From passive diffusion to active cellular migration in mathematical models of tumour invasion

Modeling the extracellular matrix in cell migration and morphogenesis: a guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.

Semi-autonomous wound invasion via matrix-deposited, haptotactic cues

Proper wound healing relies on invasion of fibroblasts via directed migration. While the related experimental and mathematical modeling literature has mainly focused on cell migration directed by soluble cues (chemotaxis), there is ample evidence that fibroblast migration is also directed by insoluble, matrix-bound cues (haptotaxis). Furthermore, numerous studies indicate that fibronectin (FN), a haptotactic ligand for fibroblasts, is present and dynamic in the provisional matrix throughout the proliferative phase of wound healing. In the present work, we show the plausibility of a hypothesis that fibroblasts themselves form and maintain haptotactic gradients in a semi-autonomous fashion. As a precursor to this, we examine the positive control scenario where FN is pre-deposited in the wound matrix, and fibroblasts maintain haptotaxis by removing FN at an appropriate rate. After developing conceptual and quantitative understanding of this scenario, we consider two cases in which fibroblasts activate the latent form of a matrix-loaded cytokine, TGFβ, which upregulates the fibroblasts' own secretion of FN. In the first of these, the latent cytokine is pre-patterned and released by the fibroblasts. In the second, fibroblasts in the wound produce the latent TGFβ, with the presence of the wound providing the only instruction. In all cases, wound invasion is more effective than a negative control model with haptotaxis disabled; however, there is a trade-off between the degree of fibroblast autonomy and the rate of invasion.

A stochastic hierarchical model for low grade glioma evolution

A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using a piecewise diffusion Markov process (PDifMP) at the cellular level, we derive an equation for the density of the transition probability of this Markov process based on the generalised Fokker-Planck equation. Then, a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells and to the onset of malignancy, i.e., the transition from low-grade to high-grade gliomas.

Modeling of glioma growth using modified reaction-diffusion equation on brain MR images

BACKGROUND AND OBJECTIVE

Modeling of glioma growth and evolution is of key importance for cancer diagnosis, predicting clinical progression and improving treatment outcomes of neurosurgery. However, existing models are unable to characterize spatial variations of the proliferation and infiltration of tumor cells, making it difficult to achieve accurate prediction of tumor growth.

METHODS

In this paper, a new growth model of brain tumor using a reaction-diffusion equation on brain magnetic resonance images is proposed. Both the heterogeneity of brain tissue and the density of tumor cells are used to estimate the proliferation and diffusion coefficients of brain tumor cells. The diffusion coefficient that characterizes tumor diffusion and infiltration is calculated based on the ratio of tissues (white and gray matter), while the proliferation coefficient is evaluated using the spatial gradient of tumor cells. In addition, a parameter space is constructed using inverse distance weighted interpolation to describe the spatial distribution of proliferation coefficient.

RESULTS

The glioma growth predicted by the proposed model were tested by comparing with the real magnetic resonance images of the patients. Experiments and simulation results show that the proposed method achieves accurate modeling of glioma growth. The interpolation-based growth model has higher average dice score of 0.0647 and 0.0545, and higher average Jaccard index of 0.0673 and 0.0573, respectively, compared to the uniform- and gradient-based growth models.

CONCLUSIONS

The experimental results demonstrate the feasibility of calculating the proliferation and diffusion coefficients of the growth model based on patient-specific anatomy. The parameter space that characterizes spatial distribution of proliferation and diffusion coefficients is established and incorporated into the simulation of glioma growth. It enables to obtain patient-specific models about glioma growth by estimating and calibrating only a few model parameters.

Using mathematics in MRI data management for glioma assesment

Our aim is to review the mathematical tools usefulness in MR data management for glioma diagnosis and treatment optimization. MRI does not give access to organs variations in hours or days. However a lot of multiparametric data are generated. Mathematics could help to override this paradox, the aim of this article is to show how. We first make a review on mathematical modelling using equations. Afterwards we present statistical analysis. We provide detailed examples in both sections. We finally conclude, giving some clues on in silico models.

A Coupled Modified Reaction Diffusion and Biomechanical Models for Cerebral Tumor Growth Modeling in Presence of Treatment

Tumor growth modeling at macroscopic level from multimodal images can help in predicting the future evolution of tumor and the treatment planning. This can be achieved using mathematical models where multi time-point images are available. In this paper, we propose a coupled modified reaction diffusion model that measures tumor invasion and infiltration with biomechanical model to consider tumor mass effect. In addition, our model considers treatment effects from radiotherapy and/or chemotherapy if any. The chemotherapy effect is included via a modified log-kill method to consider tissue heterogeneity while radiotherapy effect is considered using the linear quadratic model. We test the proposed model on both synthetic and 6 real datasets of low grade glioma cases with and without treatments. Experimental results of the proposed model on the clinical magnetic resonance images show that our model can simulate the tumor growth with good accuracy and effectively include the treatment effects.

Comparative study between spatio-temporal models for brain tumor growth

Modeling of brain tumor growth simulator can estimate life expectancy for individual patients, estimate future effect of brain damages toward human senses and attitude and help in evaluating the efficiency of applied treatments. Brain tumor growth can be calculated based on Spatio-Temporal mathematical models namely the isotropic reaction-diffusion model and the anisotropic reaction-diffusion model where the second model produces more realistic results. Tumor normally grows in White Matter (WM) five times faster than in Gray Matter (GM) which makes brain tissues modeled as inhomogeneous-anisotropic material to assign different parameters to each tissue. In this research a comparative study between several tumor growth models has been achieved to clarify the effect of different algorithms on modeling tumor grow.

Low grade glioma growth modeling considering chemotherapy and radiotherapy effects from magnetic resonance images

Studying tumor growth using mathematical models from magnetic resonance (MR) images is an important application that is believed to play a major role in cancer treatment by predicting tumor evolution, quantifying the response to therapy, and treatment planning. Reaction diffusion is the most popular model because of its simplicity and consistency with the biological growth process. However, most of the current growth models focus on presurgical images and likely without treatment. In this paper, we propose a new reaction diffusion model to consider the chemotherapy and radiotherapy effects on the tumor growth modelling for patients with low grade glioma. The proposed model does not consider the tensor information from diffusion tensor imaging. Instead it uses a weighted parameter to promote higher diffusivity in white matter. The radiotherapy and chemotherapy effects are considered as a loss terms in the proposed model. The preliminary results of the proposed model on synthetic and 2 real MR images show that, our model can effectively simulate tumor growth with high accuracies when treatments are administrated to low grade glioma patients.

Post-Surgery Glioma Growth Modeling from Magnetic Resonance Images for Patients with Treatment

Reaction diffusion is the most common growth modelling methodology due to its simplicity and consistency with the biological tumor growth process. However, current extensions of the reaction diffusion model lack one or more of the following: efficient inclusion of treatments' effects, taking into account the viscoelasticity of brain tissues, and guaranteed stability of the numerical solution. We propose a new model to overcome the aforementioned drawbacks. Guided by directional information derived from diffusion tensor imaging, our model relates tissue heterogeneity with the absorption of the chemotherapy, adopts the linear-quadratic term to simulate the radiotherapy effect, employs Maxwell-Weichert model to incorporate brain viscoelasticity, and ensures the stability of the numerical solution. The performance is verified through experiments on synthetic and real MR images. Experiments on 9 MR datasets of patients with low grade gliomas undergoing surgery with different treatment regimens are carried out and validated using Jaccard score and Dice coefficient. The growth simulation accuracies of the proposed model are in ranges of [0.673 0.822] and [0.805 0.902] for Jaccard scores and Dice coefficients, respectively. The accuracies decrease up to 4% and 2.4% when ignoring treatment effects and the tensor information, while brain viscoelasticity has no significant impact on the accuracies.

Towards the Personalized Treatment of Glioblastoma: Integrating Patient-Specific Clinical Data in a Continuous Mechanical Model

Glioblastoma multiforme (GBM) is the most aggressive and malignant among brain tumors. In addition to uncontrolled proliferation and genetic instability, GBM is characterized by a diffuse infiltration, developing long protrusions that penetrate deeply along the fibers of the white matter. These features, combined with the underestimation of the invading GBM area by available imaging techniques, make a definitive treatment of GBM particularly difficult. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of GBM evolution in every single patient throughout his/her oncological history, in order to target therapeutic weapons in a patient-specific manner. In this work, we propose a continuous mechanical model and we perform numerical simulations of GBM invasion combining the main mechano-biological characteristics of GBM with the micro-structural information extracted from radiological images, i.e. by elaborating patient-specific Diffusion Tensor Imaging (DTI) data. The numerical simulations highlight the influence of the different biological parameters on tumor progression and they demonstrate the fundamental importance of including anisotropic and heterogeneous patient-specific DTI data in order to obtain a more accurate prediction of GBM evolution. The results of the proposed mathematical model have the potential to provide a relevant benefit for clinicians involved in the treatment of this particularly aggressive disease and, more importantly, they might drive progress towards improving tumor control and patient’s prognosis.

An agent-based model approach to multi-phase life-cycle for contact inhibited, anchorage dependent cells

Cellular agent-based models are a technique that can be easily adapted to describe nuances of a particular cell type. Within we have concentrated on the cellular particularities of the human Endothelial Cell, explicitly the effects both of anchorage dependency and of heightened scaffold binding on the total confluence time of a system. By expansion of a discrete, homogeneous, asynchronous cellular model to account for several states per cell (phases within a cell's life); we accommodate and track dependencies of confluence time and population dynamics on these factors. Increasing the total motility time, analogous to weakening the binding between lattice and cell, affects the system in unique ways from increasing the average cellular velocity; each degree of freedom allows for control over the time length the system achieves logistic growth and confluence. These additional factors may allow for greater control over behaviors of the system. Examinations of system's dependence on both seed state velocity and binding are also enclosed.

Glioma follow white matter tracts: a multiscale DTI-based model

Gliomas are a class of rarely curable tumors arising from abnormal glia cells in the human brain. The understanding of glioma spreading patterns is essential for both radiological therapy as well as surgical treatment. Diffusion tensor imaging (DTI) allows to infer the white matter fibre structure of the brain in a noninvasive way. Painter and Hillen (J Theor Biol 323:25-39, 2013) used a kinetic partial differential equation to include DTI data into a class of anisotropic diffusion models for glioma spread. Here we extend this model to explicitly include adhesion mechanisms between glioma cells and the extracellular matrix components which are associated to white matter tracts. The mathematical modelling follows the multiscale approach proposed by Kelkel and Surulescu (Math Models Methods Appl Sci 23(3), 2012). We use scaling arguments to deduce a macroscopic advection-diffusion model for this process. The tumor diffusion tensor and the tumor drift velocity depend on both, the directions of the white matter tracts as well as the binding dynamics of the adhesion molecules. The advanced computational platform DUNE enables us to accurately solve our macroscopic model. It turns out that the inclusion of cell binding dynamics on the microlevel is an important factor to explain finger-like spread of glioma.

Simulating radiotherapy effect in high-grade glioma by using diffusive modeling and brain atlases

Applying diffusive models for simulating the spatiotemporal change of concentration of tumour cells is a modern application of predictive oncology. Diffusive models are used for modelling glioblastoma, the most aggressive type of glioma. This paper presents the results of applying a linear quadratic model for simulating the effects of radiotherapy on an advanced diffusive glioma model. This diffusive model takes into consideration the heterogeneous velocity of glioma in gray and white matter and the anisotropic migration of tumor cells, which is facilitated along white fibers. This work uses normal brain atlases for extracting the proportions of white and gray matter and the diffusion tensors used for anisotropy. The paper also presents the results of applying this glioma model on real clinical datasets.

In silico modelling of tumour margin diffusion and infiltration: review of current status

As a result of advanced treatment techniques, requiring precise target definitions, a need for more accurate delineation of the Clinical Target Volume (CTV) has arisen. Mathematical modelling is found to be a powerful tool to provide fairly accurate predictions for the Microscopic Extension (ME) of a tumour to be incorporated in a CTV. In general terms, biomathematical models based on a sequence of observations or development of a hypothesis assume some links between biological mechanisms involved in cancer development and progression to provide quantitative or qualitative measures of tumour behaviour as well as tumour response to treatment. Generally, two approaches are taken: deterministic and stochastic modelling. In this paper, recent mathematical models, including deterministic and stochastic methods, are reviewed and critically compared. It is concluded that stochastic models are more promising to provide a realistic description of cancer tumour behaviour due to being intrinsically probabilistic as well as discrete, which enables incorporation of patient-specific biomedical data such as tumour heterogeneity and anatomical boundaries.

In-depth analysis and evaluation of diffusive glioma models

Glioma is one of the most aggressive types of brain tumor. Several mathematical models have been developed during the past two decades, toward simulating the mechanisms that govern the development of glioma. The most common models use the diffusion-reaction equation (DRE) for simulating the spatiotemporal variation of tumor cell concentration. Nevertheless, despite the applications presented, there has been little work on studying the details of the mathematical solution and implementation of the 3-D diffusion model and presenting a qualitative analysis of the algorithmic results. This paper presents a complete mathematical framework on the solution of the DRE using different numerical schemes. This framework takes into account all characteristics of the latest models, such as brain tissue heterogeneity, anisotropic tumor cell migration, chemotherapy, and resection modeling. The different numerical schemes presented have been evaluated based upon the degree to which the DRE exact solution is approximated. Experiments have been conducted both on real datasets and a test case for which there is a known algebraic expression of the solution. Thus, it is possible to calculate the accuracy of the different models.

Historical and conceptual background of self-organization by reactive processes

Order, form, pattern and organization are properties central to much living matter. The physicochemical processes by which an initially homogeneous solution of reacting chemicals or biochemicals might self-organize is hence a question of fundamental biological importance. In most cases, solutions of reacting chemicals in a test-tube do not self-organize. Because of this, for many years, it was not thought possible that reactive processes could result in self-organization. However, progressively over the last hundred years, it has been shown that this is not always the case, and under certain conditions, the combination of reaction with molecular diffusion can lead to macroscopic self-organization. In 'complex' systems comprised of populations of strongly coupled elements, new 'emergent' properties, such as self-organization, arise by way of the dynamics of the system. Self-organizing reaction-diffusion systems form a specific type of complex system. Here, I will give a personal overview of the conceptual and historical background to this approach with an emphasis on biological self-organization.

High-grade glioma diffusive modeling using statistical tissue information and diffusion tensors extracted from atlases

Glioma, especially glioblastoma, is a leading cause of brain cancer fatality involving highly invasive and neoplastic growth. Diffusive models of glioma growth use variations of the diffusion-reaction equation in order to simulate the invasive patterns of glioma cells by approximating the spatiotemporal change of glioma cell concentration. The most advanced diffusive models take into consideration the heterogeneous velocity of glioma in gray and white matter, by using two different discrete diffusion coefficients in these areas. Moreover, by using diffusion tensor imaging (DTI), they simulate the anisotropic migration of glioma cells, which is facilitated along white fibers, assuming diffusion tensors with different diffusion coefficients along each candidate direction of growth. Our study extends this concept by fully exploiting the proportions of white and gray matter extracted by normal brain atlases, rather than discretizing diffusion coefficients. Moreover, the proportions of white and gray matter, as well as the diffusion tensors, are extracted by the respective atlases; thus, no DTI processing is needed. Finally, we applied this novel glioma growth model on real data and the results indicate that prognostication rates can be improved.

Density-dependent quiescence in glioma invasion: instability in a simple reaction-diffusion model for the migration/proliferation dichotomy

Gliomas are very aggressive brain tumours, in which tumour cells gain the ability to penetrate the surrounding normal tissue. The invasion mechanisms of this type of tumour remain to be elucidated. Our work is motivated by the migration/proliferation dichotomy (go-or-grow) hypothesis, i.e. the antagonistic migratory and proliferating cellular behaviours in a cell population, which may play a central role in these tumours. In this paper, we formulate a simple go-or-grow model to investigate the dynamics of a population of glioma cells for which the switch from a migratory to a proliferating phenotype (and vice versa) depends on the local cell density. The model consists of two reaction-diffusion equations describing cell migration, proliferation and a phenotypic switch. We use a combination of numerical and analytical techniques to characterize the development of spatio-temporal instabilities and travelling wave solutions generated by our model. We demonstrate that the density-dependent go-or-grow mechanism can produce complex dynamics similar to those associated with tumour heterogeneity and invasion.

Comparing finite elements and finite differences for developing diffusive models of glioma growth

Glioma is the most aggressive type of brain tumor. Several mathematical models have been developed during the last two decades, towards simulating the mechanisms that govern the development of glioma. The most common models use the diffusion-reaction equation (DRE) for simulating the spatiotemporal variation of tumor cell concentration. The proposed diffusive models have mainly used finite differences (FDs) or finite elements (FEs) for the approximation of the solution of the partial differential DRE. This paper presents experimental results on the comparison of the FEs and FDs, especially focused on the glioma model case. It is studied how the different meshes of brain can affect computational consistency, simulation time and efficiency of the model. The experiments have been studied on a test case, for which there is a known algebraic expression of the solution. Thus, it is possible to calculate the error that the different models yield.

In-silico oncology: an approximate model of brain tumor mass effect based on directly manipulated free form deformation

PURPOSE

The present work introduces a novel method for approximating mass effect of primary brain tumors.

METHODS

The spatio-temporal dynamics of cancerous cells are modeled by means of a deterministic reaction-diffusion equation. Diffusion tensor information obtained from a probabilistic diffusion tensor imaging atlas is incorporated into the model to simulate anisotropic diffusion of cancerous cells. To account for the expansive nature of the tumor, the computed net cell density of malignant cells is linked to a parametric deformation model. This mass effect model is based on the so-called directly manipulated free form deformation. Spatial correspondence between two successive simulation steps is established by tracking landmarks, which are attached to the boundary of the gross tumor volume. The movement of these landmarks is used to compute the new configuration of the control points and, hence, determines the resulting deformation. To prevent a deformation of rigid structures (i.e. the skull), fixed shielding landmarks are introduced. In a refinement step, an adaptive landmark scheme ensures a dense sampling of the tumor isosurface, which in turn allows for an appropriate representation of the tumor shape.

RESULTS

The influence of different parameters on the model is demonstrated by a set of simulations. Additionally, simulation results are qualitatively compared to an exemplary set of clinical magnetic resonance images of patients diagnosed with high-grade glioma.

CONCLUSIONS

Careful visual inspection of the results demonstrates the potential of the implemented model and provides first evidence that the computed approximation of tumor mass effect is sensible. The shape of diffusive brain tumors (glioblastoma multiforme) can be recovered and approximately matches the observations in real clinical data.

Front instabilities and invasiveness of simulated 3D avascular tumors

We use the Glazier-Graner-Hogeweg model to simulate three-dimensional (3D), single-phenotype, avascular tumors growing in an homogeneous tissue matrix (TM) supplying a single limiting nutrient. We study the effects of two parameters on tumor morphology: a diffusion-limitation parameter defined as the ratio of the tumor-substrate consumption rate to the substrate-transport rate, and the tumor-TM surface tension. This initial model omits necrosis and oxidative/hypoxic metabolism effects, which can further influence tumor morphology, but our simplified model still shows significant parameter dependencies. The diffusion-limitation parameter determines whether the growing solid tumor develops a smooth (noninvasive) or fingered (invasive) interface, as in our earlier two-dimensional (2D) simulations. The sensitivity of 3D tumor morphology to tumor-TM surface tension increases with the size of the diffusion-limitation parameter, as in 2D. The 3D results are unexpectedly close to those in 2D. Our results therefore may justify using simpler 2D simulations of tumor growth, instead of more realistic but more computationally expensive 3D simulations. While geometrical artifacts mean that 2D sections of connected 3D tumors may be disconnected, the morphologies of 3D simulated tumors nevertheless correlate with the morphologies of their 2D sections, especially for low-surface-tension tumors, allowing the use of 2D sections to partially reconstruct medically-important 3D-tumor structures.

A complete mathematical study of a 3D model of heterogeneous and anisotropic glioma evolution

Glioma is the most aggressive type of brain cancer. Several mathematical models have been developed towards identifying the mechanism of tumor growth. The most successful models have used variations of the diffusion-reaction equation, with the recent ones taking into account brain tissue heterogeneity and anisotropy. However, to the best of our knowledge, there hasn't been any work studying in detail the mathematical solution and implementation of the 3D diffusion model, addressing related heterogeneity and anisotropy issues. To this end, this paper introduces a complete mathematical framework on how to derive the solution of the equation using different numerical approximation of finite differences. It indicates how different proliferation rate schemes can be incorporated in this solution and presents a comparative study of different numerical approaches.

Front instabilities and invasiveness of simulated avascular tumors

We study the interface morphology of a 2D simulation of an avascular tumor composed of identical cells growing in an homogeneous healthy tissue matrix (TM), in order to understand the origin of the morphological changes often observed during real tumor growth. We use the Glazier-Graner-Hogeweg model, which treats tumor cells as extended, deformable objects, to study the effects of two parameters: a dimensionless diffusion-limitation parameter defined as the ratio of the tumor consumption rate to the substrate transport rate, and the tumor-TM surface tension. We model TM as a nondiffusing field, neglecting the TM pressure and haptotactic repulsion acting on a real growing tumor; thus, our model is appropriate for studying tumors with highly motile cells, e.g., gliomas. We show that the diffusion-limitation parameter determines whether the growing tumor develops a smooth (noninvasive) or fingered (invasive) interface, and that the sensitivity of tumor morphology to tumor-TM surface tension increases with the size of the dimensionless diffusion-limitation parameter. For large diffusion-limitation parameters, we find a transition (missed in previous work) between dendritic structures, produced when tumor-TM surface tension is high, and seaweed-like structures, produced when tumor-TM surface tension is low. This observation leads to a direct analogy between the mathematics and dynamics of tumors and those observed in nonbiological directional solidification. Our results are also consistent with the biological observation that hypoxia promotes invasive growth of tumor cells by inducing higher levels of receptors for scatter factors that weaken cell-cell adhesion and increase cell motility. These findings suggest that tumor morphology may have value in predicting the efficiency of antiangiogenic therapy in individual patients.

Modeling tumor cell migration: From microscopic to macroscopic models

It has been shown experimentally that contact interactions may influence the migration of cancer cells. Previous works have modelized this thanks to stochastic, discrete models (cellular automata) at the cell level. However, for the study of the growth of real-size tumors with several million cells, it is best to use a macroscopic model having the form of a partial differential equation (PDE) for the density of cells. The difficulty is to predict the effect, at the macroscopic scale, of contact interactions that take place at the microscopic scale. To address this, we use a multiscale approach: starting from a very simple, yet experimentally validated, microscopic model of migration with contact interactions, we derive a macroscopic model. We show that a diffusion equation arises, as is often postulated in the field of glioma modeling, but it is nonlinear because of the interactions. We give the explicit dependence of diffusivity on the cell density and on a parameter governing cell-cell interactions. We discuss in detail the conditions of validity of the approximations used in the derivation, and we compare analytic results from our PDE to numerical simulations and to some in vitro experiments. We notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses, supercooled liquids, and jamming systems.

A model for short- and long-range interactions of migrating tumour cell

We examine the consequences of long-range effects on tumour cell migration. Our starting point are previous results of ours where we have shown that the migration patterns of glioma cells are best interpreted if one assumes attractive interactions between cells. Here we complement the cellular automaton model previously introduced by the assumption of the existence of a chemorepellent produced by the main bulk of large spheroids (in the hypoxic/necrotic areas). Visible effects due to the presence of such a substance can be found in the density profiles of cells migrating out of a single spheroid as well as in the angular distribution of cells coming from two close-lying spheroids. These effects depend crucially on the diffusion speed of the chemorepellent. A comparison of the simulation results to experimental data of Werbowetski et al. allows to draw (tentative) conclusions on the existence of a chemorepellent and its properties.

Biocomputing: numerical simulation of glioblastoma growth using diffusion tensor imaging

Glioblastoma multiforma (GBM) is one of the most aggressive tumors of the central nervous system. It can be represented by two components: a proliferative component with a mass effect on brain structures and an invasive component. GBM has a distinct pattern of spread showing a preferential growth in the white fiber direction for the invasive component. By using the architecture of white matter fibers, we propose a new model to simulate the growth of GBM. This architecture is estimated by diffusion tensor imaging in order to determine the preferred direction for the diffusion component. It is then coupled with a mechanical component. To set up our growth model, we make a brain atlas including brain structures with a distinct response to tumor aggressiveness, white fiber diffusion tensor information and elasticity. In this atlas, we introduce a virtual GBM with a mechanical component coupled with a diffusion component. These two components are complementary, and can be tuned independently. Then, we tune the parameter set of our model with an MRI patient. We have compared simulated growth (initialized with the MRI patient) with observed growth six months later. The average and the odd ratio of image difference between observed and simulated images are computed. Displacements of reference points are compared to those simulated by the model. The results of our simulation have shown a good correlation with tumor growth, as observed on an MRI patient. Different tumor aggressiveness can also be simulated by tuning additional parameters. This work has demonstrated that modeling the complex behavior of brain tumors is feasible and will account for further validation of this new conceptual approach.

A coupled finite element model of tumor growth and vascularization

We present a model of solid tumor growth which can account for several stages of tumorigenesis, from the early avascular phase to the angiogenesis driven proliferation. The model combines several previously identified components in a consistent framework, including neoplastic tissue growth, blood and oxygen transport, and angiogenic sprouting. First experiments with the framework and comparisons with observations made on solid tumors in vivo illustrate the plausibility of the approach. Explanations of several experimental observations are naturally provided by the model. To the best of our knowledge this is the first report of a model coupling tumor growth and angiogenesis.

Overall survival in patients with malignant glioma may be significantly longer with tumors located in deep grey matter

BACKGROUND

The aim of this study was to assess the correlation of overall survival with tumor location (lobar vs. deep grey matter) and with other clinical and imaging variables in a cohort of patients with high grade gliomas.

METHODS

Adult patients with newly diagnosed supratentorial WHO grade 3 and 4 gliomas were evaluated. Clinical data included demographics, symptoms at presentation, treatment variables, and overall survival. Radiological data included tumor side, site (deep vs. lobar) and size, extent of peritumoral edema, and presence of midline shift. Biostatistics were carried out using log rank tests and univariate and multivariate Cox regression models.

RESULTS

A total of 121 patients were investigated, 23 (19.0%) with WHO grade 3 and 98 (81.0%) with WHO grade 4 gliomas. Tumors had lobar location in 96 cases (79.3%) and deep grey matter location in 25 cases (20.7%). Median survival time for all patients was 26 weeks (IQR: 14-53). Patients with deep tumors survived significantly longer than those with lobar gliomas (log rank test, p=0.0083). Extensive brain edema significantly shortened survival (log rank test, p=0.0003). Presence of midline shift (>1 cm) was a statistically significant risk factor for shorter survival (log rank test, p<0.0001). The univariate Cox regression model demonstrated statistical significance for the variables age, side, site and size of tumor, presence of extensive edema, and presence of mass effect (>1 cm). In the multivariate Cox models, tumor grade, site and size showed statistical significance.

CONCLUSIONS

This study suggests that patients with deep grey matter gliomas may survive significantly longer after the initial diagnosis than those with tumors in a lobar location. This is a potentially novel finding, which may corroborate the theory of differential invasion of glioma cells in different microenvironments of the brain, but remains to be confirmed in future prospective studies.

QxDB: a generic database to support mathematical modelling in biology

QxDB (quantitative x-modelling database) is a web-based generic database package designed especially to house quantitative and structural information. Its development was motivated by the need for centralized access to such results for development of mathematical models, but its usefulness extends to the general research community of both modellers and experimentalists. Written in PHP (Hyper Preprocessor) and MYSQL, the database is easily adapted to new fields of research and ported to Apache-based web servers. Unlike most existing databases, experimental and observational results curated in QxDB are supplemented by comments from the experts who contribute input to the database, giving their evaluations of experimental techniques, breadth of validity of results, experimental conditions, and the like, thus providing the visitor with a basis for gauging the quality (or appropriateness) of each item for his/her needs. QxDB can be easily customized by adapting the contents of the database table containing the descriptors that characterize each data record according to an informal ontology of the research domain. We will illustrate this adaptability of QxDB by presenting two examples, the first dealing with modelling in oncology and the second with mechanical properties of cells and tissues.

A mathematical model of integrin-mediated haptotactic cell migration

Haptotactic cell migration, a directed response to gradients of cell-extracellular matrix adhesion, is an important process in a number of biological phenomena such as wound healing and tumour cell invasion. Previously, mathematical models of haptotaxis have been developed on the premise that cells migrate in response to gradients in the density of the extracellular matrix. In this paper, we develop a novel mathematical model of haptotaxis which includes the adhesion receptors known as integrins and a description of their functional activation, local recruitment and protrusion as part of lamellipodia. Through the inclusion of integrins, the modelled cell matter is able to respond to a true gradient of cell-matrix adhesion, represented by functionally active integrins. We also show that previous matrix-mediated models are in fact a subset of the novel integrin-mediated models, characterised by specific choices of diffusion and haptotaxis coefficients in their model equations. Numerical solutions suggest the existence of travelling waves of cell migration that are confirmed via a phase plane analysis of a simplified model.

Mathematical modeling of cancer: the future of prognosis and treatment

BACKGROUND

Cancer research has undergone radical changes in the past few years. Producing information both at the basic and clinical levels is no longer the issue. Rather, how to handle this information has become the major obstacle to progress. Intuitive approaches are no longer feasible. The next big step will be to implement mathematical modeling approaches to interrogate the enormous amount of data being produced and extract useful answers (a "top-down" approach to biology and medicine).

METHODS

Quantitative simulation of clinically relevant cancer situations-based on experimentally validated mathematical modeling-provides an opportunity for the researcher, and eventually the clinician, to address data and information in the context of well-formulated questions and "what if" scenarios.

RESULTS AND CONCLUSIONS

At the Vanderbilt Integrative Cancer Biology Center (VICBC), we are integrating cancer researchers, oncologists, chemical and biological engineers, computational biologists, computer modelers, theoretical and applied mathematicians, and imaging scientists, in order to implement a vision for a combined web site and computational server that will be a home for our mathematical modeling of cancer invasion. The web site (www.vanderbilt.edu/VICBC/) will serve as a portal to our code, which simulates tumor growth by calculating the dynamics of individual cancer cells (an experimental "bottom-up" approach to complement the top-down model). Eventually, cancer researchers outside of Vanderbilt will be able to initiate a simulation based on providing individual cell data through a web page. We envision placing the web site and computer cluster directly in the hands of biological researchers involved in data mining and mathematical modeling. Furthermore, the web site will also contain teaching props for a new generation of biomedical researchers fluent in both mathematics and biology. This is unconventional bioinformatics: We will be incorporating biological data and functional information into a unified community-based mathematical framework. The result will be a tool for cancer modeling that will ultimately have basic research, therapeutic and educational value.

Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation

We propose a new model to simulate the three-dimensional (3-D) growth of glioblastomas multiforma (GBMs), the most aggressive glial tumors. The GBM speed of growth depends on the invaded tissue: faster in white than in gray matter, it is stopped by the dura or the ventricles. These different structures are introduced into the model using an atlas matching technique. The atlas includes both the segmentations of anatomical structures and diffusion information in white matter fibers. We use the finite element method (FEM) to simulate the invasion of the GBM in the brain parenchyma and its mechanical interaction with the invaded structures (mass effect). Depending on the considered tissue, the former effect is modeled with a reaction-diffusion or a Gompertz equation, while the latter is based on a linear elastic brain constitutive equation. In addition, we propose a new coupling equation taking into account the mechanical influence of the tumor cells on the invaded tissues. The tumor growth simulation is assessed by comparing the in-silico GBM growth with the real growth observed on two magnetic resonance images (MRIs) of a patient acquired with 6 mo difference. Results show the feasibility of this new conceptual approach and justifies its further evaluation.

Origins and evolution of cell phenotypes in breast tumors

This study presents a stochastic model that correlates genomic instability with tumor formation. The model describes the time- and space-variant volumetric concentrations of cancer cells of various phenotypes in a breast tumor. The cells of epithelial origin in the cancerous breast tissue are classified into four different phenotypes, normal epithelial cells and the grade 1, grade 2 and grade 3 cancer cell types with increasing potential for growth and invasion. Equations governing the time course of volumetric concentrations of cell phenotypes are derived by using the principle of conservation of mass. Cell migration into and from the stroma is taken into account. The transformations between cell phenotypes are due to genetic inheritance and chromosome aberrations. These transformations are assumed to be stochastic functions of the local cell concentration. The simulations of the model for planar geometry replicate the shapes of human breast tumors and capture the time history of tumor growth in animal models. Simulations point to transformation of tumor cell population from heterogeneous compositions to a single phenotype at advanced stages of invasive tumors. Systematic variations of model parameters in the computations indicate the important roles the migration capacity, proliferation rate, and phenotype transition probability play in tumor growth. The model developed provides realistic simulations for standard breast cancer therapies and can be used in the optimization studies of chemotherapy, radiotherapy, hormone therapy and emerging individualized therapies for cancer.

Simulating the impact of a molecular 'decision-process' on cellular phenotype and multicellular patterns in brain tumors

Experimental evidence indicates that human brain cancer cells proliferate or migrate, yet do not display both phenotypes at the same time. Here, we present a novel computational model simulating this cellular decision-process leading up to either phenotype based on a molecular interaction network of genes and proteins. The model's regulatory network consists of the epidermal growth factor receptor (EGFR), its ligand transforming growth factor-alpha (TGF alpha), the downstream enzyme phospholipaseC-gamma (PLC gamma) and a mitosis-associated response pathway. This network is activated by autocrine TGF alpha secretion, and the EGFR-dependent downstream signaling this step triggers, as well as modulated by an extrinsic nutritive glucose gradient. Employing a framework of mass action kinetics within a multiscale agent-based environment, we analyse both the emergent multicellular behavior of tumor growth and the single-cell molecular profiles that change over time and space. Our results show that one can indeed simulate the dichotomy between cell migration and proliferation based solely on an EGFR decision network. It turns out that these behavioral decisions on the single cell level impact the spatial dynamics of the entire cancerous system. Furthermore, the simulation results yield intriguing experimentally testable hypotheses also on the sub-cellular level such as spatial cytosolic polarization of PLC gamma towards an extrinsic chemotactic gradient. Implications of these results for future works, both on the modeling and experimental side are discussed.

Traveling Wave Model to Interpret a Wound-Healing Cell Migration Assay for Human Peritoneal Mesothelial Cells

The critical determinants of the speed of an invading cell front are not well known. We performed a "wound-healing" experiment that quantifies the migration of human peritoneal mesothelial cells over components of the extracellular matrix. Results were interpreted in terms of Fisher's equation, which includes terms for the modeling of random cell motility (diffusion) and proliferation. The model predicts that, after a short transient, the invading cell front will move as a traveling wave at constant speed. This is consistent with the experimental findings. Using the model, a relationship between the rate of cell proliferation and the diffusion coefficient was obtained. We used the model to deduce the cell diffusion coefficients under control conditions and in the presence of collagen IV and compared these with other published data. The model may be useful in analyzing the invasive capacity of cancer cells as well in predicting the efficacy of growth factors in tissue reconstruction, including the development of monolayer sheets of cells in skin engineering or the repair of injured corneas using grafts of cultured cells.

The impact of "search precision" in an agent-based tumor model

Several cell surface receptors are overexpressed in malignant brain tumors and reportedly involved in tumor progression and invasion. It is unclear, however, whether such an improvement of cellular signal reception leads to a monotonic increase in the tumor system's average velocity during invasion or whether there is a maximum threshold beyond which the average velocity starts to decelerate. To gain more insight into the systemic effects of such cellular search precision modulations, this study proposes a two-dimensional agent-based model in which the spatio-temporal expansion of malignant brain tumor cells is guided by environmental heterogeneities in mechanical confinement, toxic metabolites and nutrient sources. Here, the spatial field of action is represented by an adaptive grid lattice, which corresponds to the experimental finding that tumor cells are more likely to follow each other along preformed pathways. Another prominent feature is the dual threshold concept for both nutrient level and toxicity, which determine whether cells proliferate, migrate, remain quiescent or die in the next period. The numerical results from varying the key parameters encoding the capability of tumor cells to invade and their ability to proliferate indicate an emergent behavior. Specifically, increasing invasiveness not only leads to an increase in maximum expansion velocity, but also requires a more precise spatial search process, corresponding to an improved cell signal reception, in order to obtain maximum velocity. To increase cellular invasiveness beyond the maximum that can be achieved by exclusively tuning the motility parameter, it requires an additional reduction in the cells' proliferation rate and prompts an even more biased search process. Most interestingly, however, a prominent phase transition suggests that tumor cells do not employ a 100 percent search precision to attain maximum spatial velocity. These findings argue for a selection advantage conferred by limited randomness in processing spatial search and indicate that our computational platform may prove valuable in investigating emergent, multicellular tumor patterns caused by alterations on the molecular level.

Emerging patterns in tumor systems: simulating the dynamics of multicellular clusters with an agent-based spatial agglomeration model

Brain cancer cells invade early on surrounding parenchyma, which makes it impossible to surgically remove all tumor cells and thus significantly worsens the prognosis of the patient. Specific structural elements such as multicellular clusters have been seen in experimental settings to emerge within the invasive cell system and are believed to express the systems' guidance toward nutritive sites in a heterogeneous environment. Based on these observations, we developed a novel agent-based model of spatio-temporal search and agglomeration to investigate the dynamics of cell motility and aggregation with the assumption that tumors behave as complex dynamic self-organizing biosystems. In this model, virtual cells migrate because they are attracted by higher nutrient concentrations and to avoid overpopulated areas with high levels of toxic metabolites. A specific feature of our model is the capability of cells to search both globally and locally. This concept is applied to simulate cell-surface receptor-mediated information processing of tumor cells such that a cell searching for a more growth-permissive place "learns" the information content of a brain tissue region within a two-dimensional lattice in two stages, processing first the global and then the local input. In both stages, differences in microenvironment characteristics define distinctions in energy expenditure for a moving cell and thus influence cell migration, proliferation, agglomeration, and cell death. Numerical results of our model show a phase transition leading to the emergence of two distinct spatio-temporal patterns depending on the dominant search mechanism. If global search is dominant, the result is a small number of large clusters exhibiting rapid spatial expansion but shorter lifetime of the tumor system. By contrast, if local search is dominant, the trade-off is many small clusters with longer lifetime but much slower velocity of expansion. Furthermore, in the case of such dominant local search, the model reveals an expansive advantage for tumor cell populations with a lower nutrient-depletion rate. Important implications of these results for cancer research are discussed.

Evaluation of microscopic tumor extension in non-small-cell lung cancer for three-dimensional conformal radiotherapy planning

PURPOSE

One of the most difficult steps of the three-dimensional conformal radiotherapy (3DCRT) is to define the clinical target volume (CTV) according to the degree of local microscopic extension (ME). In this study, we tried to quantify this ME in non-small-cell lung cancer (NSCLC).

MATERIAL AND METHODS

Seventy NSCLC surgical resection specimens for which the border between tumor and adjacent lung parenchyma were examined on routine sections. This border was identified with the naked eye, outlined with a marker pen, and the value of the local ME outside of this border was measured with an eyepiece micrometer. The pattern of histologic spread was also determined.

RESULTS

A total of 354 slides were examined, corresponding to 176 slides for adenocarcinoma (ADC) and 178 slides for squamous cell carcinoma (SCC). The mean value of ME was 2.69 mm for ADC and 1.48 mm for SCC (p = 0.01). The usual 5-mm margin covers 80% of the ME for ADC and 91% for SCC. To take into account 95% of the ME, a margin of 8 mm and 6 mm must be chosen for ADC and SCC, respectively. Aerogenous dissemination was the most frequent pattern observed for all groups, followed by lymphatic invasion for ADC and interstitial extension for SCC.

CONCLUSION

The ME was different between ADC and SCC. The usual CTV margin of 5 mm appears inadequate to cover the ME for either group, and it must be increased to 8 mm and 6 mm for ADC and SCC, respectively, to cover 95% of the ME. This approach is obviously integrated into the overall 3DCRT procedure and with other margins.

Simulated brain tumor growth dynamics using a three-dimensional cellular automaton

We have developed a novel and versatile three-dimensional cellular automaton model of brain tumor growth. We show that macroscopic tumor behavior can be realistically modeled using microscopic parameters. Using only four parameters, this model simulates Gompertzian growth for a tumor growing over nearly three orders of magnitude in radius. It also predicts the composition and dynamics of the tumor at selected time points in agreement with medical literature. We also demonstrate the flexibility of the model by showing the emergence, and eventual dominance, of a second tumor clone with a different genotype. The model incorporates several important and novel features, both in the rules governing the model and in the underlying structure of the model. Among these are a new definition of how to model proliferative and non-proliferative cells, an isotropic lattice, and an adaptive grid lattice.

Mechanical signalling and angiogenesis. The integration of cell-extracellular matrix couplings

In vitro angiogenesis assays have shown that the couplings between fibrin gel and cell traction forces trigger biogel pre-patterning, consisting, in the formation of lacunae which evolve toward capillary-like structures (CLS) networks. Depending on the experimental conditions (number of seeded cells, gel elasticity,...), this pre-patterning can be enhanced or inhibited. A theoretical model based on a description of the cell-biogel biochemical and mechanical interactions is proposed as a basis for understanding how integrating these interactions can lead to the pre-patterning of the biogel. We showed that the critical parameter values corresponding to the bifurcation of the model solutions correspond to threshold values of the experimental variables. Furthermore, simulations of the mechanocellular model give rise to dynamic remodelling patterns of the biogel which are in good agreement both with the lacunae morphologies and with the time and space scales derived from the in vitro angiogenesis assays. Special attention has been paid in the simulations to cell proteolytic activity and to the amplitude of cell traction forces. We finally discussed how modelling guided experiments can be inferred from these results.

Some perspectives on modeling leukemia

A diffusion model of leukemia is presented. The space-occupying effects of leukemic cells during leukemic expansion is investigated. The analyses and simulations of the model suggest that acute leukemia is a state in which positions inhabited by colonies of normal cells are invaded by emerging colonies of abnormal cells. Normal cells are then driven to a state of extinction as leukemic cells evolve toward high and dominant steady state levels.