Intratumor heterogeneity alters most effective drugs in designed combinations

Abstract

The substantial spatial and temporal heterogeneity observed in patient tumors poses considerable challenges for the design of effective drug combinations with predictable outcomes. Currently, the implications of tissue heterogeneity and sampling bias during diagnosis are unclear for selection and subsequent performance of potential combination therapies. Here, we apply a multiobjective computational optimization approach integrated with empirical information on efficacy and toxicity for individual drugs with respect to a spectrum of genetic perturbations, enabling derivation of optimal drug combinations for heterogeneous tumors comprising distributions of subpopulations possessing these perturbations. Analysis across probabilistic samplings from the spectrum of various possible distributions reveals that the most beneficial (considering both efficacy and toxicity) set of drugs changes as the complexity of genetic heterogeneity increases. Importantly, a significant likelihood arises that a drug selected as the most beneficial single agent with respect to the predominant subpopulation in fact does not reside within the most broadly useful drug combinations for heterogeneous tumors. The underlying explanation appears to be that heterogeneity essentially homogenizes the benefit of drug combinations, reducing the special advantage of a particular drug on a specific subpopulation. Thus, this study underscores the importance of considering heterogeneity in choosing drug combinations and offers a principled approach toward designing the most likely beneficial set, even if the subpopulation distribution is not precisely known.

Keywords: cancer; combination therapy; systems biology.

Fig. 1.

Schematic of computational model. (A) Current clinical practice for diagnosis may result in sampling bias, with analysis based on only/mostly the predominant subpopulation. As such, a key question we try to address here is how intratumor heterogeneity (and specifically, consideration of the entire heterogeneity vs. just the predominant subpopulation) affects the resulting optimized drug combinations. (B) To examine the effects of intratumor heterogeneity on optimal drug combinations, Monte Carlo sampling was applied to sample 10,000 heterogeneous tumor compositions. For each tumor composition, we mathematically optimized for a set of drug combinations. Statistical and sensitivity analyses were applied to the sampling results. (C) Schematic for the optimization model. For a given tumor composition, drug combinations were optimized to maximize efficacy and minimize toxicity using a multiobjective optimization approach, scalarized using an augmented weighted Tchebycheff method, and solved iteratively using linear programming (SI Appendix). Additional drug or tumor properties (derived from prior knowledge or machine learning) also may be incorporated into this framework. Solving the multiobjective optimization model leads to a set of solutions, or a Pareto optimal set, on the Pareto frontier, which represents a surface for which any increase in one objective results in a decrease in the other objective.

Fig. 2.

Intratumor heterogeneity linearizes the Pareto frontier and homogenizes drug combination efficacy. (A) Representative objective space showing the Pareto frontier and the tradeoff between toxicity and efficacy for a homogeneous population of murine Eμ-myc; p19^Arf−/− lymphoma cells infected with shp53 hairpin. The plot was generated based on the optimization described in Fig. 1C, using efficacy data composed of single-drug efficacy for individual subpopulations (Fig. S1) and a symmetric toxicity profile for each drug. (B) Representative objective space showing the effects of a heterogeneous population containing a predominant shp53 subpopulation and 13 minor subpopulations on the Pareto frontier. (C and D) Representative objective and solution space (shown for the same population as in B), with a maximum toxicity constraint of 6, which is set based on the number of drugs present in commonly used combination regimens. In the context of the mathematical model, this refers to the value for the parameter a_max (SI Appendix). Compromise solution (colored green) refers to the point closest to the utopia based on an L₁ norm distance metric.

Fig. 3.

Optimal drugs dominate at higher tumor complexity. (A) Frequency distribution of component drugs in optimal drug combinations based on Monte Carlo sampling results, specifically for six-drug combinations (regimen 6) and optimized using efficacy data (Fig. S1) and a symmetric toxicity profile. 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 (i.e., the number of subpopulations in the heterogeneous tumor). The trend for each scatter plot is quantified with Spearman correlation, the values of which are shown below the corresponding component drug title. (C) Heat map of the Spearman correlations (in B) between component drug frequency and tumor complexity. Together, B and C reveal that selecting a set of drugs (e.g., rapamycin, 5-FU, vincristine, sunitinib) strongly depends on tumor complexity. (D) Spearman correlation of component drug frequencies across different drug regimens in the Pareto optimal set. The high correlation illustrates that the frequency distribution of drugs for the six-drug combination (shown in A) is comparable to other regimens (with N-drug combinations).

Fig. 4.

Drug optimization based on consideration of only the predominant subpopulation abrogates drug dominance at greater tumor complexity. Statistical analyses are formatted similar to those in Fig 3. However, here the analyses are based on drug optimization by considering only the predominant subpopulation in each heterogeneous tumor population. (A) Frequency distribution of component drugs in optimal drug combinations, based on 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 trend for each scatter plot is quantified with Spearman correlation, the values of which are shown below the corresponding component drug title. (C) Heat map of the Spearman correlations (in B) between component drug frequency and tumor complexity. Together, B and C reveal that in contrast to Fig. 3 B and C, the drugs no longer depend on tumor complexity when only the predominant subpopulation is considered in the drug optimization. (D) Frequency of component drugs in relation to other drug regimens in the Pareto optimal set. In contrast to Fig. 3C, the lower correlation values suggest that the drug frequencies are less consistent across the different regimens.

Fig. 5.

Drug combinations optimized based on consideration of the entire heterogeneity may be nonintuitive. (A) Breakdown of the optimal drug combination regimens for 10,000 Monte Carlo-sampled heterogeneous tumor populations, showing proportions of solutions that are the same or different depending on whether the entire heterogeneity is considered vs. just the predominant subpopulation. (B) Distribution of the difference in efficacy between drug combinations optimized based on the entire heterogeneity vs. just the predominant subpopulation for the six-drug combination. (C) For each regimen in the Pareto optimal set, breakdown of solutions that are different based on the two optimization approaches, showing the proportions for which drug combinations optimized based on entire heterogeneity still contains the single best drug for the predominant subpopulation.

Fig. 6.

Sensitivity analysis reveals optimal drugs are most robust and, on average, most efficacious. (A) Point-biserial correlation (r_pb) showing the dependence of component drug choice on subpopulation. The drugs are ordered 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 frequency is strongly positively correlated with mean efficacy and negatively correlated with kurtosis of r_pb, a metric used here for robustness.