Modeling the Effects of Space Structure and Combination Therapies on Phenotypic Heterogeneity and Drug Resistance in Solid Tumors

Mathematical modeling and quantitative analysis of phenotypic plasticity during tumor evolution based on single-cell data

Tumor is a complex and aggressive type of disease that poses significant health challenges. Understanding the cellular mechanisms underlying its progression is crucial for developing effective treatments. In this study, we develop a novel mathematical framework to investigate the role of cellular plasticity and heterogeneity in tumor progression. By leveraging temporal single-cell data, we propose a reaction-convection-diffusion model that effectively captures the spatiotemporal dynamics of tumor cells and macrophages within the tumor microenvironment. Through theoretical analysis, we obtain the estimate of the pulse wave speed and analyze the stability of the homogeneous steady state solutions. Notably, we employe the AddModuleScore function to quantify cellular plasticity. One of the highlights of our approach is the introduction of pulse wave speed as a quantitative measure to precisely gauge the rate of cell phenotype transitions, as well as the novel implementation of the high-plasticity cell state/low-plasticity cell state ratio as an indicator of tumor malignancy. Furthermore, the bifurcation analysis reveals the complex dynamics of tumor cell populations. Our extensive analysis demonstrates that an increased rate of phenotype transition is associated with heightened malignancy, attributable to the tumor's ability to explore a wider phenotypic space. The study also investigates how the proliferation rate and the death rate of tumor cells, phenotypic convection velocity, and the midpoint of the phenotype transition stage affect the speed of tumor cell phenotype transitions and the progression to adenocarcinoma. These insights and quantitative measures can help guide the development of targeted therapeutic strategies to regulate cellular plasticity and control tumor progression effectively.

On minimising tumoural growth under treatment resistance

Drug resistance is a major challenge for curative cancer treatment, representing the main reason of death in patients. Evolutionary biology suggests pauses between treatment rounds as a way to delay or even avoid resistance emergence. Indeed, this approach has already shown promising preclinical and early clinical results, and stimulated the development of mathematical models for finding optimal treatment protocols. Due to their complexity, however, these models do not lend themself to a rigorous mathematical analysis, hence so far clinical recommendations generally relied on numerical simulations and ad-hoc heuristics. Here, we derive two mathematical models describing tumour growth under genetic and epigenetic treatment resistance, respectively, which are simple enough for a complete analytical investigation. First, we find key differences in response to treatment protocols between the two modes of resistance. Second, we identify the optimal treatment protocol which leads to the largest possible tumour shrinkage rate. Third, we fit the "epigenetic model" to previously published xenograft experiment data, finding excellent agreement, underscoring the biological validity of our approach. Finally, we use the fitted model to calculate the optimal treatment protocol for this specific experiment, which we demonstrate to cause curative treatment, making it superior to previous approaches which generally aimed at stabilising tumour burden. Overall, our approach underscores the usefulness of simple mathematical models and their analytical examination, and we anticipate our findings to guide future preclinical and, ultimately, clinical research in optimising treatment regimes.

Spatio-temporal modelling of phenotypic heterogeneity in tumour tissues and its impact on radiotherapy treatment

We present a mathematical model that describes how tumour heterogeneity evolves in a tissue slice that is oxygenated by a single blood vessel. Phenotype is identified with the stemness level of a cell and determines its proliferative capacity, apoptosis propensity and response to treatment. Our study is based on numerical bifurcation analysis and dynamical simulations of a system of coupled, non-local (in phenotypic "space") partial differential equations that link the phenotypic evolution of the tumour cells to local tissue oxygen levels. In our formulation, we consider a 1D geometry where oxygen is supplied by a blood vessel located on the domain boundary and consumed by the tumour cells as it diffuses through the tissue. For biologically relevant parameter values, the system exhibits multiple steady states; in particular, depending on the initial conditions, the tumour is either eliminated ("tumour-extinction") or it persists ("tumour-invasion"). We conclude by using the model to investigate tumour responses to radiotherapy, and focus on identifying radiotherapy strategies which can eliminate the tumour. Numerical simulations reveal how phenotypic heterogeneity evolves during treatment and highlight the critical role of tissue oxygen levels on the efficacy of radiation protocols that are commonly used in the clinic.

A combination therapy of oncolytic viruses and chimeric antigen receptor T cells: a mathematical model proof-of-concept

Combining chimeric antigen receptor T (CAR-T) cells with oncolytic viruses (OVs) has recently emerged as a promising treatment approach in preclinical studies that aim to alleviate some of the barriers faced by CAR-T cell therapy. In this study, we address by means of mathematical modeling the main question of whether a single dose or multiple sequential doses of CAR-T cells during the OVs therapy can have a synergetic effect on tumor reduction. To that end, we propose an ordinary differential equations-based model with virus-induced synergism to investigate potential effects of different regimes that could result in efficacious combination therapy against tumor cell populations. Model simulations show that, while the treatment with a single dose of CAR-T cells is inadequate to eliminate all tumor cells, combining the same dose with a single dose of OVs can successfully eliminate the tumor in the absence of virus-induced synergism. However, in the presence of virus-induced synergism, the same combination therapy fails to eliminate the tumor. Furthermore, it is shown that if the intensity of virus-induced synergy and/or virus oncolytic potency is high, then the induced CAR-T cell response can inhibit virus oncolysis. Additionally, the simulations show a more robust synergistic effect on tumor cell reduction when OVs and CAR-T cells are administered simultaneously compared to the combination treatment where CAR-T cells are administered first or after OV injection. Our findings suggest that the combination therapy of CAR-T cells and OVs seems unlikely to be effective if the virus-induced synergistic effects are included when genetically engineering oncolytic viral vectors.

Optimal chemotherapy counteracts cancer adaptive resistance in a cell-based, spatially-extended, evolutionary model

Most aggressive cancers are incurable due to their fast evolution of drug resistance. We model cancer growth and adaptive response in a simplified cell-based (CB) setting, assuming a genetic resistance to two chemotherapeutic drugs. We show that optimal administration protocols can steer cells resistance and turned it into a weakness for the disease. Our work extends the population-based (PB) model proposed by Orlando et al. (Physical Biology, 2012), in which a homogeneous population of cancer cells evolves according to a fitness landscape. The landscape models three types of trade-offs, differing on whether the cells are more, less, or equal effective when generalizing resistance to two drugs as opposed to specializing to a single one. The CB framework allows us to include genetic heterogeneity, spatial competition, and drugs diffusion, as well as realistic administration protocols. By calibrating our model on Orlando et al.'s assumptions, we show that dynamical protocols that alternate the two drugs minimize the cancer size at the end of (or at mid-points during) treatment. These results significantly differ from those obtained with the homogeneous model---suggesting static protocols under the pro-generalizing and neutral allocation trade-offs---highlighting the important role of spatial and genetic heterogeneities. Our work is the first attempt to search for optimal treatments in a CB setting, a step forward toward realistic clinical applications.

OncoVee™-MiniPDX-Guided Anticancer Treatment for Gastric Cancer Patients With Synchronous Liver Metastases: A Retrospective Cohort Analysis

Background

It is estimated that 35% of gastric cancer patients appear with synchronous distant metastases—the vast majority of patients presenting with metastatic hepatic disease. How to choose the most appropriate drugs or regimens is crucial to improve the prognosis of patients. We conducted this retrospective cohort analysis to evaluate the efficacy of OncoVee™-MiniPDX-guided treatment for these patients.

Methods

Gastric cancer patients with liver metastases (GCLM) were enrolled. Patients were divided into MiniPDX and control group according to their wishes. In the observation group, the OncoVee™-MiniPDX model was conducted to screen the most sensitive drug or regimens to determine the clinical administration. Meanwhile, patients were treated with regular medications in the control group according to the guidelines without the MiniPDX model. The primary endpoint was overall survival (OS), and the secondary outcomes included objective response rate (ORR), disease control rate (DCR), and progression-free survival (PFS).

Results

A total of 68 patients with GCLM were included, with the observation and control groups of 21 and 47 patients, respectively. The baseline characteristics of patients were balanced between these two groups. MiniPDX drug sensitivity tests were associated with the increased use of targeted drugs when compared with the control group (33.3 vs. 0%, p=0.032). Median OS was estimated to be 9.4 (95% CI, 7.9–11.2) months and 7.9 (95% CI, 7.2–8.7) months in the observation and control group, respectively. Both univariate (control group vs. MiniPDX group: HR=2.586, 95% CI= 1.362–4.908, p=0.004) and multivariate regression analyses (Control group vs. MiniPDX group: adjusted HR (aHR)=4.288, 95% CI= 1.452–12.671, p=0.008) showed the superiority of the observation group on OS. Similarly, MiniPDX-based regiments significantly improve the PFS of these cases (median PFS 6.7 months vs. 4.2 months, aHR=2.773, 95% CI=1.532–3.983, p=0.029). ORR and DCR were also improved in MiniPDX group comparing with control group (ORR, 57.14 vs. 25.53%, p=0.029; DCR: 85.71 vs. 68.08%, p=0.035).

Conclusion

OncoVee™-MiniPDX model, which was used to select drugs to guide antitumor treatment, was promising to prolong survival and improve the response rate of patients with GCLM. Further well-designed studies are needed to confirm the clinical benefits of MiniPDX.

Improving cancer treatments via dynamical biophysical models

Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.

Mini-patient-derived xenograft assay based on microfluidic technology promises to be an effective tool for screening individualized chemotherapy regimens for advanced non-small cell lung cancer

Patient-derived xenograft (PDX) assay has been widely used in preclinical research in patients with multidrug-resistant lung cancer. One hundred patients with non-small cell lung cancer (NSCLC) were divided into MiniPDX group and conventional group, with 50 cases in each group. The MiniPDX assay was established by enriching high-purity tumor cells using microfluidic technology to detect the drug sensitivity of NSCLC cells. All patients underwent conventional computed tomography (CT) scans of lung and mediastinum at baseline and during follow-up. Kaplan-Meier method was used to compare the overall survival and progression-free survival of two groups. The sensitivity of the same drug in different tumor xenograft varied greatly. The overall survival, progression-free survival, and clinical benefit rate of patients in the MiniPDX-guided chemotherapy group were significantly longer than those in the conventional chemotherapy group. MiniPDX assay may be an effective tool for screening chemotherapy regimens in NSCLC patients.

A Mathematical Study of the Influence of Hypoxia and Acidity on the Evolutionary Dynamics of Cancer

Hypoxia and acidity act as environmental stressors promoting selection for cancer cells with a more aggressive phenotype. As a result, a deeper theoretical understanding of the spatio-temporal processes that drive the adaptation of tumour cells to hypoxic and acidic microenvironments may open up new avenues of research in oncology and cancer treatment. We present a mathematical model to study the influence of hypoxia and acidity on the evolutionary dynamics of cancer cells in vascularised tumours. The model is formulated as a system of partial integro-differential equations that describe the phenotypic evolution of cancer cells in response to dynamic variations in the spatial distribution of three abiotic factors that are key players in tumour metabolism: oxygen, glucose and lactate. The results of numerical simulations of a calibrated version of the model based on real data recapitulate the eco-evolutionary spatial dynamics of tumour cells and their adaptation to hypoxic and acidic microenvironments. Moreover, such results demonstrate how nonlinear interactions between tumour cells and abiotic factors can lead to the formation of environmental gradients which select for cells with phenotypic characteristics that vary with distance from intra-tumour blood vessels, thus promoting the emergence of intra-tumour phenotypic heterogeneity. Finally, our theoretical findings reconcile the conclusions of earlier studies by showing that the order in which resistance to hypoxia and resistance to acidity arise in tumours depend on the ways in which oxygen and lactate act as environmental stressors in the evolutionary dynamics of cancer cells.

Phenotypic variation modulates the growth dynamics and response to radiotherapy of solid tumours under normoxia and hypoxia

In cancer, treatment failure and disease recurrence have been associated with small subpopulations of cancer cells with a stem-like phenotype. In this paper, we develop and investigate a phenotype-structured model of solid tumour growth in which cells are structured by a stemness level, which varies continuously between stem-like and terminally differentiated behaviours. Cell evolution is driven by proliferation and death, as well as advection and diffusion with respect to the stemness structure variable. Here, the magnitude and sign of the advective flux are allowed to vary with the oxygen level. We use the model to investigate how the environment, in particular oxygen levels, affects the tumour's population dynamics and composition, and its response to radiotherapy. We use a combination of numerical and analytical techniques to quantify how under physiological oxygen levels the cells evolve to a differentiated phenotype and under low oxygen level (i.e., hypoxia) they de-differentiate. Under normoxia, the proportion of cancer stem cells is typically negligible and the tumour may ultimately become extinct whereas under hypoxia cancer stem cells comprise a dominant proportion of the tumour volume, enhancing radio-resistance and favouring the tumour's long-term survival. We then investigate how such phenotypic heterogeneity impacts the tumour's response to treatment with radiotherapy under normoxia and hypoxia. Of particular interest is establishing how the presence of radio-resistant cancer stem cells can facilitate a tumour's regrowth following radiotherapy. We also use the model to show how radiation-induced changes in tumour oxygen levels can give rise to complex re-growth dynamics. For example, transient periods of hypoxia induced by damage to tumour blood vessels may rescue the cancer cell population from extinction and drive secondary regrowth.

Engineering in Medicine To Address the Challenge of Cancer Drug Resistance: From Micro- and Nanotechnologies to Computational and Mathematical Modeling

Drug resistance has profoundly limited the success of cancer treatment, driving relapse, metastasis, and mortality. Nearly all anticancer drugs and even novel immunotherapies, which recalibrate the immune system for tumor recognition and destruction, have succumbed to resistance development. Engineers have emerged across mechanical, physical, chemical, mathematical, and biological disciplines to address the challenge of drug resistance using a combination of interdisciplinary tools and skill sets. This review explores the developing, complex, and under-recognized role of engineering in medicine to address the multitude of challenges in cancer drug resistance. Looking through the "lens" of intrinsic, extrinsic, and drug-induced resistance (also referred to as "tolerance"), we will discuss three specific areas where active innovation is driving novel treatment paradigms: (1) nanotechnology, which has revolutionized drug delivery in desmoplastic tissues, harnessing physiochemical characteristics to destroy tumors through photothermal therapy and rationally designed nanostructures to circumvent cancer immunotherapy failures, (2) bioengineered tumor models, which have benefitted from microfluidics and mechanical engineering, creating a paradigm shift in physiologically relevant environments to predict clinical refractoriness and enabling platforms for screening drug combinations to thwart resistance at the individual patient level, and (3) computational and mathematical modeling, which blends in silico simulations with molecular and evolutionary principles to map mutational patterns and model interactions between cells that promote resistance. On the basis that engineering in medicine has resulted in discoveries in resistance biology and successfully translated to clinical strategies that improve outcomes, we suggest the proliferation of multidisciplinary science that embraces engineering.

Cell plasticity in cancer cell populations

In this review, we propose a recension of biological observations on plasticity in cancer cell populations and discuss theoretical considerations about their mechanisms.

A Mathematical Dissection of the Adaptation of Cell Populations to Fluctuating Oxygen Levels

The disordered network of blood vessels that arises from tumour angiogenesis results in variations in the delivery of oxygen into the tumour tissue. This brings about regions of chronic hypoxia (i.e. sustained low oxygen levels) and regions with alternating periods of low and relatively higher oxygen levels, and makes it necessary for cancer cells to adapt to fluctuating environmental conditions. We use a phenotype-structured model to dissect the evolutionary dynamics of cell populations exposed to fluctuating oxygen levels. In this model, the phenotypic state of every cell is described by a continuous variable that provides a simple representation of its metabolic phenotype, ranging from fully oxidative to fully glycolytic, and cells are grouped into two competing populations that undergo heritable, spontaneous phenotypic variations at different rates. Model simulations indicate that, depending on the rate at which oxygen is consumed by the cells, dynamic nonlinear interactions between cells and oxygen can stimulate chronic hypoxia and cycling hypoxia. Moreover, the model supports the idea that under chronic-hypoxic conditions lower rates of phenotypic variation lead to a competitive advantage, whereas higher rates of phenotypic variation can confer a competitive advantage under cycling-hypoxic conditions. In the latter case, the numerical results obtained show that bet-hedging evolutionary strategies, whereby cells switch between oxidative and glycolytic phenotypes, can spontaneously emerge. We explain how these results can shed light on the evolutionary process that may underpin the emergence of phenotypic heterogeneity in vascularised tumours.

Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment

Purpose

Drug resistance is a major impediment to the success of cancer treatment. Resistance is typically thought to arise from random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests that progression to drug resistance need not occur randomly, but instead may be induced by the treatment itself via either genetic changes or epigenetic alterations. This relatively novel notion of resistance complicates the already challenging task of designing effective treatment protocols.

Materials and Methods

To better understand resistance, we have developed a mathematical modeling framework that incorporates both spontaneous and drug-induced resistance.

Results

Our model demonstrates that the ability of a drug to induce resistance can result in qualitatively different responses to the same drug dose and delivery schedule. We have also proven that the induction parameter in our model is theoretically identifiable and propose an in vitro protocol that could be used to determine a treatment’s propensity to induce resistance.

Reveal the Regulation Patterns of Prognosis-Related miRNAs and lncRNAs Across Solid Tumors in the Cancer Genome Atlas

BACKGROUND

The dysregulation of non-coding RNAs (ncRNAs) such as miRNAs and lncRNAs are associated with the pathogenesis and progression in multiple cancers including solid tumors. Comprehensive investigations of prognosis-related ncRNA markers could promote the development of therapeutic strategies for solid tumors, but rarely reported.

METHODS

By taking advantage of The Cancer Genome Atlas (TCGA), pan-cancer prognosis analysis (PCPA) models were firstly constructed based on miRNA and lncRNA expression profiles of 8,450 samples in 19 solid tumors. Further, the co-occurrence and exclusivity among ncRNA markers were systematically analyzed for different cancers.

RESULTS

In identified ncRNA makers, 71% of the miRNA markers were shared in multiple cancers, whereas 96% of the lncRNA markers were cancer-specific. Moreover, to analyze the regulation patterns of prognosis-related ncRNAs at the pan-cancer level, miRNA markers were further annotated into eight carcinogenic pathways. Results represented that approximately 86% of these miRNA markers could regulate the PI3K-Akt signaling pathway, while only 48% for the Notch signaling pathway. Finally, among 126 common genes that participated in eight carcinogenic pathways, BCL2, CSNK2A1, EGFR, PDGFRA, and VEGFA were proposed as potential drug targets for multiple cancers.

CONCLUSION

The prognosis analysis and regulation characteristics of ncRNAs presented in this study may help to facilitate the discovery of anti-cancer drugs for multiple solid tumors.

Evolutionary dynamics of competing phenotype-structured populations in periodically fluctuating environments

Living species, ranging from bacteria to animals, exist in environmental conditions that exhibit spatial and temporal heterogeneity which requires them to adapt. Risk-spreading through spontaneous phenotypic variations is a known concept in ecology, which is used to explain how species may survive when faced with the evolutionary risks associated with temporally varying environments. In order to support a deeper understanding of the adaptive role of spontaneous phenotypic variations in fluctuating environments, we consider a system of non-local partial differential equations modelling the evolutionary dynamics of two competing phenotype-structured populations in the presence of periodically oscillating nutrient levels. The two populations undergo heritable, spontaneous phenotypic variations at different rates. The phenotypic state of each individual is represented by a continuous variable, and the phenotypic landscape of the populations evolves in time due to variations in the nutrient level. Exploiting the analytical tractability of our model, we study the long-time behaviour of the solutions to obtain a detailed mathematical depiction of the evolutionary dynamics. The results suggest that when nutrient levels undergo small and slow oscillations, it is evolutionarily more convenient to rarely undergo spontaneous phenotypic variations. Conversely, under relatively large and fast periodic oscillations in the nutrient levels, which bring about alternating cycles of starvation and nutrient abundance, higher rates of spontaneous phenotypic variations confer a competitive advantage. We discuss the implications of our results in the context of cancer metabolism.

Contrasting the impact of cytotoxic and cytostatic drug therapies on tumour progression

A tumour grows when the total division (birth) rate of its cells exceeds their total mortality (death) rate. The capability for uncontrolled growth within the host tissue is acquired via the accumulation of driver mutations which enable the tumour to progress through various hallmarks of cancer. We present a mathematical model of the penultimate stage in such a progression. We assume the tumour has reached the limit of its present growth potential due to cell competition that either results in total birth rate reduction or death rate increase. The tumour can then progress to the final stage by either seeding a metastasis or acquiring a driver mutation. We influence the ensuing evolutionary dynamics by cytotoxic (increasing death rate) or cytostatic (decreasing birth rate) therapy while keeping the effect of the therapy on net growth reduction constant. Comparing the treatments head to head we derive conditions for choosing optimal therapy. We quantify how the choice and the related gain of optimal therapy depends on driver mutation, metastasis, intrinsic cell birth and death rates, and the details of cell competition. We show that detailed understanding of the cell population dynamics could be exploited in choosing the right mode of treatment with substantial therapy gains.

Cells and organisms evolve to better survive in their environments and to adapt to new challenges. Such dynamics manifest in a particularly problematic way with the evolution of drug resistance, which is increasingly recognized as a key challenge for global health. Thus, developing therapy paradigms that factor in evolutionary dynamics is an important goal. Using a minimal mathematical model of a cancer cell population we contrast cytotoxic (increasing death rate) and cytostatic (decreasing birth rate) treatments while keeping the effect of the therapy on the net growth reduction constant. We then quantify how the choice and the related gain of optimal therapy depends on driver mutation, metastasis, intrinsic cell birth and death rates and the details of cell competition. Most importantly, we identify specific cell population dynamics under which a certain treatment could be significantly better than the alternative.

Investigation of solid tumor progression with account of proliferation/migration dichotomy via Darwinian mathematical model

A new continuous spatially-distributed model of solid tumor growth and progression is presented. The model explicitly accounts for mutations/epimutations of tumor cells which take place upon their division. The tumor grows in normal tissue and its progression is driven only by competition between populations of malignant cells for limited nutrient supply. Two reasons for the motion of tumor cells in space are taken into consideration, i.e., their intrinsic motility and convective fluxes, which arise due to proliferation of tumor cells. The model is applied to investigation of solid tumor progression under phenotypic alterations that inversely affect cell proliferation rate and cell motility by increasing the value of one of the parameters at the expense of another.It is demonstrated that the crucial feature that gives evolutionary advantage to a cell population is the speed of its intergrowth into surrounding normal tissue. Of note, increase in tumor intergrowth speed in not always associated with increase in motility of tumor cells. Depending on the parameters of functions, that describe phenotypic alterations, tumor cellular composition may evolve towards: (1) maximization of cell proliferation rate, (2) maximization of cell motility, (3) non-extremum values of cell proliferation rate and motility. Scenarios are found, where after initial tendency for maximization of cell proliferation rate, the direction of tumor progression sharply switches to maximization of cell motility, which is accompanied by decrease in total speed of tumor growth.

A Mathematical Framework for Modelling the Metastatic Spread of Cancer

Cancer is a complex disease that starts with mutations of key genes in one cell or a small group of cells at a primary site in the body. If these cancer cells continue to grow successfully and, at some later stage, invade the surrounding tissue and acquire a vascular network, they can spread to distant secondary sites in the body. This process, known as metastatic spread, is responsible for around 90% of deaths from cancer and is one of the so-called hallmarks of cancer. To shed light on the metastatic process, we present a mathematical modelling framework that captures for the first time the interconnected processes of invasion and metastatic spread of individual cancer cells in a spatially explicit manner-a multigrid, hybrid, individual-based approach. This framework accounts for the spatiotemporal evolution of mesenchymal- and epithelial-like cancer cells, membrane-type-1 matrix metalloproteinase (MT1-MMP) and the diffusible matrix metalloproteinase-2 (MMP-2), and for their interactions with the extracellular matrix. Using computational simulations, we demonstrate that our model captures all the key steps of the invasion-metastasis cascade, i.e. invasion by both heterogeneous cancer cell clusters and by single mesenchymal-like cancer cells; intravasation of these clusters and single cells both via active mechanisms mediated by matrix-degrading enzymes (MDEs) and via passive shedding; circulation of cancer cell clusters and single cancer cells in the vasculature with the associated risk of cell death and disaggregation of clusters; extravasation of clusters and single cells; and metastatic growth at distant secondary sites in the body. By faithfully reproducing experimental results, our simulations support the evidence-based hypothesis that the membrane-bound MT1-MMP is the main driver of invasive spread rather than diffusible MDEs such as MMP-2.

Modeling three-dimensional invasive solid tumor growth in heterogeneous microenvironment under chemotherapy

A systematic understanding of the evolution and growth dynamics of invasive solid tumors in response to different chemotherapy strategies is crucial for the development of individually optimized oncotherapy. Here, we develop a hybrid three-dimensional (3D) computational model that integrates pharmacokinetic model, continuum diffusion-reaction model and discrete cell automaton model to investigate 3D invasive solid tumor growth in heterogeneous microenvironment under chemotherapy. Specifically, we consider the effects of heterogeneous environment on drug diffusion, tumor growth, invasion and the drug-tumor interaction on individual cell level. We employ the hybrid model to investigate the evolution and growth dynamics of avascular invasive solid tumors under different chemotherapy strategies. Our simulations indicate that constant dosing is generally more effective in suppressing primary tumor growth than periodic dosing, due to the resulting continuous high drug concentration. In highly heterogeneous microenvironment, the malignancy of the tumor is significantly enhanced, leading to inefficiency of chemotherapies. The effects of geometrically-confined microenvironment and non-uniform drug dosing are also investigated. Our computational model, when supplemented with sufficient clinical data, could eventually lead to the development of efficient in silico tools for prognosis and treatment strategy optimization.

The role of spatial variations of abiotic factors in mediating intratumour phenotypic heterogeneity

We present here a space- and phenotype-structured model of selection dynamics between cancer cells within a solid tumour. In the framework of this model, we combine formal analyses with numerical simulations to investigate in silico the role played by the spatial distribution of abiotic components of the tumour microenvironment in mediating phenotypic selection of cancer cells. Numerical simulations are performed both on the 3D geometry of an in silico multicellular tumour spheroid and on the 3D geometry of an in vivo human hepatic tumour, which was imaged using computerised tomography. The results obtained show that inhomogeneities in the spatial distribution of oxygen, currently observed in solid tumours, can promote the creation of distinct local niches and lead to the selection of different phenotypic variants within the same tumour. This process fosters the emergence of stable phenotypic heterogeneity and supports the presence of hypoxic cells resistant to cytotoxic therapy prior to treatment. Our theoretical results demonstrate the importance of integrating spatial data with ecological principles when evaluating the therapeutic response of solid tumours to cytotoxic therapy.

Signal Propagation in Sensing and Reciprocating Cellular Systems with Spatial and Structural Heterogeneity

Sensing and reciprocating cellular systems (SARs) are important for the operation of many biological systems. Production in interferon (IFN) SARs is achieved through activation of the Jak-Stat pathway, and downstream upregulation of IFN regulatory factor (IRF)-7 and IFN transcription, but the role that high- and low-affinity IFNs play in this process remains unclear. We present a comparative between a minimal spatio-temporal partial differential equation model and a novel spatio-structural-temporal (SST) model for the consideration of receptor, binding, and metabolic aspects of SAR behaviour. Using the SST framework, we simulate single- and multi-cluster paradigms of IFN communication. Simulations reveal a cyclic process between the binding of IFN to the receptor, and the consequent increase in metabolism, decreasing the propensity for binding due to the internal feedback mechanism. One observes the effect of heterogeneity between cellular clusters, allowing them to individualise and increase local production, and within clusters, where we observe 'subpopular quiescence'; a process whereby intra-cluster subpopulations reduce their binding and metabolism such that other such subpopulations may augment their production. Finally, we observe the ability for low-affinity IFN to communicate a long range signal, where high affinity cannot, and the breakdown of this relationship through the introduction of cell motility. Biological systems may utilise cell motility where environments are unrestrictive and may use fixed system, with low-affinity communication, where a localised response is desirable.

Application of mathematical models to metronomic chemotherapy: What can be inferred from minimal parameterized models?

Metronomic chemotherapy refers to the frequent administration of chemotherapy at relatively low, minimally toxic doses without prolonged treatment interruptions. Different from conventional or maximum-tolerated-dose chemotherapy which aims at an eradication of all malignant cells, in a metronomic dosing the goal often lies in the long-term management of the disease when eradication proves elusive. Mathematical modeling and subsequent analysis (theoretical as well as numerical) have become an increasingly more valuable tool (in silico) both for determining conditions under which specific treatment strategies should be preferred and for numerically optimizing treatment regimens. While elaborate, computationally-driven patient specific schemes that would optimize the timing and drug dose levels are still a part of the future, such procedures may become instrumental in making chemotherapy effective in situations where it currently fails. Ideally, mathematical modeling and analysis will develop into an additional decision making tool in the complicated process that is the determination of efficient chemotherapy regimens. In this article, we review some of the results that have been obtained about metronomic chemotherapy from mathematical models and what they infer about the structure of optimal treatment regimens.

Synergistic chemotherapeutic effect of sorafenib-loaded pullulan-Dox conjugate nanoparticles against murine breast carcinoma

pH-Sensitive pullulan-doxorubicin conjugates encapsulating sorafenib (P-Dox/S) nanoparticles were developed as a synergistic combinatorial delivery system against murine breast carcinoma. The nanoparticles can encapsulate Dox and sorafenib with ultra-high loading capacity (65.34 wt%) through chemical conjugation and physical loading, whereas can remain stable under physiological conditions and gradually release Dox and sorafenib with the decreasing pH. These conjugates can be effectively internalized and clearly suppress 4T1 cell growth in vitro. Furthermore, research data of in vivo animal models revealed that the synergistic combinatorial P-Dox/S nanoparticles heavily accumulated in solid tumor tissue sites to maximize therapeutic efficacy; they also significantly inhibited solid tumor growth, even remarkably reduced solid tumor volume in comparison to the initial volume, and obviously diminished adverse effects. The anti-tumor therapeutic effect obviously outperformed the delivery of combinational chemotherapy of free drugs or single drug-loaded P-Dox nanoparticles at the same concentration. These promising results indicate the high-efficiency synergistic chemotherapeutic effects of these nanoparticles. Combinational chemotherapy using P-Dox/S nanoparticles has important potential in the clinical treatment of malignancy for overcoming drug resistance and heterogeneity.

On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach

Effects that tumor heterogeneity and drug resistance have on the structure of chemotherapy protocols are discussed from a mathematical modeling and optimal control point of view. In the case when two compartments consisting of sensitive and resistant cells are considered, optimal protocols consist of full dose chemotherapy as long as the relative proportion of sensitive cells is high. When resistant cells become more dominant, optimal controls switch to lower dose regimens defined by so-called singular controls. The role that singular controls play in the structure of optimal therapy protocols for cell populations with a large number of traits is explored in mathematical models.

Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells

Accumulating evidence indicates that the interaction between epithelial and mesenchymal cells plays a pivotal role in cancer development and metastasis formation. Here we propose an integro-differential model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Our model takes into account the effects of chemotaxis, adhesive interactions between epithelial-like cells, proliferation and competition for nutrients. We present a sample of numerical results which display the emergence of spots, stripes and honeycomb patterns, depending on parameters and initial data. These simulations also suggest that epithelial-like and mesenchymal-like cells can segregate when there is little competition for nutrients. Furthermore, our computational results provide a possible explanation for how the concerted action between epithelial-cell adhesion and mesenchymal-cell spreading can precipitate the formation of ring-like structures, which resemble the fibrous capsules frequently observed in hepatic tumours.

Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity

We consider cancer chemotherapy as an optimal control problem with the aim to minimize a combination of the tumor volume and side effects over an a priori specified therapy horizon when the tumor consists of a heterogeneous agglomeration of many subpopulations. The mathematical model, which accounts for different growth and apoptosis rates in the presence of cell densities, is a finite-dimensional approximation of a model originally formulated by Lorz et al. [18,19] and Greene et al. [10,11] with a continuum of possible traits. In spite of an arbitrarily high dimension, for this problem singular controls (which correspond to time-varying administration schedules at less than maximum doses) can be computed explicitly in feedback form. Interestingly, these controls have the property to keep the entire tumor population constant. Numerical computations and simulations that explore the optimality of bang-bang and singular controls are given. These point to the optimality of protocols that combine a full dose therapy segment with a period of lower dose drug administration.

Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases

While chemoresistance in primary tumors is well-studied, much less is known about the influence of systemic chemotherapy on the development of drug resistance at metastatic sites. In this work, we use a hybrid spatial model of tumor response to a DNA damaging drug to study how the development of chemoresistance in micrometastases depends on the drug dosing schedule. We separately consider cell populations that harbor pre-existing resistance to the drug, and those that acquire resistance during the course of treatment. For each of these independent scenarios, we consider one hypothetical cell line that is responsive to metronomic chemotherapy, and another that with high probability cannot be eradicated by a metronomic protocol. Motivated by experimental work on ovarian cancer xenografts, we consider all possible combinations of a one week treatment protocol, repeated for three weeks, and constrained by the total weekly drug dose. Simulations reveal a small number of fractionated-dose protocols that are at least as effective as metronomic therapy in eradicating micrometastases with acquired resistance (weak or strong), while also being at least as effective on those that harbor weakly pre-existing resistant cells. Given the responsiveness of very different theoretical cell lines to these few fractionated-dose protocols, these may represent more effective ways to schedule chemotherapy with the goal of limiting metastatic tumor progression.

Optimization of an in vitro chemotherapy to avoid resistant tumours

Chemotherapy use against solid tumours often results in the resistance of the cancer cells to the molecule used. In this paper, we will set up and analyse an ODE model for heterogeneous in vitro tumours, consisting of cells that are sensitive or resistant to a certain drug. We will then use this model to develop different protocols, that aim at reducing the tumour volume while preserving its heterogeneity. These drug administration schedules are determined through analysis of the system dynamics, and optimal control theory.

Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations

Background

A thorough understanding of the ecological and evolutionary mechanisms that drive the phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer research community. In this respect, mathematical modelling can complement experimental cancer research by offering alternative means of understanding the results of in vitro and in vivo experiments, and by allowing for a quick and easy exploration of a variety of biological scenarios through in silico studies.

Results

To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here a phenotype-structured model of evolutionary dynamics in a cancer cell population which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size of the cell population are shaped by the strength of natural selection, the rate of random epimutations, the intensity of the competition for limited resources between cells, and the drug dose in use.

Conclusions

Our analytical results clarify the conditions for the successful adaptation of cancer cells faced with environmental changes. Furthermore, the results of our analyses demonstrate that the same cell population exposed to different concentrations of the same cytotoxic drug can take different evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents is more complex than a simple binary outcome, i.e., extinction of sensitive cells and selection of highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents at high doses can act as a double-edged sword by promoting the outgrowth of drug resistant cellular clones. Overall, our theoretical work offers a formal basis for the development of anti-cancer therapeutic protocols that go beyond the ‘maximum-tolerated-dose paradigm’, as they may be more effective than traditional protocols at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones.

Reviewers

This article was reviewed by Angela Pisco, Sébastien Benzekry and Heiko Enderling.

Electronic supplementary material

The online version of this article (doi:10.1186/s13062-016-0143-4) contains supplementary material, which is available to authorized users.

Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation

BACKGROUND

Drug-induced drug resistance in cancer has been attributed to diverse biological mechanisms at the individual cell or cell population scale, relying on stochastically or epigenetically varying expression of phenotypes at the single cell level, and on the adaptability of tumours at the cell population level.

SCOPE OF REVIEW

We focus on intra-tumour heterogeneity, namely between-cell variability within cancer cell populations, to account for drug resistance. To shed light on such heterogeneity, we review evolutionary mechanisms that encompass the great evolution that has designed multicellular organisms, as well as smaller windows of evolution on the time scale of human disease. We also present mathematical models used to predict drug resistance in cancer and optimal control methods that can circumvent it in combined therapeutic strategies.

MAJOR CONCLUSIONS

Plasticity in cancer cells, i.e., partial reversal to a stem-like status in individual cells and resulting adaptability of cancer cell populations, may be viewed as backward evolution making cancer cell populations resistant to drug insult. This reversible plasticity is captured by mathematical models that incorporate between-cell heterogeneity through continuous phenotypic variables. Such models have the benefit of being compatible with optimal control methods for the design of optimised therapeutic protocols involving combinations of cytotoxic and cytostatic treatments with epigenetic drugs and immunotherapies.

GENERAL SIGNIFICANCE

Gathering knowledge from cancer and evolutionary biology with physiologically based mathematical models of cell population dynamics should provide oncologists with a rationale to design optimised therapeutic strategies to circumvent drug resistance, that still remains a major pitfall of cancer therapeutics. This article is part of a Special Issue entitled "System Genetics" Guest Editor: Dr. Yudong Cai and Dr. Tao Huang.

Mass concentration in a nonlocal model of clonal selection

Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To better understand this impact, we propose a mathematical model describing the dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling which describes regulatory feedback loops of cell proliferation and differentiation. We show that this coupling leads to mass concentration in points corresponding to the maxima of the self-renewal potential and the solutions of the model tend asymptotically to Dirac measures multiplied by positive constants. Furthermore, using a Lyapunov function constructed for the finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in the space of positive Radon measures equipped with the flat metric (bounded Lipschitz distance). Analytical results are illustrated by numerical simulations.

Microenvironmental Niches and Sanctuaries: A Route to Acquired Resistance

A tumor vasculature that is functionally abnormal results in irregular gradients of metabolites and drugs within the tumor tissue. Recently, significant efforts have been committed to experimentally examine how cellular response to anti-cancer treatments varies based on the environment in which the cells are grown. In vitro studies point to specific conditions in which tumor cells can remain dormant and survive the treatment. In vivo results suggest that cells can escape the effects of drug therapy in tissue regions that are poorly penetrated by the drugs. Better understanding how the tumor microenvironments influence the emergence of drug resistance in both primary and metastatic tumors may improve drug development and the design of more effective therapeutic protocols. This chapter presents a hybrid agent-based model of the growth of tumor micrometastases and explores how microenvironmental factors can contribute to the development of acquired resistance in response to a DNA damaging drug. The specific microenvironments of interest in this work are tumor hypoxic niches and tumor normoxic sanctuaries with poor drug penetration. We aim to quantify how spatial constraints of limited drug transport and quiescent cell survival contribute to the development of drug resistant tumors.

Optimizing Chemotherapeutic Anti-cancer Treatment and the Tumor Microenvironment: An Analysis of Mathematical Models

We review results about the structure of administration of chemotherapeutic anti-cancer treatment that we have obtained from an analysis of minimally parameterized mathematical models using methods of optimal control. This is a branch of continuous-time optimization that studies the minimization of a performance criterion imposed on an underlying dynamical system subject to constraints. The scheduling of anti-cancer treatments has all the features of such a problem: treatments are administered in time and the interactions of the drugs with the tumor and its microenvironment determine the efficacy of therapy. At the same time, constraints on the toxicity of the treatments need to be taken into account. The models we consider are low-dimensional and do not include more refined details, but they capture the essence of the underlying biology and our results give robust and rather conclusive qualitative information about the administration of optimal treatment protocols that strongly correlate with approaches taken in medical practice. We describe the changes that arise in optimal administration schedules as the mathematical models are increasingly refined to progress from models that only consider the cancerous cells to models that include the major components of the tumor microenvironment, namely the tumor vasculature and tumor-immune system interactions.

Spatial evolution of tumors with successive driver mutations

We study the influence of driver mutations on the spatial evolutionary dynamics of solid tumors. We start with a cancer clone that expands uniformly in three dimensions giving rise to a spherical shape. We assume that cell division occurs on the surface of the growing tumor. Each cell division has a chance to give rise to a mutation that activates an additional driver gene. The resulting clone has an enhanced growth rate, which generates a local ensemble of faster growing cells, thereby distorting the spherical shape of the tumor. We derive formulas for the abundance and diversity of additional driver mutations as function of time. Our model is semi-deterministic: the spatial growth of cancer clones is deterministic, while mutants arise stochastically.