Drug independence and the curability of cancer by combination chemotherapy

Abstract

Combination chemotherapy can cure certain leukemias and lymphomas, but most solid cancers are only curable at early stages. We review quantitative principles that explain the benefits of combining independently active cancer therapies in both settings. Understanding the mechanistic principles underlying curative treatments, including those developed many decades ago, is valuable for improving future combination therapies. We discuss contemporary evidence for long-established but currently neglected ideas of how combination therapy overcomes tumor heterogeneity. We show that a unified model of interpatient and intratumor heterogeneity describes historical progress in the treatment of pediatric acute lymphocytic leukemia (ALL), in which increasingly intensive combination regimens ultimately achieved high cure rates. We also describe three distinct aspects of drug independence that apply at different biological scales. The ability of these principles to quantitatively explain curative regimens suggests that supra-additive (synergistic) drug interactions are not required for successful combination therapy.

Keywords: combination therapy; drug independence; tumor heterogeneity.

Figure 1.. Conceptual model of how multi-agent chemotherapy overcomes inter- and intra-tumor heterogeneity to cure childhood ALL.

(A) ALGB Protocol 2 showed that single-agent chemotherapy for childhood ALL (methotrexate or 6-mercaptopurine) produces a survival distribution that is approximately normal, though truncated at zero (left); this is visualized by the probability density function of survival (right). (B) Response to chemotherapy can be quantified as ‘log-kills’, the reduction in the logarithm of the number of cancer cells. Log-kills can be estimated from therapy-induced increase in survival time, because when fewer cancer cells remain they take longer to grow back [16]. (C) Independently acting drugs are expected to produce additive log-kills, but in a heterogeneous human population where patients have different responsiveness to different chemotherapies (as in panel A), ‘drug additivity’ involves a different sum of effects in each patient. Here patient responses are illustrated for six independently acting chemotherapies: for each drug in each patient, the number of log-kills is randomly sampled from the single-drug distribution (panel A), and the net effect of the combination therapy is additive. Half of the illustrated patients achieve over 12 log-kills from combination therapy, which cures a cancer with initial population 10¹² (orange patients), and half have a complete response but are not cured (blue). (D) Combinations of 1 to 8 chemotherapies were simulated by the principles in panel C. Distributions of patient outcomes are plotted for each number of chemotherapies, where orange represents cured patients, and blue represents patients with a complete remission. (E) The historically measured rates of complete remission (blue band) and/or cure (orange band) achieved by increasing numbers of chemotherapies are compared to the rates expected according to independent drug action (dashed blue and orange lines).

Figure 2.. The Addition Law for Probability and its application to three distinct meanings of drug independence.

(A) The Addition Law for probability states that the probability of one or both of two events A and B occurring is the sum of the probability of event A and event B minus the probability of both events occurring. (B) Bliss independence applies the Addition Law to the toxicity of drug combinations. In the case of cancer chemotherapy, this is equivalent to adding log-kills. The Bliss independence model was experimentally observed to describe the cytotoxicity of multiple drugs (2, 3, 4 or 5) from the RCHOP combination in Diffuse Large B-Cell Lymphoma (DLBCL) cultures (replotted from [28]). (C) Law independence describes the expected fraction of cells in a tumor that are resistant to two or more drugs. Whereas Bliss independence describes the net toxicity of multiple drugs (how the ‘arrows’ add up), Law independence concerns how many cells belong to the subpopulation with multi-drug resistance (the group in which both ‘arrows’ are smaller). Law independence was tested by comparing the observed and expected number of DLBCL clones with resistance to multiple drugs in RCHOP (replotted from [28]); a higher observed rate of resistance indicated a modest degree of cross-resistance. (D) Frei Independence applies the Addition Law to the probability that a patient with cancer will respond to a drug combination; it can also be applied to progression free survival (PFS) versus time. The Frei independence model was tested for twenty clinical trials of combinations of cancer therapies; two had greater PFS than expected at 12 months (green points) (data replotted from [13,31]).

Figure 3.. Implications of drug additivity in different scenarios in oncology.

(A) In chemo-sensitive cancers such as pediatric ALL, the use of many highly active and non-cross-resistant therapies cures some patients. (B) When available drugs have ‘crossresistance’, cancer cells that survive one therapy have a greater likelihood of also surviving other therapies; this is schematized as ‘less than additive’ log-kills. Substantial cross-resistance can limit depth of response and could be an obstacle to cure. (C) In a cancer with limited sensitivity to available therapies, combinations of many agents may increase response rate and median survival time, but depth of response is unlikely to produce cure, unless used as adjuvant / neo-adjuvant therapy for microscopic disease. (D) In a cancer where few active therapies are available, none of which have a high response rate, the advantage of combination therapy may be to increase the chance that at least agent is active for a patient.