Modelling the role of angiogenesis and vasculogenesis in solid tumour growth

Analysis of tumor-immune functional responses in a mathematical model of neoantigen cancer vaccines

Cancer neoantigen vaccines have emerged as a promising approach to stimulating the immune system to fight cancer. We propose a simple model including key elements of cancer-immune interactions and conduct a phase plane analysis to understand the immunological mechanisms of cancer neoantigen vaccines. Analytical results are obtained for two widely used functional forms that represent the killing rate of tumor cells by immune cells: the law of mass action (LMA) and the dePillis-Radunskaya Law (LPR). Using the LMA, our results reveal that a slowly growing tumor can escape the immune surveillance and that there is a unique periodic solution. The LPR offers richer dynamics, in which tumor elimination and uncontrolled tumor growth are both present. We show that tumor elimination requires sufficient number of initial activated T cells in relationship to the malignant cells, which lends support to using the neoantigen cancer vaccine as an adjuvant therapy after the primary tumor is surgically removed or treated using radiotherapy. We also derive a sufficient condition for uncontrolled tumor growth under the assumption of the LPR. The juxtaposition of analyses with these two different choices for the killing rate function highlights their importance on model behavior and biological implications, by which we hope to spur further theoretical and experimental work to understand mechanisms underlying different functional forms for the killing rate.

A simulation of parental and glycolytic tumor phenotype competition predicts observed responses to pH changes and increased glycolysis after anti-VEGF therapy

Clinical cancers are typically spatially and temporally heterogeneous, containing multiple microenvironmental habitats and diverse phenotypes and/or genotypes, which can interact through resource competition and direct or indirect interference. A common intratumoral evolutionary pathway, probably initiated as adaptation to hypoxia, leads to the "Warburg phenotype" which maintains high glycolytic rates and acid production, even in normoxic conditions. Since individual cancer cells are the unit of Darwinian selection, intraspecific competition dominates intratumoral evolution. Thus, elements of the Warburg phenotype become key "strategies" in competition with cancer cell populations that retain the metabolism of the parental normal cells. Here we model the complex interactions of cell populations with Warburg and parental phenotypes as they compete for access to vasculature, while subject to direct interference by Warburg-related acidosis. In this competitive environment, vasculature delivers nutrients, removes acid and necrotic detritus, and responds to signaling molecules (VEGF and TNF-α). The model is built in a nested fashion and growth parameters are derived from monolayer, spheroid, and xenograft experiments on prostate cancer. The resulting model of in vivo tumor growth reaches a steady state, displaying linear growth and coexistence of both glycolytic and parental phenotypes consistent with experimental observations. The model predicts that increasing tumor pH sufficiently early can arrest the development of the glycolytic phenotype, while decreasing tumor pH accelerates this evolution and increases VEGF production. The model's predicted dual effects of VEGF blockers in decreasing tumor growth while increasing the glycolytic fraction of tumor cells has potential implications for optimizing angiogenic inhibitors.

Enhanced antitumour effect for hepatocellular carcinoma in the advanced stage using a cyclodextrin-sorafenib-chaperoned inclusion complex

Hepatocellular carcinoma (HCC) is a hypervascular tumour characterized by tumour-driven neovascularization. The degrees of blood oxygen saturation (DBOS), microvessel density (MVD) and tumour size (TS) are indicators in identifying the development stage of HCC. Herein, we proposed an HCC staging model using HepG2 tumour-bearing mice based on DBOS, MVD and TS. According to the patterns of these three criteria, HCC was classified into four stages: early, intermediate, advanced and end stages. The advanced stage was characterized by MVD of 50-90 (number per mm²), DBOS of 12-16% and TS of 250-600 mm³, which poses a critical challenge in HCC therapy. In order to efficiently control and treat HCC in the advanced stage, we developed a cyclodextrin (CD)-based chaperoned inclusion complex using Sorafenib (Sor), β-CD and γ-CD (SCD) via the co-crystallization method. The structural study manifested that CDs could encapsulate Sor with the hydrophobic cavities at a 1 : 1 stoichiometry ratio. The crystallographic analysis indicated that Sor-β-CD presented a diagonal stacking pattern, while Sor-γ-CD possessed a channel-type structure. The resultant chaperoned inclusion complexes significantly improved the solubility, dissolution rate and drug release of Sor, leading to superior pharmacokinetics, biodistribution and biosafety through oral administration. The antitumour effect was then evaluated on a mouse model with advanced HCC through oral administration and intratumour injection. The treatment involving the oral administration of SCDs showed a promising therapeutic effect on advanced HCC, which efficiently blocked angiogenesis and inhibited tumour progression. For the treatments using intratumour injections, only Sor-γ-CD exhibited a satisfactory anti-tumour effect with reduction in TS, MVD and DBOS. The enhanced therapeutic performance of Sor-γ-CD was attributed to its channel-type structure, which had an impact on the dissociation and release of the drug. Thus, Sor-γ-CD can be used as a potential pro-drug for clinical medicine and basic research to treat HCC.

Calibrating a Predictive Model of Tumor Growth and Angiogenesis with Quantitative MRI

The spatiotemporal variations in tumor vasculature inevitably alters cell proliferation and treatment efficacy. Thus, rigorous characterization of tumor dynamics must include a description of this phenomenon. We have developed a family of biophysical models of tumor growth and angiogenesis that are calibrated with diffusion-weighted magnetic resonance imaging (DW-MRI) and dynamic contrast-enhanced (DCE-) MRI data to provide individualized tumor growth forecasts. Tumor and blood volume fractions were evolved using two, coupled partial differential equations consisting of proliferation, diffusion, and death terms. To evaluate these models, rats (n = 8) with C6 gliomas were imaged seven times. The tumor volume fraction was estimated using DW-MRI, while DCE-MRI provided estimates of the blood volume fraction. The first three time points were used to calibrate model parameters, which were then used to predict growth at the remaining four time points and compared directly to the measurements. The best performing model predicted tumor growth with less than 10.3% error in tumor volume and with less than 9.4% error at the voxel-level at all prediction time points. The best performing model resulted in less than 9.3% error in blood volume at the voxel-level. This pre-clinical study demonstrates the potential for image-based, mechanistic modeling of tumor growth and angiogenesis.

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

INTRODUCTION

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

The tumor as an organ: comprehensive spatial and temporal modeling of the tumor and its microenvironment

BACKGROUND

Research related to cancer is vast, and continues in earnest in many directions. Due to the complexity of cancer, a better understanding of tumor growth dynamics can be gleaned from a dynamic computational model. We present a comprehensive, fully executable, spatial and temporal 3D computational model of the development of a cancerous tumor together with its environment.

RESULTS

The model was created using Statecharts, which were then connected to an interactive animation front-end that we developed especially for this work, making it possible to visualize on the fly the on-going events of the system's execution, as well as the effect of various input parameters. We were thus able to gain a better understanding of, e.g., how different amounts or thresholds of oxygen and VEGF (vascular endothelial growth factor) affect the progression of the tumor. We found that the tumor has a critical turning point, where it either dies or recovers. If minimum conditions are met at that time, it eventually develops into a full, active, growing tumor, regardless of the actual amount; otherwise it dies.

CONCLUSIONS

This brings us to the conclusion that the tumor is in fact a very robust system: changing initial values of VEGF and oxygen can increase the time it takes to become fully developed, but will not necessarily completely eliminate it.

Toward precision medicine of breast cancer

In this review, we report on breast cancer's molecular features and on how high throughput technologies are helping in understanding the dynamics of tumorigenesis and cancer progression with the aim of developing precision medicine methods. We first address the current state of the art in breast cancer therapies and challenges in order to progress towards its cure. Then, we show how the interaction of high-throughput technologies with in silico modeling has led to set up useful inferences for promising strategies of target-specific therapies with low secondary effect incidence for patients. Finally, we discuss the challenge of pharmacogenetics in the clinical practice of cancer therapy. All these issues are explored within the context of precision medicine.

Spatial Metrics of Tumour Vascular Organisation Predict Radiation Efficacy in a Computational Model

Intratumoural heterogeneity is known to contribute to poor therapeutic response. Variations in oxygen tension in particular have been correlated with changes in radiation response in vitro and at the clinical scale with overall survival. Heterogeneity at the microscopic scale in tumour blood vessel architecture has been described, and is one source of the underlying variations in oxygen tension. We seek to determine whether histologic scale measures of the erratic distribution of blood vessels within a tumour can be used to predict differing radiation response. Using a two-dimensional hybrid cellular automaton model of tumour growth, we evaluate the effect of vessel distribution on cell survival outcomes of simulated radiation therapy. Using the standard equations for the oxygen enhancement ratio for cell survival probability under differing oxygen tensions, we calculate average radiation effect over a range of different vessel densities and organisations. We go on to quantify the vessel distribution heterogeneity and measure spatial organization using Ripley’s L function, a measure designed to detect deviations from complete spatial randomness. We find that under differing regimes of vessel density the correlation coefficient between the measure of spatial organization and radiation effect changes sign. This provides not only a useful way to understand the differences seen in radiation effect for tissues based on vessel architecture, but also an alternate explanation for the vessel normalization hypothesis.

In this paper we use a mathematical model, called a hybrid cellular automaton, to study the effect of different vessel distributions on radiation therapy outcomes at the cellular level. We show that the correlation between radiation outcome and spatial organization of vessels changes signs between relatively low and high vessel density. Specifically, that for relatively low vessel density, radiation efficacy is decreased when vessels are more homogeneously distributed, and the opposite is true, that radiation efficacy is improved, when vessel organisation is normalised in high densities. This result suggests an alteration to the vessel normalization hypothesis which states that normalisation of vascular beds should improve radio- and chemo-therapeutic response, but has failed to be validated in clinical studies. In this alteration, we show that Ripley’s L function allows discrimination between vascular architectures in different density regimes in which the standard hypothesis holds and does not hold. Further, we find that this information can be used to augment quantitative histologic analysis of tumours to aid radiation dose personalisation.

Computer Simulations of the Tumor Vasculature: Applications to Interstitial Fluid Flow, Drug Delivery, and Oxygen Supply

Tumor vasculature, the blood vessel network supplying a growing tumor with nutrients such as oxygen or glucose, is in many respects different from the hierarchically organized arterio-venous blood vessel network in normal tissues. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature. Integrative models, based on detailed experimental data and physical laws, implement, in silico, the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. This chapter provides an overview over the current status of computer simulations of vascular remodeling during tumor growth including interstitial fluid flow, drug delivery, and oxygen supply within the tumor. The model predictions are compared with experimental and clinical data and a number of longstanding physiological paradigms about tumor vasculature and intratumoral solute transport are critically scrutinized.

Computational systems biology approaches to anti-angiogenic cancer therapeutics

Angiogenesis is an exquisitely regulated process that is required for physiological processes and is also important in numerous diseases. Tumors utilize angiogenesis to generate the vascular network needed to supply the cancer cells with nutrients and oxygen, and many cancer drugs aim to inhibit tumor angiogenesis. Anti-angiogenic therapy involves inhibiting multiple cell types, molecular targets, and intracellular signaling pathways. Computational tools are useful in guiding treatment strategies, predicting the response to treatment, and identifying new targets of interest. Here, we describe progress that has been made in applying mathematical modeling and bioinformatics approaches to study anti-angiogenic therapeutics in cancer.

Effect of treatment on the global dynamics of delayed pathological angiogenesis models

For three different types of angiogenesis models with variable delays, we consider either continuous or impulse therapy that eradicates tumor cells and suppresses angiogenesis. For the cancer-free solution, explicit conditions of global stability for the continuous and impulsive systems are obtained, together with delay-dependent estimates for the rates of decay for the tumor volume and pathological angiogenesis.

Cellular potts modeling of tumor growth, tumor invasion, and tumor evolution

Despite a growing wealth of available molecular data, the growth of tumors, invasion of tumors into healthy tissue, and response of tumors to therapies are still poorly understood. Although genetic mutations are in general the first step in the development of a cancer, for the mutated cell to persist in a tissue, it must compete against the other, healthy or diseased cells, for example by becoming more motile, adhesive, or multiplying faster. Thus, the cellular phenotype determines the success of a cancer cell in competition with its neighbors, irrespective of the genetic mutations or physiological alterations that gave rise to the altered phenotype. What phenotypes can make a cell "successful" in an environment of healthy and cancerous cells, and how? A widely used tool for getting more insight into that question is cell-based modeling. Cell-based models constitute a class of computational, agent-based models that mimic biophysical and molecular interactions between cells. One of the most widely used cell-based modeling formalisms is the cellular Potts model (CPM), a lattice-based, multi particle cell-based modeling approach. The CPM has become a popular and accessible method for modeling mechanisms of multicellular processes including cell sorting, gastrulation, or angiogenesis. The CPM accounts for biophysical cellular properties, including cell proliferation, cell motility, and cell adhesion, which play a key role in cancer. Multiscale models are constructed by extending the agents with intracellular processes including metabolism, growth, and signaling. Here we review the use of the CPM for modeling tumor growth, tumor invasion, and tumor progression. We argue that the accessibility and flexibility of the CPM, and its accurate, yet coarse-grained and computationally efficient representation of cell and tissue biophysics, make the CPM the method of choice for modeling cellular processes in tumor development.

Properties of tumor spheroid growth exhibited by simple mathematical models

Solid tumors, whether in vitro or in vivo, are not an undifferentiated mass of cells. They include necrotic regions, regions of cells that are in a quiescent state (either slowly growing or not growing at all), and regions where cells proliferate rapidly. The decision of a cell to become quiescent or proliferating is thought to depend on both nutrient and oxygen availability and on the presence of tumor necrosis factor, a substance produced by necrotic cells that somehow inhibits the further growth of the tumor. Several different models have been suggested for the basic growth rate of in vitro tumor spheroids, and several different mechanisms are possible by which tumor necrosis factor might halt growth. The models predict the trajectory of growth for a virtual tumor, including proportions of the various components during its time evolution. In this paper we look at a range of hypotheses about basic rates tumor growth and the role of tumor necrotic factor, and determine what possible tumor growth patterns follow from each of twenty-five reasonable models. Proliferating, quiescent and necrotic cells are included, along with tumor necrosis factor as a potential inhibitor of growth in the proliferating pool and two way exchange between the quiescent and proliferating pools. We show that a range of observed qualitative properties of in vitro tumor spheroids at equilibrium are exhibited by one particular simple mathematical model, and discuss implications of this model for tumor growth.

Using Fractal Geometry and Universal Growth Curves as Diagnostics for Comparing Tumor Vasculature and Metabolic Rate With Healthy Tissue and for Predicting Responses to Drug Therapies

Healthy vasculature exhibits a hierarchical branching structure in which, on average, vessel radius and length change systematically with branching order. In contrast, tumor vasculature exhibits less hierarchy and more variability in its branching patterns. Although differences in vasculature have been highlighted in the literature, there has been very little quantification of these differences. Fractal analysis is a natural tool for comparing tumor and healthy vasculature, especially because it has already been used extensively to model healthy tissue. In this paper, we provide a fractal analysis of existing vascular data, and we present a new mathematical framework for predicting tumor growth trajectories by coupling: (1) the fractal geometric properties of tumor vascular networks, (2) metabolic properties of tumor cells and host vascular systems, and (3) spatial gradients in resources and metabolic states within the tumor. First, we provide a new analysis for how the mean and variation of scaling exponents for ratios of vessel radii and lengths in tumors differ from healthy tissue. Next, we use these characteristic exponents to predict metabolic rates for tumors. Finally, by combining this analysis with general growth equations based on energetics, we derive universal growth curves that enable us to compare tumor and ontogenetic growth. We also extend these growth equations to include necrotic, quiescent, and proliferative cell states and to predict novel growth dynamics that arise when tumors are treated with drugs. Taken together, this mathematical framework will help to anticipate and understand growth trajectories across tumor types and drug treatments.

Phase I trial of OTS11101, an anti-angiogenic vaccine targeting vascular endothelial growth factor receptor 1 in solid tumor

OTS11101 is a novel peptide vaccine that acts as an angiogenesis inhibitor by inducing cytotoxic T lymphocyte (CTL) cells that specifically target vascular endothelial cells expressing vascular endothelial growth factor (VEGF) receptor 1. We conducted a phase I study to evaluate the safety, tolerability, maximum tolerated dose, and pharmacodynamic biomarker status of this vaccine. Nine patients with advanced solid tumors received 1.0, 2.0, or 3.0 mg of OTS11101 subcutaneously, once a week in a 28-day cycle. Three patients experienced grade 1 injection site reactions, which were the most frequent adverse events. Grade 2 proteinuria and hypertension each occurred in one patient. As other toxicities were generally mild, the maximum tolerated dose was not reached. Furthermore, we explored the induction of specific activated CTLs, and biomarkers related to angiogenesis. A pharmacodynamics study revealed that induction of specific CTLs was observed for a dose of 2.0 and 3.0 mg. The serum concentrations of soluble VEGF receptor 1 and 2 after vaccination increased significantly compared with baseline. A microarray was performed to give a comprehensive analysis of gene expression, suggesting that OTS11101 vaccination resulted in T cell activation in a clinical setting. In conclusion, OTS11101 was well tolerated in patients up to 3.0 mg once weekly and our biomarker analysis suggested that this anti-angiogenesis vaccine is biologically active.

Modeling stem/progenitor cell-induced neovascularization and oxygenation around solid implants

Tissue engineering constructs and other solid implants with biomedical applications, such as drug delivery devices or bioartificial organs, need oxygen (O(2)) to function properly. To understand better the vascular integration of such devices, we recently developed a novel model sensor containing O(2)-sensitive crystals, consisting of a polymeric capsule limited by a nanoporous filter. The sensor was implanted in mice with hydrogel alone (control) or hydrogel embedded with mouse CD117/c-kit+ bone marrow progenitor cells in order to stimulate peri-implant neovascularization. The sensor provided local partial O(2) pressure (pO(2)) using noninvasive electron paramagnetic resonance signal measurements. A consistently higher level of peri-implant oxygenation was observed in the cell-treatment case than in the control over a 10-week period. To provide a mechanistic explanation of these experimental observations, we present in this article a mathematical model, formulated as a system of coupled partial differential equations, that simulates peri-implant vascularization. In the control case, vascularization is considered to be the result of a foreign body reaction, while in the cell-treatment case, adipogenesis in response to paracrine stimuli produced by the stem cells is assumed to induce neovascularization. The model is validated by fitting numerical predictions of local pO(2) to measurements from the implanted sensor. The model is then used to investigate further the potential for using stem cell treatment to enhance the vascular integration of biomedical implants. We thus demonstrate how mathematical modeling combined with experimentation can be used to infer how vasculature develops around biomedical implants in control and stem cell-treated cases.

Cancer stem cells in solid tumors: is 'evading apoptosis' a hallmark of cancer?

Conventional wisdom has long held that once a cancer cell has developed it will inevitably progress to clinical disease. Updating this paradigm, it has more recently become apparent that the tumor interacts with its microenvironment and that some environmental bottlenecks, such as the angiogenic switch, must be overcome for the tumor to progress. In parallel, attraction has been drawn to the concept that there is a minority population of cells - the cancer stem cells - bestowed with the exclusive ability to self-renew and regenerate the tumor. With therapeutic targeting issues at stake, much attention has shifted to the identification of cancer stem cells, the thinking being that the remaining non-stem population, already fated to die, will play a negligible role in tumor development. In fact, the newly appreciated importance of intercellular interactions in cancer development also extends in a unique and unexpected way to interactions between the stem and non-stem compartments of the tumor. Here we discuss recent findings drawn from a hybrid mathematical-cellular automaton model that simulates growth of a heterogeneous solid tumor comprised of cancer stem cells and non-stem cancer cells. The model shows how the introduction of cell fate heterogeneity paradoxically influences the tumor growth dynamic in response to apoptosis, to reveal yet another bottleneck to tumor progression potentially exploitable for disease control.

Tumor-initiating cells and tumor vascularization

Tumor-initiating cells (TICs) with stem-like cell properties initiate and sustain progressive growth, resulting in a heterogeneous tumor mass. The survival and growth of tumors rely on the development of a vasculature to provide nutrients and oxygen. Crosstalk between TICs and vascularization may be one of the central players in the initiation, long-term maintenance, and progression of tumors. This review surveys current evidence concerning the crosstalk that occurs in tumor/stromal interactions, including genetic change, vascular niche, hypoxia, and dormancy of tumors. A better understanding of this crosstalk might help provide the basis for developing more effective therapeutic drug targets.

Phase I safety, pharmacokinetic, and biomarker study of BIBF 1120, an oral triple tyrosine kinase inhibitor in patients with advanced solid tumors

BIBF 1120 is an oral multitargeted tyrosine kinase inhibitor that blocks the activity of vascular endothelial growth factor (VEGF) and other growth factor receptors. We have done a phase I study to evaluate the safety, pharmacokinetics, and pharmacodynamic biomarkers of BIBF 1120. Patients with advanced refractory solid tumors were treated with BIBF 1120 at oral doses of 150 to 250 mg twice daily. Drug safety and pharmacokinetics were evaluated, as were baseline and post-treatment levels of circulating CD117-positive bone marrow-derived progenitor cells and plasma soluble VEGF receptor 2 as potential biomarkers for BIBF 1120. Twenty-one patients were treated at BIBF 1120 doses of 150 (n = 3), 200 (n = 12), or 250 mg twice daily (n = 6). Dose-limiting toxicities of reversible grade 3 or 4 elevations of liver enzymes occurred in 3 of 12 patients at 200 mg twice daily and 3 of 6 patients at 250 mg twice daily. Stable disease was achieved in 16 (76.2%) patients, and median progression-free survival was 113 days (95% confidence interval, 77-119 d). Pharmacokinetic analysis indicated that the maximum plasma concentration and area under the curve for BIBF 1120 increased with the dose within the dose range tested. Levels of CD117-positive bone marrow-derived progenitors and soluble VEGF receptor 2 decreased significantly during treatment over all BIBF 1120 dose cohorts. In conclusion, the maximum tolerated dose of BIBF 1120 in the current study was determined to be 200 mg twice daily, and our biomarker analysis indicated that this angiokinase inhibitor is biologically active.

Dissecting cancer through mathematics: from the cell to the animal model

This Timeline article charts progress in mathematical modelling of cancer over the past 50 years, highlighting the different theoretical approaches that have been used to dissect the disease and the insights that have arisen. Although most of this research was conducted with little involvement from experimentalists or clinicians, there are signs that the tide is turning and that increasing numbers of those involved in cancer research and mathematical modellers are recognizing that by working together they might more rapidly advance our understanding of cancer and improve its treatment.

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.

Mathematical modelling of tissue-engineered angiogenesis

We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay.

A three species model to simulate application of Hyperbaric Oxygen Therapy to chronic wounds

Chronic wounds are a significant socioeconomic problem for governments worldwide. Approximately 15% of people who suffer from diabetes will experience a lower-limb ulcer at some stage of their lives, and 24% of these wounds will ultimately result in amputation of the lower limb. Hyperbaric Oxygen Therapy (HBOT) has been shown to aid the healing of chronic wounds; however, the causal reasons for the improved healing remain unclear and hence current HBOT protocols remain empirical. Here we develop a three-species mathematical model of wound healing that is used to simulate the application of hyperbaric oxygen therapy in the treatment of wounds. Based on our modelling, we predict that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds. Furthermore, treatment should continue until healing is complete, and HBOT will not stimulate healing under all circumstances, leading us to conclude that finding the right protocol for an individual patient is crucial if HBOT is to be effective. We provide constraints that depend on the model parameters for the range of HBOT protocols that will stimulate healing. More specifically, we predict that patients with a poor arterial supply of oxygen, high consumption of oxygen by the wound tissue, chronically hypoxic wounds, and/or a dysfunctional endothelial cell response to oxygen are at risk of nonresponsiveness to HBOT. The work of this paper can, in some way, highlight which patients are most likely to respond well to HBOT (for example, those with a good arterial supply), and thus has the potential to assist in improving both the success rate and hence the cost-effectiveness of this therapy.

Structural adaptation and heterogeneity of normal and tumor microvascular networks

Relative to normal tissues, tumor microcirculation exhibits high structural and functional heterogeneity leading to hypoxic regions and impairing treatment efficacy. Here, computational simulations of blood vessel structural adaptation are used to explore the hypothesis that abnormal adaptive responses to local hemodynamic and metabolic stimuli contribute to aberrant morphological and hemodynamic characteristics of tumor microcirculation. Topology, vascular diameter, length, and red blood cell velocity of normal mesenteric and tumor vascular networks were recorded by intravital microscopy. Computational models were used to estimate hemodynamics and oxygen distribution and to simulate vascular diameter adaptation in response to hemodynamic, metabolic and conducted stimuli. The assumed sensitivity to hemodynamic and conducted signals, the vascular growth tendency, and the random variability of vascular responses were altered to simulate ‘normal’ and ‘tumor’ adaptation modes. The heterogeneous properties of vascular networks were characterized by diameter mismatch at vascular branch points (d³_var) and deficit of oxygen delivery relative to demand (O_2def). In the tumor, d³_var and O_2def were higher (0.404 and 0.182) than in normal networks (0.278 and 0.099). Simulated remodeling of the tumor network with ‘normal’ parameters gave low values (0.288 and 0.099). Conversely, normal networks attained tumor-like characteristics (0.41 and 0.179) upon adaptation with ‘tumor’ parameters, including low conducted sensitivity, increased growth tendency, and elevated random biological variability. It is concluded that the deviant properties of tumor microcirculation may result largely from defective structural adaptation, including strongly reduced responses to conducted stimuli.

Blood vessels of tumors have abnormal structures, being irregular and tortuous. Oxygen supply to tumors is heterogeneous, with regions of low oxygen that resist radiation treatment and some types of chemotherapy. Blood vessels undergo continual structural change (adaptation) in response to blood flow and metabolite levels. Our hypothesis is that abnormal adaptation of tumor microvessels causes their heterogeneous structure and impaired function. We used computational models to estimate blood flow and oxygen delivery and to simulate diameter adaptation in networks of microvessels, using network structures derived from microscopic observations of living normal and tumor tissues. The simulation of adaptation depends on several parameters that describe vessel sensitivity to fluid shear stress, to blood pressure, to oxygen levels, and to signals propagated along vessel walls (conducted response). We found that structural adaptation of a tumor network using parameters derived from normal tissues could ‘normalize’ the network, giving it properties similar to a normal tissue. Conversely, adaptation of normal networks using parameters derived from the tumor network, including reduced conducted response, gave tumor-like properties. We conclude that the deviant properties of tumor microcirculation may result largely from defective structural adaptation, including reduced conducted responses.

The dynamics of tumour-vasculature interaction suggests low-dose, time-dense anti-angiogenic schedulings

OBJECTIVES

The administration schedule appears to be a particularly relevant factor in determining the effectiveness of an antiangiogenic drug. A better quantitative knowledge of the interactions between tumour growth and the development of its vasculature could help to design effective therapies.

MATERIAL AND METHODS

Biological and clinical inferences were derived from the analysis of a mathematical model proposed by Hahnfeldt et al. (1999), and some of its variants. In particular, we compared the effect of constant continuous infusion of an anti-angiogenic drug that induces vascular loss, to the effect of periodic, bolus-based therapy.

RESULTS AND CONCLUSIONS

The role of drug elimination rate and of dose fractionation was investigated, and we show that different schedulings, guaranteeing the same mean value of drug concentration, may exhibit very different long-term responses according to their concentration vs. time profile. For a large class of tumour growth laws, the profiles that approach the constant one are the most effective. This behaviour appears to depend on the 'cooperativity' of the tumour-vasculature interaction and on the functional form of the relationship between tumour growth and vasculature extent. Moreover, we suggest that a therapy approaching constant drug infusion might be advantageous also in the case of cytostatic anti-angiogenic drugs.

Elongation, proliferation & migration differentiate endothelial cell phenotypes and determine capillary sprouting

BACKGROUND

Angiogenesis, the growth of capillaries from preexisting blood vessels, has been extensively studied experimentally over the past thirty years. Molecular insights from these studies have lead to therapies for cancer, macular degeneration and ischemia. In parallel, mathematical models of angiogenesis have helped characterize a broader view of capillary network formation and have suggested new directions for experimental pursuit. We developed a computational model that bridges the gap between these two perspectives, and addresses a remaining question in angiogenic sprouting: how do the processes of endothelial cell elongation, migration and proliferation contribute to vessel formation?

RESULTS

We present a multiscale systems model that closely simulates the mechanisms underlying sprouting at the onset of angiogenesis. Designed by agent-based programming, the model uses logical rules to guide the behavior of individual endothelial cells and segments of cells. The activation, proliferation, and movement of these cells lead to capillary growth in three dimensions. By this means, a novel capillary network emerges out of combinatorially complex interactions of single cells. Rules and parameter ranges are based on literature data on endothelial cell behavior in vitro. The model is designed generally, and will subsequently be applied to represent species-specific, tissue-specific in vitro and in vivo conditions. Initial results predict tip cell activation, stalk cell development and sprout formation as a function of local vascular endothelial growth factor concentrations and the Delta-like 4 Notch ligand, as it might occur in a three-dimensional in vitro setting. Results demonstrate the differential effects of ligand concentrations, cell movement and proliferation on sprouting and directional persistence.

CONCLUSION

This systems biology model offers a paradigm closely related to biological phenomena and highlights previously unexplored interactions of cell elongation, migration and proliferation as a function of ligand concentration, giving insight into key cellular mechanisms driving angiogenesis.