An Adaptive Multigrid Algorithm for Simulating Solid Tumor Growth Using Mixture Models

Predicting the effect of a combination drug therapy on the prostate tumor growth via an improvement of a direct radial basis function partition of unity technique for a diffuse-interface model

Chemotherapy is usually applied to treat advanced prostate cancer that cancer cells spread outside the prostate gland. The treatment uses cytotoxic drugs to target cells that grow and divide quickly. On the other hand, the growth of such cancerous tumors depends on angiogenesis. In this paper, we numerically study a diffuse-interface model in a two-dimensional space related to the therapies of prostate cancer. The proposed model describes the tumor growth driven by a generic nutrient and producing the prostate-specific antigen. More precisely, the effect of cytotoxic chemotherapy in the model is evaluated by considering a time-dependent function in the tumor dynamics. Also, another function related to the antiangiogenic therapy is considered to show the reducing intratumoral nutrient supply in the nutrient dynamics. Here, a meshless approximation, i.e., a generalized form of the direct radial basis function partition of unity (D-RBF-PU) method is presented for finding the numerical simulations of this model utilizing in medical oncology. The method uses the lower number of trial points in each patch than the original D-RBF-PU scheme for approximating the trial function per test point. Hence, the time complexity of the method is less than the D-RBF-PU technique. Besides, a semi-implicit time discretization of order 1 has been used to deal with the time variable. Consequently, a linear system of algebraic equations could be solved iteratively per time step by the use of the biconjugate gradient stabilized method with zero-fill incomplete lower-upper preconditioner. Finally, the obtained results without using any adaptive algorithm demonstrate the response of the prostate tumor growth to the chemotherapy, antiangiogenic therapy and a combined therapy.

Data-driven spatio-temporal modelling of glioblastoma

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Evaluation of innate and adaptive immune system interactions in the tumor microenvironment via a 3D continuum model

Immune cells in the tumor microenvironment (TME) are known to affect tumor growth, vascularization, and extracellular matrix (ECM) deposition. Marked interest in system-scale analysis of immune species interactions within the TME has encouraged progress in modeling tumor-immune interactions in silico. Due to the computational cost of simulating these intricate interactions, models have typically been constrained to representing a limited number of immune species. To expand the capability for system-scale analysis, this study develops a three-dimensional continuum mixture model of tumor-immune interactions to simulate multiple immune species in the TME. Building upon a recent distributed computing implementation that enables efficient solution of such mixture models, major immune species including monocytes, macrophages, natural killer cells, dendritic cells, neutrophils, myeloid-derived suppressor cells (MDSC), cytotoxic, helper, regulatory T-cells, and effector and regulatory B-cells and their interactions are represented in this novel implementation. Immune species extravasate from blood vasculature, undergo chemotaxis toward regions of high chemokine concentration, and influence the TME in proportion to locally defined levels of stimulation. The immune species contribute to the production of angiogenic and tumor growth factors, promotion of myofibroblast deposition of ECM, upregulation of angiogenesis, and elimination of living and dead tumor species. The results show that this modeling approach offers the capability for quantitative insight into the modulation of tumor growth by diverse immune-tumor interactions and immune-driven TME effects. In particular, MDSC-mediated effects on tumor-associated immune species' activation levels, volume fraction, and influence on the TME are explored. Longer term, linking of the model parameters to particular patient tumor information could simulate cancer-specific immune responses and move toward a more comprehensive evaluation of immunotherapeutic strategies.

An Explicit Adaptive Finite Difference Method for the Cahn-Hilliard Equation

In this study, we propose an explicit adaptive finite difference method (FDM) for the Cahn-Hilliard (CH) equation which describes the process of phase separation. The CH equation has been successfully utilized to model and simulate diverse field applications such as complex interfacial fluid flows and materials science. To numerically solve the CH equation fast and efficiently, we use the FDM and time-adaptive narrow-band domain. For the adaptive grid, we define a narrow-band domain including the interfacial transition layer of the phase field based on an undivided finite difference and solve the numerical scheme on the narrow-band domain. The proposed numerical scheme is based on an alternating direction explicit (ADE) method. To make the scheme conservative, we apply a mass correction algorithm after each temporal iteration step. To demonstrate the superior performance of the proposed adaptive FDM for the CH equation, we present two- and three-dimensional numerical experiments and compare them with those of other previous methods.

Simulation of 3D centimeter-scale continuum tumor growth at sub-millimeter resolution via distributed computing

Simulation of cm-scale tumor growth has generally been constrained by the computational cost to numerically solve the associated equations, with models limited to representing mm-scale or smaller tumors. While the work has proven useful to the study of small tumors and micro-metastases, a biologically-relevant simulation of cm-scale masses as would be typically detected and treated in patients has remained an elusive goal. This study presents a distributed computing (parallelized) implementation of a mixture model of tumor growth to simulate 3D cm-scale vascularized tissue at sub-mm resolution. The numerical solving scheme utilizes a two-stage parallelization framework. The solution is written for GPU computation using the CUDA framework, which handles all Multigrid-related computations. Message Passing Interface (MPI) handles distribution of information across multiple processes, freeing the program from RAM and the processing limitations found on single systems. On each system, Nvidia's CUDA library allows for fast processing of model data using GPU-bound computing on fewer systems. The results show that a combined MPI-CUDA implementation enables the continuum modeling of cm-scale tumors at reasonable computational cost. Further work to calibrate model parameters to particular tumor conditions could enable simulation of patient-specific tumors for clinical application.

Indirect Contributions to Tumor Dynamics in the First Stage of the Avascular Phase

A continuum model for tumor invasion in a two-dimensional spatial domain based on the interaction of the urokinase plasminogen activation system with a model for cancer cell dynamics is proposed. The arising system of partial differential equations is numerically solved using the finite element method. We simulated a portion of biological tissue imposing no flux boundary conditions. We monitored the cancer cell dynamics, as well the degradation of an extra cellular matrix representative, vitronectin, and the evolution of a specific degrading enzyme, plasmin, inside the biological tissue. The computations were parameterized as a function of the indirect cell proliferation induced by a plasminogen activator inhibitor binding to vitronectin and of the indirect plasmin deactivation due to the plasminogen activator inhibitor binding to the urokinase plasminogen activator. Their role during the cancer dynamical evolution was identified, together with a possible marker helping the mapping of the cancer invasive front. Our results indicate that indirect cancer cell proliferation biases the speed of the tumor invasive front as well as the heterogeneity of the cancer cell clustering and networking, as it ultimately acts on the proteolytic activity supporting cancer formation. Because of the initial conditions imposed, the numerical solutions of the model show a symmetrical dynamical evolution of heterogeneities inside the simulated domain. Moreover, an increase of up to about 12% in the invasion speed was observed, increasing the rate of indirect cancer cell proliferation, while increasing the plasmin deactivation rate inhibits heterogeneities and networking. As cancer cell proliferation causes vitronectin consumption and plasmin formation, the intensities of the concentration maps of both vitronectin and plasmin are superimposable to the cancer cell concentration maps. The qualitative imprinting that cancer cells leave on the extra cellular matrix during the time evolution as well their activity area is identified, framing the numerical results in the context of a methodology aimed at diagnostic and therapeutic improvement.

Generalized moving least squares approximation for the solution of local and non-local models of cancer cell invasion of tissue under the effect of adhesion in one- and two-dimensional spaces

The main aim of this study is to solve numerically the mathematical models showing cancer cell invasion of tissue with/without considering the effect of cell-cell and cell-matrix adhesion. The mathematical models studied here are the systems of time-dependent reaction-diffusion-taxis equations in one- and two-dimensional spaces, which are formulated in the local and non-local forms. There are some difficulties in finding their solutions via numerical methods. The main difficulty is to compute the non-local term appearing in one of the studied models, which causes more CPU time during simulations. The current paper aims to overcome this problem, where a new meshless method, namely generalized moving least squares (GMLS) approximation in space and a semi-implicit backward differential formula of first-order (SBDF1) in time have been applied. Based on GMLS theory, the non-local term is approximated without any difficulties. Moreover, a simple method based on the GMLS technique is presented to implement the boundary conditions. The obtained discrete scheme for both mathematical models is a linear system of algebraic equations per time step. The biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower-upper (ILU) preconditioner is used to solve the obtained linear system at each time step. At the end of this paper, some simulation results are reported to show the behavior of cancer cell invasion in the local model, and the non-local model due to reduction of cell-cell adhesion and increasing cell-matrix adhesion in one- and two-dimensional spaces, where two different types of distribution points have been considered in the square domain. The computational algorithms of the GMLS approximation and the developed numerical method for solving the non-local (local) model are included in the Appendix.

Tumor growth and calcification in evolving microenvironmental geometries

In this paper, we apply the diffuse domain framework developed in Chen and Lowengrub (Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach, J. Theor. Biol. 361 (2014) 14-30) to study the effects of a deformable basement membrane (BM) on the growth of a tumor in a confined, ductal geometry, such as ductal carcinoma in situ (DCIS). We use a continuum model of tumor microcalcification and investigate the tumor extent beyond the microcalcification. In order to solve the governing equations efficiently, we develop a stable nonlinear multigrid finite difference method. Two dimensional simulations are performed where the adhesion between tumor cells and the basement membrane is varied. Additional simulations considering the variation of duct radius and membrane stiffness are also conducted. The results demonstrate that enhanced membrane deformability promotes tumor growth and tumor calcification. When the duct radius is small, the cell-BM adhesion is weak or when the membrane is slightly deformed, the mammographic and pathologic tumor extents are linearly correlated, as predicted by Macklin et al. (J. Theor. Biol. 301 (2012) 122-140) using an agent-based model that does not account for the deformability of the basement membrane and the active forces that the membrane imparts on the tumor cells. Interestingly, we predict that when the duct radius is large, there is strong cell-BM adhesion or the membrane is highly deformed, the extents of the mammographic and pathologic tumors are instead quadratically correlated. The simulations can help surgeons to measure DCIS surgical margins while removing less non-cancerous tissue, and can improve targeting of intra- and post-operative radiotherapy.

A Visually Apparent and Quantifiable CT Imaging Feature Identifies Biophysical Subtypes of Pancreatic Ductal Adenocarcinoma

PURPOSE

Pancreatic ductal adenocarcinoma (PDAC) is a heterogeneous disease with variable presentations and natural histories of disease. We hypothesized that different morphologic characteristics of PDAC tumors on diagnostic computed tomography (CT) scans would reflect their underlying biology.

EXPERIMENTAL DESIGN

We developed a quantitative method to categorize the PDAC morphology on pretherapy CT scans from multiple datasets of patients with resectable and metastatic disease and correlated these patterns with clinical/pathologic measurements. We modeled macroscopic lesion growth computationally to test the effects of stroma on morphologic patterns, hypothesizing that the balance of proliferation and local migration rates of the cancer cells would determine tumor morphology.

RESULTS

In localized and metastatic PDAC, quantifying the change in enhancement on CT scans at the interface between tumor and parenchyma (delta) demonstrated that patients with conspicuous (high-delta) tumors had significantly less stroma, higher likelihood of multiple common pathway mutations, more mesenchymal features, higher likelihood of early distant metastasis, and shorter survival times compared with those with inconspicuous (low-delta) tumors. Pathologic measurements of stromal and mesenchymal features of the tumors supported the mathematical model's underlying theory for PDAC growth.

CONCLUSIONS

At baseline diagnosis, a visually striking and quantifiable CT imaging feature reflects the molecular and pathological heterogeneity of PDAC, and may be used to stratify patients into distinct subtypes. Moreover, growth patterns of PDAC may be described using physical principles, enabling new insights into diagnosis and treatment of this deadly disease.

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.

Three-Dimensional Spatiotemporal Modeling of Colon Cancer Organoids Reveals that Multimodal Control of Stem Cell Self-Renewal is a Critical Determinant of Size and Shape in Early Stages of Tumor Growth

We develop a three-dimensional multispecies mathematical model to simulate the growth of colon cancer organoids containing stem, progenitor and terminally differentiated cells, as a model of early (prevascular) tumor growth. Stem cells (SCs) secrete short-range self-renewal promoters (e.g., Wnt) and their long-range inhibitors (e.g., Dkk) and proliferate slowly. Committed progenitor (CP) cells proliferate more rapidly and differentiate to produce post-mitotic terminally differentiated cells that release differentiation promoters, forming negative feedback loops on SC and CP self-renewal. We demonstrate that SCs play a central role in normal and cancer colon organoids. Spatial patterning of the SC self-renewal promoter gives rise to SC clusters, which mimic stem cell niches, around the organoid surface, and drive the development of invasive fingers. We also study the effects of externally applied signaling factors. Applying bone morphogenic proteins, which inhibit SC and CP self-renewal, reduces invasiveness and organoid size. Applying hepatocyte growth factor, which enhances SC self-renewal, produces larger sizes and enhances finger development at low concentrations but suppresses fingers at high concentrations. These results are consistent with recent experiments on colon organoids. Because many cancers are hierarchically organized and are subject to feedback regulation similar to that in normal tissues, our results suggest that in cancer, control of cancer stem cell self-renewal should influence the size and shape in similar ways, thereby opening the door to novel therapies.

Activation of the HGF/c-Met axis in the tumor microenvironment: A multispecies model

The tumor microenvironment is an integral component in promoting tumor development. Cancer-associated fibroblasts (CAFs), which reside in the tumor stroma, produce Hepatocyte Growth Factor (HGF), an important trigger for invasive and metastatic tumor behavior. HGF contributes to a pro-tumorigenic environment by activating its cognate receptor, c-Met, on tumor cells. Tumor cells, in turn, secrete growth factors that upregulate HGF production in CAFs, thereby establishing a dynamic tumor-host signaling program. Using a spatiotemporal multispecies model of tumor growth, we investigate how the development and spread of a tumor is impacted by the initiation of a dynamic interaction between tumor-derived growth factors and CAF-derived HGF. We show that establishment of such an interaction results in increased tumor growth and morphological instability, the latter due in part to increased cell species heterogeneity at the tumor-host boundary. Invasive behavior is further increased if the tumor lowers responsiveness to paracrine pro-differentiation signals, which is a hallmark of neoplastic development. By modeling anti-HGF and anti-c-Met therapy, we show how disruption of the HGF/c-Met axis can reduce tumor invasiveness and growth, thereby providing theoretical evidence that targeting tumor-microenvironment interactions is a promising avenue for therapeutic development.

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.

Progress Towards Computational 3-D Multicellular Systems Biology

Tumors cannot be understood in isolation from their microenvironment. Tumor and stromal cells change phenotype based upon biochemical and biophysical inputs from their surroundings, even as they interact with and remodel the microenvironment. Cancer should be investigated as an adaptive, multicellular system in a dynamical microenvironment. Computational modeling offers the potential to detangle this complex system, but the modeling platform must ideally account for tumor heterogeneity, substrate and signaling factor biotransport, cell and tissue biophysics, tissue and vascular remodeling, microvascular and interstitial flow, and links between all these sub-systems. Such a platform should leverage high-throughput experimental data, while using open data standards for reproducibility. In this chapter, we review advances by our groups in these key areas, particularly in advanced models of tissue mechanics and interstitial flow, open source simulation software, high-throughput phenotypic screening, and multicellular data standards. In the future, we expect a transformation of computational cancer biology from individual groups modeling isolated parts of cancer, to coalitions of groups combining compatible tools to simulate the 3-D multicellular systems biology of cancer tissues.

Multiscale Modeling of Glioblastoma Suggests that the Partial Disruption of Vessel/Cancer Stem Cell Crosstalk Can Promote Tumor Regression Without Increasing Invasiveness

OBJECTIVE

In glioblastoma, the crosstalk between vascular endothelial cells (VECs) and glioma stem cells (GSCs) has been shown to enhance tumor growth. We propose a multiscale mathematical model to study this mechanism, explore tumor growth under various initial and microenvironmental conditions, and investigate the effects of blocking this crosstalk.

METHODS

We develop a hybrid continuum-discrete model of highly organized vascularized tumors. VEC-GSC crosstalk is modeled via vascular endothelial growth factor (VEGF) production by tumor cells and by secretion of soluble factors by VECs that promote GSC self-renewal and proliferation.

RESULTS

VEC-GSC crosstalk increases both tumor size and GSC fraction by enhancing GSC activity and neovascular development. VEGF promotes vessel formation, and larger VEGF sources typically increase vessel numbers, which enhances tumor growth and stabilizes the tumor shape. Increasing the initial GSC fraction has a similar effect. Partially disrupting the crosstalk by blocking VEC secretion of GSC promoters reduces tumor size but does not increase invasiveness, which is in contrast to antiangiogenic therapies, which reduce tumor size but may significantly increase tumor invasiveness.

SIGNIFICANCE

Multiscale modeling supports the targeting of VEC-GSC crosstalk as a promising approach for cancer therapy.

Feedback, Lineages and Self-Organizing Morphogenesis

Feedback regulation of cell lineage progression plays an important role in tissue size homeostasis, but whether such feedback also plays an important role in tissue morphogenesis has yet to be explored. Here we use mathematical modeling to show that a particular feedback architecture in which both positive and negative diffusible signals act on stem and/or progenitor cells leads to the appearance of bistable or bi-modal growth behaviors, ultrasensitivity to external growth cues, local growth-driven budding, self-sustaining elongation, and the triggering of self-organization in the form of lamellar fingers. Such behaviors arise not through regulation of cell cycle speeds, but through the control of stem or progenitor self-renewal. Even though the spatial patterns that arise in this setting are the result of interactions between diffusible factors with antagonistic effects, morphogenesis is not the consequence of Turing-type instabilities.

Mathematical model and its fast numerical method for the tumor growth

In this paper, we reformulate the diffuse interface model of the tumor growth (S.M. Wise et al., Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524--543). In the new proposed model, we use the conservative second-order Allen--Cahn equation with a space--time dependent Lagrange multiplier instead of using the fourth-order Cahn--Hilliard equation in the original model. To numerically solve the new model, we apply a recently developed hybrid numerical method. We perform various numerical experiments. The computational results demonstrate that the new model is not only fast but also has a good feature such as distributing excess mass from the inside of tumor to its boundary regions.

Strategies for efficient numerical implementation of hybrid multi-scale agent-based models to describe biological systems

Biologically related processes operate across multiple spatiotemporal scales. For computational modeling methodologies to mimic this biological complexity, individual scale models must be linked in ways that allow for dynamic exchange of information across scales. A powerful methodology is to combine a discrete modeling approach, agent-based models (ABMs), with continuum models to form hybrid models. Hybrid multi-scale ABMs have been used to simulate emergent responses of biological systems. Here, we review two aspects of hybrid multi-scale ABMs: linking individual scale models and efficiently solving the resulting model. We discuss the computational choices associated with aspects of linking individual scale models while simultaneously maintaining model tractability. We demonstrate implementations of existing numerical methods in the context of hybrid multi-scale ABMs. Using an example model describing Mycobacterium tuberculosis infection, we show relative computational speeds of various combinations of numerical methods. Efficient linking and solution of hybrid multi-scale ABMs is key to model portability, modularity, and their use in understanding biological phenomena at a systems level.

Tumor growth in complex, evolving microenvironmental geometries: a diffuse domain approach

We develop a mathematical model of tumor growth in complex, dynamic microenvironments with active, deformable membranes. Using a diffuse domain approach, the complex domain is captured implicitly using an auxiliary function and the governing equations are appropriately modified, extended and solved in a larger, regular domain. The diffuse domain method enables us to develop an efficient numerical implementation that does not depend on the space dimension or the microenvironmental geometry. We model homotypic cell-cell adhesion and heterotypic cell-basement membrane (BM) adhesion with the latter being implemented via a membrane energy that models cell-BM interactions. We incorporate simple models of elastic forces and the degradation of the BM and ECM by tumor-secreted matrix degrading enzymes. We investigate tumor progression and BM response as a function of cell-BM adhesion and the stiffness of the BM. We find tumor sizes tend to be positively correlated with cell-BM adhesion since increasing cell-BM adhesion results in thinner, more elongated tumors. Prior to invasion of the tumor into the stroma, we find a negative correlation between tumor size and BM stiffness as the elastic restoring forces tend to inhibit tumor growth. In order to model tumor invasion of the stroma, we find it necessary to downregulate cell-BM adhesiveness, which is consistent with experimental observations. A stiff BM promotes invasiveness because at early stages the opening in the BM created by MDE degradation from tumor cells tends to be narrower when the BM is stiffer. This requires invading cells to squeeze through the narrow opening and thus promotes fragmentation that then leads to enhanced growth and invasion. In three dimensions, the opening in the BM was found to increase in size even when the BM is stiff because of pressure induced by growing tumor clusters. A larger opening in the BM can increase the potential for further invasiveness by increasing the possibility that additional tumor cells could invade the stroma.

A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane

In this paper, we extend the 3D multispecies diffuse-interface model of the tumor growth, which was derived in Wise et al. (Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524-543), and incorporate the effect of a stiff membrane to model tumor growth in a confined microenvironment. We then develop accurate and efficient numerical methods to solve the model. When the membrane is endowed with a surface energy, the model is variational, and the numerical scheme, which involves adaptive mesh refinement and a nonlinear multigrid finite difference method, is demonstrably shown to be energy stable. Namely, in the absence of cell proliferation and death, the discrete energy is a nonincreasing function of time for any time and space steps. When a simplified model of membrane elastic energy is used, the resulting model is derived analogously to the surface energy case. However, the elastic energy model is actually nonvariational because certain coupling terms are neglected. Nevertheless, a very stable numerical scheme is developed following the strategy used in the surface energy case. 2D and 3D simulations are performed that demonstrate the accuracy of the algorithm and illustrate the shape instabilities and nonlinear effects of membrane elastic forces that may resist or enhance growth of the tumor. Compared with the standard Crank-Nicholson method, the time step can be up to 25 times larger using the new approach.

A multiple time-scale computational model of a tumor and its micro environment

Experimental evidence suggests that a tumor's environment may be critical to designing successful therapeutic protocols: Modeling interactions between a tumor and its environment could improve our understanding of tumor growth and inform approaches to treatment. This paper describes an efficient, flexible, hybrid cellular automaton-based implementation of numerical solutions to multiple time-scale reaction-diffusion equations, applied to a model of tumor proliferation. The growth and maintenance of cells in our simulation depend on the rate of cellular energy (ATP) metabolized from nearby nutrients such as glucose and oxygen. Nutrient consumption rates are functions of local pH as well as local concentrations of oxygen and other fuels. The diffusion of these nutrients is modeled using a novel variation of random-walk techniques. Furthermore, we detail the effects of three boundary update rules on simulations, describing their effects on computational efficiency and biological realism. Qualitative and quantitative results from simulations provide insight on how tumor growth is affected by various environmental changes such as micro-vessel density or lower pH, both of high interest in current cancer research.

Multispecies model of cell lineages and feedback control in solid tumors

We develop a multispecies continuum model to simulate the spatiotemporal dynamics of cell lineages in solid tumors. The model accounts for protein signaling factors produced by cells in lineages, and nutrients supplied by the microenvironment. Together, these regulate the rates of proliferation, self-renewal and differentiation of cells within the lineages, and control cell population sizes and distributions. Terminally differentiated cells release proteins (e.g., from the TGFβ superfamily) that feedback upon less differentiated cells in the lineage both to promote differentiation and decrease rates of proliferation (and self-renewal). Stem cells release a short-range factor that promotes self-renewal (e.g., representative of Wnt signaling factors), as well as a long-range inhibitor of this factor (e.g., representative of Wnt inhibitors such as Dkk and SFRPs). We find that the progression of the tumors and their response to treatment is controlled by the spatiotemporal dynamics of the signaling processes. The model predicts the development of spatiotemporal heterogeneous distributions of the feedback factors (Wnt, Dkk and TGFβ) and tumor cell populations with clusters of stem cells appearing at the tumor boundary, consistent with recent experiments. The nonlinear coupling between the heterogeneous expressions of growth factors and the heterogeneous distributions of cell populations at different lineage stages tends to create asymmetry in tumor shape that may sufficiently alter otherwise homeostatic feedback so as to favor escape from growth control. This occurs in a setting of invasive fingering, and enhanced aggressiveness after standard therapeutic interventions. We find, however, that combination therapy involving differentiation promoters and radiotherapy is very effective in eradicating such a tumor.

Numerical simulation of a thermodynamically consistent four-species tumor growth model

In this paper, we develop a thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures. Unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models. A mixed finite element spatial discretization is developed and implemented to provide numerical results demonstrating the range of solutions this model can produce. A time-stepping algorithm is then presented for this system, which is shown to be first order accurate and energy gradient stable. The results of an array of numerical experiments are presented, which demonstrate a wide range of solutions produced by various choices of model parameters.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Three-dimensional multispecies nonlinear tumor growth--I Model and numerical method

This is the first paper in a two-part series in which we develop, analyze, and simulate a diffuse interface continuum model of multispecies tumor growth and tumor-induced angiogenesis in two and three dimensions. Three-dimensional simulations of nonlinear tumor growth and neovascularization using this diffuse interface model were recently presented in Frieboes et al. [2007. Computer simulation of glioma growth and morphology. NeuroImage S59-S70], but that paper did not describe the details of the model or the numerical algorithm. This is done here. In this diffuse interface approach, sharp interfaces are replaced by narrow transition layers that arise due to differential adhesive forces among the cell species. Accordingly, a continuum model of adhesion is introduced. The model is thermodynamically consistent, is related to recently developed mixture models, and thus is capable of providing a detailed description of tumor progression. The model is well-posed and consists of fourth-order nonlinear advection-reaction-diffusion equations (of Cahn-Hilliard-type) for the cell species coupled with reaction-diffusion equations for the substrate components. We demonstrate analytically and numerically that when the diffuse interface thickness tends to zero, the system reduces to a classical sharp interface model. Using a new fully adaptive and nonlinear multigrid/finite difference method, the system is simulated efficiently. In this first paper, we present simulations of unstable avascular tumor growth in two and three dimensions and demonstrate that our techniques now make large-scale three-dimensional simulations of tumors with complex morphologies computationally feasible. In part II of this study, we will investigate multispecies tumor invasion, tumor-induced angiogenesis, and focus on the morphological instabilities that may underlie invasive phenotypes.