffective drug combinations with predictable outcomes. Currently,the implications of tissue heterogeneity and sampling bias duringdiagnosis are unclear for selection and subsequent performance ofpotential combination therapies. Here, we apply a multiobjectivecomputational optimization approach integrated with empiricalinformation on efficacy and toxicity for individual drugs withrespect to a spectrum of genetic perturbations, enabling deriva-tion of optimal drug combinations for heterogeneous tumorscomprising distributions of subpopulations possessing these per-turbations. Analysis across probabilistic samplings from the spec-trum of various possible distributions reveals that the mostbeneficial (considering both efficacy and toxicity) set of drugschanges as the complexity of genetic heterogeneity increases.Importantly, a significant likelihood arises that a drug selected asthe most beneficial single agent with respect to the predominantsubpopulation in fact does not reside within the most broadlyuseful drug combinations for heterogeneous tumors. The un-derlying explanation appears to be that heterogeneity essentiallyhomogenizes the benefit of drug combinations, reducing thespecial advantage of a particular drug on a specific subpopulation.Thus, this study underscores the importance of consideringheterogeneity in choosing drug combinations and offers a princi-pled approach toward designing the most likely beneficial set,even if the subpopulation distribution is not precisely known.

systems biology | cancer | combination therapy

Genetic intratumor heterogeneity has long been appreciatedas present in cancer patients (1). Recent sequencing studies

further revealed the extent of this tumor diversity, arising fromhighly complex clonal evolutionary processes (2). This phe-nomenon has been observed in many solid (3–5) and hemato-poietic cancers (6–8). Moreover, treatments also may havedramatic effects on tumor composition—with a preexisting sub-clone at diagnosis often becoming dominant at relapse (9–12).To meet this challenge, rational drug combination treatmentsmust be designed so as to account for intratumor heterogeneityto better predict reoccurrence.The history of theoretical studies aimed at drug optimization

in cancer therapy is long. Some studies used differential equationmodels formulated as deterministic optimal control problems(13–16), whereas others used stochastic birth–death processmodels (17–21), to examine treatment regimens for tumorscomprising a sensitive population along with a few resistantsubpopulations. However, these studies, many of which dealtonly with generic scenarios, focused primarily on drug schedul-ing, investigating the effects of frequency and dose intensity ofdrugs on resistance potential. Practical applications of suchresults have been limited, although some of these strategies re-cently were combined with experimental validation to examinesingle-drug scheduling in vitro and in xenograft models (22, 23).The extensive number of chemotherapy and targeted thera-

peutics in clinical and preclinical use presents a large searchspace to determine even the choice of component drugs in a drug

combination for best treatment of a heterogeneous tumor. Per-haps the closest endeavor was a theoretical analysis of a minimalset of drugs that maximize the coverage of molecular targetvariants, with respect to structural considerations (24). None-theless, to our knowledge, there are no previous systematicstudies of combination drug design examining efficacy and tox-icity in the context of tumor heterogeneity and aimed at (i) howheterogeneity affects the utility of drug combinations and (ii)how the current clinical practice of tumor diagnosis, which biasestoward focus on the predominant subpopulation, affects thedesign of drug combinations (Fig. 1A).To address these questions, we use a computational optimi-

zation approach in concert with previously reported experi-mental data across a range of specific drugs effects on a spectrumof particular RNAi perturbations characterizing genetic varia-tions in model tumors. Our goal is to determine the best drugcombinations to minimize all subpopulations significantly pres-ent in a heterogeneous tumor. We previously experimentallyvalidated our model and the computational optimization pre-dictions regarding a two-drug combination with three-compo-nent heterogeneous tumors in vitro and in vivo in the presence ofinformation about the subpopulation proportions (25). Here, wedramatically extend our model to examine more complex het-erogeneous tumor compositions and a greater number of com-ponent drugs and, most importantly, to integrate over samplingsfrom all possible subpopulation distributions via computationalsimulation. The latter integration facilitates our understandingof which drugs should be included in the combination treat-ment, even in the absence of quantitative information on the

Significance

Tumors within each cancer patient have been found to be ex-tensively heterogeneous both spatially across distinct regionsand temporally in response to treatment. This poses challengesfor prognostic/diagnostic biomarker identification and rationaldesign of optimal drug combinations to minimize reoccurrence.Here we present a computational approach incorporating drugefficacy and drug side effects to derive effective drug combi-nations and study how tumor heterogeneity affects drugselection. We find that considering subpopulations beyondjust the predominant subpopulation in a heterogeneous tu-mor may result in nonintuitive drug combinations. Additionalanalyses reveal general properties of effective drugs. Thisstudy highlights the importance of optimizing drug combina-tions in the context of intratumor heterogeneity and offersa principled approach toward their rational design.

Author contributions: B.Z., M.T.H., and D.A.L. designed research; B.Z. performed research;B.Z. and D.A.L. analyzed data; and B.Z., M.T.H., and D.A.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. G.B.M. is a guest editor invited by the EditorialBoard.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: lauffen@mit.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1323934111/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1323934111 PNAS | July 22, 2014 | vol. 111 | no. 29 | 10773–10778

SYST

EMSBIOLO

GY

ENGINEE

RING

Dow

nloa

ded

by g

uest

on

July

25,

202

0

subpopulation distributions for a given tumor, as would be typ-ical in clinical situations. Among the insights gained, a keyprinciple is that significant differences may exist between thecombinations predicted to be most beneficial for the pre-dominant single subpopulation and those more beneficial whenmultiple subpopulations are taken into account. This suggeststhat gaining at least some degree of information concerning theheterogeneity of a primary tumor upon diagnosis, even if notquantitative or complete, may lead to an indicated drug regimendifferent from what would be viewed as best if only the pre-dominant genetic signature was obtained. Moreover, consistentwith our recent experimental findings (26), our simulations dem-onstrate that drug effects are homogenized by tumor heteroge-neity over essentially a full range of conceivable subpopulationdistributions. Consequently, the effectiveness of drugs most likelyto be found beneficial in combination for heterogeneous tumorsmay be characterized by average efficacy and robustness acrossthe subpopulations rather than by extreme efficacy in a partic-ular subpopulation. Taken together, our results offer concep-tual principles for designing drug combinations in the contextof intratumor heterogeneity.

ResultsConceptual and Computational Approach. Our premise is that atypical tumor comprises multiple diverse subclones along witha dominant subpopulation, and that dynamic changes in thedistribution of these populations may be influenced by drugtreatments. Currently, the prevalent clinical practice for di-agnosing tumor type and drug regimen is based on histologicaland/or biomarker identification and generally is biased towardthe predominant subpopulation as a result of its greater rep-resentation in the population (2, 27). However, minor sub-populations are important in determining how a tumor responds

to treatments and may be responsible for resistance and relapse(28). An immediate critical question is, how should consider-ation of heterogeneity, instead of simply the predominant sub-population, affect drug combination design (Fig. 1A)?Our approach integrates a theoretical framework with em-

pirical experimental information. The theoretical frameworkis a multiobjective linear optimization algorithm that simul-taneously incorporates efficacy (the desired tumor cell killing)and toxicity (adverse side effects in patients) for drug combina-tion effects on many heterogeneous tumor compositions (see SIAppendix for methodological details). The calculations involvedin the optimization algorithm incorporate the effects of eachdrug on each tumor subpopulation, sampling from among theconceivable compositions of subpopulations and combinationsof drugs (Fig. 1B). These compositions and drugs come fromprevious experimental information (26, 29). The tumor sub-populations are taken from a battery of shRNA knockdowns ofgenes in the DNA damage repair and apoptosis pathways in themurine Eμ-myc; p19Arf−/− lymphoma cell line [derived from awell-established preclinical Eμ-myc mouse model of humanBurkitt lymphoma (30, 31)], and the drugs are taken from a se-ries of commonly used chemotherapeutics and targeted thera-peutics (Fig. S1). A key finding in the previous work is that theoverall efficacy of multiple drugs in combinations typically isapproximated by linear combinations of the individual drugefficacies when they are applied together at overall LD80–90concentration. Such additivity allows for the use of a linearobjective function for efficacy with linear constraints in ouroptimization algorithm.Of course, the design of drug combinations involves con-

straints and conflicting objectives, resulting in multiple tradeoffsto consider simultaneously. For instance, the tradeoff betweenefficacy and toxicity prohibits maximizing the number of drugswithout exceeding the tolerable level of toxicity. Previous anal-yses of optimal control theory-based design of chemotherapeuticregimens have long used toxicity (e.g., in terms of maximal/cumulative drug concentrations) as constraints or secondaryobjectives in their mathematical formulations (14). Here, ourframework instead posits multiobjective optimization to maxi-mize overall efficacy and minimize overall toxicity concomitantly(Fig. 1C and SI Appendix).As one of the primary considerations in clinical drug combi-

nation regimens is nonoverlapping toxicity (again, adverse sideeffects), we formulated our drug toxicity model as a linearmodel, with the goal of minimizing overall toxicity with con-straints on a maximal allowable toxicity for each side effect (e.g.,myelosuppression, gastrointestinal effects). This formulationeffectively captures nonlinear behaviors as a result of non-overlapping drug combinations. Subsequent analyses first arebased on a symmetric toxicity profile—that is, each drug has thesame toxicity unit of 1. Later, this is relaxed to include the actualasymmetric toxicity profile of drugs in our efficacy dataset.Solving a multiobjective optimization problem, concomitant-

ly incorporating efficacy and toxicity, results in a discrete set(known as a “Pareto optimal set”) of feasible solutions (i.e.,optimal drug combinations) rather than a single “absolute best”solution (Fig. S2). The set is discrete, because each solutionrepresents an integer number of drugs included in the combi-nation, but it resides upon a continuous curve termed the“Pareto frontier.” Each solution in the Pareto optimal set has theproperty that an increase in one objective (say, efficacy) resultsin a decrease in at least one or more objectives (then, toxicity).We enumerate the Pareto optimal solutions via linear pro-gramming calculations (see SI Appendix for technical details).

Effects of Tumor Heterogeneity on Drug Combination Efficacy andToxicity Tradeoffs. As the first manifestation of our analysis, weused our empirical efficacy dataset (26, 29) in concert with asymmetric toxicity profile. In this circumstance, we expect thealgorithm to produce as optimal solutions drug combinationswith the most efficacious drug plus additional drugs incorporated

static deterministic linearmulti-objective optimizationefficacy

toxicity

tumor composition

drug

tum

or

drug combinationsat Pareto frontier

machine learning prior knowledge

given: tumor compositionmaximize: overall efficacyminimize: overall toxicity

Monte Carlosampling

drug optimization

Heterogeneoustumor

biopsy

How different are themost efficacious drugcombinations?

predominantsubpopulation only

all subpopulations

optim

al d

rug

com

bina

tions

statisticalanalyses

augmented weightedTchebycheff method

e.g. PLSR, k-NN, SVM elastic net, naive Bayes

C

B

...

A

...

Fig. 1. Schematic of computational model. (A) Current clinical practice fordiagnosis may result in sampling bias, with analysis based on only/mostly thepredominant subpopulation. As such, a key question we try to addresshere is how intratumor heterogeneity (and specifically, consideration ofthe entire heterogeneity vs. just the predominant subpopulation) affectsthe resulting optimized drug combinations. (B) To examine the effects ofintratumor heterogeneity on optimal drug combinations, Monte Carlosampling was applied to sample 10,000 heterogeneous tumor compositions.For each tumor composition, we mathematically optimized for a set of drugcombinations. Statistical and sensitivity analyses were applied to the sam-pling results. (C) Schematic for the optimization model. For a given tumorcomposition, drug combinations were optimized to maximize efficacy andminimize toxicity using a multiobjective optimization approach, scalarizedusing an augmented weighted Tchebycheff method, and solved iterativelyusing linear programming (SI Appendix). Additional drug or tumor proper-ties (derived from prior knowledge or machine learning) also may be in-corporated into this framework. Solving the multiobjective optimizationmodel leads to a set of solutions, or a Pareto optimal set, on the Paretofrontier, which represents a surface for which any increase in one objectiveresults in a decrease in the other objective.

10774 | www.pnas.org/cgi/doi/10.1073/pnas.1323934111 Zhao et al.

Dow

nloa

ded

by g

uest

on

July

25,

202

0

into the combination in order of decreasing efficacy. Partneringa symmetric toxicity profile with decreasing efficacy would sug-gest a deviation from linearity from the Pareto frontier curve asthe number of drugs in the combination is increased. Fig. 2Ashows a representative example of this anticipated tradeoffbetween toxicity and efficacy on the Pareto frontier for thehomogeneous tumor population (100% shp53). Interestingly,however, as heterogeneity is introduced to the population (55%shp53 plus 13 minor subpopulations), the shape of the Paretofrontier becomes surprisingly linear (Fig. 2B and Table S2). Thisbehavior, the shift from nonlinear Pareto frontier curve to linearas a tumor becomes more heterogeneous, also was observed withother subpopulation distributions (Fig. S4 and Table S3). Themeaning of this in tumor treatment terms is that heterogeneityhomogenizes the benefit of drug combinations, because eachadditional drug has differential effects on each subpopulation.In fact, most clinically used chemotherapeutic regimens con-

sist of multiple drugs, such as ABVD (doxorubicin, vinblastine,bleomycin, and dacarbazine), Stanford V (vinblastine, doxorubicin,vincristine, bleomycin, mechlorethamine, etoposide, and pred-nisone), and BEACOPP (bleomycin, etoposide, doxorubicin,cyclophosphamide, vincristine, procarbazine, and prednisone)for Hodgkin lymphoma. Accordingly, we imposed an upper limitof six drugs in our subsequent analyses. The consequence of thislimit for the same heterogeneous tumor in Fig. 2B is shown inFig. 2C (the corresponding Pareto frontier) for direct compari-son. Fig. 2D enumerates the drugs elicited by the algorithm foreach of the regimens, from one drug to six drugs.

Monte Carlo Analyses of Drug Combinations in Diverse HeterogeneousTumor Populations. The results above were obtained for singleheterogeneous tumors possessing a specified subpopulation dis-tribution. In practice, from a clinical tumor biopsy, one mightgain data on the genetic variants present but not with explicitquantitative distribution proportions. To deal with this challenge,we performed Monte Carlo sampling of heterogeneous tumorsfrom among the possible quantitative distributions for a givenset of genetic variants and analyzed the resulting frequency

distribution of component drugs in six-drug combinations. Overall,10,000 heterogeneous populations were generated with each pop-ulation sampled, based first on the number of subpopulations,followed by the subpopulation genetic variants, and subsequentlythe subpopulation proportions. Upon drug combination op-timization, we obtain predictions of the six-drug combinationsmost likely to be beneficial for a heterogeneous tumor with thespecified genetic variants, averaged over potential unknownquantitative proportions. These genetic variants are character-ized by a set of up to 30 shRNA knockdowns, targeting genes inthe DNA damage repair and apoptosis pathways (Fig. S1). Thecomponent drugs most frequently included in the optimal drugcombination for this class of heterogeneous tumor were rapa-mycin, 5-fluorouracil (5-FU), vincristine, and sunitinib (Fig. 3A);we term these the “dominant” drugs. Although these drugs donot covary based on clustering analysis over all populations (Fig.S5), when we analyzed the frequency of drug inclusion as afunction of tumor complexity (i.e., the number of subpopulationsin the heterogeneous tumor), we observed a strong dependencefor a select number of drugs: some positively and some nega-tively correlating with tumor complexity (Fig. 3 B and C). Forinstance, with increasing tumor complexity, we observed an in-crease in the frequencies of inclusion of the four drugs notedabove but a decrease in the frequencies of some other drugs(such as NSC3852, temozolomide, roscovitine, cisplatin, mito-mycin C, and camptothecin). In Fig. 3B, these relationships areillustrated with positive or negative trends, respectively, and Fig.3C quantifies them in terms of Spearman correlation coefficientvalues. Furthermore, tumor complexity is a crucial characteristic,because this strong correlation is abrogated when we control fortumor complexity (by examining the subsequent partial correla-tion) (Fig. S6).To expand the generalizability of our insights further, we

also analyzed combinations of numbers of drugs other than six;i.e., representing other regimens along a Pareto frontier. The

A B

C

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Compromise

Utopia

Nadir100% shp53

Efficacy

Toxi

city

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Compromise

Utopia

Nadir55% shp53 + others

Efficacy

Toxi

city

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Compromise

Utopia

Nadir Objective space

Efficacy

Toxi

city

Drugs

Reg

imen

sSolution space

So Su Dox EtopCDDPMMC CBL CPT 6−TG TMZ MTX 5−FU HU VCR Taxol ActD RapaSAHA DACRosco NSC

0123456

D

0

1

2

3

4

5

6

Fig. 2. Intratumor heterogeneity linearizes the Pareto frontier andhomogenizes drug combination efficacy. (A) Representative objective spaceshowing the Pareto frontier and the tradeoff between toxicity and efficacyfor a homogeneous population of murine Eμ-myc; p19Arf−/− lymphoma cellsinfected with shp53 hairpin. The plot was generated based on the optimi-zation described in Fig. 1C, using efficacy data composed of single-drugefficacy for individual subpopulations (Fig. S1) and a symmetric toxicityprofile for each drug. (B) Representative objective space showing the effectsof a heterogeneous population containing a predominant shp53 sub-population and 13 minor subpopulations on the Pareto frontier. (C and D)Representative objective and solution space (shown for the same populationas in B), with a maximum toxicity constraint of 6, which is set based on thenumber of drugs present in commonly used combination regimens. In thecontext of the mathematical model, this refers to the value for the param-eter amax (SI Appendix). Compromise solution (colored green) refers to thepoint closest to the utopia based on an L1 norm distance metric.

A

B

C D

0 2000 4000 6000 8000CPT

MMCRosco

NSCCDDP

DACTMZCBL

TaxolDoxHU

MTX6-TGEtop

SAHASo

ActDSu

VCR5-FURapa

Frequency

Dru

gs

no. of subpop.

freq.

0

100

200

So0.21

150

200

250

Su0.65

50

100

150

Dox0.1

50

100

150

Etop0.2

20

40

60

CDDP-0.73

0

20

40

MMC-0.66

40

60

80

CBL-0.42

0

10

20

CPT-0.64

80

100

120

6-TG-0.43

0

100

200

TMZ-0.86

80

100

120

MTX0.19

150

200

250

5-FU0.63

50

100

150

HU0.14

100

200

300

VCR0.77

50

100

150

Taxol0.02

100

150

200

ActD0.25

100

200

300

Rapa0.79

50

100

150

SAHA0.17

0

50

100

DAC-0.45

0

50

100

Rosco-0.77

0

50

100

NSC-0.87

SoSu

DoxEtop

CDDPMMCCBLCPT

6−TGTMZMTX

5−FUHU

VCRTaxolActDRapa

SAHADAC

RoscoNSC

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Spearm

an correlation

Regimens1 2 3 4 5 6

1

2

34

5

6

Regim

ens

Spearman correlation0.5 1

Fig. 3. Optimal drugs dominate at higher tumor complexity. (A) Frequencydistribution of component drugs in optimal drug combinations based onMonte Carlo sampling results, specifically for six-drug combinations (regi-men 6) and optimized using efficacy data (Fig. S1) and a symmetric toxicityprofile. Analyses shown in B and C also are based on the six-drug combi-nation. (B) Frequency of component drugs as a function of tumor complexity(i.e., the number of subpopulations in the heterogeneous tumor). The trendfor each scatter plot is quantified with Spearman correlation, the values ofwhich are shown below the corresponding component drug title. (C) Heatmap of the Spearman correlations (in B) between component drug fre-quency and tumor complexity. Together, B and C reveal that selecting a setof drugs (e.g., rapamycin, 5-FU, vincristine, sunitinib) strongly depends ontumor complexity. (D) Spearman correlation of component drug frequenciesacross different drug regimens in the Pareto optimal set. The high correla-tion illustrates that the frequency distribution of drugs for the six-drugcombination (shown in A) is comparable to other regimens (with N-drugcombinations).

Zhao et al. PNAS | July 22, 2014 | vol. 111 | no. 29 | 10775

SYST

EMSBIOLO

GY

ENGINEE

RING

Dow

nloa

ded

by g

uest

on

July

25,

202

0

frequency distributions of the component drugs were found to besimilar among the various N-drug combinations (annotated as“Regimen N”), as calculated in terms of the correlation betweenthe drug frequency distribution in Regimen N vs. that in Regi-men M (Fig. 3D and Fig. S8). In other words, the drugs more andless dominant in a regimen combining N drugs are similarly moreand less dominant in a regimen combining M drugs. Thus, thebehavior of drug dominance among a constellation of potentialdrugs as a function of heterogeneous tumor complexity is con-sistent across a broad range of multidrug regimens.We next asked how the drug frequency distributions might

change if we determine the optimal combinations based solely onthe predominant tumor subpopulation instead of consideringall the subpopulations present in each heterogeneous tumor. Usingthe same set of 10,000 theoretical tumors, we found only smallchanges in the overall distribution of drug frequencies withinoptimal combinations under this approach (Fig. 4A, in compar-ison with Fig. 3A). However, quite surprisingly, the correlationbetween drug dominance and tumor complexity is abrogateddramatically (Fig. 4B in comparison with Fig. 3B, and Fig. 4C incomparison with Fig. 3C). For example, the choice of whether toinclude drugs such as rapamycin, 5-FU, vincristine, and sunitinibno longer strongly depends on tumor complexity, but insteadvaries sensitively with the specific individual predominant sub-population. As a consequence, the drug distributions found inoptimal combinations across N-drug regimens may be very dif-ferent from those for M-drug regimens (Fig. 4D and Fig. S9, instriking contrast to Fig. 3D and Fig. S8). In addition, changes incorrelations are minimal when compared before and after con-trolling for tumor complexity, further suggesting that tumorcomplexity no longer is an important factor (Fig. S7). Hence, theoptimal design of combination drugs based on only a predominantsubpopulation—i.e., a tumor assumed to be homogeneous—

potentially may exclude drugs that work effectively for highlyheterogeneous tumors.We therefore proceeded to analyze in detail the effects of

these two alternative optimization perspectives—considerationof the entire heterogeneous tumor vs. just the predominantsubpopulation—on the drug combinations predicted to be opti-mal for tumors whose complexity resides within the sampled10,000 populations. We observed that there was a large pro-portion of tumors for which the optimal drug combinations weredisparate between the two perspectives (Fig. 5A), with the frac-tion of optimal drug combinations for a given tumor found to bedifferent when taking heterogeneity into account vs. not taking itinto account, increasing from about 40% for one-drug regimensto almost 70% for six-drug regimens. Quantitatively, the differ-ence in efficacy between the drug combinations optimized basedon the two perspectives followed a roughly exponential profile(Fig. 5B). This means that cases for which the efficacy differencefalls toward the tail of the profile will exhibit dramatic dis-parities in therapeutic outcome when only the predominantsubpopulation is considered in determining the best drug.Now, it is conceivable that a drug most frequently deemed

worthy of inclusion within a regimen optimized for a pre-dominant subpopulation might generally also be found fre-quently included within a regimen determined from consideringthe entire tumor complexity. Therefore, we probed this notionagain using our 10,000 heterogeneous tumor sample, for eachcase asking whether the drug combination optimized based onthe entire tumor complexity also contained the single-best drugfor the predominant subpopulation. Notably, there were manyregimens for which the drug combination optimized based on theentire heterogeneity did not contain the single-best drug for thepredominant subpopulation (Fig. 5C). This proportion expect-edly decreased with an increasing number of drugs in the com-binations; for instance, for two- and three-drug regimens, 39%and 23% of the cases, respectively, did not include the drug thatwould have been best for the predominant subpopulation. Weobtained similar conclusions when we performed the sameanalyses on drug combinations optimized based also on thesecond largest subpopulations (Fig. S10). To confirm the ro-bustness of these conclusions, we performed the same analysesbut decomposed the results with respect to the predominant andsecond largest subpopulation proportions (Fig. S11). We foundthat the proportion of regimens with different drug combina-tions, as well as those containing single-best drugs for pre-dominant (and second largest) subpopulations, is robust nearboundary cases across different predominant subpopulationproportions—and begins to decrease expectedly only when thepredominant subpopulation becomes very large compared withthe remaining subpopulations (Fig. S12). We also performedMonte Carlo sampling with more than 13,000 heterogeneouspopulations near the boundaries, with similar results (Fig. S13).This firmly demonstrates that selecting drug combinations fora heterogeneous tumor often may not follow from simple in-tuition—which seemingly would involve including the best drugfor the predominant subpopulation.Taken together, these analyses indicate that intratumor het-

erogeneity may influence drug combinations dramatically suchthat the chosen regimen may have strongly disparate outcomesbased on whether we examine the entire heterogeneity or just thepredominant subpopulation. This insight argues for the necessityof considering at least even qualitative features of heterogeneityin selecting combination regimens.

Sensitivity Analysis of Monte Carlo Sampling. Next we sought toexamine in greater detail the specific dependencies of a par-ticular drug choice on the tumor subpopulations present. Toaccomplish this, we performed a global sensitivity analysis bycalculating the correlation between a given tumor subpopula-tion (i.e., shRNA knockdown) and the categorical outcome ofwhether a particular drug was chosen by the optimization al-gorithm to be included in the combination regimen for the

A

B

C D Regimens1 2 3 4 5 6

1

234

5

6

Regim

ens

Spearman correlation0.5 10 2000 4000 6000

CPTMMCNSC

CDDPDAC

RoscoCBL

TaxolHU

DoxSo

MTXSAHA

TMZEtop6-TGVCRActDRapa

Su5-FU

Frequency

Dru

gs

SoSu

DoxEtop

CDDPMMCCBLCPT

6−TGTMZMTX

5−FUHU

VCRTaxolActDRapa

SAHADAC

RoscoNSC

Spearm

an correlation

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

no. of subpop.

freq.

050

100

150

So0.01

100

150

200

Su-0.01

50

100

150

Dox0.12

50

100

150

Etop0.16

40

60

80

CDDP-0.02

0

20

40

MMC-0.01

60

80

100

CBL0.09

0

10

20

CPT-0.09

100

150

200

6-TG0.08

50

100

150

TMZ0.12

50

100

150

MTX0.06

150

200

250

5-FU0.22

60

80

100

HU-0.17

100

150

200

VCR0.11

60

80

100

Taxol0.16

100

150

200

ActD0.3

100

150

200

Rapa0.22

50

100

150

SAHA0.15

0

50

100

DAC0.25

0

50

100

Rosco-0.09

0

50

100

NSC0.22

Fig. 4. Drug optimization based on consideration of only the predominantsubpopulation abrogates drug dominance at greater tumor complexity. Sta-tistical analyses are formatted similar to those in Fig 3. However, here theanalyses are based on drug optimization by considering only the pre-dominant subpopulation in each heterogeneous tumor population. (A) Fre-quency distribution of component drugs in optimal drug combinations, basedon Monte Carlo sampling results, specifically for six-drug combinations.Analyses shown in B and C also are based on the six-drug combination. (B)Frequency of component drugs as a function of tumor complexity. The trendfor each scatter plot is quantified with Spearman correlation, the values ofwhich are shown below the corresponding component drug title. (C) Heatmap of the Spearman correlations (in B) between component drug frequencyand tumor complexity. Together, B and C reveal that in contrast to Fig. 3 Band C, the drugs no longer depend on tumor complexity when only thepredominant subpopulation is considered in the drug optimization. (D) Fre-quency of component drugs in relation to other drug regimens in the Paretooptimal set. In contrast to Fig. 3C, the lower correlation values suggest thatthe drug frequencies are less consistent across the different regimens.

10776 | www.pnas.org/cgi/doi/10.1073/pnas.1323934111 Zhao et al.

Dow

nloa

ded

by g

uest

on

July

25,

202

0

10,000 sampled tumors; this is termed a point-biserial correla-tion. In general, there were varying degrees of subpopulationdependency for the different drugs, negative as well as positive(Fig. 6A). In addition, the more frequently chosen drugs exhibitbroader distributions of point-biserial correlation values than dothe less frequently chosen drugs (Fig. 6B). Interestingly, the lessfrequently chosen drugs typically were characterized by a verytight range of dependencies, and were characterized most dis-tinctly by outliers with respect to a particular tumor. In otherwords, the more frequently chosen drugs are relatively robustto the uncertainty concerning quantitative features of tumorheterogeneity, whereas the less frequently chosen drugs areproblematically sensitive to precisely which subpopulations arepresent and in what quantitative proportions. This insight can becaptured by the kurtosis of the distribution of the point-biserialcorrelation values, which was found to be strongly negativelycorrelated with the drug’s frequency (Fig. 6C). In addition torobustness, we also observed that, as expected, drug frequencywas correlated with mean efficacy averaged over the entire set of10,000 tumor samples. Taken together, all the results from thesensitivity analysis indicate that the most frequently chosen drugsfor optimal drug combinations for heterogeneous tumors ofuncertain subpopulation distribution are characterized by theirmean and robustness in efficacy.

Multiobjective Optimization Comprising Particular Drug ToxicityAlong with Particular Drug Efficacy. Our multiobjective optimiza-tion model allows for the incorporation of multiple objectives inderiving optimal drug combinations, with whatever sophistica-tion in objective might be desired. As such, we extended ouranalysis with greater nuance by recognizing that the toxicitiesdiffer—that is, are asymmetric—across systemic tissue types foreach given drug and include this recognition in our objectivefunctions. Accordingly, we incorporated actual dose-limitingtoxicity information for each drug in the dataset (Fig. S15A). Weobserved that now a larger set of drug choices could be in-corporated for each multidrug regimen along the Pareto frontier(Fig. S14) and that for myelosuppression and gastrointestinaleffects, two of the most common side effects of the drugs in ourdataset, the distribution in the number of overlaps between drugsin toxicity was effectively shifted to minimize such overlaps (Fig.S15 B and C and S16). Hence, a multiobjective optimizationapproach enables the incorporation of multiple drug charac-

teristics in examining tradeoffs in the rational design of drugcombinations in the context of intratumor heterogeneity.

DiscussionThe substantial spatial and temporal intratumor heterogeneitythat inevitably exists in cancer patients presents a fundamentalchallenge for the rational design of effective drug combina-tions. Here, we applied a multiobjective optimization approach,grounded in empirical experimental data. These data comprisedquantitative effects of commonly used chemotherapeutics andtargeted therapeutics on subpopulations of the murine Eμ-myc;p19Arf−/− lymphoma cell line generated via shRNA knockdownsof genes in the DNA damage repair and apoptosis pathways. Thecell line was derived from the well-established Eμ-myc mousemodel of human Burkitt lymphoma (30), and the existing p19Arf−/−

potentially captures the early events in tumor evolution (31). Ourcomputational analysis on these data generated predictions ofoptimal drug combinations for tumors in which the geneticheterogeneity is qualitatively characterized (in terms of theshRNA knockdowns) but quantitatively uncertain with respect tothe various relative proportions in the tumor. An initial key re-sult is that tumor heterogeneity can effectively homogenize drugefficacy, so the most effective drug combinations are those thatbest kill the broadest range of subpopulations. Notably, we foundthat optimizing drug combinations based on consideration ofthe entire tumor heterogeneity instead of just the predominantsubpopulation may result in nonintuitive optimal drug combi-nations. As such, knowledge of the single “best agent” for eachsubpopulation does not promise that it will be an optimal choicefor inclusion within the overall drug combination when the entireheterogeneous tumor is considered. We recently demonstratedsuccessful validation of our optimization approach for selectedtumor subpopulation distributions in both in vitro cell cultureand in vivo mouse contexts (25).The substantial tumor complexity that exists in patients and

the detection limit of diagnostic tools undoubtedly cast un-certainty and incomplete information on the underlying tumorcomposition for each patient. However, our statistical samplingand sensitivity analyses offer guiding principles for the charac-teristics of effective drugs under tumor diversity. In particular,we discovered that a certain set of drugs dominated the solutions

1 2 3 4 5 60

2000

4000

6000

8000

10000

Freq

uenc

y

100% 61%

39%

77%

23%

85%

15%

89%

11%

92%

8%

0 1 2 3 4 5 60

500

1000

1500

2000

2500

3000

A B

C

No. of drugs in combination

Different drug combinationSame drug combination

Contains single-best drug forpredominant subpopulation

Does not contain single-best drug forpredominant subpopulation

Difference in efficacy

Freq

uenc

y

optimized based on entire heterogeneity versus predominant subpop. only}

Fig. 5. Drug combinations optimized based on consideration of the entireheterogeneity may be nonintuitive. (A) Breakdown of the optimal drugcombination regimens for 10,000 Monte Carlo-sampled heterogeneous tu-mor populations, showing proportions of solutions that are the same ordifferent depending on whether the entire heterogeneity is considered vs.just the predominant subpopulation. (B) Distribution of the difference inefficacy between drug combinations optimized based on the entire het-erogeneity vs. just the predominant subpopulation for the six-drug combi-nation. (C) For each regimen in the Pareto optimal set, breakdown ofsolutions that are different based on the two optimization approaches,showing the proportions for which drug combinations optimized based onentire heterogeneity still contains the single best drug for the predominantsubpopulation.

A B

C-0.96

-0.96

-0.68

-0.68

-0.86

0.087

0.087-0.042

-0.0420.69

0.69

0.94

0.94

1

1

1

1

1

-0.86 -0.63

-0.63

0.15

0.150.01

0.01

shC

trlsh

p53

shA

TMsh

Chk

2sh

Pum

ash

ATR

shC

hk1

shA

TXsh

DN

AP

Ksh

JNK

1sh

p38

shB

axsh

Bak

shB

oksh

Bim

shB

idsh

Nox

ash

Bad

shB

clx

shB

mf

shbc

lbsh

Bik

shB

PR

shH

rksh

BN

IPsh

Mil1

shB

clg

shB

clw

shM

ule

shA

1

Rapa5−FUVCR

SuActD

SoSAHA

Etop6−TGMTX

HUDox

TaxolCBLTMZDAC

CDDPNSC

RoscoMMCCPT

−1 0 1Point−biserial correlation (rpb)

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

CPTMMC

RoscoNSC

CDDPDACTMZCBL

TaxolDoxHU

MTX6−TGEtop

SAHASo

ActDSu

VCR5−FURapa

Point−biserial correlation (rpb)

kurto

sis

r pb

rang

e r p

b

mea

n r p

b

mea

n ef

ficac

yof

dru

gs

freq.

of d

rugs

−1 0 1Spearman correlation

kurtosis rpb

range rpb

mean rpb

mean efficacyof drugs

freq. of drugs-0.96

-0.96

-0.68

-0.68

-0.86

0.087

0.087-0.042

-0.0420.69

0.69

0.94

0.94

1

1

1

1

1

-0.86 -0.63

-0.63

0.15

0.150.01

0.01

Fig. 6. Sensitivity analysis reveals optimal drugs are most robust and, onaverage, most efficacious. (A) Point-biserial correlation (rpb) showing thedependence of component drug choice on subpopulation. The drugs areordered according to their frequency in the optimal drug combination (Fig.3A). (B) Distribution of point-biserial correlations for each component drug.(C) Correlation matrix of various drug characteristics. Component drug fre-quency is strongly positively correlated with mean efficacy and negativelycorrelated with kurtosis of rpb, a metric used here for robustness.

Zhao et al. PNAS | July 22, 2014 | vol. 111 | no. 29 | 10777

SYST

EMSBIOLO

GY

ENGINEE

RING

Dow

nloa

ded

by g

uest

on

July

25,

202

0

for optimal drug combination at increasing tumor complexity,and such drugs were associated with greater average efficacyand robustness. This indicates that in the absence of completeknowledge of a tumor’s composition, we nonetheless can ap-ply a de novo design of optimal drug combinations based ondrugs’ average efficacy and robustness properties. Ultimatelythis requires experimental validation in a clinically relevantmodel. Although our dataset, acquired based on in vitro assaysusing clinically relevant genes and chemotherapies, is compre-hensive with regard to knowledge of the efficacy of single agentsin single subpopulations, such complete knowledge has yet to berealized in the clinical setting for experimental validation. Nev-ertheless, recent large-scale efforts, such as The Cancer GenomeAtlas and the Cancer Cell Line Encyclopedia (32), are beginningto elucidate patient tumor composition and subpopulationresponses. The accumulation of this type of information mayprovide opportunities to deconvolute and derive efficacy data fora single drug on individuals subpopulations, allowing this meth-odology to be used in the derivation of drug combinations sampledbased on a prior distribution of known genetic variants of a cancertype. This would enable the experimental validation of many clinicalsamples, with responses tracked longitudinally following treatment.Additional considerations of drug–drug interactions, including

nonlinear drug–drug efficacy (e.g., drug synergy or antagonism)and toxicity interactions, and tumor dynamics undoubtedly addto the complexity of drug combination optimization. Specificgenetic alterations also may affect subpopulation interactionsand mutation rates, with consequences on the evolutionary tra-jectories of the tumor, in the absence of drug selection. Theseadditional rate parameters, generally used in stochastic and de-terministic differential equation-based models of tumor evolu-tionary dynamics, would be particularly useful to consider in thedesign of sequential treatment strategies. Moreover, although

our model optimized for overall efficacy by targeting all sub-populations, an optimal drug combination would require furtherunderstanding of how differential selective pressures imposed bydrugs affect the potential for secondary resistance. This issue hasbeen studied particularly in antibiotic resistance, in which strongselection by synergistic drug combinations actually may increasethe risk of resistance (33, 34). Ultimately, just as multiobjectiveoptimization approaches may be applied to derive small mol-ecule structures with the desired polypharmacological profiles(35), a multiobjective optimization with considerations of theseadditional properties provides a potentially promising approach tooptimizing drug combination in the context of tumor heterogeneity.

MethodsFull details of methods and mathematical models may be found in SI Ap-pendix. Briefly, the efficacy dataset was derived previously by using an invitro competition assay with murine Eμ-myc; p19Arf−/− lymphoma cells par-tially infected with a specific shRNA and treated with a broad range ofchemotherapies (26, 29). The toxicity dataset was a binary matrix of knowndose-limiting toxicities for each drug. The optimization was formulated asa multiobjective optimization model with the objective of maximizing effi-cacy while minimizing toxicity. The optimization was solved iteratively withlinear programming using MATLAB 2012b (MathWorks) and CPLEX 12.5.Monte Carlo sampling was performed with simulation of 10,000 heteroge-neous tumors drawn from a specified distribution of subpopulations, followedby drug combination optimization on each tumor composition. Statistical andsensitivity analyses were performed using standard packages in MATLAB. AllMATLAB codes are publicly available at http://icbp.mit.edu/data-models.

ACKNOWLEDGMENTS. We thank Dane Wittrup and Leona Samson for theirinsightful comments. Funding was provided by Integrative Cancer BiologyProgram Grant U54-CA112967 (to M.T.H. and D.A.L.). B.Z. is supported byNational Institutes of Health/National Institute of General Medical SciencesInterdepartmental Biotechnology Training Program 5T32GM008334.

1. Nowell PC (1976) The clonal evolution of tumor cell populations. Science 194(4260):23–28.

2. Marusyk A, Almendro V, Polyak K (2012) Intra-tumour heterogeneity: A looking glassfor cancer? Nat Rev Cancer 12(5):323–334.

3. Snuderl M, et al. (2011) Mosaic amplification of multiple receptor tyrosine kinasegenes in glioblastoma. Cancer Cell 20(6):810–817.

4. Navin N, et al. (2011) Tumour evolution inferred by single-cell sequencing. Nature472(7341):90–94.

5. Gerlinger M, et al. (2012) Intratumor heterogeneity and branched evolution revealedby multiregion sequencing. N Engl J Med 366(10):883–892.

6. Notta F, et al. (2011) Evolution of human BCR-ABL1 lymphoblastic leukaemia-initi-ating cells. Nature 469(7330):362–367.

7. Anderson K, et al. (2011) Genetic variegation of clonal architecture and propagatingcells in leukaemia. Nature 469(7330):356–361.

8. Campbell PJ, et al. (2008) Subclonal phylogenetic structures in cancer revealed byultra-deep sequencing. Proc Natl Acad Sci USA 105(35):13081–13086.

9. Ding L, et al. (2012) Clonal evolution in relapsed acute myeloid leukaemia revealed bywhole-genome sequencing. Nature 481(7382):506–510.

10. Keats JJ, et al. (2012) Clonal competition with alternating dominance in multiplemyeloma. Blood 120(5):1067–1076.

11. Misale S, et al. (2012) Emergence of KRAS mutations and acquired resistance to anti-EGFR therapy in colorectal cancer. Nature 486(7404):532–536.

12. Mullighan CG, et al. (2008) Genomic analysis of the clonal origins of relapsed acutelymphoblastic leukemia. Science 322(5906):1377–1380.

13. Swan GW (1990) Role of optimal control theory in cancer chemotherapy. Math Biosci101(2):237–284.

14. Shi J, Alagoz O, Erenay FS, Su Q (2011) A survey of optimization models on cancerchemotherapy treatment planning. Ann Oper Res, 10.1007/s10479-011-0869-4.

15. Martin RB (1992) Optimal control drug scheduling of cancer chemotherapy. Auto-matica 28(6):1113–1123.

16. Swan GW, Vincent TL (1977) Optimal control analysis in the chemotherapy of IgGmultiple myeloma. Bull Math Biol 39(3):317–337.

17. Coldman AJ, Goldie JH (1983) A model for the resistance of tumor cells to cancerchemotherapeutic agents. Math Biosci 65(2):291–307.

18. Beckman RA, Schemmann GS, Yeang C-H (2012) Impact of genetic dynamics andsingle-cell heterogeneity on development of nonstandard personalized medicinestrategies for cancer. Proc Natl Acad Sci USA 109(36):14586–14591.

19. Bozic I, et al. (2013) Evolutionary dynamics of cancer in response to targeted com-bination therapy. eLife 2:e00747.

20. Foo J, Michor F (2009) Evolution of resistance to targeted anti-cancer therapies during

continuous and pulsed administration strategies. PLOS Comput Biol 5(11):e1000557.21. Komarova NL, Wodarz D (2005) Drug resistance in cancer: principles of emergence

and prevention. Proc Natl Acad Sci USA 102(27):9714–9719.22. Das Thakur M, et al. (2013) Modelling vemurafenib resistance in melanoma reveals

a strategy to forestall drug resistance. Nature 494(7436):251–255.23. Chmielecki J, et al. (2011) Optimization of dosing for EGFR-mutant non-small cell lung

cancer with evolutionary cancer modeling. Sci Transl Med 3(90):90ra59.24. Radhakrishnan ML, Tidor B (2008) Optimal drug cocktail design: Methods for tar-

geting molecular ensembles and insights from theoretical model systems. J Chem Inf

Model 48(5):1055–1073.25. Zhao B, Pritchard JR, Lauffenburger DA, Hemann MT (2014) Addressing genetic tu-

mor heterogeneity through computationally predictive combination therapy. Cancer

Discov 4(2):166–174.26. Pritchard JR, et al. (2013) Defining principles of combination drug mechanisms of

action. Proc Natl Acad Sci USA 110(2):E170–E179.27. Swanton C (2012) Intratumor heterogeneity: Evolution through space and time.

Cancer Res 72(19):4875–4882.28. Su K-Y, et al. (2012) Pretreatment epidermal growth factor receptor (EGFR) T790M

mutation predicts shorter EGFR tyrosine kinase inhibitor response duration in pa-

tients with non-small-cell lung cancer. J Clin Oncol 30(4):433–440.29. Jiang H, Pritchard JR, Williams RT, Lauffenburger DA, Hemann MT (2011) A mam-

malian functional-genetic approach to characterizing cancer therapeutics. Nat Chem

Biol 7(2):92–100.30. Adams JM, et al. (1985) The c-myc oncogene driven by immunoglobulin enhancers

induces lymphoid malignancy in transgenic mice. Nature 318(6046):533–538.31. Schmitt CA, McCurrach ME, de Stanchina E, Wallace-Brodeur RR, Lowe SW (1999)

INK4a/ARF mutations accelerate lymphomagenesis and promote chemoresistance by

disabling p53. Genes Dev 13(20):2670–2677.32. Barretina J, et al. (2012) The Cancer Cell Line Encyclopedia enables predictive mod-

elling of anticancer drug sensitivity. Nature 483(7391):603–607.33. Pena-Miller R, et al. (2013) When the most potent combination of antibiotics selects

for the greatest bacterial load: The smile-frown transition. PLoS Biol 11(4):e1001540.34. Hegreness M, Shoresh N, Damian D, Hartl D, Kishony R (2008) Accelerated evolution

of resistance in multidrug environments. Proc Natl Acad Sci USA 105(37):13977–13981.35. Besnard J, et al. (2012) Automated design of ligands to polypharmacological profiles.

Nature 492(7428):215–220.

10778 | www.pnas.org/cgi/doi/10.1073/pnas.1323934111 Zhao et al.

Dow

nloa

ded

by g

uest

on

July

25,

202

0