Capillary networks in tumor angiogenesis: from discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis

Microsphere-Enabled Modular Formation of Miniaturized In Vitro Breast Cancer Models

In search of effective therapeutics for breast cancers, establishing physiologically relevant in vitro models is of great benefit to facilitate the clinical translation. Despite extensive progresses, it remains to develop the tumor models maximally recapturing the key pathophysiological attributes of their native counterparts. Therefore, the current study aimed to develop a microsphere-enabled modular approach toward the formation of in vitro breast tumor models with the capability of incorporating various selected cells while retaining spatial organization. Poly (lactic-co-glycolic acid) microspheres (150-200 mm) with tailorable pore size and surface topography are fabricated and used as carriers to respectively lade with breast tumor-associated cells. Culture of cell-laden microspheres assembled within a customized microfluidic chamber allowed to form 3D tumor models with spatially controlled cell distribution. The introduction of endothelial cell-laden microspheres into cancer-cell laden microspheres at different ratios would induce angiogenesis within the culture to yield vascularized tumor. Evaluation of anticancer drugs such as doxorubicin and Cediranib on the tumor models do demonstrate corresponding physiological responses. Clearly, with the ability to modulate microsphere morphology, cell composition and spatial distribution, microsphere-enabled 3D tumor tissue formation offers a high flexibility to satisfy the needs for pathophysiological study, anticancer drug screening or design of personalized treatment.

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.

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.

Mocafe: a comprehensive Python library for simulating cancer development with Phase Field Models

SUMMARY

Mathematical models are effective in studying cancer development at different scales from metabolism to tissue. Phase Field Models (PFMs) have been shown to reproduce accurately cancer growth and other related phenomena, including expression of relevant molecules, extracellular matrix remodeling and angiogenesis. However, implementations of such models are rarely published, reducing access to these techniques. To reduce this gap, we developed Mocafe, a modular open-source Python package that implements some of the most important PFMs reported in the literature. Mocafe is designed to handle both PFMs purely based on differential equations and hybrid agent-based PFMs. Moreover, Mocafe is meant to be extensible, allowing the inclusion of new models in future releases.

AVAILABILITY AND IMPLEMENTATION

Mocafe is a Python package based on FEniCS, a popular computing platform for solving partial differential equations. The source code, extensive documentation and demos are provided on GitHub at URL: https://github.com/BioComputingUP/mocafe. Moreover, we uploaded on Zenodo an archive of the package, which is available at https://doi.org/10.5281/zenodo.6366052.

Inverting angiogenesis with interstitial flow and chemokine matrix-binding affinity

The molecular signaling pathways that orchestrate angiogenesis have been widely studied, but the role of biophysical cues has received less attention. Interstitial flow is unavoidable in vivo, and has been shown to dramatically change the neovascular patterns, but the mechanisms by which flow regulates angiogenesis remain poorly understood. Here, we study the complex interactions between interstitial flow and the affinity for matrix binding of different chemokine isoforms. Using a computational model, we find that changing the matrix affinity of the chemokine isoform can invert the effect of interstitial flow on angiogenesis—from preferential growth in the direction of the flow when the chemokine is initially matrix-bound to preferential flow against the flow when it is unbound. Although fluid forces signal endothelial cells directly, our data suggests a mechanism for the inversion based on biotransport arguments only, and offers a potential explanation for experimental results in which interstitial flow produced preferential vessel growth with and against the flow. Our results point to a particularly intricate effect of interstitial flow on angiogenesis in the tumor microenvironment, where the vessel network geometry and the interstitial flow patterns are complex.

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.

A preliminary study of endothelial cell migration during angiogenesis using a meshless method approach

Angiogenesis, the development of new blood capillaries, is crucial for the wound healing process. This biological process allows the proper blood supply to the tissue, essential for cell proliferation and viability. Several biological factors modulate angiogenesis, however the vascular endothelial growth factor (VEGF) is the main one. Given the complexity of angiogenesis, in the last years, computational modelling aroused the interest of scientists since it allows to model this process with different, more economic and faster methodologies, comparatively to experimental approaches. In this work, a mathematical model motivated by the analysis of the effect of VEGF diffusion gradient in endothelial cell migration is presented. This is the process that allows capillary formation and it is essential for angiogenesis. The proposed mathematical model is combined with the Radial Point Interpolation Method, being the area discretized considering an unorganized nodal cloud and a background mesh of integration points, without predefined relations. The nodal connectivity was achieved using the "influence-domain" approach. The interpolation functions were constructed using the Radial Point Interpolators techniques. This method combines a radial basis functions with a polynomial functions to obtain the approximation. This preliminary work does not account for the whole complexity of cell and tissue biology, and numerical results are presented for an idealised two-dimensional setting. Nevertheless, the developed RPIM software is a valid numerical tool that can be adjusted to biological problems and may also be able to complement the biological and medical subjects.

Sprouting Angiogenesis: A Numerical Approach with Experimental Validation

A functional vascular network is essential to the correct wound healing. In sprouting angiogenesis, vascular endothelial growth factor (VEGF) regulates the formation of new capillaries from pre-existing vessels. This is a very complex process and mathematical formulation permits to study angiogenesis using less time-consuming, reproducible and cheaper methodologies. This study aimed to mimic the chemoattractant effect of VEGF in stimulating sprouting angiogenesis. We developed a numerical model in which endothelial cells migrate according to a diffusion-reaction equation for VEGF. A chick chorioallantoic membrane (CAM) bioassay was used to obtain some important parameters to implement in the model and also to validate the numerical results. We verified that endothelial cells migrate following the highest VEGF concentration. We compared the parameters-total branching number, total vessel length and branching angle-that were obtained in the in silico and the in vivo methodologies and similar results were achieved (p-value smaller than 0.5; n = 6). For the difference between the total capillary volume fractions assessed using both methodologies values smaller than 15% were obtained. In this study we simulated, for the first time, the capillary network obtained during the CAM assay with a realistic morphology and structure.

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.

Parallel splitting solvers for the isogeometric analysis of the Cahn-Hilliard equation

Modeling tumor growth in biological systems is a challenging problem with important consequences for diagnosis and treatment of various forms of cancer. This growth process requires large simulation complexity due to evolving biological and chemical processes in living tissue and interactions of cellular and vascular constituents in living organisms. Herein, we describe with a phase-field model, namely the Cahn-Hilliard equation the intricate interactions between the tumors and their host tissue. The spatial discretization uses highly-continuous isogeometric elements. For fast simulation of the time-dependent Cahn-Hilliard equation, we employ an alternating directions implicit methodology. Thus, we reduce the original problems to Kronecker products of 1 D matrices that can be factorized in a linear computational cost. The implementation enables parallel multi-core simulations and shows good scalability on shared-memory multi-core machines. Combined with the high accuracy of isogeometric elements, our method shows high efficiency in solving the Cahn-Hilliard equation on tensor-product meshes.

Modelling of endothelial cell migration and angiogenesis in microfluidic cell culture systems

Tumour-induced angiogenesis is a complex biological process that involves growth of new blood vessels within the tumour microenvironment and is an important target for cancer therapies. Significant efforts have been undertaken to develop theoretical models as well as in vitro experimental models to study angiogenesis in a highly controllable and accessible manner. Various mathematical models have been developed to study angiogenesis, but these have mostly been applied to in vivo assays. Recently, microfluidic cell culture systems have emerged as useful and powerful tools for studying cell migration and angiogenesis processes, but thus far, mathematical angiogenesis models have not yet been applied to microfluidic geometries. Integrating mathematical and computational modelling with microfluidic-based assays has potential to enable greater control over experimental parameters, provide new insights into fundamental angiogenesis processes and assist in accelerating design and optimization of operating conditions. Here, we describe the development and application of a combined mathematical and computational modelling approach tailored specifically for microfluidic cell culture systems. The objective was to allow optimization of the engineering design of microfluidic systems, where the model may be used to test the impact of various geometric parameters on cell migration and angiogenesis processes, and assist in identifying optimal device dimensions to achieve desired readouts. We employed two separate continuum mathematical models that treated cell density, vessel length density and vascular endothelial growth factor (VEGF) concentration as continuous average variables, and we implemented these models numerically using finite difference discretization and a Method of Lines approach. We examined the average response of cells to VEGF gradients inside our microfluidic device, including the time-dependent changes in cell density and vessel density, and how they were affected by changes in device geometries including the migration port width and length. Our study demonstrated that mathematical modelling can be integrated with microfluidics to offer new perspectives on emerging problems in biomicrofluidics and cancer biology.

Soluble VEGFR1 signaling guides vascular patterns into dense branching morphologies

Vascular patterning is a key process during development and disease. The diffusive decoy receptor sVEGFR1 (sFlt1) is a known regulator of endothelial cell behavior, yet the mechanism by which it controls vascular structure is little understood. We propose computational models to shed light on how vascular patterning is guided by self-organized gradients of the VEGF/sVEGFR1 factors. We demonstrate that a diffusive inhibitor can generate structures with a dense branching morphology in models where the activator elicits directed growth. Inadequate presence of the inhibitor leads to compact growth, while excessive production of the inhibitor blocks expansion and stabilizes existing structures. Model predictions were compared with time-resolved experimental data obtained from endothelial sprout kinetics in fibrin gels. In the presence of inhibitory antibodies against VEGFR1 vascular sprout density increases while the speed of sprout expansion remains unchanged. Thus, the rate of secretion and stability of extracellular sVEGFR1 can modulate vascular sprout density.

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.

Phase-field boundary conditions for the voxel finite cell method: Surface-free stress analysis of CT-based bone structures

The voxel finite cell method uses unfitted finite element meshes and voxel quadrature rules to seamlessly transfer computed tomography data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field-based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then used to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, ie, the interface width of the phase field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body.

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

Cancerous tumours have the ability to recruit new blood vessels through a process called angiogenesis. By stimulating vascular growth, tumours get connected to the circulatory system, receive nutrients and open a way to colonize distant organs. Tumour-induced vascular networks become unstable in the absence of tumour angiogenic factors (TAFs). They may undergo alternating stages of growth, regression and regrowth. Following a phase-field methodology, we propose a model of tumour angiogenesis that reproduces the aforementioned features and highlights the importance of vascular regression and regrowth. In contrast with previous theories which focus on vessel remodelling due to the absence of flow, we model an alternative regression mechanism based on the dependency of tumour-induced vascular networks on TAFs. The model captures capillaries at full scale, the plastic dynamics of tumour-induced vessel networks at long time scales, and shows the key role played by filopodia during angiogenesis. The predictions of our model are in agreement with in vivo experiments and may prove useful for the design of antiangiogenic therapies.

Modelling Tumor-induced Angiogenesis: Combination of Stochastic Sprout Spacing and Sprout Progression

BACKGROUND

Angiogenesis initiated by cancerous cells is the process by which new blood vessels are formed to enhance oxygenation and growth of tumor.

OBJECTIVE

In this paper, we present a new multiscale mathematical model for the formation of a vascular network in tumor angiogenesis process.

METHODS

Our model couples an improved sprout spacing model as a stochastic mathematical model of sprouting along an existing parent blood vessel, with a mathematical model of sprout progression in the extracellular matrix (ECM) in response to some tumor angiogenic factors (TAFs). We perform simulations of the siting of capillary sprouts on an existing blood vessel using finite difference approximation of the dynamic equations of some angiogenesis activators and inhibitors. Angiogenesis activators are chemicals secreted by hypoxic tumor cells for initiating angiogenesis, and inhibitors of the angiogenesis are chemicals that are produced around every new sprout during tumor angiogenesis to inhibit the formation of further sprouts as a feedback of sprouting in angiogenesis. Moreover, for modelling sprout progression in ECM, we use three equations for the motility of endothelial cells at the tip of the activated sprouts, the consumption of TAF and the production and uptake of Fibronectin by endothelial cells.

RESULTS

Coupling these two basic models not only does provide a better time estimation of angiogenesis process, but also it is more compatible with reality.

CONCLUSION

This model can be used to provide basic information for angiogenesis in the related studies. Related simulations can estimate the position and number of sprouts along parent blood vessel during the initial steps of angiogenesis and models the process of sprout progression in ECM until they vascularize a tumor.

Effects of endothelial cell proliferation and migration rates in a computational model of sprouting angiogenesis

Angiogenesis, the recruitment of new blood vessels, is a critical process for the growth, expansion, and metastatic dissemination of developing tumors. Three types of cells make up the new vasculature: tip cells, which migrate in response to gradients of vascular endothelial growth factor (VEGF), stalk cells, which proliferate and extend the vessels, and phalanx cells, which are quiescent and support the sprout. In this study we examine the contribution of tip cell migration rate and stalk cell proliferation rate on the formation of new vasculature. We calculate several vascular metrics, such as the number of vascular bifurcations per unit volume, vascular segment length per unit volume, and vascular tortuosity. These measurements predict that proliferation rate has a greater effect on the spread and extent of vascular growth compared to migration rate. Together, these findings provide strong implications for designing anti-angiogenic therapies that may differentially target endothelial cell proliferation and migration. Computational models can be used to predict optimal anti-angiogenic therapies in combination with other therapeutics to improve outcome.

Simulation of tumor induced angiogenesis using an analytical adaptive modeling including dynamic sprouting and blood flow modeling

In this paper, an adaptive model for tumor induced angiogenesis is developed that integrates generation and diffusion of a growth factor originated from hypoxic cells, adaptive sprouting from a parent vessel, blood flow and structural adaptation. The proposed adaptive sprout spacing model (ASS) determines position, time and number of sprouts which are activated from a parent vessel and also the developed vascular network is modified by a novel sprout branching prediction algorithm. This algorithm couples local vascular endothelial growth factor (VEGF) concentrations, stresses due to the blood flow and stochastic branching to the structural reactions of each vessel segment in response to mechanical and biochemical stimuli. The results provide predictions for the time-dependent development of the network structure, including the position and diameters of each segment and the resulting distributions of blood flow and VEGF. Considering time delays between sprout progressions and number of sprouts activated at different time durations provides information about micro-vessel density in the network. Resulting insights could be useful for motivating experimental investigations of vascular pattern in tumor induced angiogenesis and development of therapies targeting angiogenesis.

Tumour angiogenesis as a chemo-mechanical surface instability

The hypoxic conditions within avascular solid tumours may trigger the secretion of chemical factors, which diffuse to the nearby vasculature and promote the formation of new vessels eventually joining the tumour. Mathematical models of this process, known as tumour angiogenesis, have mainly investigated the formation of the new capillary networks using reaction-diffusion equations. Since angiogenesis involves the growth dynamics of the endothelial cells sprouting, we propose in this work an alternative mechanistic approach, developing a surface growth model for studying capillary formation and network dynamics. The model takes into account the proliferation of endothelial cells on the pre-existing capillary surface, coupled with the bulk diffusion of the vascular endothelial growth factor (VEGF). The thermo-dynamical consistency is imposed by means of interfacial and bulk balance laws. Finite element simulations show that both the morphology and the dynamics of the sprouting vessels are controlled by the bulk diffusion of VEGF and the chemo-mechanical and geometric properties at the capillary interface. Similarly to dendritic growth processes, we suggest that the emergence of tree-like vessel structures during tumour angiogenesis may result from the free boundary instability driven by competition between chemical and mechanical phenomena occurring at different length-scales.

A Mathematical Model Coupling Tumor Growth and Angiogenesis

We present a mathematical model for vascular tumor growth. We use phase fields to model cellular growth and reaction-diffusion equations for the dynamics of angiogenic factors and nutrients. The model naturally predicts the shift from avascular to vascular growth at realistic scales. Our computations indicate that the negative regulation of the Delta-like ligand 4 signaling pathway slows down tumor growth by producing a larger density of non-functional capillaries. Our results show good quantitative agreement with experiments.

A computational model of drug delivery through microcirculation to compare different tumor treatments

Starting from the fundamental laws of filtration and transport in biological tissues, we develop a computational model to capture the interplay between blood perfusion, fluid exchange with the interstitial volume, mass transport in the capillary bed, through the capillary walls and into the surrounding tissue. These phenomena are accounted at the microscale level, where capillaries and interstitial volume are viewed as two separate regions. The capillaries are described as a network of vessels carrying blood flow. We apply the model to study drug delivery to tumors. The model can be adapted to compare various treatment options. In particular, we consider delivery using drug bolus injection and nanoparticle injection into the blood stream. The computational approach is suitable for a systematic quantification of the treatment performance, enabling the analysis of interstitial drug concentration levels, metabolization rates and cell surviving fractions. Our study suggests that for the treatment based on bolus injection, the drug dose is not optimally delivered to the tumor interstitial volume. Using nanoparticles as intermediate drug carriers overrides the shortcomings of the previous delivery approach. This work shows that the proposed theoretical and computational framework represents a promising tool to compare the efficacy of different cancer treatments.