A mathematical model for the glucose-lactate metabolism of in vitro cancer cells

DataXflow: Synergizing data-driven modeling with best parameter fit and optimal control - An efficient data analysis for cancer research

Building data-driven models is an effective strategy for information extraction from empirical data. Adapting model parameters specifically to data with a best fitting approach encodes the relevant information into a mathematical model. Subsequently, an optimal control framework extracts the most efficient targets to steer the model into desired changes via external stimuli. The DataXflow software framework integrates three software pipelines, D2D for model fitting, a framework solving optimal control problems including external stimuli and JimenaE providing graphical user interfaces to employ the other frameworks lowering the barriers for the need of programming skills, and simultaneously automating reoccurring modeling tasks. Such tasks include equation generation from a graph and script generation allowing also to approach systems with many agents, like complex gene regulatory networks. A desired state of the model is defined, and therapeutic interventions are modeled as external stimuli. The optimal control framework purposefully exploits the model-encoded information by providing those external stimuli that effect the desired changes most efficiently. The implementation of DataXflow is available under https://github.com/MarvelousHopefull/DataXflow. We showcase its application by detecting specific drug targets for a therapy of lung cancer from measurement data to lower proliferation and increase apoptosis. By an iterative modeling process refining the topology of the model, the regulatory network of the tumor is generated from the data. An application of the optimal control framework in our example reveals the inhibition of AURKA and the activation of CDH1 as the most efficient drug target combination. DataXflow paves the way to an agile interplay between data generation and its analysis potentially accelerating cancer research by an efficient drug target identification, even in complex networks.

Computational model of the cancer necrotic core formation in a tumor-on-a-chip device

The mechanisms underlying the formation of necrotic regions within avascular tumors are complex and poorly understood. In this paper, we investigate the formation of a necrotic core in a 3D tumor cell culture within a microfluidic device, considering oxygen, nutrients, and the microenvironment acidification by means of a computational-mathematical model. Our objective is to simulate cell processes, including proliferation and death inside a microfluidic device, according to the microenvironmental conditions. We employed approximation utilizing finite element models taking into account glucose, oxygen, and hydrogen ions diffusion, consumption and production, as well as cell proliferation, migration and death, addressing how tumor cells evolve under different conditions. The resulting mathematical model was examined under different scenarios, being capable of reproducing cell death and proliferation under different cell concentrations, and the formation of a necrotic core, in good agreement with experimental data reported in the literature. This approach not only advances our fundamental understanding of necrotic core formation but also provides a robust computational platform to study personalized therapeutic strategies, offering an important tool in cancer research and treatment design.

Metabolic symbiosis between oxygenated and hypoxic tumour cells: An agent-based modelling study

Deregulated metabolism is one of the hallmarks of cancer. It is well-known that tumour cells tend to metabolize glucose via glycolysis even when oxygen is available and mitochondrial respiration is functional. However, the lower energy efficiency of aerobic glycolysis with respect to mitochondrial respiration makes this behaviour, namely the Warburg effect, counter-intuitive, although it has now been recognized as source of anabolic precursors. On the other hand, there is evidence that oxygenated tumour cells could be fuelled by exogenous lactate produced from glycolysis. We employed a multi-scale approach that integrates multi-agent modelling, diffusion-reaction, stoichiometric equations, and Boolean networks to study metabolic cooperation between hypoxic and oxygenated cells exposed to varying oxygen, nutrient, and inhibitor concentrations. The results show that the cooperation reduces the depletion of environmental glucose, resulting in an overall advantage of using aerobic glycolysis. In addition, the oxygen level was found to be decreased by symbiosis, promoting a further shift towards anaerobic glycolysis. However, the oxygenated and hypoxic populations may gradually reach quasi-equilibrium. A sensitivity analysis using Latin hypercube sampling and partial rank correlation shows that the symbiotic dynamics depends on properties of the specific cell such as the minimum glucose level needed for glycolysis. Our results suggest that strategies that block glucose transporters may be more effective to reduce tumour growth than those blocking lactate intake transporters.

Evolutionary dynamics of glucose-deprived cancer cells: insights from experimentally informed mathematical modelling

Glucose is a primary energy source for cancer cells. Several lines of evidence support the idea that monocarboxylate transporters, such as MCT1, elicit metabolic reprogramming of cancer cells in glucose-poor environments, allowing them to re-use lactate, a by-product of glucose metabolism, as an alternative energy source with serious consequences for disease progression. We employ a synergistic experimental and mathematical modelling approach to explore the evolutionary processes at the root of cancer cell adaptation to glucose deprivation, with particular focus on the mechanisms underlying the increase in MCT1 expression observed in glucose-deprived aggressive cancer cells. Data from in vitro experiments on breast cancer cells are used to inform and calibrate a mathematical model that comprises a partial integro-differential equation for the dynamics of a population of cancer cells structured by the level of MCT1 expression. Analytical and numerical results of this model suggest that environment-induced changes in MCT1 expression mediated by lactate-associated signalling pathways enable a prompt adaptive response of glucose-deprived cancer cells, while fluctuations in MCT1 expression due to epigenetic changes create the substrate for environmental selection to act upon, speeding up the selective sweep underlying cancer cell adaptation to glucose deprivation, and may constitute a long-term bet-hedging mechanism.

Comparing mechanism-based and machine learning models for predicting the effects of glucose accessibility on tumor cell proliferation

Glucose plays a central role in tumor metabolism and development and is a target for novel therapeutics. To characterize the response of cancer cells to blockade of glucose uptake, we collected time-resolved microscopy data to track the growth of MDA-MB-231 breast cancer cells. We then developed a mechanism-based, mathematical model to predict how a glucose transporter (GLUT1) inhibitor (Cytochalasin B) influences the growth of the MDA-MB-231 cells by limiting access to glucose. The model includes a parameter describing dose dependent inhibition to quantify both the total glucose level in the system and the glucose level accessible to the tumor cells. Four common machine learning models were also used to predict tumor cell growth. Both the mechanism-based and machine learning models were trained and validated, and the prediction error was evaluated by the coefficient of determination (R²). The random forest model provided the highest accuracy predicting cell dynamics (R² = 0.92), followed by the decision tree (R² = 0.89), k-nearest-neighbor regression (R² = 0.84), mechanism-based (R² = 0.77), and linear regression model (R² = 0.69). Thus, the mechanism-based model has a predictive capability comparable to machine learning models with the added benefit of elucidating biological mechanisms.

Evaluation of Lung Cancer Patient Response to First-Line Chemotherapy by Integration of Tumor Core Biopsy Metabolomics with Multiscale Modeling

The standard of care for intermediate (Stage II) and advanced (Stages III and IV) non-small cell lung cancer (NSCLC) involves chemotherapy with taxane/platinum derivatives, with or without radiation. Ideally, patients would be screened a priori to allow non-responders to be initially treated with second-line therapies. This evaluation is non-trivial, however, since tumors behave as complex multiscale systems. To address this need, this study employs a multiscale modeling approach to evaluate first-line chemotherapy response of individual patient tumors based on metabolomic analysis of tumor core biopsies obtained during routine clinical evaluation. Model parameters were calculated for a patient cohort as a function of these metabolomic profiles, previously obtained from high-resolution 2DLC-MS/MS analysis. Evaluation metrics were defined to classify patients as Disease-Control (DC) [encompassing complete-response (CR), partial-response (PR), and stable-disease (SD)] and Progressive-Disease (PD) following first-line chemotherapy. Response was simulated for each patient and compared to actual response. The results show that patient classifications were significantly separated from each other, and also when grouped as DC vs. PD and as CR/PR vs. SD/PD, by fraction of initial tumor radius metric at 6 days post simulated bolus drug injection. This study shows that patient first-line chemotherapy response can in principle be evaluated from multiscale modeling integrated with tumor tissue metabolomic data, offering a first step towards individualized lung cancer treatment prognosis.

On a tumor growth model with brain lactate kinetics

Our aim in this paper is to study a mathematical model for high grade gliomas, taking into account lactates kinetics, as well as chemotherapy and antiangiogenic treatment. In particular, we prove the existence and uniqueness of biologically relevant solutions. We also perform numerical simulations based on different therapeutical situations that can be found in the literature. These simulations are consistent with what is expected in these situations.

Modeling of Tumor Growth with Input from Patient-Specific Metabolomic Data

Advances in omic technologies have provided insight into cancer progression and treatment response. However, the nonlinear characteristics of cancer growth present a challenge to bridge from the molecular- to the tissue-scale, as tumor behavior cannot be encapsulated by the sum of the individual molecular details gleaned experimentally. Mathematical modeling and computational simulation have been traditionally employed to facilitate analysis of nonlinear systems. In this study, for the first time tumor metabolomic data are linked via mathematical modeling to the tumor tissue-scale behavior, showing the capability to mechanistically simulate cancer progression personalized to omic information obtainable from patient tumor core biopsy analysis. Generally, a higher degree of metabolic dysregulation has been correlated with more aggressive tumor behavior. Accordingly, key parameters influenced by metabolomic data in this model include tumor proliferation, vascularization, aggressiveness, lactic acid production, monocyte infiltration and macrophage polarization, and drug effect. The model enables evaluating interactions of interest between these parameters which drive tumor growth based on the metabolomic data. The results show that the model can group patients consistently with the clinically observed outcomes of response/non-response to chemotherapy. This modeling approach provides a first step towards evaluation of tumor growth based on tumor-specific metabolomic data.

Targeting lactate metabolism and glycolytic pathways in the tumor microenvironment by natural products: A promising strategy in combating cancer

Anticancer drugs are not purely effective because of their toxicity, side effects, high cost, inaccessibility, and associated resistance. On the other hand, cancer is a complex public health problem that could intelligently adopt different signaling pathways and alter the body's metabolism to escape from the immune system. One of the cancer strategies to metastasize is modifying pH in the tumor microenvironment, ranging between 6.5 and 6.9. As a powerful determiner, lactate is responsible for this acidosis. It is involved in immune stimulation, including innate and adaptive immunity, apoptotic-related factors (Bax/Bcl-2, caspase), and glycolysis pathways (e.g., GLUT-1, PKM2, PFK, HK2, MCT-1, and LDH). Lactate metabolism, in turn, is interconnected with several dysregulated signaling mediators, including PI3K/Akt/mTOR, AMPK, NF-κB, Nrf2, JAK/STAT, and HIF-1α. Because of lactate's emerging and critical role, targeting lactate production and its transporters is important for preventing and managing tumorigenesis. Hence, exploring and developing novel promising anticancer agents to minimize human cancers is urgent. Based on numerous studies, natural secondary metabolites as multi-target alternative compounds with health-promoting properties possess more high effectiveness and low side effects than conventional agents. Besides, the mechanism of multi-targeted natural sources is related to lactate production and cancer-associated cross-talked factors. This review focuses on targeting the lactate metabolism/transporters, and lactate-associated mediators, including glycolytic pathways. Besides, interconnected mediators to lactate metabolism are also targeted by natural products. Accordingly, plant-derived secondary metabolites are introduced as alternative therapies in combating cancer through modulating lactate metabolism and glycolytic pathways.

An experimental-mathematical approach to predict tumor cell growth as a function of glucose availability in breast cancer cell lines

We present the development and validation of a mathematical model that predicts how glucose dynamics influence metabolism and therefore tumor cell growth. Glucose, the starting material for glycolysis, has a fundamental influence on tumor cell growth. We employed time-resolved microscopy to track the temporal change of the number of live and dead tumor cells under different initial glucose concentrations and seeding densities. We then constructed a family of mathematical models (where cell death was accounted for differently in each member of the family) to describe overall tumor cell growth in response to the initial glucose and confluence conditions. The Akaikie Information Criteria was then employed to identify the most parsimonious model. The selected model was then trained on 75% of the data to calibrate the system and identify trends in model parameters as a function of initial glucose concentration and confluence. The calibrated parameters were applied to the remaining 25% of the data to predict the temporal dynamics given the known initial glucose concentration and confluence, and tested against the corresponding experimental measurements. With the selected model, we achieved an accuracy (defined as the fraction of measured data that fell within the 95% confidence intervals of the predicted growth curves) of 77.2 ± 6.3% and 87.2 ± 5.1% for live BT-474 and MDA-MB-231 cells, respectively.

Lactate Metabolism and Signaling in Tuberculosis and Cancer: A Comparative Review

Infection with Mycobacterium tuberculosis (Mtb) leading to tuberculosis (TB) disease continues to be a major global health challenge. Critical barriers, including but not limited to the development of multi-drug resistance, lack of diagnostic assays that detect patients with latent TB, an effective vaccine that prevents Mtb infection, and infectious and non-infectious comorbidities that complicate active TB, continue to hinder progress toward a TB cure. To complement the ongoing development of new antimicrobial drugs, investigators in the field are exploring the value of host-directed therapies (HDTs). This therapeutic strategy targets the host, rather than Mtb, and is intended to augment host responses to infection such that the host is better equipped to prevent or clear infection and resolve chronic inflammation. Metabolic pathways of immune cells have been identified as promising HDT targets as more metabolites and metabolic pathways have shown to play a role in TB pathogenesis and disease progression. Specifically, this review highlights the potential role of lactate as both an immunomodulatory metabolite and a potentially important signaling molecule during the host response to Mtb infection. While long thought to be an inert end product of primarily glucose metabolism, the cancer research field has discovered the importance of lactate in carcinogenesis and resistance to chemotherapeutic drug treatment. Herein, we discuss similarities between the TB granuloma and tumor microenvironments in the context of lactate metabolism and identify key metabolic and signaling pathways that have been shown to play a role in tumor progression but have yet to be explored within the context of TB. Ultimately, lactate metabolism and signaling could be viable HDT targets for TB; however, critical additional research is needed to better understand the role of lactate at the host-pathogen interface during Mtb infection before adopting this HDT strategy.

Integrating Quantitative Assays with Biologically Based Mathematical Modeling for Predictive Oncology

We provide an overview on the use of biological assays to calibrate and initialize mechanism-based models of cancer phenomena. Although artificial intelligence methods currently dominate the landscape in computational oncology, mathematical models that seek to explicitly incorporate biological mechanisms into their formalism are of increasing interest. These models can guide experimental design and provide insights into the underlying mechanisms of cancer progression. Historically, these models have included a myriad of parameters that have been difficult to quantify in biologically relevant systems, limiting their practical insights. Recently, however, there has been much interest calibrating biologically based models with the quantitative measurements available from (for example) RNA sequencing, time-resolved microscopy, and in vivo imaging. In this contribution, we summarize how a variety of experimental methods quantify tumor characteristics from the molecular to tissue scales and describe how such data can be directly integrated with mechanism-based models to improve predictions of tumor growth and treatment response.

Optimal Control Theory for Personalized Therapeutic Regimens in Oncology: Background, History, Challenges, and Opportunities

Optimal control theory is branch of mathematics that aims to optimize a solution to a dynamical system. While the concept of using optimal control theory to improve treatment regimens in oncology is not novel, many of the early applications of this mathematical technique were not designed to work with routinely available data or produce results that can eventually be translated to the clinical setting. The purpose of this review is to discuss clinically relevant considerations for formulating and solving optimal control problems for treating cancer patients. Our review focuses on two of the most widely used cancer treatments, radiation therapy and systemic therapy, as they naturally lend themselves to optimal control theory as a means to personalize therapeutic plans in a rigorous fashion. To provide context for optimal control theory to address either of these two modalities, we first discuss the major limitations and difficulties oncologists face when considering alternate regimens for their patients. We then provide a brief introduction to optimal control theory before formulating the optimal control problem in the context of radiation and systemic therapy. We also summarize examples from the literature that illustrate these concepts. Finally, we present both challenges and opportunities for dramatically improving patient outcomes via the integration of clinically relevant, patient-specific, mathematical models and optimal control theory.

Metabolic reprogramming dynamics in tumor spheroids: Insights from a multicellular, multiscale model

Mathematical modeling provides the predictive ability to understand the metabolic reprogramming and complex pathways that mediate cancer cells’ proliferation. We present a mathematical model using a multiscale, multicellular approach to simulate avascular tumor growth, applied to pancreatic cancer. The model spans three distinct spatial and temporal scales. At the extracellular level, reaction diffusion equations describe nutrient concentrations over a span of seconds. At the cellular level, a lattice-based energy driven stochastic approach describes cellular phenomena including adhesion, proliferation, viability and cell state transitions, occurring on the timescale of hours. At the sub-cellular level, we incorporate a detailed kinetic model of intracellular metabolite dynamics on the timescale of minutes, which enables the cells to uptake and excrete metabolites and use the metabolites to generate energy and building blocks for cell growth. This is a particularly novel aspect of the model. Certain defined criteria for the concentrations of intracellular metabolites lead to cancer cell growth, proliferation or death. Overall, we model the evolution of the tumor in both time and space. Starting with a cluster of tumor cells, the model produces an avascular tumor that quantitatively and qualitatively mimics experimental measurements of multicellular tumor spheroids. Through our model simulations, we can investigate the response of individual intracellular species under a metabolic perturbation and investigate how that response contributes to the response of the tumor as a whole. The predicted response of intracellular metabolites under various targeted strategies are difficult to resolve with experimental techniques. Thus, the model can give novel predictions as to the response of the tumor as a whole, identifies potential therapies to impede tumor growth, and predicts the effects of those therapeutic strategies. In particular, the model provides quantitative insight into the dynamic reprogramming of tumor cells at the intracellular level in response to specific metabolic perturbations. Overall, the model is a useful framework to study targeted metabolic strategies for inhibiting tumor growth.

Cancer cells expertly alter their metabolism in order to sustain growth, a hallmark of cancer. Quantitative details about this metabolic reprogramming are difficult to obtain without the use of predictive mathematical models. Here, we present a robust computational model of avascular tumor growth. The novel aspect of this work lies in the incorporation of a detailed model of the dynamics of metabolism within each individual cell, which directly influence growth of the multicellular tumor as a whole. We apply the model to simulate how the tumor grows in space and time and to predict how the tumor responds to targeted inhibition of specific intracellular metabolic reactions. Our results show, first-hand, the dynamic metabolic reprogramming that occurs in cancer cells. Specifically, the model provides insight into how the cells alter their metabolism to compensate for the loss of a nutrient by exploiting alternative pathways for continued tumor growth. Our work provides a quantitative tool for identifying the impact of cellular and sub-cellular features on the evolution of a tumor. This framework is useful for developing potential cancer therapies, complementing experimental studies.

Mathematical Modeling of Substrates Fluxes and Tumor Growth in the Brain

The aim of this article is to show how a tumor can modify energy substrates fluxes in the brain to support its own growth. To address this question we use a modeling approach to explain brain nutrient kinetics. In particular we set up a system of 17 equations for oxygen, lactate, glucose concentrations and cells number in the brain. We prove the existence and uniqueness of nonnegative solutions and give bounds on the solutions. We also provide numerical simulations.

Multi-scale computational study of the Warburg effect, reverse Warburg effect and glutamine addiction in solid tumors

Cancer metabolism has received renewed interest as a potential target for cancer therapy. In this study, we use a multi-scale modeling approach to interrogate the implications of three metabolic scenarios of potential clinical relevance: the Warburg effect, the reverse Warburg effect and glutamine addiction. At the intracellular level, we construct a network of central metabolism and perform flux balance analysis (FBA) to estimate metabolic fluxes; at the cellular level, we exploit this metabolic network to calculate parameters for a coarse-grained description of cellular growth kinetics; and at the multicellular level, we incorporate these kinetic schemes into the cellular automata of an agent-based model (ABM), iDynoMiCS. This ABM evaluates the reaction-diffusion of the metabolites, cellular division and motion over a simulation domain. Our multi-scale simulations suggest that the Warburg effect provides a growth advantage to the tumor cells under resource limitation. However, we identify a non-monotonic dependence of growth rate on the strength of glycolytic pathway. On the other hand, the reverse Warburg scenario provides an initial growth advantage in tumors that originate deeper in the tissue. The metabolic profile of stromal cells considered in this scenario allows more oxygen to reach the tumor cells in the deeper tissue and thus promotes tumor growth at earlier stages. Lastly, we suggest that glutamine addiction does not confer a selective advantage to tumor growth with glutamine acting as a carbon source in the tricarboxylic acid (TCA) cycle, any advantage of glutamine uptake must come through other pathways not included in our model (e.g., as a nitrogen donor). Our analysis illustrates the importance of accounting explicitly for spatial and temporal evolution of tumor microenvironment in the interpretation of metabolic scenarios and hence provides a basis for further studies, including evaluation of specific therapeutic strategies that target metabolism.

Cancer metabolism is an emerging hallmark of cancer. In the past decade, a renewed focus on cancer metabolism has led to several distinct hypotheses describing the role of metabolism in cancer. To complement experimental efforts in this field, a scale-bridging computational framework is needed to allow rapid evaluation of emerging hypotheses in cancer metabolism. In this study, we present a multi-scale modeling platform and demonstrate the distinct outcomes in population-scale growth dynamics under different metabolic scenarios: the Warburg effect, the reverse Warburg effect and glutamine addiction. Within this modeling framework, we confirmed population-scale growth advantage enabled by the Warburg effect, provided insights into the symbiosis between stromal cells and tumor cells in the reverse Warburg effect and argued that the anaplerotic role of glutamine is not exploited by tumor cells to gain growth advantage under resource limitations. We point to the opportunity for this framework to help understand tissue-scale response to therapeutic strategies that target cancer metabolism while accounting for the tumor complexity at multiple scales.

Mathematical models of tumor cell proliferation: A review of the literature

INTRODUCTION

A defining hallmark of cancer is aberrant cell proliferation. Efforts to understand the generative properties of cancer cells span all biological scales: from genetic deviations and alterations of metabolic pathways to physical stresses due to overcrowding, as well as the effects of therapeutics and the immune system. While these factors have long been studied in the laboratory, mathematical and computational techniques are being increasingly applied to help understand and forecast tumor growth and treatment response. Advantages of mathematical modeling of proliferation include the ability to simulate and predict the spatiotemporal development of tumors across multiple experimental scales. Central to proliferation modeling is the incorporation of available biological data and validation with experimental data. Areas covered: We present an overview of past and current mathematical strategies directed at understanding tumor cell proliferation. We identify areas for mathematical development as motivated by available experimental and clinical evidence, with a particular emphasis on emerging, non-invasive imaging technologies. Expert commentary: The data required to legitimize mathematical models are often difficult or (currently) impossible to obtain. We suggest areas for further investigation to establish mathematical models that more effectively utilize available data to make informed predictions on tumor cell proliferation.

Glioma growth modeling based on the effect of vital nutrients and metabolic products

Glioma brain tumors exhibit considerably aggressive behavior leading to high mortality rates. Mathematical modeling of tumor growth aims to explore the interactions between glioma cells and tissue microenvironment, which affect tumor evolution. Leveraging this concept, we present a three-dimensional model of glioma spatio-temporal evolution based on existing continuum approaches, yet incorporating novel factors of the phenomenon. The proposed model involves the interactions between different tumor cell phenotypes and their microenvironment, investigating how tumor growth is affected by complex biological exchanges. It focuses on the separate and combined effect of vital nutrients and cellular wastes on tumor expansion, leading to the formation of cell populations with different metabolic, proliferative, and diffusive profiles. Several simulations were performed on a virtual and a real glioma, using combinations of proliferation and diffusion rates for different evolution times. The model results were validated on a glioma model available in the literature and a real case of tumor progression. The experimental observations indicate that our model estimates quite satisfactorily the expansion of each region and the overall tumor growth. Based on the individual results, the proposed model may provide an important research tool for patient-specific simulation of different tumor evolution scenarios and reliable estimation of glioma evolution. Graphical Abstract Outline of the mathematical model functionality and application to glioma growth with indicative results.

Computational Model Predicts the Effects of Targeting Cellular Metabolism in Pancreatic Cancer

Reprogramming of energy metabolism is a hallmark of cancer that enables the cancer cells to meet the increased energetic requirements due to uncontrolled proliferation. One prominent example is pancreatic ductal adenocarcinoma, an aggressive form of cancer with an overall 5-year survival rate of 5%. The reprogramming mechanism in pancreatic cancer involves deregulated uptake of glucose and glutamine and other opportunistic modes of satisfying energetic demands in a hypoxic and nutrient-poor environment. In the current study, we apply systems biology approaches to enable a better understanding of the dynamics of the distinct metabolic alterations in KRAS-mediated pancreatic cancer, with the goal of impeding early cell proliferation by identifying the optimal metabolic enzymes to target. We have constructed a kinetic model of metabolism represented as a set of ordinary differential equations that describe time evolution of the metabolite concentrations in glycolysis, glutaminolysis, tricarboxylic acid cycle and the pentose phosphate pathway. The model is comprised of 46 metabolites and 53 reactions. The mathematical model is fit to published enzyme knockdown experimental data. We then applied the model to perform in silico enzyme modulations and evaluate the effects on cell proliferation. Our work identifies potential combinations of enzyme knockdown, metabolite inhibition, and extracellular conditions that impede cell proliferation. Excitingly, the model predicts novel targets that can be tested experimentally. Therefore, the model is a tool to predict the effects of inhibiting specific metabolic reactions within pancreatic cancer cells, which is difficult to measure experimentally, as well as test further hypotheses toward targeted therapies.

Mathematical modeling links Wnt signaling to emergent patterns of metabolism in colon cancer

Cell-intrinsic metabolic reprogramming is a hallmark of cancer that provides anabolic support to cell proliferation. How reprogramming influences tumor heterogeneity or drug sensitivities is not well understood. Here, we report a self-organizing spatial pattern of glycolysis in xenograft colon tumors where pyruvate dehydrogenase kinase (PDK1), a negative regulator of oxidative phosphorylation, is highly active in clusters of cells arranged in a spotted array. To understand this pattern, we developed a reaction-diffusion model that incorporates Wnt signaling, a pathway known to upregulate PDK1 and Warburg metabolism. Partial interference with Wnt alters the size and intensity of the spotted pattern in tumors and in the model. The model predicts that Wnt inhibition should trigger an increase in proteins that enhance the range of Wnt ligand diffusion. Not only was this prediction validated in xenograft tumors but similar patterns also emerge in radiochemotherapy-treated colorectal cancer. The model also predicts that inhibitors that target glycolysis or Wnt signaling in combination should synergize and be more effective than each treatment individually. We validated this prediction in 3D colon tumor spheroids.

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.

A Glycolysis-Based In Silico Model for the Solid Tumor Growth

Cancer-tumor growth is a complex process depending on several biological factors, such as the chemical microenvironment of the tumor, the cellular metabolic profile, and its proliferation rate. Several mathematical models have been developed for identifying the interactions between tumor cells and tissue microenvironment, since they play an important role in tumor formation and progression. Toward this direction we propose a new continuum model of avascular glioma-tumor growth, which incorporates a new factor, namely, the glycolytic potential of cancer cells, to express the interactions of three different tumor-cell populations (proliferative, hypoxic, and necrotic) with their tissue microenvironment. The glycolytic potential engages three vital nutrients, i.e., oxygen, glucose, and lactate, which provide cells with the necessary energy for their survival and proliferation. Extensive simulations are performed for different evolution times and various proliferation rates, in order to investigate how the tumor growth is affected. According to medical experts, the experimental observations indicate that the model predicts quite satisfactorily the overall tumor growth as well as the expansion of each region separately. Following extensive evaluation, the proposed model may provide an essential tool for patient-specific tumor simulation and reliable prediction of glioma spatiotemporal expansion.

Lactate does not activate NF-κB in oxidative tumor cells

The lactate anion is currently emerging as an oncometabolite. Lactate, produced and exported by glycolytic and glutaminolytic cells in tumors, can be recycled as an oxidative fuel by oxidative tumors cells. Independently of hypoxia, it can also activate transcription factor hypoxia-inducible factor-1 (HIF-1) in tumor and endothelial cells, promoting angiogenesis. These protumoral activities of lactate depend on lactate uptake, a process primarily facilitated by the inward, passive lactate-proton symporter monocarboxylate transporter 1 (MCT1); the conversion of lactate and NAD(+) to pyruvate, NADH and H(+) by lactate dehydrogenase-1 (LDH-1); and a competition between pyruvate and α-ketoglutarate that inhibits prolylhydroxylases (PHDs). Endothelial cells do not primarily use lactate as an oxidative fuel but, rather, as a signaling agent. In addition to HIF-1, lactate can indeed activate transcription factor nuclear factor-κB (NF-κB) in these cells, through a mechanism not only depending on PHD inhibition but also on NADH alimenting NAD(P)H oxidases to generate reactive oxygen species (ROS). While NF-κB activity in endothelial cells promotes angiogenesis, NF-κB activation in tumor cells is known to stimulate tumor progression by conferring resistance to apoptosis, stemness, pro-angiogenic and metastatic capabilities. In this study, we therefore tested whether exogenous lactate could activate NF-κB in oxidative tumor cells equipped for lactate signaling. We report that, precisely because they are oxidative, HeLa and SiHa human tumor cells do not activate NF-κB in response to lactate. Indeed, while lactate-derived pyruvate is well-known to inhibit PHDs in these cells, we found that NADH aliments oxidative phosphorylation (OXPHOS) in mitochondria rather than NAD(P)H oxidases in the cytosol. These data were confirmed using oxidative human Cal27 and MCF7 tumor cells. This new information positions the malate-aspartate shuttle as a key player in the oxidative metabolism of lactate: similar to glycolysis that aliments OXPHOS with pyruvate produced by pyruvate kinase and NADH produced by glyceraldehyde-3-phosphate dehydrogenase (GAPDH), oxidative lactate metabolism aliments OXPHOS in oxidative tumor cells with pyruvate and NADH produced by LDH1.

Knockdown of PKM2 induces apoptosis and autophagy in human A549 alveolar adenocarcinoma cells

The M2 isoform of pyruvate kinase (PKM2), which has been identified as the predominant cause of the Warburg effect in cancer cells, is essential in tumor metabolism and growth. However, the role of PKM2 in autophagy remains to be elucidated. The present study investigated the effect of PKM2 knockdown on autophagy and apoptotic cell death in human A549 alveolar adenocarcinoma cells. Two short hairpin (sh)RNAs targeting human PKM2 mRNA were designed and lentiviral vectors were constructed. The A549 cells were infected with lentiviruses, containing shRNAs against PKM2, and the expression of PKM2 was examined by reverse transcription-quantitative polymerase chain reaction (RT‑qPCR) and immonoblotting analysis. A lactose dehydrogenase (LDH)‑coupled enzyme assay was used to detect the pyruvate kinase activity. RT‑qPCR was used to detect the mRNA expression level of glycolysis‑associated enzymes. The quantification of cells with punctate LC3 and expression of LC3II were examined to demonstrate autophagy. An MTT assay was used to detect cell viability and flow cytometry was used to determine cell apoptosis. The activity of caspase 3/7 and the expression of Bcl‑2 were also detected in A549 cells with PKM2 knockdown. The present study demonstrated that the two shRNAs efficiently downregulated the mRNA and protein expression levels of PKM2 in A549 cells. The knockdown of PKM2 decreased pyruvate kinase activity and glycolysis. Autophagy was induced in A549 cells with PKM2 knockdown. Inhibition of autophagy accelerated apoptotic death in PKM2‑knockdown cells and this was dependent on increased caspase 3/7 activity and decreased expression of Bcl‑2. In conclusion, the downregulation of PKM2 induced apoptosis and autophagy in A549 cells and this autophagy protected the cells from apoptotic cell death.

Glucose-lactate metabolic cooperation in cancer: insights from a spatial mathematical model and implications for targeted therapy

A recent study has hypothesised a glucose-lactate metabolic symbiosis between adjacent hypoxic and oxygenated regions of a developing tumour, and proposed a treatment strategy to target this symbiosis. However, in vivo experimental support remains inconclusive. Here we develop a minimal spatial mathematical model of glucose-lactate metabolism to examine, in principle, whether metabolic symbiosis is plausible in human tumours, and to assess the potential impact of inhibiting it. We find that symbiosis is a robust feature of our model system-although on the length scale at which oxygen supply is diffusion-limited, its occurrence requires very high cellular metabolic activity-and that necrosis in the tumour core is reduced in the presence of symbiosis. Upon simulating therapeutic inhibition of lactate uptake, we predict that targeted treatment increases the extent of tissue oxygenation without increasing core necrosis. The oxygenation effect is correlated strongly with the extent of wild-type hypoxia and only weakly with wild-type symbiotic behaviour, and therefore may be promising for radiosensitisation of hypoxic, lactate-consuming tumours even if they do not exhibit a spatially well-defined symbiosis. Finally, we conduct in vitro experiments on the U87 glioblastoma cell line to facilitate preliminary speculation as to where highly malignant tumours might fall in our parameter space, and find that these experiments suggest a weakly symbiotic regime for U87 cells, thus raising the new question of what relationship might exist between symbiosis and tumour malignancy.

Modeling the connection between primary and metastatic tumors

We put forward a model for cancer metastasis as a migration phenomenon between tumor cell populations coexisting and evolving in two different habitats. One of them is a primary tumor and the other one is a secondary or metastatic tumor. The evolution of the different cell phenotype populations in each habitat is described by means of a simple quasispecies model allowing for a cascade of mutations between the different phenotypes in each habitat. The cell migration event is supposed to be unidirectional and take place continuously in time. The possible clinical outcomes of the model depending on the parameter space are analyzed and the effect of the resection of the primary tumor is studied.