Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids

A seven-step guide to spatial, agent-based modelling of tumour evolution

Spatial agent-based models are frequently used to investigate the evolution of solid tumours subject to localized cell-cell interactions and microenvironmental heterogeneity. As spatial genomic, transcriptomic and proteomic technologies gain traction, spatial computational models are predicted to become ever more necessary for making sense of complex clinical and experimental data sets, for predicting clinical outcomes, and for optimizing treatment strategies. Here we present a non-technical step by step guide to developing such a model from first principles. Stressing the importance of tailoring the model structure to that of the biological system, we describe methods of increasing complexity, from the basic Eden growth model up to off-lattice simulations with diffusible factors. We examine choices that unavoidably arise in model design, such as implementation, parameterization, visualization and reproducibility. Each topic is illustrated with examples drawn from recent research studies and state of the art modelling platforms. We emphasize the benefits of simpler models that aim to match the complexity of the phenomena of interest, rather than that of the entire biological system. Our guide is aimed at both aspiring modellers and other biologists and oncologists who wish to understand the assumptions and limitations of the models on which major cancer studies now so often depend.

Optimization of Size of Nanosensitizers for Antitumor Radiotherapy Using Mathematical Modeling

The efficacy of antitumor radiotherapy can be enhanced by utilizing nonradioactive nanoparticles that emit secondary radiation when activated by a primary beam. They consist of small volumes of a radiosensitizing substance embedded within a polymer layer, which is coated with tumor-specific antibodies. The efficiency of nanosensitizers relies on their successful delivery to the tumor, which depends on their size. Increasing their size leads to a higher concentration of active substance; however, it hinders the penetration of nanosensitizers through tumor capillaries, slows down their movement through the tissue, and accelerates their clearance. In this study, we present a mathematical model of tumor growth and radiotherapy with the use of intravenously administered tumor-specific nanosensitizers. Our findings indicate that their optimal size for achieving maximum tumor radiosensitization following a single injection of their fixed total volume depends on the permeability of the tumor capillaries. Considering physiologically plausible spectra of capillary pore radii, with a nanoparticle polymer layer width of 7 nm, the optimal radius of nanoparticles falls within the range of 13-17 nm. The upper value is attained when considering an extreme spectrum of capillary pores.

A reduced model of cell metabolism to revisit the glycolysis-OXPHOS relationship in the deregulated tumor microenvironment

Cancer cells metabolism focuses the interest of the cancer research community. Although this process is intensely studied experimentally, there are very few theoretical models that address this issue. One of the main reasons is the extraordinary complexity of the metabolism that involves numerous interdependent regulatory networks which makes the computational recreation of this complexity illusory. In this study we propose a reduced model of the metabolism which focuses on the interrelation of the three main energy metabolites which are oxygen, glucose and lactate in order to better understand the dynamics of the core system of the glycolysis-OXPHOS relationship. So simple as it is, the model highlights the main rules allowing the cell to dynamically adapt its metabolism to its changing environment. It also makes it possible to address this impact at the tissue scale. The simulations carried out in a spheroid show non-trivial spatial heterogeneity of energy metabolism. It further suggests that the metabolic features that are commonly attributed to cancer cells are not necessarily due to an intrinsic abnormality of the cells. They can emerge spontaneously due to the deregulated over-acidic environment.

Optimization of antitumor radiotherapy fractionation via mathematical modeling with account of 4 R's of radiobiology

A spatially-distributed continuous mathematical model of solid tumor growth and treatment by fractionated radiotherapy is presented. The model explicitly accounts for the factors, widely referred to as 4 R's of radiobiology, which influence the efficacy of radiotherapy fractionation protocols: tumor cell repopulation, their redistribution in cell cycle, reoxygenation and repair of sublethal damage of both tumor and normal tissues. With the use of special algorithm the fractionation protocols that provide increased tumor control probability, compared to standard clinical protocol, are found for various physiologically-based values of model parameters under the constraints of fixed overall normal tissue damage and maximum admissible fractional dose. In particular, it is shown that significant gain in treatment efficacy can be achieved for tumors of low malignancy by the use of protracted hyperfractionated protocols. The optimized non-uniform protocols are characterized by gradual escalation of fractional doses in their last parts, which start after the levels of oxygen and nutrients significantly elevate throughout the tumor and accelerated tumor proliferation manifests itself, which is a well-known experimental phenomenon.

A hybrid spatiotemporal model of PCa dynamics and insights into optimal therapeutic strategies

Using a hybrid cellular automaton with stochastic elements, we investigate the effectiveness of multiple drug therapies on prostate cancer (PCa) growth. The ability of Androgen Deprivation Therapy to reduce PCa growth represents a milestone in prostate cancer treatment, nonetheless most patients eventually become refractory and develop castration-resistant prostate cancer. In recent years, a "second generation" drug called enzalutamide has been used to treat advanced PCa, or patients already exposed to chemotherapy that stopped responding to it. However, tumour resistance to enzalutamide is not well understood, and in this context, preclinical models and in silico experiments (numerical simulations) are key to understanding the mechanisms of resistance and to assessing therapeutic settings that may delay or prevent the onset of resistance. In our mathematical system, we incorporate cell phenotype switching to model the development of increased drug resistance, and consider the effect of the micro-environment dynamics on necrosis and apoptosis of the tumour cells. The therapeutic strategies that we explore include using a single drug (enzalutamide), and drug combinations (enzalutamide and everolimus or cabazitaxel) with different treatment schedules. Our results highlight the effectiveness of alternating therapies, especially alternating enzalutamide and cabazitaxel over a year, and a comparison is made with data taken from TRAMP mice to verify our findings.

Searching for the Metabolic Signature of Cancer: A Review from Warburg's Time to Now

This review focuses on the evolving understanding that we have of tumor cell metabolism, particularly glycolytic and oxidative metabolism, and traces back its evolution through time. This understanding has developed since the pioneering work of Otto Warburg, but the understanding of tumor cell metabolism continues to be hampered by misinterpretation of his work. This has contributed to the use of the new concepts of metabolic switch and metabolic reprogramming， that are out of step with reality. The Warburg effect is often considered to be a hallmark of cancer, but is it really? More generally, is there a metabolic signature of cancer? We draw the conclusion that the signature of cancer cannot be reduced to a single factor, but is expressed at the tissue level in terms of the capacity of cells to dynamically explore a vast metabolic landscape in the context of significant environmental heterogeneities.

Asymmetric Growth of Tumor Spheroids in a Symmetric Environment

In this work, we studied the stability of radially symmetric growth in tumor spheroids using a reaction-diffusion model. In this model, nutrient concentration and internal pressure are local variables that implicitly relate the proliferation of cells to the growth of the tumor. The analytical solution of the governing model was presented in an orthonormal spherical harmonic basis. It was shown that the radially symmetric steady-state solution to the growth of tumor spheroids, under symmetric growth conditions, was unstable with respect to small asymmetric perturbations. Such perturbations excited the asymmetric modes of growth, which could grow in time and change the spherical configuration of the tumor. The number of such modes and their rates of growth depended on parameters such as surface tension, external energy and the rate of nutrient consumption. This analysis indicated that the spherical configuration of tumor spheroids, even under experimentally controlled symmetric growth conditions, were naturally unstable. This was confirmed by a comparison between the shapes of in vitro human glioblastoma (hGB) spheroids and the configuration of the first few asymmetric modes predicted by the model.

Designing and interpreting 4D tumour spheroid experiments

Tumour spheroid experiments are routinely used to study cancer progression and treatment. Various and inconsistent experimental designs are used, leading to challenges in interpretation and reproducibility. Using multiple experimental designs, live-dead cell staining, and real-time cell cycle imaging, we measure necrotic and proliferation-inhibited regions in over 1000 4D tumour spheroids (3D space plus cell cycle status). By intentionally varying the initial spheroid size and temporal sampling frequencies across multiple cell lines, we collect an abundance of measurements of internal spheroid structure. These data are difficult to compare and interpret. However, using an objective mathematical modelling framework and statistical identifiability analysis we quantitatively compare experimental designs and identify design choices that produce reliable biological insight. Measurements of internal spheroid structure provide the most insight, whereas varying initial spheroid size and temporal measurement frequency is less important. Our general framework applies to spheroids grown in different conditions and with different cell types.

A 3D Mathematical Model of Coupled Stem Cell-Nutrient Dynamics in Myocardial Regeneration Therapy

Stem cell therapy is a promising treatment for the regeneration of myocardial tissue injured by an ischemic event. Mathematical modeling of myocardial regeneration via stem cell therapy is a challenging task, since the mechanisms underlying the processes involved in the treatment are not yet fully understood. Many aspects must be accounted for, such as the spread of stem cells and nutrients, chemoattraction, cell proliferation, stages of cell maturation, differentiation, angiogenesis, stochastic effects, just to name a few. In this paper we propose a 3D mathematical model with a free boundary that aims to provide a qualitative description of some main aspects of the stem cell regenerative therapy in a simplified scenario. The paper mainly focuses on the description of the shrinking of the necrotic core during treatment. The stem cell and nutrients dynamics are described through coupled reaction-diffusion problems. Proliferation, chemoattraction, tissue regeneration and nutrient consumption are included in the model.

Self-Organization Formation of Multicellular Spheroids Mediated by Mechanically Tunable Hydrogel Platform: Toward Revealing the Synergy of Chemo- and Noninvasive Photothermal Therapy against Colon Microtumor

Three-dimensional (3D) tumor cell culture offers a more tissue-recapitulating model in cancer treatment evaluation. However, conventional models based on cell-substrate adhesion deprivation are still of insufficient real tumor mimic. In this work, a novel method is proposed for inducing multicellular spheroids (MCSs) formation based on hydrogel with tunable microenvironmental properties. Colon tumor cells DLD1 cultured on hydrogel substrate with proper physical stimulation form MCSs via self-organization. Chemotherapy based on clinical drug and far-infrared photothermal therapy is evaluated with DLD1 MCSs obtained by this method. The synergism of chemotherapy and noninvasive photothermal therapy based on graphene device is further verified in MCSs model and it is believed this method holds potential in in vitro anti-tumor strategies evaluation for precision medicine.

Model of Drug Delivery to Populations Composed of Two Cell Types

The rate of drug delivery to cells and the subsequent rate of drug metabolism are dependent on the cell membrane permeability to the drug. In some cases, tissue may be composed of different types of cells that exhibit order of magnitude differences in their membrane permeabilities. This paper presents a brief review of the components of the tissue scale three-compartment pharmacokinetic model of drug delivery to single-cell-type populations. The existing model is extended to consider tissue composed of two different cell types. A case study is presented of infusion mediated delivery of doxorubicin to a tumor that is composed a drug reactive cell type and of a drug resistive cell type. The membrane permeabilities of the two cell types differ by an order of magnitude. A parametric investigation of the population composition is conducted and it is shown that the drug metabolism of the low permeability cells are negatively influenced by the fraction of the tissue composed of the permeable drug reactive cells. This is because when the population is composed mostly of drug permeable cells, the extracellular space is rapidly depleted of the drug. This has two compounding effects: (i) locally there is simply less drug available to the neighboring drug resistant cells, and (ii) the depletion of the drug from the extracellular space near the vessel-tissue interface leaves less drug to be transported to booth cell types farther away from the vessel.

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.

Combined Influence of Nutrient Supply Level and Tissue Mechanical Properties on Benign Tumor Growth as Revealed by Mathematical Modeling

A continuous mathematical model of non-invasive avascular tumor growth in tissue is presented. The model considers tissue as a biphasic material, comprised of a solid matrix and interstitial fluid. The convective motion of tissue elements happens due to the gradients of stress, which change as a result of tumor cells proliferation and death. The model accounts for glucose as the crucial nutrient, supplied from the normal tissue, and can reproduce both diffusion-limited and stress-limited tumor growth. Approximate tumor growth curves are obtained semi-analytically in the limit of infinite tissue hydraulic conductivity, which implies instantaneous equalization of arising stress gradients. These growth curves correspond well to the numerical solutions and represent classical sigmoidal curves with a short initial exponential phase, subsequent almost linear growth phase and a phase with growth deceleration, in which tumor tends to reach its maximum volume. The influence of two model parameters on tumor growth curves is investigated: tissue hydraulic conductivity, which links the values of stress gradient and convective velocity of tissue phases, and tumor nutrient supply level, which corresponds to different permeability and surface area density of capillaries in the normal tissue that surrounds the tumor. In particular, it is demonstrated, that sufficiently low tissue hydraulic conductivity (intrinsic, e.g., to tumors arising from connective tissue) and sufficiently high nutrient supply can lead to formation of giant benign tumors, reaching tens of centimeters in diameter, which are indeed observed clinically.

Digital twinning of Cellular Capsule Technology: Emerging outcomes from the perspective of porous media mechanics

Spheroids encapsulated within alginate capsules are emerging as suitable in vitro tools to investigate the impact of mechanical forces on tumor growth since the internal tumor pressure can be retrieved from the deformation of the capsule. Here we focus on the particular case of Cellular Capsule Technology (CCT). We show in this contribution that a modeling approach accounting for the triphasic nature of the spheroid (extracellular matrix, tumor cells and interstitial fluid) offers a new perspective of analysis revealing that the pressure retrieved experimentally cannot be interpreted as a direct picture of the pressure sustained by the tumor cells and, as such, cannot therefore be used to quantify the critical pressure which induces stress-induced phenotype switch in tumor cells. The proposed multiphase reactive poro-mechanical model was cross-validated. Parameter sensitivity analyses on the digital twin revealed that the main parameters determining the encapsulated growth configuration are different from those driving growth in free condition, confirming that radically different phenomena are at play. Results reported in this contribution support the idea that multiphase reactive poro-mechanics is an exceptional theoretical framework to attain an in-depth understanding of CCT experiments, to confirm their hypotheses and to further improve their design.

In-Silico Modeling of Tumor Spheroid Formation and Growth

Mathematical modeling has significant potential for understanding of biological models of cancer and to accelerate the progress in cross-disciplinary approaches of cancer treatment. In mathematical biology, solid tumor spheroids are often studied as preliminary in vitro models of avascular tumors. The size of spheroids and their cell number are easy to track, making them a simple in vitro model to investigate tumor behavior, quantitatively. The growth of solid tumors is comprised of three main stages: transient formation, monotonic growth and a plateau phase. The last two stages are extensively studied. However, the initial transient formation phase is typically missing from the literature. This stage is important in the early dynamics of growth, formation of clonal sub-populations, etc. In the current work, this transient formation is modeled by a reaction–diffusion partial differential equation (PDE) for cell concentration, coupled with an ordinary differential equation (ODE) for the spheroid radius. Analytical and numerical solutions of the coupled equations were obtained for the change in the radius of tumor spheroids over time. Human glioblastoma (hGB) cancer cells (U251 and U87) were spheroid cultured to validate the model prediction. Results of this study provide insight into the mechanism of development of solid tumors at their early stage of formation.

Development of a stacked, porous silk scaffold neuroblastoma model for investigating spatial differences in cell and drug responsiveness

Development of in vitro, preclinical cancer models that contain cell-driven microenvironments remains a challenge. Engineering of millimeter-scale, in vitro tumor models with spatially distinct regions that can be independently assessed to study tumor microenvironments has been limited. Here, we report the use of porous silk scaffolds to generate a high cell density neuroblastoma (NB) model that can spatially recapitulate changes resulting from cell and diffusion driven changes. Using COMSOL modeling, a scaffold holder design that facilitates stacking of thin, 200 μm silk scaffolds into a thick, bulk millimeter-scale tumor model (2, 4, 6, and 8 stacked scaffolds) and supports cell-driven oxygen gradients was developed. Cell-driven oxygen gradients were confirmed through pimonidazole staining. Post-culture, the stacked scaffolds were separated for analysis on a layer-by-layer basis. The analysis of each scaffold layer demonstrated decreasing DNA and increasing expression of hypoxia related genes (VEGF, CAIX, and GLUT1) from the exterior scaffolds to the interior scaffolds. Furthermore, the expression of hypoxia related genes at the interior of the stacks was comparable to that of a single scaffold cultured under 1% O₂ and at the exterior of the stacks was comparable to that of a single scaffold cultured under 21% O₂. The four-stack scaffold model underwent further evaluation to determine if a hypoxia activated drug, tirapazamine, induced reduced cell viability within the internal stacks (region of reduced oxygen) as compared with the external stacks. Decreased DNA content was observed in the internal stacks as compared to the external stacks when treated with tirapazamine, which suggests the internal scaffold stacks had higher levels of hypoxia than the external scaffolds. This stacked silk scaffold system presents a method for creating a single culture model capable of generating controllable cell-driven microenvironments through different stacks that can be individually assessed and used for drug screening.

Angiogenesis and vessel co-option in a mathematical model of diffusive tumor growth: The role of chemotaxis

This work considers the propagation of a tumor from the stage of a small avascular sphere in a host tissue and the progressive onset of a tumor neovasculature stimulated by a pro-angiogenic factor secreted by hypoxic cells. The way new vessels are formed involves cell sprouting from pre-existing vessels and following a trail via a chemotactic mechanism (CM). Namely, it is first proposed a detailed general family of models of the CM, based on a statistical mechanics approach. The key hypothesis is that the CM is composed by two components: i) the well-known bias induced by the angiogenic factor gradient; ii) the presence of stochastic changes of the velocity direction, thus giving rise to a diffusive component. Then, some further assumptions and simplifications are applied in order to derive a specific model to be used in the simulations. The tumor progression is favored by its acidic aggression towards the healthy cells. The model includes the evolution of many biological and chemical species. Numerical simulations show the onset of a traveling wave eventually replacing the host tissue with a fully vascularized tumor. The results of simulations agree with experimental measures of the vasculature density in tumors, even in the case of particularly hypoxic tumors.

An on-lattice agent-based Monte Carlo model simulating the growth kinetics of multicellular tumor spheroids

PURPOSE

To develop an on-lattice agent-based model describing the growth of multicellular tumor spheroids using simple Monte Carlo tools.

METHODS

Cells are situated on the vertices of a cubic grid. Different cell states (proliferative, hypoxic or dead) and cell evolution rules, driven by 10 parameters, and the effects of the culture medium are included. About twenty spheroids of MCF-7 human breast cancer were cultivated and the experimental data were used for tuning the model parameters.

RESULTS

Simulated spheroids showed adequate sizes of the necrotic nuclei and of the hypoxic and proliferative cell phases as a function of the growth time, mimicking the overall characteristics of the experimental spheroids. The relation between the radii of the necrotic nucleus and the whole spheroid obtained in the simulations was similar to the experimental one and the number of cells, as a function of the spheroid volume, was well reproduced. The statistical variability of the Monte Carlo model described the whole volume range observed for the experimental spheroids. Assuming that the model parameters vary within Gaussian distributions it was obtained a sample of spheroids that reproduced much better the experimental findings.

CONCLUSIONS

The model developed allows describing the growth of in vitro multicellular spheroids and the experimental variability can be well reproduced. Its flexibility permits to vary both the agents involved and the rules that govern the spheroid growth. More general situations, such as, e. g., tumor vascularization, radiotherapy effects on solid tumors, or the validity of the tumor growth mathematical models can be studied.

Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity

A spatially-distributed continuous mathematical model of solid tumor growth and treatment by fractionated radiotherapy is presented. The model explicitly accounts for three time and space-dependent factors that influence the efficiency of radiotherapy fractionation schemes—tumor cell repopulation, reoxygenation and redistribution of proliferative states. A special algorithm is developed, aimed at finding the fractionation schemes that provide increased tumor cure probability under the constraints of maximum normal tissue damage and maximum fractional dose. The optimization procedure is performed for varied radiosensitivity of tumor cells under the values of model parameters, corresponding to different degrees of tumor malignancy. The resulting optimized schemes consist of two stages. The first stages are aimed to increase the radiosensitivity of the tumor cells, remaining after their end, sparing the caused normal tissue damage. This allows to increase the doses during the second stages and thus take advantage of the obtained increased radiosensitivity. Such method leads to significant expansions in the curative ranges of the values of tumor radiosensitivity parameters. Overall, the results of this study represent the theoretical proof of concept that non-uniform radiotherapy fractionation schemes may be considerably more effective that uniform ones, due to the time and space-dependent effects.

Multiscale modeling of tumor response to vascular endothelial growth factor (VEGF) inhibitor

Vascular endothelial growth factor (VEGF) has been known as a key mediator of angiogenesis in cancer. Bevacizumab is anti-VEGF monoclonal antibody that has been approved by the FDA as a first-line treatment in many types of cancer. In this paper, we extend a previously validated multiscale tumor model to comprehensively include the multiple roles of VEGF during the course of angiogenesis and its binding mechanism with bevacizumab. We use the model to simulate tumor system response under various bevacizumab concentrations, both in stand-alone treatment and in combination with chemotherapy. Our simulation indicates that periodic administration of bevacizumab with lower concentration can achieve greater efficacy than a single treatment with higher concentration. The simulation of the combined therapy also shows that the continuous administration of bevacizumab during the maintenance phase can lead to antitumor activity which further suppresses its growth. Agreement with experimental results indicates the potential of the model in predicting the efficacy of anti-VEGF therapies and could therefore contribute to developing prospective clinical trials.

A hybrid multiscale model for investigating tumor angiogenesis and its response to cell-based therapy

Angiogenesis, a formation of blood vessels from an existing vasculature, plays a key role in tumor growth and its progression into cancer. The lining of blood vessels consists of endothelial cells (ECs) which proliferate and migrate, allowing the capillaries to sprout towards the tumor to deliver the needed oxygen. Various treatments aiming to suppress or even inhibit angiogenesis have been explored. Mesenchymal stem cells (MSCs) have recently been undergoing development in cell-based therapy for cancer due to their ability to migrate towards the capillaries and induce the apoptosis of the ECs, causing capillary degeneration. However, further investigations in this direction are needed as it is usually difficult to preclinically assess the efficacy of such therapy. We develop a hybrid multiscale model that integrates molecular, cellular, tissue and extracellular components of tumor system to investigate angiogenesis and tumor growth under MSC-mediated therapy. Our simulations produce angiogenesis and vascular tumor growth profiles as observed in the experiments. Furthermore, the simulations show that the effectiveness of MSCs in inducing EC apoptosis is density dependent and its full effect is reached within several days after MSCs application. Quantitative agreements with experimental data indicate the predictive potential of our model for evaluating the efficacy of cell-based therapies targeting angiogenesis.

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.

Three-Dimensional Modeling of Avascular Tumor Growth in Both Static and Dynamic Culture Platforms

Microfluidic cell culture platforms are ideal candidates for modeling the native tumor microenvironment because they can precisely reconstruct in vivo cellular behavior. Moreover, mathematical modeling of tumor growth can pave the way toward description and prediction of growth pattern as well as improving cancer treatment. In this study, a modified mathematical model based on concentration distribution is applied to tumor growth in both conventional static culture and dynamic microfluidic cell culture systems. Apoptosis and necrosis mechanisms are considered as the main inhibitory factors in the model, while tumor growth rate and nutrient consumption rate are modified in both quiescent and proliferative zones. We show that such modification can better predict the experimental results of tumor growth reported in the literature. Using numerical simulations, the effects of the concentrations of the nutrients as well as the initial tumor radius on the tumor growth are investigated and discussed. Furthermore, tumor growth is simulated by taking into account the dynamic perfusion into the proposed model. Subsequently, tumor growth kinetics in a three-dimensional (3D) microfluidic device containing a U-shaped barrier is numerically studied. For this case, the effect of the flow rate of culture medium on tumor growth is investigated as well. Finally, to evaluate the impact of the trap geometry on the tumor growth, a comparison is made between the tumor growth kinetics in two frequently used traps in microfluidic cell culture systems, i.e., the U-shaped barrier and microwell structures. The proposed model can provide insight into better predicting the growth and development of avascular tumor in both static and dynamic cell culture platforms.

Agent-based model of multicellular tumor spheroid evolution including cell metabolism

Computational models aiming at the spatio-temporal description of cancer evolution are a suitable framework for testing biological hypotheses from experimental data, and generating new ones. Building on our recent work (J. Theor. Biol. 389, 146 (2016)) we develop a 3D agent-based model, capable of tracking hundreds of thousands of interacting cells, over time scales ranging from seconds to years. Cell dynamics is driven by a Monte Carlo solver, incorporating partial differential equations to describe chemical pathways and the activation/repression of "genes", leading to the up- or down-regulation of specific cell markers. Each cell-agent of different kind (stem, cancer, stromal etc.) runs through its cycle, undergoes division, can exit to a dormant, senescent, necrotic state, or apoptosis, according to the inputs from its systemic network. The basic network at this stage describes glucose/oxygen/ATP cycling, and can be readily extended to cancer-cell specific markers. Eventual accumulation of chemical/radiation damage to each cell's DNA is described by a Markov chain of internal states, and by a damage-repair network, whose evolution is linked to the cell systemic network. Aimed at a direct comparison with experiments of tumorsphere growth from stem cells, the present model will allow to quantitatively study the role of transcription factors involved in the reprogramming and variable radio-resistance of simulated cancer-stem cells, evolving in a realistic computer simulation of a growing multicellular tumorsphere.

Mathematical modeling of the proliferation gradient in multicellular tumor spheroids

MultiCellular Tumor Spheroids are 3D cell cultures that can accurately reproduce the behavior of solid tumors. It has been experimentally observed that large spheroids exhibit a decreasing gradient of proliferation from the periphery to the center of these multicellular 3D models: the proportion of proliferating cells is higher in the periphery while the non-proliferating quiescent cells increase in depth. In this paper, we propose to investigate the key mechanisms involved in the establishment of this gradient with a Partial Differential Equations model that mimics the experimental set-up of growing spheroids under different nutrients supply conditions. The model consists of mass balance equations on the two cell populations observed in the data: the proliferating cells and the quiescent cells. The spherical symmetry is used to rewrite the model in radial and relative coordinates. Thanks to a rigorous data postprocessing the model is then fit and compared quantitatively with the experimental quantification of the percentage of proliferating cells from EdU immunodetection on 2D spheroid cryosection images. The results of this calibration show that the proliferation gradient observed in spheroids can be quantitatively reproduced by our model.

Transient alleviation of tumor hypoxia during first days of antiangiogenic therapy as a result of therapy-induced alterations in nutrient supply and tumor metabolism - Analysis by mathematical modeling

A number of experiments on mouse tumor models, as well as certain clinical data, have demonstrated, that antiangiogenic therapy can lead to transient improvement in tumor oxygenation, that allows to increase efficiency of following radiotherapy. In the majority of works, this phenomenon has been explained by enhanced tumor perfusion due to normalization of capillaries' structure, that results in elevated oxygen inflow in tumor. However, changes in tumor perfusion often haven't been directly measured in relevant works, moreover, antiangiogenic therapy has been proven to have ambiguous effect on tumor perfusion both in mouse tumor models and in clinics. Herein, we suggest that elevation of blood perfusion may be not the only reason for transient alleviation of tumor hypoxia, and that it may manifest itself even under unchanged tumor blood flow. We propose that it may be as well caused by the decrease in tumor oxygen consumption rate (OCR) due to the reduction of tumor proliferation level, caused by nutrient shortage in result of antiangiogenic treatment. We provide detailed explanation of this hypothesis and visualize it using a specially developed mathematical model, which takes into account basic features of tumor growth and antiangiogenic therapy. We investigate the influence of the model parameters on oxygen dynamics; demonstrate, that transient alleviation of tumor hypoxia occurs in a fairly wide range of physiologically justified values of parameters; and point out the major factors, that determine oxygen dynamics during antiangiogenic therapy.

Simulation of Multispecies Desmoplastic Cancer Growth via a Fully Adaptive Non-linear Full Multigrid Algorithm

A fully adaptive non-linear full multigrid (FMG) algorithm is implemented to computationally simulate a model of multispecies desmoplastic tumor growth in three spatial dimensions. The algorithm solves a thermodynamic mixture model employing a diffuse interface approach with Cahn-Hilliard-type fourth-order equations that are coupled, non-linear, and numerically stiff. The tumor model includes extracellular matrix (ECM) as a major component with elastic energy contribution in its chemical potential term. Blood and lymphatic vasculatures are simulated via continuum representations. The model employs advection-reaction-diffusion partial differential equations (PDEs) for the cell, ECM, and vascular components, and reaction-diffusion PDEs for the elements diffusing from the vessels. This study provides the details of the numerical solution obtained by applying the fully adaptive non-linear FMG algorithm with finite difference method to solve this complex system of PDEs. The results indicate that this type of computational model can simulate the extracellular matrix-rich desmoplastic tumor microenvironment typical of fibrotic tumors, such as pancreatic adenocarcinoma.

A multiscale model for heterogeneous tumor spheroid in vitro

In this paper, a novel multiscale method is proposed for the study of heterogeneous tumor spheroid growth in vitro. The entire tumor spheroid is described by an ellipsoid-based model while nutrient and other environmental factors are treated as continua. The ellipsoid-based discrete component is capable of incorporating mechanical effects and deformability, while keeping a minimum set of free variables to describe complex shape variations. Moreover, our purely cell-based description of tumor avoids the complex mutual conversion between a cell-based model and continuum model within a tumor, such as force and mass transformation. This advantage makes it highly suitable for the study of tumor spheroids in vitro whose size are normally less than 800 μm in diameter. In addition, our numerical scheme provides two computational options depending on tumor size. For a small or medium tumor spheroid, a three-dimensional (3D) numerical model can be directly applied. For a large spheroid, we suggest the use of a 3D-adapted 2D cross section configuration, which has not yet been explored in the literature, as an alternative for the theoretical investigation to bridge the gap between the 2D and 3D models. Our model and its implementations have been validated and applied to various studies given in the paper. The simulation results fit corresponding in vitro experimental observations very well.

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.

Translational models of tumor angiogenesis: A nexus of in silico and in vitro models

Emerging evidence shows that endothelial cells are not only the building blocks of vascular networks that enable oxygen and nutrient delivery throughout a tissue but also serve as a rich resource of angiocrine factors. Endothelial cells play key roles in determining cancer progression and response to anti-cancer drugs. Furthermore, the endothelium-specific deposition of extracellular matrix is a key modulator of the availability of angiocrine factors to both stromal and cancer cells. Considering tumor vascular network as a decisive factor in cancer pathogenesis and treatment response, these networks need to be an inseparable component of cancer models. Both computational and in vitro experimental models have been extensively developed to model tumor-endothelium interactions. While informative, they have been developed in different communities and do not yet represent a comprehensive platform. In this review, we overview the necessity of incorporating vascular networks for both in vitro and in silico cancer models and discuss recent progresses and challenges of in vitro experimental microfluidic cancer vasculature-on-chip systems and their in silico counterparts. We further highlight how these two approaches can merge together with the aim of presenting a predictive combinatorial platform for studying cancer pathogenesis and testing the efficacy of single or multi-drug therapeutics for cancer treatment.

Model of vascular desmoplastic multispecies tumor growth

We present a three-dimensional nonlinear tumor growth model composed of heterogeneous cell types in a multicomponent-multispecies system, including viable, dead, healthy host, and extra-cellular matrix (ECM) tissue species. The model includes the capability for abnormal ECM dynamics noted in tumor development, as exemplified by pancreatic ductal adenocarcinoma, including dense desmoplasia typically characterized by a significant increase of interstitial connective tissue. An elastic energy is implemented to provide elasticity to the connective tissue. Cancer-associated fibroblasts (myofibroblasts) are modeled as key contributors to this ECM remodeling. The tumor growth is driven by growth factors released by these stromal cells as well as by oxygen and glucose provided by blood vasculature which along with lymphatics are stimulated to proliferate in and around the tumor based on pro-angiogenic factors released by hypoxic tissue regions. Cellular metabolic processes are simulated, including respiration and glycolysis with lactate fermentation. The bicarbonate buffering system is included for cellular pH regulation. This model system may be of use to simulate the complex interactions between tumor and stromal cells as well as the associated ECM and vascular remodeling that typically characterize malignant cancers notorious for poor therapeutic response.

A Mathematical Model for the Onset of Avascular Tumor Growth in Response to the Loss of P53 Function

We present a mathematical model for the formation of an avascular tumor based on the loss by gene mutation of the tumor suppressor function of p53. The wild type p53 protein regulates apoptosis, cell expression of growth factor and matrix metalloproteinase, which are regulatory functions that many mutant p53 proteins do not possess. The focus is on a description of cell movement as the transport of cell population density rather than as the movement of individual cells. In contrast to earlier works on solid tumor growth, a model is proposed for the initiation of tumor growth. The central idea, taken from the mathematical theory of dynamical systems, is to view the loss of p53 function in a few cells as a small instability in a rest state for an appropriate system of differential equations describing cell movement. This instability is shown (numerically) to lead to a second, spatially inhomogeneous, solution that can be thought of as a solid tumor whose growth is nutrient diffusion limited. In this formulation, one is led to a system of nine partial differential equations. We show computationally that there can be tumor states that coexist with benign states and that are highly unstable in the sense that a slight increase in tumor size results in the tumor occupying the sample region while a slight decrease in tumor size results in its ultimate disappearance.

Finite Element Modeling of Avascular Tumor Growth Using a Stress-Driven Model

Tumor growth being a multistage process has been investigated from different aspects. In the present study, an attempt is made to represent a constitutive-structure-based model of avascular tumor growth in which the effects of tensile stresses caused by collagen fibers are considered. Collagen fibers as a source of anisotropy in the structure of tissue are taken into account using a continuous fiber distribution formulation. To this end, a finite element modeling is implemented in which a neo-Hookean hyperelastic material is assigned to the tumor and its surrounding host. The tumor is supplied with a growth term. The growth term includes the effect of parameters such as nutrient concentration on the tumor growth and the tumor's solid phase content in the formulation. Results of the study revealed that decrease of solid phase is indicative of decrease in growth rate and the final steady-state value of tumor's radius. Moreover, fiber distribution affects the final shape of the tumor, and it could be used to control the shape and geometry of the tumor in complex morphologies. Finally, the findings demonstrated that the exerted stresses on the tumor increase as time passes. Compression of tumor cells leads to the reduction of tumor growth rate until it gradually reaches an equilibrium radius. This finding is in accordance with experimental data. Hence, this formulation can be deployed to evaluate both the residual stresses induced by growth and the mechanical interactions with the host tissue.

Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-[Formula: see text]B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53-Mdm2, NF-[Formula: see text]B) and through the use of high-performance computing be capable of simulating up to [Formula: see text] cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.

Evaluating the influence of mechanical stress on anticancer treatments through a multiphase porous media model

Drug resistance is one of the leading causes of poor therapy outcomes in cancer. As several chemotherapeutics are designed to target rapidly dividing cells, the presence of a low-proliferating cell population contributes significantly to treatment resistance. Interestingly, recent studies have shown that compressive stresses acting on tumor spheroids are able to hinder cell proliferation, through a mechanism of growth inhibition. However, studies analyzing the influence of mechanical compression on therapeutic treatment efficacy have still to be performed. In this work, we start from an existing mathematical model for avascular tumors, including the description of mechanical compression. We introduce governing equations for transport and uptake of a chemotherapeutic agent, acting on cell proliferation. Then, model equations are adapted for tumor spheroids and the combined effect of compressive stresses and drug action is investigated. Interestingly, we find that the variation in tumor spheroid volume, due to the presence of a drug targeting cell proliferation, considerably depends on the compressive stress level of the cell aggregate. Our results suggest that mechanical compression of tumors may compromise the efficacy of chemotherapeutic agents. In particular, a drug dose that is effective in reducing tumor volume for stress-free conditions may not perform equally well in a mechanically compressed environment.

A multiscale agent-based framework integrated with a constraint-based metabolic network model of cancer for simulating avascular tumor growth

The integration of an agent-based framework with a constraint-based metabolic network model of cancer for simulating avascular tumor growth.

Identifying the necrotic zone boundary in tumour spheroids with pair-correlation functions

Automatic identification of the necrotic zone boundary is important in the assessment of treatments on in vitro tumour spheroids. This has been difficult especially when the difference in cell density between the necrotic and viable zones of a tumour spheroid is small. To help overcome this problem, we developed novel one-dimensional pair-correlation functions (PCFs) to provide quantitative estimates of the radial distance of the necrotic zone boundary from the centre of a tumour spheroid. We validate our approach on synthetic tumour spheroids in which the position of the necrotic zone boundary is known a priori It is then applied to nine real tumour spheroids imaged with light sheet-based fluorescence microscopy. PCF estimates of the necrotic zone boundary are compared with those of a human expert and an existing standard computational method.

On the morphological stability of multicellular tumour spheroids growing in porous media

Multicellular tumour spheroids (MCTSs) are extensively used as in vitro system models for investigating the avascular growth phase of solid tumours. In this work, we propose a continuous growth model of heterogeneous MCTSs within a porous material, taking into account a diffusing nutrient from the surrounding material directing both the proliferation rate and the mobility of tumour cells. At the time scale of interest, the MCTS behaves as an incompressible viscous fluid expanding inside a porous medium. The cell motion and proliferation rate are modelled using a non-convective chemotactic mass flux, driving the cell expansion in the direction of the external nutrients' source. At the early stages, the growth dynamics is derived by solving the quasi-stationary problem, obtaining an initial exponential growth followed by an almost linear regime, in accordance with experimental observations. We also perform a linear-stability analysis of the quasi-static solution in order to investigate the morphological stability of the radially symmetric growth pattern. We show that mechano-biological cues, as well as geometric effects related to the size of the MCTS subdomains with respect to the diffusion length of the nutrient, can drive a morphological transition to fingered structures, thus triggering the formation of complex shapes that might promote tumour invasiveness. The results also point out the formation of a retrograde flow in the MCTS close to the regions where protrusions form, that could describe the initial dynamics of metastasis detachment from the in vivo tumour mass. In conclusion, the results of the proposed model demonstrate that the integration of mathematical tools in biological research could be crucial for better understanding the tumour's ability to invade its host environment.

Modelling tumour cell proliferation from vascular structure using tissue decomposition into avascular elements

Computer models allow the mechanistically detailed study of tumour proliferation and its dependency on nutrients. However, the computational study of large vascular tumours requires detailed information on the 3-dimensional vessel network and rather high computation times due to complex geometries. This study puts forward the idea of partitioning vascularised tissue into connected avascular elements that can exchange cells and nutrients between each other. Our method is able to rapidly calculate the evolution of proliferating as well as dead and quiescent cells, and hence a proliferative index, from a given amount and distribution of vascularisation of arbitrary complexity. Applying our model, we found that a heterogeneous vessel distribution provoked a higher proliferative index, suggesting increased malignancy, and increased the amount of dead cells compared to a more static tumour environment when a homogenous vessel distribution was assumed. We subsequently demonstrated that under certain amounts of vascularisation, cell proliferation may even increase when vessel density decreases, followed by a subsequent decrease of proliferation. This effect was due to a trade-off between an increase in compensatory proliferation for replacing dead cells and a decrease of cell population due to lack of oxygen supply in lowly vascularised tumours. Findings were illustrated by an ectopic colorectal cancer mouse xenograft model. Our presented approach can be in the future applied to study the effect of cytostatic, cytotoxic and anti-angiogenic chemotherapy and is ideally suited for translational systems biology, where rapid interaction between theory and experiment is essential.

Predicting the growth of glioblastoma multiforme spheroids using a multiphase porous media model

Tumor spheroids constitute an effective in vitro tool to investigate the avascular stage of tumor growth. These three-dimensional cell aggregates reproduce the nutrient and proliferation gradients found in the early stages of cancer and can be grown with a strict control of their environmental conditions. In the last years, new experimental techniques have been developed to determine the effect of mechanical stress on the growth of tumor spheroids. These studies report a reduction in cell proliferation as a function of increasingly applied stress on the surface of the spheroids. This work presents a specialization for tumor spheroid growth of a previous more general multiphase model. The equations of the model are derived in the framework of porous media theory, and constitutive relations for the mass transfer terms and the stress are formulated on the basis of experimental observations. A set of experiments is performed, investigating the growth of U-87MG spheroids both freely growing in the culture medium and subjected to an external mechanical pressure induced by a Dextran solution. The growth curves of the model are compared to the experimental data, with good agreement for both the experimental settings. A new mathematical law regulating the inhibitory effect of mechanical compression on cancer cell proliferation is presented at the end of the paper. This new law is validated against experimental data and provides better results compared to other expressions in the literature.

A Computational Model for Investigating Tumor Apoptosis Induced by Mesenchymal Stem Cell-Derived Secretome

Apoptosis is a programmed cell death that occurs naturally in physiological and pathological conditions. Defective apoptosis can trigger the development and progression of cancer. Experiments suggest the ability of secretome derived from mesenchymal stem cells (MSC) to induce apoptosis in cancer cells. We develop a hybrid discrete-continuous multiscale model to further investigate the effect of MSC-derived secretome in tumor growth. The model encompasses three biological scales. At the molecular scale, a system of ordinary differential equations regulate the expression of proteins involved in apoptosis signaling pathways. At the cellular scale, discrete equations control cellular migration, phenotypic switching, and proliferation. At the extracellular scale, a system of partial differential equations are employed to describe the dynamics of microenvironmental chemicals concentrations. The simulation is able to produce both avascular tumor growth rate and phenotypic patterns as observed in the experiments. In addition, we obtain good quantitative agreements with the experimental data on the apoptosis of HeLa cancer cells treated with MSC-derived secretome. We use this model to predict the growth of avascular tumor under various secretome concentrations over time.

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.