Angiogenesis, the growth of a network of blood vessels, is a crucial component of solid tumor growth, linking the relatively harmless avascular and the potentially fatal vascular growth phases of the tumor. As a process, angiogenesis is a well-orchestrated sequence of events involving endothelial cell migration and proliferation; degradation of tissue; new capillary vessel formation; loop formation (anastomosis) and, crucially, blood flow through the network. Once there is flow associated with the nascent network, subsequent growth evolves both temporally and spatially in response to the combined effects of angiogenic factors, migratory cues via the extracellular matrix, and perfusion-related hemodynamic forces in a manner that may be described as both adaptive and dynamic. In this article, we first present a review of previous theoretical and computational models of angiogenesis and then indicate how recent developments in flow models are providing insight into antiangiogenic and chemotherapeutic drug treatment of solid tumors.

https://www.annualreviews.org/doi/pdf/10.1146/annurev.bioeng.8.061505.095807

Mathematical Modeling of Tumor-Induced Angiogenesis

We present a 3D multi-cell simulation of a generic simplification of vascular tumor growth which can be easily extended and adapted to describe more specific vascular tumor types and host tissues. Initially, tumor cells proliferate as they take up the oxygen which the pre-existing vasculature supplies. The tumor grows exponentially. When the oxygen level drops below a threshold, the tumor cells become hypoxic and start secreting pro-angiogenic factors. At this stage, the tumor reaches a maximum diameter characteristic of an avascular tumor spheroid. The endothelial cells in the pre-existing vasculature respond to the pro-angiogenic factors both by chemotaxing towards higher concentrations of pro-angiogenic factors and by forming new blood vessels via angiogenesis. The tumor-induced vasculature increases the growth rate of the resulting vascularized solid tumor compared to an avascular tumor, allowing the tumor to grow beyond the spheroid in these linear-growth phases. First, in the linear-spherical phase of growth, the tumor remains spherical while its volume increases. Second, in the linear-cylindrical phase of growth the tumor elongates into a cylinder. Finally, in the linear-sheet phase of growth, tumor growth accelerates as the tumor changes from cylindrical to paddle-shaped. Substantial periods during which the tumor grows slowly or not at all separate the exponential from the linear-spherical and the linear-spherical from the linear-cylindrical growth phases. In contrast to other simulations in which avascular tumors remain spherical, our simulated avascular tumors form cylinders following the blood vessels, leading to a different distribution of hypoxic cells within the tumor. Our simulations cover time periods which are long enough to produce a range of biologically reasonable complex morphologies, allowing us to study how tumor-induced angiogenesis affects the growth rate, size and morphology of simulated tumors.

/pdf/3d-multi-cell-simulation-of-tumor-growth-and-angiogenesis-eixlm8enx0.pdf

3D Multi-Cell Simulation of Tumor Growth and Angiogenesis

Vascular development and homeostasis are underpinned by two fundamental features: the generation of new vessels to meet the metabolic demands of under-perfused regions and the elimination of vessels that do not sustain flow. In this paper we develop the first multiscale model of vascular tissue growth that combines blood flow, angiogenesis, vascular remodelling and the subcellular and tissue scale dynamics of multiple cell populations. Simulations show that vessel pruning, due to low wall shear stress, is highly sensitive to the pressure drop across a vascular network, the degree of pruning increasing as the pressure drop increases. In the model, low tissue oxygen levels alter the internal dynamics of normal cells, causing them to release vascular endothelial growth factor (VEGF), which stimulates angiogenic sprouting. Consequently, the level of blood oxygenation regulates the extent of angiogenesis, with higher oxygenation leading to fewer vessels. Simulations show that network remodelling (and de novo network formation) is best achieved via an appropriate balance between pruning and angiogenesis. An important factor is the strength of endothelial tip cell chemotaxis in response to VEGF. When a cluster of tumour cells is introduced into normal tissue, as the tumour grows hypoxic regions form, producing high levels of VEGF that stimulate angiogenesis and cause the vascular density to exceed that for normal tissue. If the original vessel network is sufficiently sparse then the tumour may remain localised near its parent vessel until new vessels bridge the gap to an adjacent vessel. This can lead to metastable periods, during which the tumour burden is approximately constant, followed by periods of rapid growth.

/pdf/angiogenesis-and-vascular-remodelling-in-normal-and-4zw9g4tk09.pdf

Angiogenesis and vascular remodelling in normal and cancerous tissues

Whether the nom de guerre is Mathematical Oncology, Computational or Systems Biology, Theoretical Biology, Evolutionary Oncology, Bioinformatics, or simply Basic Science, there is no denying that mathematics continues to play an increasingly prominent role in cancer research. Mathematical Oncology-defined here simply as the use of mathematics in cancer research-complements and overlaps with a number of other fields that rely on mathematics as a core methodology. As a result, Mathematical Oncology has a broad scope, ranging from theoretical studies to clinical trials designed with mathematical models. This Roadmap differentiates Mathematical Oncology from related fields and demonstrates specific areas of focus within this unique field of research. The dominant theme of this Roadmap is the personalization of medicine through mathematics, modelling, and simulation. This is achieved through the use of patient-specific clinical data to: develop individualized screening strategies to detect cancer earlier; make predictions of response to therapy; design adaptive, patient-specific treatment plans to overcome therapy resistance; and establish domain-specific standards to share model predictions and to make models and simulations reproducible. The cover art for this Roadmap was chosen as an apt metaphor for the beautiful, strange, and evolving relationship between mathematics and cancer.

https://iopscience.iop.org/article/10.1088/1478-3975/ab1a09/pdf

The 2019 mathematical oncology roadmap.

Tumor spheroids grown in vitro have been widely used as models of in vivo tumor growth because they display many of the characteristics of in vivo growth, including the effects of nutrient limitations and perhaps the effect of stress on growth. In either case there are numerous biochemical and biophysical processes involved whose interactions can only be understood via a detailed mathematical model. Previous models have focused on either a continuum description or a cell-based description, but both have limitations. In this paper we propose a new mathematical model of tumor spheroid growth that incorporates both continuum and cell-level descriptions, and thereby retains the advantages of each while circumventing some of their disadvantages. In this model the cell-based description is used in the region where the majority of growth and cell division occurs, at the periphery of a tumor, while a continuum description is used for the quiescent and necrotic zones of the tumor and for the extracellular matrix. Reaction-diffusion equations describe the transport and consumption of two important nutrients, oxygen and glucose, throughout the entire domain. The cell-based component of this hybrid model allows us to examine the effects of cell–cell adhesion and variable growth rates at the cellular level rather than at the continuum level. We show that the model can predict a number of cellular behaviors that have been observed experimentally.

/pdf/a-hybrid-model-for-tumor-spheroid-growth-in-vitro-i-1jvgy80fpn.pdf

A hybrid model for tumor spheroid growth in vitro i: theoretical development and early results

Herbal remedies are increasing in popularity as treatments for metabolic conditions such as obesity and Type 2 Diabetes. One potential therapeutic option is fenugreek seeds (Trigonella foenum-graecum), which have been used for treating high cholesterol and Type 2 diabetes. A proposed mechanism for these benefits is through alterations in the microbiome, which impact mammalian host metabolic function. This study used untargeted metabolomics to investigate the fenugreek-induced alterations in the intestinal, liver, and serum profiles of mice fed either a 60% high-fat or low-fat control diet each with or without fenugreek supplementation (2% w/w) for 14 weeks. Metagenomic analyses of intestinal contents found significant alterations in the relative composition of the gut microbiome resulting from fenugreek supplementation. Specifically, Verrucomicrobia, a phylum containing beneficial bacteria which are correlated with health benefits, increased in relative abundance with fenugreek. Metabolomics partial least squares discriminant analysis revealed substantial fenugreek-induced changes in the large intestines. However, it was observed that while the magnitude of changes was less, significant modifications were present in the liver tissues resulting from fenugreek supplementation. Further analyses revealed metabolic processes affected by fenugreek and showed broad ranging impacts in multiple pathways, including carnitine biosynthesis, cholesterol and bile acid metabolism, and arginine biosynthesis. These pathways may play important roles in the beneficial effects of fenugreek.

/pdf/cross-omics-analysis-of-fenugreek-supplementation-reveals-jmascp6w.pdf

Cross-Omics Analysis of Fenugreek Supplementation Reveals Beneficial Effects Are Caused by Gut Microbiome Changes Not Mammalian Host Physiology

. The global importance of eﬀective and aﬀordable pesticides to optimise crop yield and to support health of our growing population cannot be understated. But to develop new products or reﬁne existing ones in response to climate and environmental changes is both time-intensive and expensive which is why the agrochemical industry is increasingly interested in using mechanistic models as part of their formulation development toolbox. In this work, we develop such a model to describe uptake of pesticide spray droplets across the leaf surface. We simplify the leaf structure by identifying the outer cuticle as the main barrier to uptake; the result is a novel, hybrid model in which two well-mixed compartments are separated by a membrane in which we describe the spatio-temporal distribution of the pesticide. This leads to a boundary value partial diﬀerential equation problem coupled to a pair of ordinary diﬀerential equation systems which we solve numerically. We also simplify the pesticide formulation into two key components: the Active Ingredient which produces the desired eﬀect of the pesticide and an Adjuvant which is present in the formulation to facilitate eﬀective absorption of the Active Ingredient into the leaf. This approach gives rise to concentration-dependent diﬀusion. We take an intuitive approach to parameter estimation using a small experimental data set and subsequently demonstrate the importance of the concentration-dependent diﬀusion in replicating the data. Finally, we demonstrate the need for further work to identify how the physicochemical properties of pesticides aﬀect ﬂow into and across the leaf surface.

Mathematical modelling of adjuvant-enhanced active ingredient leaf uptake of pesticides

Mathematical modeling of tumor-induced angiogenesis

Evaluation of a novel test design to determine uptake of chemicals by plant roots.

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Cross-Omics Analysis of Fenugreek Supplementation Reveals Beneficial Effects Are Caused by Gut Microbiome Changes Not Mammalian Host Physiology

Mathematical modelling of adjuvant-enhanced active ingredient leaf uptake of pesticides

Mathematical modeling of tumor-induced angiogenesis

Evaluation of a novel test design to determine uptake of chemicals by plant roots.