On the probability of random genetic mutations for various types of tumor growth

Analyzing the distribution of progression-free survival for combination therapies: A study of model-based translational predictive methods in oncology

Progression-free survival (PFS) is an important clinical metric in oncology and is typically illustrated and evaluated using a survival function. The survival function is often estimated post-hoc using the Kaplan-Meier estimator but more sophisticated techniques, such as population modeling using the nonlinear mixed-effects framework, also exist and are used for predictions. However, depending on the choice of population model PFS will follow different distributions both quantitatively and qualitatively. Hence the choice of model will also affect the predictions of the survival curves. In this paper, we analyze the distribution of PFS for a frequently used tumor growth inhibition model with and without drug-resistance and highlight the translational implications of this. Moreover, we explore and compare how the PFS distribution for combination therapy differs under the hypotheses of additive and independent-drug action. Furthermore, we calibrate the model to preclinical data and use a previously calibrated clinical model to show that our analytical conclusions are applicable to real-world setting. Finally, we demonstrate that independent-drug action can effectively describe the tumor dynamics of patient-derived xenografts (PDXs) given certain drug combinations.

Mutation divergence over space in tumour expansion

Mutation accumulation in tumour evolution is one major cause of intra-tumour heterogeneity (ITH), which often leads to drug resistance during treatment. Previous studies with multi-region sequencing have shown that mutation divergence among samples within the patient is common, and the importance of spatial sampling to obtain a complete picture in tumour measurements. However, quantitative comparisons of the relationship between mutation heterogeneity and tumour expansion modes, sampling distances as well as the sampling methods are still few. Here, we investigate how mutations diverge over space by varying the sampling distance and tumour expansion modes using individual-based simulations. We measure ITH by the Jaccard index between samples and quantify how ITH increases with sampling distance, the pattern of which holds in various sampling methods and sizes. We also compare the inferred mutation rates based on the distributions of variant allele frequencies under different tumour expansion modes and sampling sizes. In exponentially fast expanding tumours, a mutation rate can always be inferred for any sampling size. However, the accuracy compared with the true value decreases when the sampling size decreases, where small sampling sizes result in a high estimate of the mutation rate. In addition, such an inference becomes unreliable when the tumour expansion is slow, such as in surface growth.

Clone Wars: Quantitatively Understanding Cancer Drug Resistance

A key aim of early clinical development for new cancer treatments is to detect the potential for efficacy early and to identify a safe therapeutic dose to take forward to phase II. Because of this need, researchers have sought to build mathematical models linking initial radiologic tumor response, often assessed after 6 to 8 weeks of treatment, with overall survival. However, there has been mixed success of this approach in the literature. We argue that evolutionary selection pressure should be considered to interpret these early efficacy signals and so optimize cancer therapy.

Universal Asymptotic Clone Size Distribution for General Population Growth

Deterministically growing (wild-type) populations which seed stochastically developing mutant clones have found an expanding number of applications from microbial populations to cancer. The special case of exponential wild-type population growth, usually termed the Luria–Delbrück or Lea–Coulson model, is often assumed but seldom realistic. In this article, we generalise this model to different types of wild-type population growth, with mutants evolving as a birth–death branching process. Our focus is on the size distribution of clones—that is the number of progeny of a founder mutant—which can be mapped to the total number of mutants. Exact expressions are derived for exponential, power-law and logistic population growth. Additionally, for a large class of population growth, we prove that the long-time limit of the clone size distribution has a general two-parameter form, whose tail decays as a power-law. Considering metastases in cancer as the mutant clones, upon analysing a data-set of their size distribution, we indeed find that a power-law tail is more likely than an exponential one.

Modeling drug resistance in a conjoint normal-tumor setting

Background

In this paper, we modify our previously developed conjoint tumor-normal cell model in order to make a distinction between tumor cells that are responsive to chemotherapy and those that may show resistance.

Results

Using this newly developed core model, the evolution of three cell types: normal, tumor, and drug-resistant tumor cells, is studied through a series of numerical simulations. In addition, we illustrate critical factors that cause different dynamical patterns for normal and tumor cells. Among these factors are the co-dependency of the normal and tumor cells, the cells’ response mechanism to a single or multiple chemotherapeutic treatment, the drug administration sequence, and the treatment starting time.

Conclusion

The results provide us with a deeper understanding of the possible evolution of normal, drug-responsive, and drug-resistant tumor cells during the cancer progression, which may contribute to improving the therapeutic strategies.

Evolutionary rescue: linking theory for conservation and medicine

Evolutionary responses that rescue populations from extinction when drastic environmental changes occur can be friend or foe. The field of conservation biology is concerned with the survival of species in deteriorating global habitats. In medicine, in contrast, infected patients are treated with chemotherapeutic interventions, but drug resistance can compromise eradication of pathogens. These contrasting biological systems and goals have created two quite separate research communities, despite addressing the same central question of whether populations will decline to extinction or be rescued through evolution. We argue that closer integration of the two fields, especially of theoretical understanding, would yield new insights and accelerate progress on these applied problems. Here, we overview and link mathematical modelling approaches in these fields, suggest specific areas with potential for fruitful exchange, and discuss common ideas and issues for empirical testing and prediction.

Evolution of acquired resistance to anti-cancer therapy

Acquired drug resistance is a major limitation for the successful treatment of cancer. Resistance can emerge due to a variety of reasons including host environmental factors as well as genetic or epigenetic alterations in the cancer cells. Evolutionary theory has contributed to the understanding of the dynamics of resistance mutations in a cancer cell population, the risk of resistance pre-existing before the initiation of therapy, the composition of drug cocktails necessary to prevent the emergence of resistance, and optimum drug administration schedules for patient populations at risk of evolving acquired resistance. Here we review recent advances towards elucidating the evolutionary dynamics of acquired drug resistance and outline how evolutionary thinking can contribute to outstanding questions in the field.

Effect of dedifferentiation on time to mutation acquisition in stem cell-driven cancers

Accumulating evidence suggests that many tumors have a hierarchical organization, with the bulk of the tumor composed of relatively differentiated short-lived progenitor cells that are maintained by a small population of undifferentiated long-lived cancer stem cells. It is unclear, however, whether cancer stem cells originate from normal stem cells or from dedifferentiated progenitor cells. To address this, we mathematically modeled the effect of dedifferentiation on carcinogenesis. We considered a hybrid stochastic-deterministic model of mutation accumulation in both stem cells and progenitors, including dedifferentiation of progenitor cells to a stem cell-like state. We performed exact computer simulations of the emergence of tumor subpopulations with two mutations, and we derived semi-analytical estimates for the waiting time distribution to fixation. Our results suggest that dedifferentiation may play an important role in carcinogenesis, depending on how stem cell homeostasis is maintained. If the stem cell population size is held strictly constant (due to all divisions being asymmetric), we found that dedifferentiation acts like a positive selective force in the stem cell population and thus speeds carcinogenesis. If the stem cell population size is allowed to vary stochastically with density-dependent reproduction rates (allowing both symmetric and asymmetric divisions), we found that dedifferentiation beyond a critical threshold leads to exponential growth of the stem cell population. Thus, dedifferentiation may play a crucial role, the common modeling assumption of constant stem cell population size may not be adequate, and further progress in understanding carcinogenesis demands a more detailed mechanistic understanding of stem cell homeostasis.

Recent evidence suggests that, like many normal tissues, many cancers are maintained by a small population of immortal stem cells that divide indefinitely to produce many differentiated cells. Cancer stem cells may come directly from mutation of normal stem cells, but this route demands high mutation rates, because there are few normal stem cells. There are, however, many differentiated cells, and mutations can cause such cells to “dedifferentiate” into a stem-like state. We used mathematical modeling to study the effects of dedifferentiation on the time to cancer onset. We found that the effect of dedifferentiation depends critically on how stem cell numbers are controlled by the body. If homeostasis is very tight (due to all divisions being asymmetric), then dedifferentiation has little effect, but if homeostatic control is looser (allowing both symmetric and asymmetric divisions), then dedifferentiation can dramatically hasten cancer onset and lead to exponential growth of the cancer stem cell population. Our results suggest that dedifferentiation may be a very important factor in cancer and that more study of dedifferentiation and stem cell control is necessary to understand and prevent cancer onset.

Accumulation of neutral mutations in growing cell colonies with competition

Neutral mutations play an important role in many biological processes including cancer initiation and progression, the generation of drug resistance in bacterial and viral diseases as well as cancers, and the development of organs in multicellular organisms. In this paper we study how neutral mutants are accumulated in nonlinearly growing colonies of cells subject to growth constraints such as crowding or lack of resources. We investigate different types of growth control which range from "division-controlled" to "death-controlled" growth (and various mixtures of both). In division-controlled growth, the burden of handling overcrowding lies with the process of cell-divisions, the divisions slow down as the carrying capacity is approached. In death-controlled growth, it is death rate that increases to slow down expansion. We show that division-controlled growth minimizes the number of accumulated mutations, and death-controlled growth corresponds to the maximum number of mutants. We check that these results hold in both deterministic and stochastic settings. We further develop a general (deterministic) theory of neutral mutations and achieve an analytical understanding of the mutant accumulation in colonies of a given size in the absence of back-mutations. The long-term dynamics of mutants in the presence of back-mutations is also addressed. In particular, with equal forward- and back-mutation rates, if division-controlled and a death-controlled types are competing for space and nutrients, cells obeying division-controlled growth will dominate the population.