A mathematical model of tumour angiogenesis: growth, regression and regrowth

Skin inflammatory signatures, as measured by disordered spatial redness patterns, predict current and future skin ageing attributes

Facial skin redness can be an indicator of skin inflammation, however the physiological connection between facial redness and inflammatory status, as well as its role in age-related skin changes, remains poorly understood. This study aims to investigate the association between the pattern of facial skin redness and biological inflammatory status, as well as age-related changes occurring in the skin. Four studies were conducted recruiting healthy Northern Asian females. Disordered spatial patterns of facial skin redness signals were assessed using image analysis, i.e., the a* gradient algorithm, which quantifies the disordered shape and pattern of localized redness signals on facial skin. This redness pattern was compared with (1) inflammatory protein markers (IL-1Ra/ IL-1α and IL-8) measured from stripped corneocyte samples, (2) gene expression profiles obtained through transcriptome analysis using skin biopsy samples, and (3) the distribution pattern of blood vessel measured using a photoacoustic microscope. The association between the skin redness pattern and current and future ageing-related skin changes was examined through a longitudinal study tracking the same subjects for 10 years. A significant correlation was observed between the a* gradient and the levels of inflammatory cytokines (IL-1Ra/IL-1α and IL-8). Transcriptome analysis revealed upregulation of genes related to acute inflammation, chronic inflammation, cellular senescence, and angiogenesis in subjects with higher a* gradients. The high a* gradient group exhibited an extension of blood vessel diameter and increased blood vessel density, while the medium a* gradient group only exhibited blood vessel extension. Lastly, the 10-year longitudinal study demonstrated that the a* gradient was associated with current and future skin ageing-related attributes, such as increased skin texture and wrinkle formation. The spatial pattern of localized redness on the skin reflects the biological inflammatory status, and this inflammatory condition helps predict current and future age-related skin changes.

Simulation of Soluble and Bound VEGF-stimulated in vitro Capillary-like Network Formation on Deformed Substrate

Capillary plexus cultivation is crucial in tissue engineering and regenerative medicine. Theoretical simulations have been conducted to supplement the expensive experimental works. However, the mechanisms connecting mechanical and chemical stimuli remained undefined, and the functions of the different VEGF forms in the culture environment were still unclear. In this paper, we developed a hybrid model for simulating short-term in vitro capillary incubations. We used the Cellular Potts model to predict individual cell migration, morphology change, and continuum mechanics to quantify biogel deformation and VEGF transport dynamics. By bridging the mechanical regulation and chemical stimulation in the model, the results showed good agreement between the predicted network topology and experiments, in which elongated cells connected, forming the network cords and round cells gathered, creating cobblestone-like aggregates. The results revealed that the capillary-like networks could develop in high integrity only when the mechanical and chemical couplings worked adequately, with the cell morphology and haptotaxis driven by the soluble and bound forms of VEGF, respectively, functioning simultaneously.

Disordered spatial pattern of redness signal on facial skin and visual perception of health, stress, and hidden aging

BACKGROUND

Well-being is commonly communicated across industries; however, experimental understanding how human perceive skin health and skin stresses are not sufficient.

MATERIALS AND METHODS

Image analysis algorithm, a* gradient, was developed to evaluate spatial pattern and shape of red signal on skin. Human perception for skin health and stresses were compared with technical measurements in two visual perception studies.

RESULTS

a* gradient correlated with perceived Inflamed Skin (R = 0.73, p < 0.01), Stressed Skin (R = 0.79, p < 0.01), Sensitive Skin (R = 0.75, p < 0.01), Healthy Skin (R = -0.83, p < 0.01), and Start Aging (R = 0.75, p < 0.01).

CONCLUSIONS

Disordered spatial pattern of redness signal drives human perception of skin health, stress, and aging. This new skin index of redness signal shows higher correlation with those human perception than basal a* mean, unevenness of a*, and other conventional skin color attributes.

Computational modeling of angiogenesis: The importance of cell rearrangements during vascular growth

Angiogenesis is the process wherein endothelial cells (ECs) form sprouts that elongate from the pre-existing vasculature to create new vascular networks. In addition to its essential role in normal development, angiogenesis plays a vital role in pathologies such as cancer, diabetes and atherosclerosis. Mathematical and computational modeling has contributed to unraveling its complexity. Many existing theoretical models of angiogenic sprouting are based on the "snail-trail" hypothesis. This framework assumes that leading ECs positioned at sprout tips migrate toward low-oxygen regions while other ECs in the sprout passively follow the leaders' trails and proliferate to maintain sprout integrity. However, experimental results indicate that, contrary to the snail-trail assumption, ECs exchange positions within developing vessels, and the elongation of sprouts is primarily driven by directed migration of ECs. The functional role of cell rearrangements remains unclear. This review of the theoretical modeling of angiogenesis is the first to focus on the phenomenon of cell mixing during early sprouting. We start by describing the biological processes that occur during early angiogenesis, such as phenotype specification, cell rearrangements and cell interactions with the microenvironment. Next, we provide an overview of various theoretical approaches that have been employed to model angiogenesis, with particular emphasis on recent in silico models that account for the phenomenon of cell mixing. Finally, we discuss when cell mixing should be incorporated into theoretical models and what essential modeling components such models should include in order to investigate its functional role. This article is categorized under: Cardiovascular Diseases > Computational Models Cancer > Computational Models.

Metastasis Models: Thermodynamics and Complexity

The thermodynamic formalism of nonequilibrium systems together with the theory of complex systems and systems biology offer an appropriate theoretical framework to explain the complexity observed at the macroscopic level in physiological phenomena. In turn, they allow the establishment of an appropriate conceptual and operational framework to address the study of phenomena such as the emergence and evolution of cancer.This chapter is organized as follows: In Subheading 1, an integrated vision of these disciplines is offered for the characterization of the emergence and evolution of cancer, seen as a nonlinear dynamic system, temporally and spatially self-organized out of thermodynamic equilibrium. The development of the various mathematical models and different techniques and approaches used in the characterization of cancer metastasis is presented in Subheading 2. Subheading 3 is devoted to the time course of cancer metastasis, with particular emphasis on the epithelial-mesenchymal transition (EMT henceforth) as well as chronotherapeutic treatments. In Subheading 4, models of the spatial evolution of cancer metastasis are presented. Finally, in Subheading 5, some conclusions and remarks are presented.

Numerical investigation of the pharmacokinetics and pharmacodynamics of the chemotherapeutic drug in avascular and vascular stages of a brain tumor

One of the most commonly used approaches for treating solid tumors is the systemic delivery of chemotherapeutic drugs. However, our understanding of the factors influencing treatment efficacy through this method is still limited. This study presents a comprehensive and realistic mathematical model that incorporates the dynamics of tumor growth, capillary network extension, and drug delivery in a coupled and simultaneous manner. The model covers two stages of tumor growth: avascular and vascular. For tumor growth, a continuum model is employed using the phase field interface-capturing method. The neo-vascularization process is modeled using a hybrid discrete-continuum approach. Additionally, a multi-scale model is used to describe the pharmacokinetics of doxorubicin, considering various agents. The study investigates the effect of haptotaxis and reveals that a higher haptotaxis coefficient leads to faster tumor growth (up to 2.6 times) and a quicker progression to angiogenesis. The impact of tumor-related and drug-related parameters is also examined, including tumor size, tumor sensitivity to the drug, chemotherapy initialization, treatment cycle duration, drug affinity to cells, and drug dose. The findings indicate that chemotherapy is more effective during the angiogenesis stage when active loops have formed. Other clinical methods such as radiotherapy and surgery may be more appropriate during the avascular stage or the transition period between angiogenesis initialization and loop formation. The penetration depth of the drug decreases by approximately 50% with an increase in the drug binding rate to surface-cell receptors. As a result, high-associate-rate drugs are preferred for chemotherapy after active loops have formed, while low-associate-rate drugs are suitable for earlier stages.

Approaches to vascular network, blood flow, and metabolite distribution modeling in brain tissue

The cardiovascular system plays a key role in the transport of nutrients, ensuring a continuous supply of all cells of the body with the metabolites necessary for life. The blood supply to the brain is carried out by the large arteries located on its surface, which branch into smaller arterioles that penetrate the cerebral cortex and feed the capillary bed, thereby forming an extensive branching network. The formation of blood vessels is carried out via vasculogenesis and angiogenesis, which play an important role in both embryo and adult life. The review presents approaches to modeling various aspects of both the formation of vascular networks and the construction of the formed arterial tree. In addition, a brief description of models that allows one to study the blood flow in various parts of the circulatory system and the spatiotemporal metabolite distribution in brain tissues is given. Experimental study of these issues is not always possible due to both the complexity of the cardiovascular system and the mechanisms through which the perfusion of all body cells is carried out. In this regard, mathematical models are a good tool for studying hemodynamics and can be used in clinical practice to diagnose vascular diseases and assess the need for treatment.

Mechanistic characterization of oscillatory patterns in unperturbed tumor growth dynamics: The interplay between cancer cells and components of tumor microenvironment

Mathematical modeling of unperturbed and perturbed tumor growth dynamics (TGD) in preclinical experiments provides an opportunity to establish translational frameworks. The most commonly used unperturbed tumor growth models (i.e. linear, exponential, Gompertz and Simeoni) describe a monotonic increase and although they capture the mean trend of the data reasonably well, systematic model misspecifications can be identified. This represents an opportunity to investigate possible underlying mechanisms controlling tumor growth dynamics through a mathematical framework. The overall goal of this work is to develop a data-driven semi-mechanistic model describing non-monotonic tumor growth in untreated mice. For this purpose, longitudinal tumor volume profiles from different tumor types and cell lines were pooled together and analyzed using the population approach. After characterizing the oscillatory patterns (oscillator half-periods between 8-11 days) and confirming that they were systematically observed across the different preclinical experiments available (p<10-9), a tumor growth model was built including the interplay between resources (i.e. oxygen or nutrients), angiogenesis and cancer cells. The new structure, in addition to improving the model diagnostic compared to the previously used tumor growth models (i.e. AIC reduction of 71.48 and absence of autocorrelation in the residuals (p>0.05)), allows the evaluation of the different oncologic treatments in a mechanistic way. Drug effects can potentially, be included in relevant processes taking place during tumor growth. In brief, the new model, in addition to describing non-monotonic tumor growth and the interaction between biological factors of the tumor microenvironment, can be used to explore different drug scenarios in monotherapy or combination during preclinical drug development.

Bridging Scales: a Hybrid Model to Simulate Vascular Tumor Growth and Treatment Response

Cancer is a disease driven by random DNA mutations and the interaction of many complex phenomena. To improve the understanding and ultimately find more effective treatments, researchers leverage computer simulations mimicking the tumor growth in silico. The challenge here is to account for the many phenomena influencing the disease progression and treatment protocols. This work introduces a computational model to simulate vascular tumor growth and the response to drug treatments in 3D. It consists of two agent-based models for the tumor cells and the vasculature. Moreover, partial differential equations govern the diffusive dynamics of the nutrients, the vascular endothelial growth factor, and two cancer drugs. The model focuses explicitly on breast cancer cells over-expressing HER2 receptors and a treatment combining standard chemotherapy (Doxorubicin) and monoclonal antibodies with anti-angiogenic properties (Trastuzumab). However, large parts of the model generalize to other scenarios. We show that the model qualitatively captures the effects of the combination therapy by comparing our simulation results with previously published pre-clinical data. Furthermore, we demonstrate the scalability of the model and the associated C++ code by simulating a vascular tumor occupying a volume of 400mm3 using a total of 92.5 million agents.

Modelling of chemotactic sprouting endothelial cells through an extracellular matrix

Sprouting angiogenesis is a core biological process critical to vascular development. Its accurate simulation, relevant to multiple facets of human health, is of broad, interdisciplinary appeal. This study presents an in-silico model replicating a microfluidic assay where endothelial cells sprout into a biomimetic extracellular matrix, specifically, a large-pore, low-concentration fibrin-based porous hydrogel, influenced by chemotactic factors. We introduce a novel approach by incorporating the extracellular matrix and chemotactic factor effects into a unified term using a single parameter, primarily focusing on modelling sprouting dynamics and morphology. This continuous model naturally describes chemotactic-induced sprouting with no need for additional rules. In addition, we extended our base model to account for matrix sensing and degradation, crucial aspects of angiogenesis. We validate our model via a hybrid in-silico experimental method, comparing the model predictions with experimental results derived from the microfluidic setup. Our results underscore the intricate relationship between the extracellular matrix structure and angiogenic sprouting, proposing a promising method for predicting the influence of the extracellular matrix on angiogenesis.

Simulation of Angiogenesis in Three Dimensions: Development of the Retinal Circulation

A theoretical model is used to describe the three-dimensional development of the retinal circulation in the human eye, which occurs after the initial spread of vasculature across the inner surface of the retina. In the model, random sprouting angiogenesis is driven by a growth factor that is produced in tissue at a rate dependent on oxygen level and diffuses to existing vessels. Vessel sprouts connect to form pathways for blood flow and undergo remodeling and pruning. These processes are controlled by known or hypothesized vascular responses to hemodynamic and biochemical stimuli, including conducted responses along vessel walls. The model shows regression of arterio-venous connections on the surface of the retina, allowing perfusion of the underlying tissue. A striking feature of the retinal circulation is the formation of two vascular plexuses located at the inner and outer surfaces of the inner nuclear layer within the retina. The model is used to test hypotheses regarding the formation of these structures. A mechanism based on local production and diffusion of growth factor is shown to be ineffective. However, sprout guidance by localized structures on the boundaries of the inner nuclear layer can account for plexus formation. The resulting networks have vascular density, perfusion and oxygen transport characteristics consistent with observed properties. The model shows how stochastic generation of vascular sprouts combined with a set of biologically based response mechanisms can lead to the generation of a specialized three-dimensional vascular structure with a high degree of organization.

A mathematical model of fibrinogen-mediated erythrocyte-erythrocyte adhesion

Erythrocytes are deformable cells that undergo progressive biophysical and biochemical changes affecting the normal blood flow. Fibrinogen, one of the most abundant plasma proteins, is a primary determinant for changes in haemorheological properties, and a major independent risk factor for cardiovascular diseases. In this study, the adhesion between human erythrocytes is measured by atomic force microscopy (AFM) and its effect observed by micropipette aspiration technique, in the absence and presence of fibrinogen. These experimental data are then used in the development of a mathematical model to examine the biomedical relevant interaction between two erythrocytes. Our designed mathematical model is able to explore the erythrocyte-erythrocyte adhesion forces and changes in erythrocyte morphology. AFM erythrocyte-erythrocyte adhesion data show that the work and detachment force necessary to overcome the adhesion between two erythrocytes increase in the presence of fibrinogen. The changes in erythrocyte morphology, the strong cell-cell adhesion and the slow separation of the two cells are successfully followed in the mathematical simulation. Erythrocyte-erythrocyte adhesion forces and energies are quantified and matched with experimental data. The changes observed on erythrocyte-erythrocyte interactions may give important insights about the pathophysiological relevance of fibrinogen and erythrocyte aggregation in hindering microcirculatory blood flow.

Geometric analysis enables biological insight from complex non-identifiable models using simple surrogates

An enduring challenge in computational biology is to balance data quality and quantity with model complexity. Tools such as identifiability analysis and information criterion have been developed to harmonise this juxtaposition, yet cannot always resolve the mismatch between available data and the granularity required in mathematical models to answer important biological questions. Often, it is only simple phenomenological models, such as the logistic and Gompertz growth models, that are identifiable from standard experimental measurements. To draw insights from complex, non-identifiable models that incorporate key biological mechanisms of interest, we study the geometry of a map in parameter space from the complex model to a simple, identifiable, surrogate model. By studying how non-identifiable parameters in the complex model quantitatively relate to identifiable parameters in surrogate, we introduce and exploit a layer of interpretation between the set of non-identifiable parameters and the goodness-of-fit metric or likelihood studied in typical identifiability analysis. We demonstrate our approach by analysing a hierarchy of mathematical models for multicellular tumour spheroid growth experiments. Typical data from tumour spheroid experiments are limited and noisy, and corresponding mathematical models are very often made arbitrarily complex. Our geometric approach is able to predict non-identifiabilities, classify non-identifiable parameter spaces into identifiable parameter combinations that relate to features in the data characterised by parameters in a surrogate model, and overall provide additional biological insight from complex non-identifiable models.

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

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

Towards integration of time-resolved confocal microscopy of a 3D in vitro microfluidic platform with a hybrid multiscale model of tumor angiogenesis

The goal of this study is to calibrate a multiscale model of tumor angiogenesis with time-resolved data to allow for systematic testing of mathematical predictions of vascular sprouting. The multi-scale model consists of an agent-based description of tumor and endothelial cell dynamics coupled to a continuum model of vascular endothelial growth factor concentration. First, we calibrate ordinary differential equation models to time-resolved protein concentration data to estimate the rates of secretion and consumption of vascular endothelial growth factor by endothelial and tumor cells, respectively. These parameters are then input into the multiscale tumor angiogenesis model, and the remaining model parameters are then calibrated to time resolved confocal microscopy images obtained within a 3D vascularized microfluidic platform. The microfluidic platform mimics a functional blood vessel with a surrounding collagen matrix seeded with inflammatory breast cancer cells, which induce tumor angiogenesis. Once the multi-scale model is fully parameterized, we forecast the spatiotemporal distribution of vascular sprouts at future time points and directly compare the predictions to experimentally measured data. We assess the ability of our model to globally recapitulate angiogenic vasculature density, resulting in an average relative calibration error of 17.7% ± 6.3% and an average prediction error of 20.2% ± 4% and 21.7% ± 3.6% using one and four calibrated parameters, respectively. We then assess the model's ability to predict local vessel morphology (individualized vessel structure as opposed to global vascular density), initialized with the first time point and calibrated with two intermediate time points. In this study, we have rigorously calibrated a mechanism-based, multiscale, mathematical model of angiogenic sprouting to multimodal experimental data to make specific, testable predictions.

Angiogenesis goes computational - The future way forward to discover new angiogenic targets?

Multi-omics technologies are being increasingly utilized in angiogenesis research. Yet, computational methods have not been widely used for angiogenic target discovery and prioritization in this field, partly because (wet-lab) vascular biologists are insufficiently familiar with computational biology tools and the opportunities they may offer. With this review, written for vascular biologists who lack expertise in computational methods, we aspire to break boundaries between both fields and to illustrate the potential of these tools for future angiogenic target discovery. We provide a comprehensive survey of currently available computational approaches that may be useful in prioritizing candidate genes, predicting associated mechanisms, and identifying their specificity to endothelial cell subtypes. We specifically highlight tools that use flexible, machine learning frameworks for large-scale data integration and gene prioritization. For each purpose-oriented category of tools, we describe underlying conceptual principles, highlight interesting applications and discuss limitations. Finally, we will discuss challenges and recommend some guidelines which can help to optimize the process of accurate target discovery.

Integrating a dynamic central metabolism model of cancer cells with a hybrid 3D multiscale model for vascular hepatocellular carcinoma growth

We develop here a novel modelling approach with the aim of closing the conceptual gap between tumour-level metabolic processes and the metabolic processes occurring in individual cancer cells. In particular, the metabolism in hepatocellular carcinoma derived cell lines (HEPG2 cells) has been well characterized but implementations of multiscale models integrating this known metabolism have not been previously reported. We therefore extend a previously published multiscale model of vascular tumour growth, and integrate it with an experimentally verified network of central metabolism in HEPG2 cells. This resultant combined model links spatially heterogeneous vascular tumour growth with known metabolic networks within tumour cells and accounts for blood flow, angiogenesis, vascular remodelling and nutrient/growth factor transport within a growing tumour, as well as the movement of, and interactions between normal and cancer cells. Model simulations report for the first time, predictions of spatially resolved time courses of core metabolites in HEPG2 cells. These simulations can be performed at a sufficient scale to incorporate clinically relevant features of different tumour systems using reasonable computational resources. Our results predict larger than expected temporal and spatial heterogeneity in the intracellular concentrations of glucose, oxygen, lactate pyruvate, f16bp and Acetyl-CoA. The integrated multiscale model developed here provides an ideal quantitative framework in which to study the relationship between dosage, timing, and scheduling of anti-neoplastic agents and the physiological effects of tumour metabolism at the cellular level. Such models, therefore, have the potential to inform treatment decisions when drug response is dependent on the metabolic state of individual cancer cells.

Simulating the behaviour of glioblastoma multiforme based on patient MRI during treatments

Glioblastoma multiforme is a brain cancer that still shows poor prognosis for patients despite the active research for new treatments. In this work, the goal is to model and simulate the evolution of tumour associated angiogenesis and the therapeutic response to glioblastoma multiforme. Multiple phenomena are modelled in order to fit different biological pathways, such as the cellular cycle, apoptosis, hypoxia or angiogenesis. This leads to a nonlinear system with 4 equations and 4 unknowns: the density of tumour cells, the [Formula: see text] concentration, the density of endothelial cells and the vascular endothelial growth factor concentration. This system is solved numerically on a mesh fitting the geometry of the brain and the tumour of a patient based on a 2D slice of MRI. We show that our numerical scheme is positive, and we give the energy estimates on the discrete solution to ensure its existence. The numerical scheme uses nonlinear control volume finite elements in space and is implicit in time. Numerical simulations have been done using the different standard treatments: surgery, chemotherapy and radiotherapy, in order to conform to the behaviour of a tumour in response to treatments according to empirical clinical knowledge. We find that our theoretical model exhibits realistic behaviours.

A Fokker-Planck feedback control framework for optimal personalized therapies in colon cancer-induced angiogenesis

In this paper, a new framework for obtaining personalized optimal treatment strategies in colon cancer-induced angiogenesis is presented. The dynamics of colon cancer is given by a Itó stochastic process, which helps in modeling the randomness present in the system. The stochastic dynamics is then represented by the Fokker-Planck (FP) partial differential equation that governs the evolution of the associated probability density function. The optimal therapies are obtained using a three step procedure. First, a finite dimensional FP-constrained optimization problem is formulated that takes input individual noisy patient data, and is solved to obtain the unknown parameters corresponding to the individual tumor characteristics. Next, a sensitivity analysis of the optimal parameter set is used to determine the parameters to be controlled, thus, helping in assessing the types of treatment therapies. Finally, a feedback FP control problem is solved to determine the optimal combination therapies. Numerical results with the combination drug, comprising of Bevacizumab and Capecitabine, demonstrate the efficiency of the proposed framework.

OUP accepted manuscript

Motivation

Inferring the parameters of models describing biological systems is an important problem in the reverse engineering of the mechanisms underlying these systems. Much work has focused on parameter inference of stochastic and ordinary differential equation models using Approximate Bayesian Computation (ABC). While there is some recent work on inference in spatial models, this remains an open problem. Simultaneously, advances in topological data analysis (TDA), a field of computational mathematics, have enabled spatial patterns in data to be characterized.

Results

Here, we focus on recent work using TDA to study different regimes of parameter space for a well-studied model of angiogenesis. We propose a method for combining TDA with ABC to infer parameters in the Anderson–Chaplain model of angiogenesis. We demonstrate that this topological approach outperforms ABC approaches that use simpler statistics based on spatial features of the data. This is a first step toward a general framework of spatial parameter inference for biological systems, for which there may be a variety of filtrations, vectorizations and summary statistics to be considered.

Availability and implementation

All code used to produce our results is available as a Snakemake workflow from github.com/tt104/tabc_angio.

Simulation of angiogenesis in three dimensions: Application to cerebral cortex

The vasculature is a dynamic structure, growing and regressing in response to embryonic development, growth, changing physiological demands, wound healing, tumor growth and other stimuli. At the microvascular level, network geometry is not predetermined, but emerges as a result of biological responses of each vessel to the stimuli that it receives. These responses may be summarized as angiogenesis, remodeling and pruning. Previous theoretical simulations have shown how two-dimensional vascular patterns generated by these processes in the mesentery are consistent with experimental observations. During early development of the brain, a mesh-like network of vessels is formed on the surface of the cerebral cortex. This network then forms branches into the cortex, forming a three-dimensional network throughout its thickness. Here, a theoretical model is presented for this process, based on known or hypothesized vascular response mechanisms together with experimentally obtained information on the structure and hemodynamics of the mouse cerebral cortex. According to this model, essential components of the system include sensing of oxygen levels in the midrange of partial pressures and conducted responses in vessel walls that propagate information about metabolic needs of the tissue to upstream segments of the network. The model provides insights into the effects of deficits in vascular response mechanisms, and can be used to generate physiologically realistic microvascular network structures.

Biologically-Based Mathematical Modeling of Tumor Vasculature and Angiogenesis via Time-Resolved Imaging Data

Tumor-associated vasculature is responsible for the delivery of nutrients, removal of waste, and allowing growth beyond 2-3 mm3. Additionally, the vascular network, which is changing in both space and time, fundamentally influences tumor response to both systemic and radiation therapy. Thus, a robust understanding of vascular dynamics is necessary to accurately predict tumor growth, as well as establish optimal treatment protocols to achieve optimal tumor control. Such a goal requires the intimate integration of both theory and experiment. Quantitative and time-resolved imaging methods have emerged as technologies able to visualize and characterize tumor vascular properties before and during therapy at the tissue and cell scale. Parallel to, but separate from those developments, mathematical modeling techniques have been developed to enable in silico investigations into theoretical tumor and vascular dynamics. In particular, recent efforts have sought to integrate both theory and experiment to enable data-driven mathematical modeling. Such mathematical models are calibrated by data obtained from individual tumor-vascular systems to predict future vascular growth, delivery of systemic agents, and response to radiotherapy. In this review, we discuss experimental techniques for visualizing and quantifying vascular dynamics including magnetic resonance imaging, microfluidic devices, and confocal microscopy. We then focus on the integration of these experimental measures with biologically based mathematical models to generate testable predictions.

Topological data analysis distinguishes parameter regimes in the Anderson-Chaplain model of angiogenesis

Angiogenesis is the process by which blood vessels form from pre-existing vessels. It plays a key role in many biological processes, including embryonic development and wound healing, and contributes to many diseases including cancer and rheumatoid arthritis. The structure of the resulting vessel networks determines their ability to deliver nutrients and remove waste products from biological tissues. Here we simulate the Anderson-Chaplain model of angiogenesis at different parameter values and quantify the vessel architectures of the resulting synthetic data. Specifically, we propose a topological data analysis (TDA) pipeline for systematic analysis of the model. TDA is a vibrant and relatively new field of computational mathematics for studying the shape of data. We compute topological and standard descriptors of model simulations generated by different parameter values. We show that TDA of model simulation data stratifies parameter space into regions with similar vessel morphology. The methodologies proposed here are widely applicable to other synthetic and experimental data including wound healing, development, and plant biology.

Mathematical simulation of tumour angiogenesis: angiopoietin balance is a key factor in vessel growth and regression

Excessive tumour growth results in a hypoxic environment around cancer cells, thus inducing tumour angiogenesis, which refers to the generation of new blood vessels from pre-existing vessels. This mechanism is biologically and physically complex, with various mathematical simulation models proposing to reproduce its formation. However, although temporary vessel regression is clinically known, few models succeed in reproducing this phenomenon. Here, we developed a three-dimensional simulation model encompassing both angiogenesis and tumour growth, specifically including angiopoietin. Angiopoietin regulates both adhesion and migration between vascular endothelial cells and wall cells, thus inhibiting the cell-to-cell adhesion required for angiogenesis initiation. Simulation results showed a regression, i.e. transient decrease, in the overall length of new vessels during vascular network formation. Using our model, we also evaluated the efficacy of administering the drug bevacizumab. The results highlighted differences in treatment efficacy: (1) earlier administration showed higher efficacy in inhibiting tumour growth, and (2) efficacy depended on the treatment interval even with the administration of the same dose. After thorough validation in the future, these results will contribute to the design of angiogenesis treatment protocols.

A multiscale model of complex endothelial cell dynamics in early angiogenesis

We introduce a hybrid two-dimensional multiscale model of angiogenesis, the process by which endothelial cells (ECs) migrate from a pre-existing vascular bed in response to local environmental cues and cell-cell interactions, to create a new vascular network. Recent experimental studies have highlighted a central role of cell rearrangements in the formation of angiogenic networks. Our model accounts for this phenomenon via the heterogeneous response of ECs to their microenvironment. These cell rearrangements, in turn, dynamically remodel the local environment. The model reproduces characteristic features of angiogenic sprouting that include branching, chemotactic sensitivity, the brush border effect, and cell mixing. These properties, rather than being hardwired into the model, emerge naturally from the gene expression patterns of individual cells. After calibrating and validating our model against experimental data, we use it to predict how the structure of the vascular network changes as the baseline gene expression levels of the VEGF-Delta-Notch pathway, and the composition of the extracellular environment, vary. In order to investigate the impact of cell rearrangements on the vascular network structure, we introduce the mixing measure, a scalar metric that quantifies cell mixing as the vascular network grows. We calculate the mixing measure for the simulated vascular networks generated by ECs of different lineages (wild type cells and mutant cells with impaired expression of a specific receptor). Our results show that the time evolution of the mixing measure is directly correlated to the generic features of the vascular branching pattern, thus, supporting the hypothesis that cell rearrangements play an essential role in sprouting angiogenesis. Furthermore, we predict that lower cell rearrangement leads to an imbalance between branching and sprout elongation. Since the computation of this statistic requires only individual cell trajectories, it can be computed for networks generated in biological experiments, making it a potential biomarker for pathological angiogenesis.

Angiogenesis, the process by which new blood vessels are formed by sprouting from the pre-existing vascular bed, plays a key role in both physiological and pathological processes, including tumour growth. The structure of a growing vascular network is determined by the coordinated behaviour of endothelial cells in response to various signalling cues. Recent experimental studies have highlighted the importance of cell rearrangements as a driver for sprout elongation. However, the functional role of this phenomenon remains unclear. We formulate a new multiscale model of angiogenesis which, by accounting explicitly for the complex dynamics of endothelial cells within growing angiogenic sprouts, is able to reproduce generic features of angiogenic structures (branching, chemotactic sensitivity, cell mixing, etc.) as emergent properties of its dynamics. We validate our model against experimental data and then use it to quantify the phenomenon of cell mixing in vascular networks generated by endothelial cells of different lineages. Our results show that there is a direct correlation between the time evolution of cell mixing in a growing vascular network and its branching structure, thus paving the way for understanding the functional role of cell rearrangements in angiogenesis.

Digitally Fabricated and Naturally Augmented In Vitro Tissues

Human in vitro tissues are extracorporeal 3D cultures of human cells embedded in biomaterials, commonly hydrogels, which recapitulate the heterogeneous, multiscale, and architectural environment of the human body. Contemporary strategies used in 3D tissue and organ engineering integrate the use of automated digital manufacturing methods, such as 3D printing, bioprinting, and biofabrication. Human tissues and organs, and their intra- and interphysiological interplay, are particularly intricate. For this reason, attentiveness is rising to intersect materials science, medicine, and biology with arts and informatics. This report presents advances in computational modeling of bioink polymerization and its compatibility with bioprinting, the use of digital design and fabrication in the development of fluidic culture devices, and the employment of generative algorithms for modeling the natural and biological augmentation of in vitro tissues. As a future direction, the use of serially linked in vitro tissues as human body-mimicking systems and their application in drug pharmacokinetics and metabolism, disease modeling, and diagnostics are discussed.

Phase-field modeling of constrained interactive fungal networks

Fungi develop structures that interact with their surroundings and evolve adaptively in the presence of geometrical constraints, finding optimal solutions for complex combinatorial problems. The pathogenic fungus Ophiocordyceps constitutes a perfect model for the study of constrained interactive networks. Modeling these networks is challenging due to the highly coupled physics involved and their interaction with moving boundaries. In this work, we develop a computational phase-field model to elucidate the mechanics of the emerging properties observed in fungal networks. We use a variational approach to derive the equations governing the evolution in time of the mycelium biomass and the nutrients in the medium. We present an extensive testing of our model, reproduce growing and decaying phenomena, and capture spatial and temporal scales. We explore the variables interplay mechanism that leads to different colony morphologies, and explain abrupt changes of patterns observed in the laboratory. We apply our model to simulate analogous processes to the evolution of Ophiocordyceps as it grows through confined geometry and depletes available resources, demonstrating the suitability of the formulation to study this class of biological networks.

Adhesion modulates cell morphology and migration within dense fibrous networks

One of the most fundamental abilities required for the sustainability of complex life forms is active cell migration, since it is essential in diverse processes from morphogenesis to leukocyte chemotaxis in immune response. The movement of a cell is the result of intricate mechanisms, that involve the coordination between mechanical forces, biochemical regulatory pathways and environmental cues. In particular, epithelial cancer cells have to employ mechanical strategies in order to migrate through the tissue's basement membrane and infiltrate the bloodstream during the invasion stage of metastasis. In this work we explore how mechanical interactions such as spatial restriction and adhesion affect migration of a self-propelled droplet in dense fibrous media. We have performed a systematic analysis using a phase-field model and we propose a novel approach to simulate cell migration with dissipative particle dynamics modelling. With this purpose we have measured in our simulation the cell's velocity and quantified its morphology as a function of the fibre density and of its adhesiveness to the matrix fibres. Furthermore, we have compared our results to a previousin vitromigration assay of fibrosarcoma cells in fibrous matrices. The results show good agreement between the two methodologies and experiments in the literature, which indicates that these minimalist descriptions are able to capture the main features of the system. Our results indicate that adhesiveness is critical for cell migration, by modulating cell morphology in crowded environments and by enhancing cell velocity. In addition, our analysis suggests that matrix metalloproteinases (MMPs) play an important role as adhesiveness modulators. We propose that new assays should be carried out to address the role of adhesion and the effect of different MMPs in cell migration under confined conditions.

A hybrid model of tumor growth and angiogenesis: In silico experiments

Tumor associated angiogenesis is the development of new blood vessels in response to proteins secreted by tumor cells. These new blood vessels allow tumors to continue to grow beyond what the pre-existing vasculature could support. Here, we construct a mathematical model to simulate tumor angiogenesis by considering each endothelial cell as an agent, and allowing the vascular endothelial growth factor (VEGF) and nutrient fields to impact the dynamics and phenotypic transitions of each tumor and endothelial cell. The phenotypes of the endothelial cells (i.e., tip, stalk, and phalanx cells) are selected by the local VEGF field, and govern the migration and growth of vessel sprouts at the cellular level. Over time, these vessels grow and migrate to the tumor, forming anastomotic loops to supply nutrients, while interacting with the tumor through mechanical forces and the consumption of VEGF. The model is able to capture collapsing and breaking of vessels caused by tumor-endothelial cell interactions. This is accomplished through modeling the physical interaction between the vasculature and the tumor, resulting in vessel occlusion and tumor heterogeneity over time due to the stages of response in angiogenesis. Key parameters are identified through a sensitivity analysis based on the Sobol method, establishing which parameters should be the focus of subsequent experimental efforts. During the avascular phase (i.e., before angiogenesis is triggered), the nutrient consumption rate, followed by the rate of nutrient diffusion, yield the greatest influence on the number and distribution of tumor cells. Similarly, the consumption and diffusion of VEGF yield the greatest influence on the endothelial and tumor cell numbers during angiogenesis. In summary, we present a hybrid mathematical approach that characterizes vascular changes via an agent-based model, while treating nutrient and VEGF changes through a continuum model. The model describes the physical interaction between a tumor and the surrounding blood vessels, explicitly allowing the forces of the growing tumor to influence the nutrient delivery of the vasculature.

Computational modelling suggests complex interactions between interstitial flow and tumour angiogenesis

Angiogenesis, the growth of capillaries from pre-existing ones, plays a key role in cancer progression. Tumours release tumour angiogenic factors (TAFs) into the extracellular matrix (ECM) that trigger angiogenesis once they reach the vasculature. The neovasculature provides nutrients and oxygen to the tumour. In the ECM, the interstitial fluid moves driven by pressure differences and may affect the distribution of tumour TAFs, and, in turn, tumour vascularization. In this work, we propose a hybrid mathematical model to investigate the influence of fluid flow in tumour angiogenesis. Our model shows the impact of interstitial flow in a time-evolving capillary network using a continuous approach. The flow model is coupled to a model of angiogenesis that includes tip endothelial cells, filopodia, capillaries and TAFs. The TAF transport equation considers not only diffusive mechanisms but also the convective transport produced by interstitial flow. Our simulations predict a significant alteration of the new vascular networks, which tend to grow more prominently against the flow. The model suggests that interstitial flow may produce increased tumour malignancies and hindered treatments.

Lattice and continuum modelling of a bioactive porous tissue scaffold

A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.

Fine-grained simulations of the microenvironment of vascularized tumours

One of many important features of the tumour microenvironment is that it is a place of active Darwinian selection where different tumour clones become adapted to the variety of ecological niches that make up the microenvironment. These evolutionary processes turn the microenvironment into a powerful source of tumour heterogeneity and contribute to the development of drug resistance in cancer. Here, we describe a computational tool to study the ecology of the microenvironment and report results about the ecology of the tumour microenvironment and its evolutionary dynamics.

Methods to label, image, and analyze the complex structural architectures of microvascular networks

Microvascular networks play key roles in oxygen transport and nutrient delivery to meet the varied and dynamic metabolic needs of different tissues throughout the body, and their spatial architectures of interconnected blood vessel segments are highly complex. Moreover, functional adaptations of the microcirculation enabled by structural adaptations in microvascular network architecture are required for development, wound healing, and often invoked in disease conditions, including the top eight causes of death in the Unites States. Effective characterization of microvascular network architectures is not only limited by the available techniques to visualize microvessels but also reliant on the available quantitative metrics that accurately delineate between spatial patterns in altered networks. In this review, we survey models used for studying the microvasculature, methods to label and image microvessels, and the metrics and software packages used to quantify microvascular networks. These programs have provided researchers with invaluable tools, yet we estimate that they have collectively attained low adoption rates, possibly due to limitations with basic validation, segmentation performance, and nonstandard sets of quantification metrics. To address these existing constraints, we discuss opportunities to improve effectiveness, rigor, and reproducibility of microvascular network quantification to better serve the current and future needs of microvascular research.

A review on computational modelling of phase-transition problems

Phase-transition problems are ubiquitous in science and engineering. They have been widely studied via theory, experiments and computations. This paper reviews the main challenges associated with computational modelling of phase-transition problems, addressing both model development and numerical discretization of the resulting equations. We focus on classical phase-transition problems, including liquid-solid, gas-liquid and solid-solid transformations. Our review has a strong emphasis on the treatment of interfacial phenomena and the phase-field method. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Angiogenic Factors produced by Hypoxic Cells are a leading driver of Anastomoses in Sprouting Angiogenesis-a computational study

Angiogenesis - the growth of new blood vessels from a pre-existing vasculature - is key in both physiological processes and on several pathological scenarios such as cancer progression or diabetic retinopathy. For the new vascular networks to be functional, it is required that the growing sprouts merge either with an existing functional mature vessel or with another growing sprout. This process is called anastomosis. We present a systematic 2D and 3D computational study of vessel growth in a tissue to address the capability of angiogenic factor gradients to drive anastomosis formation. We consider that these growth factors are produced only by tissue cells in hypoxia, i.e. until nearby vessels merge and become capable of carrying blood and irrigating their vicinity. We demonstrate that this increased production of angiogenic factors by hypoxic cells is able to promote vessel anastomoses events in both 2D and 3D. The simulations also verify that the morphology of these networks has an increased resilience toward variations in the endothelial cell's proliferation and chemotactic response. The distribution of tissue cells and the concentration of the growth factors they produce are the major factors in determining the final morphology of the network.

The coordinated effects of Apatinib and Tripterine on the proliferation, invasiveness and apoptosis of human hepatoma Hep3B cells

As a novel vascular endothelial growth factor receptor-2 (VEGFR-2) tyrosine kinase inhibitor, Apatinib has exhibited antitumor effects in a variety of solid tumors. Extracts of Chinese herbal medicines have emerged as a promising alternative option to increase the sensitivity of patients to chemotherapeutics while alleviating side effects. The present study aimed to investigate the effects of Apatinib and the traditional Chinese herb Tripterine on the proliferation, invasion and apoptosis of human hepatoma Hep3B cells. The expression of VEGFR-2 in Hep3B cells was detected by western blotting and immunofluorescence assays. Hep3B cells were then divided into four different groups: Control group, Apatinib group, Tripterine group and Apatinib plus Tripterine group. The proliferation, invasion and apoptosis of these four groups of Hep3B cells were assessed by MTS, wound healing and Transwell assays, and flow cytometry, respectively. Finally, the levels of the proliferation-associated proteins phosphorylated protein kinase B (p-Akt) and phosphorylated extracellular signal-regulated kinase (p-ERK) and the apoptosis-associated proteins cleaved Caspase-3 and B-cell lymphoma-associated X protein (Bax) were detected by western blotting. The proliferation, migration and invasion of Hep3B cells were significantly inhibited by Apatinib and Tripterine, compared with the control group (P<0.01). The inhibitory effect of the combination group was markedly stronger than that of the Apatinib and Tripterine groups. The downregulation of p-Akt and p-ERK induced by Apatinib and Tripterine was further inhibited in the combination group (P<0.05), and the expression levels of Caspase-3 and Bax were also significantly increased in the combination group (P<0.05). The combination of Apatinib and Tripterine significantly inhibited the proliferation, migration and invasion ability and promoted the apoptosis of Hep3B cells by downregulating the expression of p-Akt and p-ERK, and upregulating the expression of Caspase-3 and Bax.

The importance of geometry in the corneal micropocket angiogenesis assay

The corneal micropocket angiogenesis assay is an experimental protocol for studying vessel network formation, or neovascularization, in vivo. The assay is attractive due to the ease with which the developing vessel network can be observed in the same animal over time. Measurements from the assay have been used in combination with mathematical modeling to gain insights into the mechanisms of angiogenesis. While previous modeling studies have adopted planar domains to represent the assay, the hemispherical shape of the cornea and asymmetric positioning of the angiogenic source can be seen to affect vascular patterning in experimental images. As such, we aim to better understand: i) how the geometry of the assay influences vessel network formation and ii) how to relate observations from planar domains to those in the hemispherical cornea. To do so, we develop a three-dimensional, off-lattice mathematical model of neovascularization in the cornea, using a spatially resolved representation of the assay for the first time. Relative to the detailed model, we predict that the adoption of planar geometries has a noticeable impact on vascular patterning, leading to increased vessel 'merging', or anastomosis, in particular when circular geometries are adopted. Significant differences in the dynamics of diffusible aniogenesis simulators are also predicted between different domains. In terms of comparing predictions across domains, the 'distance of the vascular front to the limbus' metric is found to have low sensitivity to domain choice, while metrics such as densities of tip cells and vessels and 'vascularized fraction' are sensitive to domain choice. Given the widespread adoption and attractive simplicity of planar tissue domains, both in silico and in vitro, the differences identified in the present study should prove useful in relating the results of previous and future theoretical studies of neovascularization to in vivo observations in the cornea.