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Abstract Background Investigations of solid tumours suggest that vessel occlusion may occur when increased pressure from the tumour mass is exerted on the vessel walls. Since immature vessels are frequently found in tumours and may be particularly sensitive, such occlusion may impair tumour blood flow and have a negative impact on therapeutic outcome. In order to study the effects that occlusion may have on tumour growth patterns and therapeutic response, in this paper we develop and investigate a continuum model of vascular tumour growth. Results By analysing a spatially uniform submodel, we identify regions of parameter space in which the combination of tumour cell proliferation and vessel occlusion give rise to sustained temporal oscillations in the tumour cell population and in the vessel density. Alternatively, if the vessels are assumed to be less prone to collapse, stable steady state solutions are observed. When spatial effects are considered, the pattern of tumour invasion depends on the dynamics of the spatially uniform submodel. If the submodel predicts a stable steady state, then steady travelling waves are observed in the full model, and the system evolves to the same stable steady state behind the invading front. When the submodel yields oscillatory behaviour, the full model produces periodic travelling waves. The stability of the waves (which can be predicted by approximating the system as one of λ-ω type) dictates whether the waves develop into regular or irregular spatio-temporal oscillations. Simulations of chemotherapy reveal that treatment outcome depends crucially on the underlying tumour growth dynamics. In particular, if the dynamics are oscillatory, then therapeutic efficacy is difficult to assess since the fluctuations in the size of the tumour cell population are enhanced, compared to untreated controls. Conclusions We have developed a mathematical model of vascular tumour growth formulated as a system of partial differential equations (PDEs). Employing a combination of numerical and analytical techniques, we demonstrate how the spatio-temporal dynamics of the untreated tumour may influence its response to chemotherapy. Reviewers This manuscript was reviewed by Professor Zvia Agur and Professor Marek Kimmel.

We present a novel mathematical model for collagen deposition and alignment during dermal wound healing, focusing on the regulatory effects of transforming growth factor-beta (TGFbeta.) Our work extends a previously developed model which considers the interactions between fibroblasts and an extracellular matrix composed of collagen and a fibrin based blood clot, by allowing fibroblasts to orient the collagen matrix, and produce and degrade the extracellular matrix, while the matrix directs the fibroblasts and control their speed. Here we extend the model by allowing a time varying concentration of TGFbeta to alter the properties of the fibroblasts. Thus we are able to simulate experiments which alter the TGFbeta profile. Within this model framework we find that most of the known effects of TGFbeta, i.e., changes in cell motility, cell proliferation and collagen production, are of minor importance to matrix alignment and cannot explain the anti-scarring properties of TGFbeta. However, we find that by changing fibroblast reorientation rates, consistent with experimental evidence, the alignment of the regenerated tissue can be significantly altered. These data provide an explanation for the experimentally observed influence of TGFbeta on scarring. [ABSTRACT FROM AUTHOR]

In vivo observations show that oxygen levels in tumours can fluctuate on fast and slow timescales. As a result, cancer cells can be periodically exposed to pathologically low oxygen levels; a phenomenon known as cyclic hypoxia. Yet, little is known about the response and adaptation of cancer cells to cyclic, rather than, constant hypoxia. Further, existing in vitro models of cyclic hypoxia fail to capture the complex and heterogeneous oxygen dynamics of tumours growing in vivo. Mathematical models can help to overcome current experimental limitations and, in so doing, offer new insights into the biology of tumour cyclic hypoxia by predicting cell responses to a wide range of cyclic dynamics. We develop an individual-based model to investigate how cell cycle progression and cell fate determination of cancer cells are altered following exposure to cyclic hypoxia. Our model can simulate standard in vitro experiments, such as clonogenic assays and cell cycle experiments, allowing for efficient screening of cell responses under a wide range of cyclic hypoxia conditions. Simulation results show that the same cell line can exhibit markedly different responses to cyclic hypoxia depending on the dynamics of the oxygen fluctuations. We also use our model to investigate the impact of changes to cell cycle checkpoint activation and damage repair on cell responses to cyclic hypoxia. Our simulations suggest that cyclic hypoxia can promote heterogeneity in cellular damage repair activity within vascular tumours., (© 2024. The Author(s).)

Purpose: Standard of care for various retinal diseases involves recurrent intravitreal injections. This motivates mathematical modeling efforts to identify influential factors for ocular drug residence time, aiming to minimize administration frequency. We sought to describe the vitreal diffusion of therapeutics in nonclinical species frequently used during drug development assessments. In human eyes, we investigated the impact of variability in vitreous cavity size and eccentricity, and in injection location, on drug disposition., Methods: Using a first-passage time approach, we modeled the transport-controlled distribution of two standard therapeutic protein formats (Fab and IgG) and elimination through anterior and posterior pathways. Anatomical three-dimensional geometries of mouse, rat, rabbit, cynomolgus monkey, and human eyes were constructed using ocular images and biometry datasets. A scaling relationship was derived for comparison with experimental ocular half-lives., Results: Model simulations revealed a dependence of residence time on ocular size and injection location. Delivery to the posterior vitreous resulted in increased vitreal half-life and retinal permeation. Interindividual variability in human eyes had a significant influence on residence time (half-life range of 5-7 days), showing a strong correlation to axial length and vitreal volume. Anterior exit was the predominant route of drug elimination. Contribution of the posterior pathway displayed a 3% difference between protein formats but varied between species (10%-30%)., Conclusions: The modeling results suggest that experimental variability in ocular half-life is partially attributed to anatomical differences and injection site location. Simulations further suggest a potential role of the posterior pathway permeability in determining species differences in ocular pharmacokinetics.

A fundamental feature of collective cell migration is phenotypic heterogeneity which, for example, influences tumour progression and relapse. While current mathematical models often consider discrete phenotypic structuring of the cell population, in-line with the 'go-or-grow' hypothesis (Hatzikirou et al., 2012; Stepien et al., 2018), they regularly overlook the role that the environment may play in determining the cells' phenotype during migration. Comparing a previously studied volume-filling model for a homogeneous population of generalist cells that can proliferate, move and degrade extracellular matrix (ECM) (Crossley et al., 2023) to a novel model for a heterogeneous population comprising two distinct sub-populations of specialist cells that can either move and degrade ECM or proliferate, this study explores how different hypothetical phenotypic switching mechanisms affect the speed and structure of the invading cell populations. Through a continuum model derived from its individual-based counterpart, insights into the influence of the ECM and the impact of phenotypic switching on migrating cell populations emerge. Notably, specialist cell populations that cannot switch phenotype show reduced invasiveness compared to generalist cell populations, while implementing different forms of switching significantly alters the structure of migrating cell fronts. This key result suggests that the structure of an invading cell population could be used to infer the underlying mechanisms governing phenotypic switching., Competing Interests: Declaration of competing interest The authors do not have any competing interests to declare., (Copyright © 2024 The Author(s). Published by Elsevier Inc. All rights reserved.)

Following the apparent final case in an Ebola virus disease (EVD) outbreak, the decision to declare the outbreak over must balance societal benefits of relaxing interventions against the risk of resurgence. Estimates of the end-of-outbreak probability (the probability that no future cases will occur) provide quantitative evidence that can inform the timing of an end-of-outbreak declaration. An existing modeling approach for estimating the end-of-outbreak probability requires comprehensive contact tracing data describing who infected whom to be available, but such data are often unavailable or incomplete during outbreaks. Here, we develop a Markov chain Monte Carlo-based approach that extends the previous method and does not require contact tracing data. Considering data from two EVD outbreaks in the Democratic Republic of the Congo, we find that data describing who infected whom are not required to resolve uncertainty about when to declare an outbreak over.

Patterned morphologies, such as segments, spirals, stripes, and spots, frequently emerge during embryogenesis through self-organized coordination between cells. Yet, complex patterns also emerge in adults, suggesting that the capacity for spontaneous self-organization is a ubiquitous property of biological tissues. We review current knowledge on the principles and mechanisms of self-organized patterning in embryonic tissues and explore how these principles and mechanisms apply to adult tissues that exhibit features of patterning. We discuss how and why spontaneous pattern generation is integral to homeostasis and healing of tissues, illustrating it with examples from regenerative biology. We examine how aberrant self-organization underlies diverse pathological states, including inflammatory skin disorders and tumors. Lastly, we posit that based on such blueprints, targeted engineering of pattern-driving molecular circuits can be leveraged for synthetic biology and the generation of organoids with intricate patterns., Competing Interests: Declaration of interests The authors declare no competing interests., (Copyright © 2024 The Author(s). Published by Elsevier Inc. All rights reserved.)

Toxicity and emerging drug resistance pose important challenges in poly-adenosine ribose polymerase inhibitor (PARPi) maintenance therapy of ovarian cancer. We propose that adaptive therapy, which dynamically reduces treatment based on the tumor dynamics, might alleviate both issues. Utilizing in vitro time-lapse microscopy and stepwise model selection, we calibrate and validate a differential equation mathematical model, which we leverage to test different plausible adaptive treatment schedules. Our model indicates that adjusting the dosage, rather than skipping treatments, is more effective at reducing drug use while maintaining efficacy due to a delay in cell kill and a diminishing dose-response relationship. In vivo pilot experiments confirm this conclusion. Although our focus is toxicity mitigation, reducing drug use may also delay resistance. This study enhances our understanding of PARPi treatment scheduling and illustrates the first steps in developing adaptive therapies for new treatment settings. A record of this paper's transparent peer review process is included in the supplemental information., Competing Interests: Declaration of interests M.A.R.S. is now a postdoctoral research fellow at the Cleveland Clinic, OH, but all work presented in this article was conducted at the H. Lee Moffitt Cancer Center & Research Institute. R.W. reports grants and consulting fees from Merck; consulting fees from Tesaro/GSK; consulting fees from Genentech; consulting fees from Legend Biotech; grants and consulting fees from AbbVie; grants and consulting fees from AstraZeneca; consulting fees from Novacure; consulting fees, grants, and stock from Ovation Diagnostics; honoraria from Clovis Oncology; consulting fees and grants from Eisai; consulting fees from Seagen; consulting fees from Shattuck Labs; consulting fees from Immunogen; and consulting fees and grants from Regeneron (all outside the submitted work)., (Copyright © 2024 Elsevier Inc. All rights reserved.)

Standard-of-care treatment regimens have long been designed for maximal cell killing, yet these strategies often fail when applied to metastatic cancers due to the emergence of drug resistance. Adaptive treatment strategies have been developed as an alternative approach, dynamically adjusting treatment to suppress the growth of treatment-resistant populations and thereby delay, or even prevent, tumor progression. Promising clinical results in prostate cancer indicate the potential to optimize adaptive treatment protocols. Here, we applied deep reinforcement learning (DRL) to guide adaptive drug scheduling and demonstrated that these treatment schedules can outperform the current adaptive protocols in a mathematical model calibrated to prostate cancer dynamics, more than doubling the time to progression. The DRL strategies were robust to patient variability, including both tumor dynamics and clinical monitoring schedules. The DRL framework could produce interpretable, adaptive strategies based on a single tumor burden threshold, replicating and informing optimal treatment strategies. The DRL framework had no knowledge of the underlying mathematical tumor model, demonstrating the capability of DRL to help develop treatment strategies in novel or complex settings. Finally, a proposed five-step pathway, which combined mechanistic modeling with the DRL framework and integrated conventional tools to improve interpretability compared with traditional "black-box" DRL models, could allow translation of this approach to the clinic. Overall, the proposed framework generated personalized treatment schedules that consistently outperformed clinical standard-of-care protocols., Significance: Generation of interpretable and personalized adaptive treatment schedules using a deep reinforcement framework that interacts with a virtual patient model overcomes the limitations of standardized strategies caused by heterogeneous treatment responses., (©2024 The Authors; Published by the American Association for Cancer Research.)

Periodic patterning requires coordinated cell-cell interactions at the tissue level. Turing showed, using mathematical modeling, how spatial patterns could arise from the reactions of a diffusive activator-inhibitor pair in an initially homogeneous 2D field. Most activators and inhibitors studied in biological systems are proteins, and the roles of cell-cell interaction, ions, bioelectricity, etc. are only now being identified. Gap junctions (GJs) mediate direct exchanges of ions or small molecules between cells, enabling rapid long-distance communications in a cell collective. They are therefore good candidates for propagating nonprotein-based patterning signals that may act according to the Turing principles. Here, we explore the possible roles of GJs in Turing-type patterning using feather pattern formation as a model. We found 7 of the 12 investigated GJ isoforms are highly dynamically expressed in the developing chicken skin. In ovo functional perturbations of the GJ isoform, connexin 30, by siRNA and the dominant-negative mutant applied before placode development led to disrupted primary feather bud formation. Interestingly, inhibition of gap junctional intercellular communication (GJIC) in the ex vivo skin explant culture allowed the sequential emergence of new feather buds at specific spatial locations relative to the existing primary buds. The results suggest that GJIC may facilitate the propagation of long-distance inhibitory signals. Thus, inhibition of GJs may stimulate Turing-type periodic feather pattern formation during chick skin development, and the removal of GJ activity would enable the emergence of new feather buds if the local environment were competent and the threshold to form buds was reached. We further propose Turing-based computational simulations that can predict the sequential appearance of these ectopic buds. Our models demonstrate how a Turing activator-inhibitor system can continue to generate patterns in the competent morphogenetic field when the level of intercellular communication at the tissue scale is modulated., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2024 Tseng et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

Many physical and biological systems rely on the progression of material through multiple independent stages. In viral replication, for example, virions enter a cell to undergo a complex process comprising several disparate stages before the eventual accumulation and release of replicated virions. While such systems may have some control over the internal dynamics that make up this progression, a challenge for many is to regulate behavior under what are often highly variable external environments acting as system inputs. In this work, we study a simple analog of this problem through a linear multicompartment model subject to a stochastic input in the form of a mean-reverting Ornstein-Uhlenbeck process, a type of Gaussian process. By expressing the system as a multidimensional Gaussian process, we derive several closed-form analytical results relating to the covariances and autocorrelations of the system, quantifying the smoothing effect discrete compartments afford multicompartment systems. Semianalytical results demonstrate that feedback and feedforward loops can enhance system robustness, and simulation results probe the intractable problem of the first passage time distribution, which has specific relevance to eventual cell lysis in the viral replication cycle. Finally, we demonstrate that the smoothing seen in the process is a consequence of the discreteness of the system, and does not manifest in systems with continuous transport. While we make progress through analysis of a simple linear problem, many of our insights are applicable more generally, and our work enables future analysis into multicompartment processes subject to stochastic inputs.

Interstitial fluid flow is a feature of many solid tumors. In vitro experiments have shown that such fluid flow can direct tumor cell movement upstream or downstream depending on the balance between the competing mechanisms of tensotaxis (cell migration up stress gradients) and autologous chemotaxis (downstream cell movement in response to flow-induced gradients of self-secreted chemoattractants). In this work we develop a probabilistic-continuum, two-phase model for cell migration in response to interstitial flow. We use a kinetic description for the cell velocity probability density function, and model the flow-dependent mechanical and chemical stimuli as forcing terms that bias cell migration upstream and downstream. Using velocity-space averaging, we reformulate the model as a system of continuum equations for the spatiotemporal evolution of the cell volume fraction and flux in response to forcing terms that depend on the local direction and magnitude of the mechanochemical cues. We specialize our model to describe a one-dimensional cell layer subject to fluid flow. Using a combination of numerical simulations and asymptotic analysis, we delineate the parameter regime where transitions from downstream to upstream cell migration occur. As has been observed experimentally, the model predicts downstream-oriented chemotactic migration at low cell volume fractions, and upstream-oriented tensotactic migration at larger volume fractions. We show that the locus of the critical volume fraction, at which the system transitions from downstream to upstream migration, is dominated by the ratio of the rate of chemokine secretion and advection. Our model also predicts that, because the tensotactic stimulus depends strongly on the cell volume fraction, upstream, tensotaxis-dominated migration occurs only transiently when the cells are initially seeded, and transitions to downstream, chemotaxis-dominated migration occur at later times due to the dispersive effect of cell diffusion., Competing Interests: Declaration of interests The authors declare no competing interests., (Copyright © 2024 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

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., (© 2023 Wiley Periodicals LLC.)

When employing mechanistic models to study biological phenomena, practical parameter identifiability is important for making accurate predictions across wide ranges of unseen scenarios, as well as for understanding the underlying mechanisms. In this work, we use a profile-likelihood approach to investigate parameter identifiability for four extensions of the Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) model, given experimental data from a cell invasion assay. We show that more complicated models tend to be less identifiable, with parameter estimates being more sensitive to subtle differences in experimental procedures, and that they require more data to be practically identifiable. As a result, we suggest that parameter identifiability should be considered alongside goodness-of-fit and model complexity as criteria for model selection.

Tumour angiogenesis leads to the formation of blood vessels that are structurally and spatially heterogeneous. Poor blood perfusion, in conjunction with increased hypoxia and oxygen heterogeneity, impairs a tumour's response to radiotherapy. The optimal strategy for enhancing tumour perfusion remains unclear, preventing its regular deployment in combination therapies. In this work, we first identify vascular architectural features that correlate with enhanced perfusion following radiotherapy, using in vivo imaging data from vascular tumours. Then, we present a novel computational model to determine the relationship between these architectural features and blood perfusion in silico. If perfusion is defined to be the proportion of vessels that support blood flow, we find that vascular networks with small mean diameters and large numbers of angiogenic sprouts show the largest increases in perfusion post-irradiation for both biological and synthetic tumours. We also identify cases where perfusion increases due to the pruning of hypoperfused vessels, rather than blood being rerouted. These results indicate the importance of considering network composition when determining the optimal irradiation strategy. In the future, we aim to use our findings to identify tumours that are good candidates for perfusion enhancement and to improve the efficacy of combination therapies., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2024 Köry et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

Longitudinal tumour volume data from head-and-neck cancer patients show that tumours of comparable pre-treatment size and stage may respond very differently to the same radiotherapy fractionation protocol. Mathematical models are often proposed to predict treatment outcome in this context, and have the potential to guide clinical decision-making and inform personalised fractionation protocols. Hindering effective use of models in this context is the sparsity of clinical measurements juxtaposed with the model complexity required to produce the full range of possible patient responses. In this work, we present a compartment model of tumour volume and tumour composition, which, despite relative simplicity, is capable of producing a wide range of patient responses. We then develop novel statistical methodology and leverage a cohort of existing clinical data to produce a predictive model of both tumour volume progression and the associated level of uncertainty that evolves throughout a patient's course of treatment. To capture inter-patient variability, all model parameters are patient specific, with a bootstrap particle filter-like Bayesian approach developed to model a set of training data as prior knowledge. We validate our approach against a subset of unseen data, and demonstrate both the predictive ability of our trained model and its limitations., (© 2024. The Author(s).)

Conditions for self-organisation via Turing's mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further., (© 2023. The Author(s).)

Complex biological processes involve collective behavior of entities (bacteria, cells, animals) over many length and time scales and can be described by discrete models that track individuals or by continuum models involving densities and fields. We consider hybrid stochastic agent-based models of branching morphogenesis and angiogenesis (new blood vessel creation from preexisting vasculature), which treat cells as individuals that are guided by underlying continuous chemical and/or mechanical fields. In these descriptions, leader (tip) cells emerge from existing branches and follower (stalk) cells build the new sprout in their wake. Vessel branching and fusion (anastomosis) occur as a result of tip and stalk cell dynamics. Coarse graining these hybrid models in appropriate limits produces continuum partial differential equations (PDEs) for endothelial cell densities that are more analytically tractable. While these models differ in nonlinearity, they produce similar equations at leading order when chemotaxis is dominant. We analyze this leading order system in a simple quasi-one-dimensional geometry and show that the numerical solution of the leading order PDE is well described by a soliton wave that evolves from vessel to source. This wave is an attractor for intermediate times until it arrives at the hypoxic region releasing the growth factor. The mathematical techniques used here thus identify common features of discrete and continuum approaches and provide insight into general biological mechanisms governing their collective dynamics.

Throughout developmental biology and ecology, transport can be driven by nonlocal interactions. Examples include cells that migrate based on contact with pseudopodia extended from other cells, and animals that move based on their awareness of other animals. Nonlocal integro-PDE models have been used to investigate contact attraction and repulsion in cell populations in 1D. In this paper, we generalise the analysis of pattern formation in such a model from 1D to higher spatial dimensions. Numerical simulations in 2D demonstrate complex behaviour in the model, including spatio-temporal patterns, multi-stability, and patterns with wavelength and shape that differ significantly depending on whether interactions are attractive or repulsive. Through linear stability analysis in N dimensions, we demonstrate how, unlike in local Turing reaction-diffusion models, the capacity for pattern formation fundamentally changes with dimensionality for this nonlocal model. Most notably, pattern formation is possible only in higher than one spatial dimension for both the single species system with repulsive interactions, and the two species system with 'run-and-chase' interactions. The latter case may be relevant to zebrafish stripe formation, which has been shown to be driven by run-and-chase dynamics between melanophore and xanthophore pigment cells., Competing Interests: Declaration of competing interest None., (Copyright © 2023 The Author(s). Published by Elsevier Inc. All rights reserved.)