A mechanical model of tumor encapsulation and transcapsular spread

Adipsin-dependent adipocyte maturation induces cancer cell invasion in breast cancer

Adipocyte-cancer cell interactions promote tumor development and progression. Previously, we identified adipsin (CFD) and its downstream effector, hepatocyte growth factor (HGF), as adipokines that enhance adipocyte-breast cancer stem cell interactions. Here, we show that adipsin-dependent adipocyte maturation and the subsequent upregulation of HGF promote tumor invasion in breast cancers. Mature adipocytes, but not their precursors, significantly induced breast tumor cell migration and invasion in an adipsin expression-dependent manner. Promoters of tumor invasion, galectin 7 and matrix metalloproteinases, were significantly upregulated in cancer cells cocultured with mature adipocytes; meanwhile, their expression levels in cancer cells cocultured with adipocytes were reduced by adipsin knockout (Cfd KO) or a competitive inhibitor of CFD. Tumor growth and distant metastasis of mammary cancer cells were significantly suppressed when syngeneic mammary cancer cells were transplanted into Cfd KO mice. Histological analyses revealed reductions in capsular formation and tumor invasion at the cancer-adipocyte interface in the mammary tumors formed in Cfd KO mice. These findings indicate that adipsin-dependent adipocyte maturation may play an important role in adipocyte-cancer cell interaction and breast cancer progression.

A mixture-like model for tumor-immune system interactions

We introduce a mathematical model based on mixture theory intended to describe the tumor-immune system interactions within the tumor microenvironment. The equations account for the geometry of the tumor expansion, and the displacement of the immune cells, driven by diffusion and chemotactic mechanisms. They also take into account the constraints in terms of nutrient and oxygen supply. The numerical investigations analyze the impact of the different modeling assumptions and parameters. Depending on the parameters, the model can reproduce elimination, equilibrium or escape phases and it identifies a critical role of oxygen/nutrient supply in shaping the tumor growth. In addition, antitumor immune cells are key factors in controlling tumor growth, maintaining an equilibrium while protumor cells favor escape and tumor expansion.

Mathematical model and computational scheme for multi-phase modeling of cellular population and microenvironmental dynamics in soft tissue

In this paper we introduce a system of partial differential equations that is capable of modeling a variety of dynamic processes in soft tissue cellular populations and their microenvironments. The model is designed to be general enough to simulate such processes as tissue regeneration, tumor growth, immune response, and many more. It also has built-in flexibility to include multiple chemical fields and/or sub-populations of cells, interstitial fluid and/or extracellular matrix. The model is derived from the conservation laws for mass and linear momentum and therefore can be classified as a continuum multi-phase model. A careful choice of state variables provides stability in solving the system of discretized equations defining advective flux terms. A concept of deviation from normal allows us to use simplified constitutive relations for stresses. We also present an algorithm for computing numerical approximations to the solutions of the system and discuss properties of these approximations. We demonstrate several examples of applications of the model. Numerical simulations show a significant potential of the model for simulating a variety of processes in soft tissues.

Artificial selection for host resistance to tumour growth and subsequent cancer cell adaptations: an evolutionary arms race

BACKGROUND

Cancer progression is governed by evolutionary dynamics in both the tumour population and its host. Since cancers die with the host, each new population of cancer cells must reinvent strategies to overcome the host's heritable defences. In contrast, host species evolve defence strategies over generations if tumour development limits procreation.

METHODS

We investigate this "evolutionary arms race" through intentional breeding of immunodeficient SCID and immunocompetent Black/6 mice to evolve increased tumour suppression. Over 10 generations, we injected Lewis lung mouse carcinoma cells [LL/2-Luc-M38] and selectively bred the two individuals with the slowest tumour growth at day 11. Their male progeny were hosts in the subsequent round.

RESULTS

The evolved SCID mice suppressed tumour growth through biomechanical restriction from increased mesenchymal proliferation, and the evolved Black/6 mice suppressed tumour growth by increasing immune-mediated killing of cancer cells. However, transcriptomic changes of multicellular tissue organisation and function genes allowed LL/2-Luc-M38 cells to adapt through increased matrix remodelling in SCID mice, and reduced angiogenesis, increased energy utilisation and accelerated proliferation in Black/6 mice.

CONCLUSION

Host species can rapidly evolve both immunologic and non-immunologic tumour defences. However, cancer cell plasticity allows effective phenotypic and population-based counter strategies.

Mathematical modelling of trastuzumab-induced immune response in an in vivo murine model of HER2+ breast cancer

The goal of this study is to develop an integrated, mathematical-experimental approach for understanding the interactions between the immune system and the effects of trastuzumab on breast cancer that overexpresses the human epidermal growth factor receptor 2 (HER2+). A system of coupled, ordinary differential equations was constructed to describe the temporal changes in tumour growth, along with intratumoural changes in the immune response, vascularity, necrosis and hypoxia. The mathematical model is calibrated with serially acquired experimental data of tumour volume, vascularity, necrosis and hypoxia obtained from either imaging or histology from a murine model of HER2+ breast cancer. Sensitivity analysis shows that model components are sensitive for 12 of 13 parameters, but accounting for uncertainty in the parameter values, model simulations still agree with the experimental data. Given theinitial conditions, the mathematical model predicts an increase in the immune infiltrates over time in the treated animals. Immunofluorescent staining results are presented that validate this prediction by showing an increased co-staining of CD11c and F4/80 (proteins expressed by dendritic cells and/or macrophages) in the total tissue for the treated tumours compared to the controls ($p < 0.03$). We posit that the proposed mathematical-experimental approach can be used to elucidate driving interactions between the trastuzumab-induced responses in the tumour and the immune system that drive the stabilization of vasculature while simultaneously decreasing tumour growth-conclusions revealed by the mathematical model that were not deducible from the experimental data alone.

A Biomechanical Model of Tumor-Induced Intracranial Pressure and Edema in Brain Tissue

Brain tumor growth and tumor-induced edema result in increased intracranial pressure (ICP), which, in turn, is responsible for conditions as benign as headaches and vomiting or as severe as seizures, neurological damage, or even death. Therefore, it has been hypothesized that tracking ICP dynamics may offer improved prognostic potential in terms of early detection of brain cancer and better delimitation of the tumor boundary. However, translating such theory into clinical practice remains a challenge, in part because of an incomplete understanding of how ICP correlates with tumor grade. Here, we propose a multiphase mixture model that describes the biomechanical response of healthy brain tissue-in terms of changes in ICP and edema-to a growing tumor. The model captures ICP dynamics within the diseased brain and accounts for the ability/inability of healthy tissue to compensate for this pressure. We propose parameter regimes that distinguish brain tumors by grade, thereby providing critical insight into how ICP dynamics vary by severity of disease. In particular, we offer an explanation for clinically observed phenomena, such as a lack of symptoms in low-grade glioma patients versus a rapid onset of symptoms in those with malignant tumors. Our model also takes into account the effects tumor-derived proteases may have on ICP levels and the extent of tumor invasion. This work represents an important first step toward understanding the mechanisms that underlie the onset of edema and ICP in cancer-afflicted brains. Continued modeling effort in this direction has the potential to make an impact in the field of brain cancer diagnostics.

How Tumor Cells Can Make Use of Interstitial Fluid Flow in a Strategy for Metastasis

Introduction

The phenomenon of lymph node metastasis has been known for a long time. However, the underlying mechanism by which malignant tumor cells are able to break loose from the primary tumor site remains unclear. In particular, two competing fluid sensitive migration mechanisms have been reported in the experimental literature: (i) autologous chemotaxis (Shields et al. in Cancer Cell 11:526-538, 2007) which gives rise to downstream migration; (ii) an integrin-mediated and strain-induced upstream mechanism (Polacheck et al. in PNAS 108:11115-11120, 2011). How can these two competing mechanisms be used as a means for metastatic behavior in a realistic tumor setting? Excessive fluid flow is typically produced from leaky intratumoral blood vessels and collected by lymphatics in the peritumoral region giving rise to a heterogeneous fluid velocity field and a corresponding heterogeneous cell migration behavior, quite different from the experimental setup.

Method

In order to shed light on this issue there is a need for tools which allow one to extrapolate the observed single cell behavior in a homogeneous microfluidic environment to a more realistic, higher-dimensional tumor setting. Here we explore this issue by using a computational multiphase model. The model has been trained with data from the experimental results mentioned above which essentially reflect one-dimensional behavior. We extend the model to an envisioned idealized two-dimensional tumor setting.

Result

A main observation from the simulation is that the autologous chemotaxis migration mechanism, which triggers tumor cells to go with the flow in the direction of lymphatics, becomes much more aggressive and effective as a means for metastasis in the presence of realistic IF flow. This is because the outwardly directed IF flow generates upstream cell migration that possibly empowers small clusters of tumor cells to break loose from the primary tumor periphery. Without this upstream stress-mediated migration, autologous chemotaxis is inclined to move cells at the rim of the tumor in a homogeneous and collective, but space-demanding style. In contrast, inclusion of realistic IF flow generates upstream migration that allows two different aspects to be synthesized: maintain the coherency and solidity of the the primary tumor and at the same time cleave the outgoing waves of tumor cells into small clusters at the front that can move collectively in a more specific direction.

The emergence of solid stress as a potent biomechanical marker of tumour progression

Cancer is a disease of dysregulated mechanics which alters cell behaviour, compromises tissue structure, and promotes tumour growth and metastasis. In the context of tumour progression, the most widely studied of biomechanical markers is matrix stiffness as tumour tissue is typically stiffer than healthy tissue. However, solid stress has recently been identified as another marker of tumour growth, with findings strongly suggesting that its role in cancer is distinct from that of stiffness. Owing to the relative infancy of the field which draws from diverse disciplines, a comprehensive knowledge of the relationships between solid stress, tumorigenesis, and metastasis is likely to provide new and valuable insights. In this review, we discuss the micro- and macro-scale biomechanical interactions that give rise to solid stresses, and also examine the techniques developed to quantify solid stress within the tumour environment. Moreover, by reviewing the effects of solid stress on tissues, cancer and stromal cells, and signalling pathways, we also detail its mode of action at each level of the cancer cascade.

A multiscale approach for determining the morphology of endothelial cells at a coronary artery

The morphology of endothelial cells (ECs) may be an indication for determining atheroprone sites. Until now, there has been no clinical imaging technique to visualize the morphology of ECs in the arteries. The present study introduces a computational technique for determining the morphology of ECs. This technique is a multiscale simulation consisting of the artery scale and the cell scale. The artery scale is a fluid-structure interaction simulation. The input for the artery scale is the geometry of the coronary artery, that is, the dynamic curvature of the artery due to the cardiac motion, blood flow, blood pressure, heart rate, and the mechanical properties of the blood and the arterial wall, the measurements of which can be obtained for a specific patient. The results of the artery scale are wall shear stress (WSS) and cyclic strains as the mechanical stimuli of ECs. The cell scale is an inventive mass-and-spring model that is able to determine the morphological response of ECs to any combination of mechanical stimuli. The results of the multiscale simulation show the morphology of ECs at different locations of the coronary artery. The results indicate that the atheroprone sites have at least 1 of 3 factors: low time-averaged WSS, high angle of WSS, and high longitudinal strain. The most probable sites for atherosclerosis are located at the bifurcation region and lie on the myocardial side of the artery. The results also indicated that a higher dynamic curvature is a negative factor and a higher pulse pressure is a positive factor for protection against atherosclerosis.

Multiscale biphasic modelling of peritumoural collagen microstructure: The effect of tumour growth on permeability and fluid flow

We present an in-silico model of avascular poroelastic tumour growth coupled with a multiscale biphasic description of the tumour–host environment. The model is specified to in-vitro data, facilitating biophysically realistic simulations of tumour spheroid growth into a dense collagen hydrogel. We use the model to first confirm that passive mechanical remodelling of collagen fibres at the tumour boundary is driven by solid stress, and not fluid pressure. The model is then used to demonstrate the influence of collagen microstructure on peritumoural permeability and interstitial fluid flow. Our model suggests that at the tumour periphery, remodelling causes the peritumoural stroma to become more permeable in the circumferential than radial direction, and the interstitial fluid velocity is found to be dependent on initial collagen alignment. Finally we show that solid stresses are negatively correlated with peritumoural permeability, and positively correlated with interstitial fluid velocity. These results point to a heterogeneous, microstructure-dependent force environment at the tumour–peritumoural stroma interface.

Halofuginone suppresses growth of human uterine leiomyoma cells in a mouse xenograft model

STUDY QUESTION

Does halofuginone (HF) inhibit the growth of human uterine leiomyoma cells in a mouse xenograft model?

SUMMARY ANSWER

HF suppresses the growth of human uterine leiomyoma cells in a mouse xenograft model through inhibiting cell proliferation and inducing apoptosis.

WHAT IS KNOWN ALREADY

Uterine leiomyomas are the most common benign tumors of the female reproductive tract. HF can suppress the growth of human uterine leiomyoma cells in vitro. The mouse xenograft model reflects the characteristics of human leiomyomas.

STUDY DESIGN, SIZE, DURATION

Primary leiomyoma smooth muscle cells from eight patients were xenografted under the renal capsule of adult, ovariectomized NOD-scid IL2Rγ(null) mice (NSG). Mice were treated with two different doses of HF or vehicle for 4 weeks with six to eight mice per group.

PARTICIPANTS/MATERIALS, SETTING, METHODS

Mouse body weight measurements and immunohistochemical analysis of body organs were carried out to assess the safety of HF treatment. Xenografted tumors were measured and analyzed for cellular and molecular changes induced by HF. Ovarian steroid hormone receptors were evaluated for possible modulation by HF.

MAIN RESULTS AND THE ROLE OF CHANCE

Treatment of mice carrying human UL xenografts with HF at 0.25 or 0.50 mg/kg body weight for 4 weeks resulted in a 35-40% (P < 0.05) reduction in tumor volume. The HF-induced volume reduction was accompanied by increased apoptosis and decreased cell proliferation. In contrast, there was no significant change in the collagen content either at the transcript or protein level between UL xenografts in control and HF groups. HF treatment did not change the expression level of ovarian steroid hormone receptors. No adverse pathological effects were observed in other tissues from mice undergoing treatment at these doses.

LIMITATIONS, REASONS FOR CAUTION

While this study did test the effects of HF on human leiomyoma cells in an in vivo model, HF was administered to mice whose tolerance and metabolism of the drug may differ from that in humans. Also, the longer term effects of HF treatment are yet unclear.

WIDER IMPLICATIONS OF THE FINDINGS

The results of this study showing the effectiveness of HF in reducing UL tumor growth by interfering with the main cellular processes regulating cell proliferation and apoptosis are in agreement with previous studies on the effects of HF on other fibrotic diseases. HF can be considered as a candidate for reducing the size of leiomyomas, particularly prior to surgery.

STUDY FUNDING/COMPETING INTERESTS

This project was funded by NIH PO1HD057877 and R01 HD064402. Authors report no competing interests.

A mechanical model for morphological response of endothelial cells under combined wall shear stress and cyclic stretch loadings

The shape and morphology of endothelial cells (ECs) lining the blood vessels are a good indicator for atheroprone and atheroprotected sites. ECs of blood vessels experience both wall shear stress (WSS) and cyclic stretch (CS). These mechanical stimuli influence the shape and morphology of ECs. A few models have been proposed for predicting the morphology of ECs under WSS or CS. In the present study, a mathematical cell population model is developed to simulate the morphology of ECs under combined WSS and CS conditions. The model considers the cytoskeletal filaments, cell-cell interactions, and cell-extracellular matrix interactions. In addition, the reorientation and polymerization of microfilaments are implemented in the model. The simulations are performed for different conditions: without mechanical stimuli, under pure WSS, under pure CS, and under combined WSS and CS. The results are represented as shape and morphology of ECs, shape index, and angle of orientation. The model is validated qualitatively and quantitatively with several experimental studies, and good agreement with experimental studies is achieved. To the best of our knowledge, it is the first model for predicting the morphology of ECs under combined WSS and CS condition. The model can be used to indicate the atheroprone regions of a patient's artery.

Multiscale modelling of solid tumour growth: the effect of collagen micromechanics

Here we introduce a model of solid tumour growth coupled with a multiscale biomechanical description of the tumour microenvironment, which facilitates the explicit simulation of fibre-fibre and tumour-fibre interactions. We hypothesise that such a model, which provides a purely mechanical description of tumour-host interactions, can be used to explain experimental observations of the effect of collagen micromechanics on solid tumour growth. The model was specified to mouse tumour data, and numerical simulations were performed. The multiscale model produced lower stresses than an equivalent continuum-like approach, due to a more realistic remodelling of the collagen microstructure. Furthermore, solid tumour growth was found to cause a passive mechanical realignment of fibres at the tumour boundary from a random to a circumferential orientation. This is in accordance with experimental observations, thus demonstrating that such a response can be explained as purely mechanical. Finally, peritumoural fibre network anisotropy was found to produce anisotropic tumour morphology. The dependency of tumour morphology on the peritumoural microstructure was reduced by adding a load-bearing non-collagenous component to the fibre network constitutive equation.

Remodeling of extracellular matrix due to solid stress accumulation during tumor growth

Solid stresses emerge as the expanding tumor displaces and deforms the surrounding normal tissue, and also as a result of intratumoral component interplay. Among other things, solid stresses are known to induce extensive extracellular matrix synthesis and reorganization. In this study, we developed a mathematical model of tumor growth that distinguishes the contribution to stress generation by collagenous and non-collagenous tumor structural components, and also investigates collagen fiber remodeling exclusively due to solid stress. To this end, we initially conducted in vivo experiments using an orthotopic mouse model for breast cancer to monitor primary tumor growth and derive the mechanical properties of the tumor. Subsequently, we fitted the mathematical model to experimental data to determine values of the model parameters. According to the model, intratumoral solid stress is compressive, whereas extratumoral stress in the tumor vicinity is compressive in the radial direction and tensile in the periphery. Furthermore, collagen fibers engaged in stress generation only in the peritumoral region, and not in the interior where they were slackened due to the compressive stress state. Peritumoral fibers were driven away from the radial direction, tended to realign tangent to the tumor-host interface, and were also significantly stretched by tensile circumferential stresses. By means of this remodeling, the model predicts that the tumor is enveloped by a progressively thickening capsule of collagen fibers. This prediction is consistent with long-standing observations of tumor encapsulation and histologic sections that we performed, and it further corroborates the expansive growth hypothesis for the capsule formation.

Mechanical control of spheroid growth: distinct morphogenetic regimes

We develop a model of transport and growth in epithelio-mesenchymal interactions. Analysis of the growth of an avascular solid spheroid inside a passive mesenchyme or gel shows that sustained volumetric growth requires four generic mechanisms: (1) growth factor, (2) protease, (3) control of cellularity, and (4) swelling. The model reveals a bifurcation delineating two distinct morphogenetic regimes: (A) steady growth, (B) growth arrested by capsule formation in the mesenchyme. In both morphogenetic regimes, growth velocity is constant unless and until a complete capsule forms. Comprehensive exploration of the large parameter space reveals that the bifurcation is determined by just two ratios representing the relative strengths of growth and proteolytic activity. Growth velocity is determined only by the ratio governing growth, independent of proteolytic activity. There is a continuum of interior versus surface growth, with fastest growth at the surface. The model provides a theoretical basis for explaining observations of growth arrest despite proteolysis of surrounding tissue, and gives a quantitative framework for the design and interpretation of experiments involving spheroids, and tissues which are locally equivalent to spheroids.

Tumour-stromal interactions in acid-mediated invasion: a mathematical model

It is well established that the tumour microenvironment can both promote and suppress tumour growth and invasion, however, most mathematical models of invasion view the normal tissue as inhibiting tumour progression via immune modulation or spatial constraint. In particular, the production of acid by tumour cells and the subsequent creation of a low extracellular pH environment has been explored in several 'acid-mediated tumour invasion' models where the acidic environment facilitates normal cell death and permits tumour invasion. In this paper, we extend the acid-invasion model developed by Gatenby and Gawlinski (1996) to include both the competitive and cooperative interactions between tumour and normal cells, by incorporating the influence of extracellular matrix and protease production at the tumour-stroma interface. Our model predicts an optimal level of tumour acidity which produces both cell death and matrix degradation. Additionally, very aggressive tumours prevent protease production and matrix degradation by excessive normal cell destruction, leading to an acellular (but matrix filled) gap between the tumour and normal tissue, a feature seen in encapsulated tumours. These results suggest, counterintuitively, that increasing tumour acidity may, in some cases, prevent tumour invasion.

In silico estimates of the free energy rates in growing tumor spheroids

The physics of solid tumor growth can be considered at three distinct size scales: the tumor scale, the cell-extracellular matrix (ECM) scale and the sub-cellular scale. In this paper we consider the tumor scale in the interest of eventually developing a system-level understanding of the progression of cancer. At this scale, cell populations and chemical species are best treated as concentration fields that vary with time and space. The cells have chemo-mechanical interactions with each other and with the ECM, consume glucose and oxygen that are transported through the tumor, and create chemical by-products. We present a continuum mathematical model for the biochemical dynamics and mechanics that govern tumor growth. The biochemical dynamics and mechanics also engender free energy changes that serve as universal measures for comparison of these processes. Within our mathematical framework we therefore consider the free energy inequality, which arises from the first and second laws of thermodynamics. With the model we compute preliminary estimates of the free energy rates of a growing tumor in its pre-vascular stage by using currently available data from single cells and multicellular tumor spheroids.

Multiphase flow models of biogels from crawling cells to bacterial biofilms

This article reviews multiphase descriptions of the fluid mechanics of cytoplasm in crawling cells and growing bacterial biofilms. These two systems involve gels, which are mixtures composed of a polymer network permeated by water. The fluid mechanics of these systems is essential to their biological function and structure. Their mathematical descriptions must account for the mechanics of the polymer, the water, and the interaction between these two phases. This review focuses on multiphase flow models because this framework is natural for including the relative motion between the phases, the exchange of material between phases, and the additional stresses within the network that arise from nonspecific chemical interactions and the action of molecular motors. These models have been successful in accounting for how different forces are generated and transmitted to achieve cell motion and biofilm growth and they have demonstrated how emergent structures develop though the interactions of the two phases. A short description of multiphase flow models of tumor growth is included to highlight the flexibility of the model in describing diverse biological applications.

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.

Predictive oncology: a review of multidisciplinary, multiscale in silico modeling linking phenotype, morphology and growth

Empirical evidence and theoretical studies suggest that the phenotype, i.e., cellular- and molecular-scale dynamics, including proliferation rate and adhesiveness due to microenvironmental factors and gene expression that govern tumor growth and invasiveness, also determine gross tumor-scale morphology. It has been difficult to quantify the relative effect of these links on disease progression and prognosis using conventional clinical and experimental methods and observables. As a result, successful individualized treatment of highly malignant and invasive cancers, such as glioblastoma, via surgical resection and chemotherapy cannot be offered and outcomes are generally poor. What is needed is a deterministic, quantifiable method to enable understanding of the connections between phenotype and tumor morphology. Here, we critically assess advantages and disadvantages of recent computational modeling efforts (e.g., continuum, discrete, and cellular automata models) that have pursued this understanding. Based on this assessment, we review a multiscale, i.e., from the molecular to the gross tumor scale, mathematical and computational "first-principle" approach based on mass conservation and other physical laws, such as employed in reaction-diffusion systems. Model variables describe known characteristics of tumor behavior, and parameters and functional relationships across scales are informed from in vitro, in vivo and ex vivo biology. We review the feasibility of this methodology that, once coupled to tumor imaging and tumor biopsy or cell culture data, should enable prediction of tumor growth and therapy outcome through quantification of the relation between the underlying dynamics and morphological characteristics. In particular, morphologic stability analysis of this mathematical model reveals that tumor cell patterning at the tumor-host interface is regulated by cell proliferation, adhesion and other phenotypic characteristics: histopathology information of tumor boundary can be inputted to the mathematical model and used as a phenotype-diagnostic tool to predict collective and individual tumor cell invasion of surrounding tissue. This approach further provides a means to deterministically test effects of novel and hypothetical therapy strategies on tumor behavior.

Modelling aspects of cancer dynamics: a review

Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. However, these data do not always reveal the mechanisms underpinning the observed phenomena. In this paper, we explain why mathematics is a powerful tool for interpreting such data by presenting case studies that illustrate the types of insight that realistic theoretical models of solid tumour growth may yield. These range from discriminating between competing hypotheses for the formation of collagenous capsules associated with benign tumours to predicting the most likely stimulus for protease production in early breast cancer. We will also illustrate the benefits that may result when experimentalists and theoreticians collaborate by considering a novel anti-cancer therapy.

Towards whole-organ modelling of tumour growth

Multiscale approaches to modelling biological phenomena are growing rapidly. We present here some recent results on the formulation of a theoretical framework which can be developed into a fully integrative model for cancer growth. The model takes account of vascular adaptation and cell-cycle dynamics. We explore the effects of spatial inhomogeneity induced by the blood flow through the vascular network and of the possible effects of p27 on the cell cycle. We show how the model may be used to investigate the efficiency of drug-delivery protocols.

A new interpretation of the Keller-Segel model based on multiphase modelling

In this paper an alternative derivation and interpretation are presented of the classical Keller-Segel model of cell migration due to random motion and chemotaxis. A multiphase modelling approach is used to describe how a population of cells moves through a fluid containing a diffusible chemical to which the cells are attracted. The cells and fluid are viewed as distinct components of a two-phase mixture. The principles of mass and momentum balance are applied to each phase, and appropriate constitutive laws imposed to close the resulting equations. A key assumption here is that the stress in the cell phase is influenced by the concentration of the diffusible chemical. By restricting attention to one-dimensional cartesian geometry we show how the model reduces to a pair of nonlinear coupled partial differential equations for the cell density and the chemical concentration. These equations may be written in the form of the Patlak-Keller-Segel model, naturally including density-dependent nonlinearities in the cell motility coefficients. There is a direct relationship between the random motility and chemotaxis coefficients, both depending in an inter-related manner on the chemical concentration. We suggest that this may explain why many chemicals appear to stimulate both chemotactic and chemokinetic responses in cell populations. After specialising our model to describe slime mold we then show how the functional form of the chemical potential that drives cell locomotion influences the ability of the system to generate spatial patterns. The paper concludes with a summary of the key results and a discussion of avenues for future research.

Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites

Poor drug delivery and low rates of cell proliferation are two factors associated with hypoxia that diminish the efficacy of many chemotherapeutic drugs. Since macrophages are known to migrate specifically towards, and localize within, hypoxic tumour regions, a promising resolution to these problems involves genetically engineering macrophages to perform such anti-tumour functions as inducing cell lysis and inhibiting angiogenesis. In this paper we outline a modelling approach to characterize macrophage infiltration into early avascular solid tumours, and extensions to study the interaction of these cells with macrophages already present within the tumour. We investigate the role of chemotaxis and chemokine production, and the efficacy of macrophages as vehicles for drug delivery to hypoxic tumour sites. The model is based upon a growing avascular tumour spheroid, in which volume is filled by tumour cells, macrophages and extracellular material, and tumour cell proliferation and death is regulated by nutrient diffusion. Crucially, macrophages occupy volume, and hence contribute to the volume balance and hence the size of the tumour. We also include oxygen-dependent production of macrophage chemokines, which can lead to accumulations in the hypoxic region of the tumour. We find that the macrophage chemotactic sensitivity is a key determinant of macrophage infiltration and tumour size. Although increased infiltration should be beneficial from the point of view of macrophage-based therapies, such infiltration in fact leads to increased tumour sizes. Finally, we include terms representing the induced death of tumour cells by hypoxic engineered macrophages. We demonstrate that reductions in tumour size can be achieved, but predict that a combination of therapies would be required for complete eradication. We also highlight some counter-intuitive predictions-for example, absolute and relative measures of tumour burden lead to different conclusions about prognosis. In summary, this paper illustrates how mathematical models may be used to investigate promising macrophage-based therapies.