In predicting metastatic potential and improving treatment outcomes in cancer research, it is crucial that we understand the dynamics of cancer cell dormancy and reactivation. In this paper we propose, study, and evaluate a cancer growth model that incorporates cell death, dormancy, reactivation, and proliferation in the secondary sites. Using experimental data on mice, we test various statistical distributions and identify models that represent the asymmetry and variability observed in dormancy durations. Notably, the estimated cancer cell death rate remained consistent across all tested distributions, supporting its biological relevance as a robust parameter for modelling dormancy survival dynamics. The most suitable among the distributions we studied, the most suitable ones exhibit heavy-tails and asymmetric skewness; this aligns with the prolonged and rare dormancy periods expected of cancer cells. Our findings stress the importance of selecting appropriate statistical models for dormancy, both in predicting cancer cell reactivation events, and in informing therapeutic strategies that focus dormancy-driven metastasis. ### Competing Interest Statement The authors have declared no competing interest.

Background Organotypic assays are three-dimensional in vitro models widely used in cancer research to mimic the in vivo extracellular matrix (ECM) and to study cancer cell invasion by allowing investigation of critical interactions between tumor cells and their microenvironment. During invasion, cancer cells undergo genetic and epigenetic changes that disrupt cell-cell adhesion, enabling detachment from the primary tumor. Subsequently, invasive cells must (i) breach the basement membrane, a dense protein meshwork that restricts cell movement, and (ii) migrate through the surrounding ECM. This process relies on proteolytic enzyme secretion to degrade structural barriers, followed by cell-matrix adhesion-mediated migration. Methods We present an in silico model of cancer cell invasion in organotypic assays. The model is formulated as a system of partial differential equations capturing spatiotemporal dynamics, with nonlocal terms representing preferential adhesion-driven movement. Key variables in the model are cancer cells, proteolytic enzymes, and the ECM. Results Computational simulations demonstrate that modulation of cell-cell and cell-matrix adhesion parameters significantly influences tumor invasiveness. This result is consistent with experimental observations, demonstrating the model’s ability to accurately reflect in vitro behavior. ### Competing Interest Statement The authors have declared no competing interest.

The transition from the epithelial to mesenchymal phenotype and its reverse (from mesenchymal to epithelial) are crucial processes necessary for the progression and spread of cancer. In this paper, we investigate how phenotypic switching at the cancer cell level impacts the behaviour at the tissue level, specifically on the emergence of isolated foci of the invading solid tumour mass leading to a multifocal tumour. To this end, we propose a new mathematical model of cancer invasion that includes the influence of cancer cell phenotype on the rate of invasion and metastasis. The implications of the model are explored through numerical simulations revealing that the plasticity of tumour cell phenotypes appears to be crucial for disease progression and local invasive spread. The computational simulations show the progression of the invasive spread of a primary cancer reminiscent of in vivo multifocal breast carcinomas, where multiple, synchronous neoplastic foci are frequently observed and are associated with a poorer patient prognosis.

In vertical inhibition treatment strategies, multiple components of an intracellular pathway are simultaneously inhibited. Vertical inhibition of the BRAFV600E-MEK-ERK signalling pathway is a standard of care for treating BRAFV600E-mutated melanoma where two targeted cancer drugs, a BRAFV600E-inhibitor, and a MEK inhibitor, are administered in combination. Targeted therapies have been linked to early onsets of drug resistance, and thus treatment strategies of higher complexities and lower doses have been proposed as alternatives to current clinical strategies. However, finding optimal complex, low-dose treatment strategies is a challenge, as it is possible to design more treatment strategies than are feasibly testable in experimental settings. To quantitatively address this challenge, we develop a mathematical model of BRAFV600E-MEK-ERK signalling dynamics in response to combinations of the BRAFV600E-inhibitor dabrafenib (DBF), the MEK inhibitor trametinib (TMT), and the ERK-inhibitor SCH772984 (SCH). From a model of the BRAFV600E-MEK-ERK pathway, and a set of molecular-level drug-protein interactions, we extract a system of chemical reactions that is parameterised by in vitro data and converted to a system of ordinary differential equations (ODEs) using the law of mass action. The ODEs are solved numerically to produce simulations of how pathway-component concentrations change over time in response to different treatment strategies, i.e., inhibitor combinations and doses. The model can thus be used to limit the search space for effective treatment strategies that target the BRAFV600E-MEK-ERK pathway and warrant further experimental investigation. The results demonstrate that DBF and DBF-TMT-SCH therapies show marked sensitivity to BRAFV600E concentrations in silico, whilst TMT and SCH monotherapies do not.

We introduce in this paper substantial enhancements to a previously proposed hybrid multiscale cancer invasion modelling framework to better reflect the biological reality and dynamics of cancer. These model updates contribute to a more accurate representation of cancer dynamics, they provide deeper insights and enhance our predictive capabilities. Key updates include the integration of porous medium-like diffusion for the evolution of Epithelial-like Cancer Cells and other essential cellular constituents of the system, more realistic modelling of Epithelial-Mesenchymal Transition and Mesenchymal-Epithelial Transition models with the inclusion of Transforming Growth Factor beta within the tumour microenvironment, and the introduction of Compound Poisson Process in the Stochastic Differential Equations that describe the migration behaviour of the Mesenchymal-like Cancer Cells. Another innovative feature of the model is its extension into a multi-organ metastatic framework. This framework connects various organs through a circulatory network, enabling the study of how cancer cells spread to secondary sites.

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Free Article from Paradoxical Dependencies of Tumor Dormancy and Progression on Basic Cell Kinetics

Invasion of the surrounding tissue is a key aspect of cancer growth and spread involving a coordinated effort between cell migration and matrix degradation, and has been the subject of mathematical modelling for almost 30 years. In this current paper we address a long-standing question in the field of cancer cell migration modelling. Namely, identify the migratory pattern and spread of individual cancer cells, or small clusters of cancer cells, when the macroscopic evolution of the cancer cell colony is dictated by a specific partial differential equation (PDE). We show that the usual heuristic understanding of the diffusion and advection terms of the PDE being one-to-one responsible for the random and biased motion of the solitary cancer cells, respectively, is not precise. On the contrary, we show that the drift term of the correct stochastic differential equation scheme that dictates the individual cancer cell migration, should account also for the divergence of the diffusion of the PDE. We support our claims with a number of numerical experiments and computational simulations.

Individual responses to SARS-CoV-2 infection vary significantly, ranging from mild courses of infection that do not require hospitalisation to the development of disease which not only requires hospitalisation but can be fatal. Whilst many immunological studies have revealed fundamental insights into SARS-CoV-2 infection and COVID-19, mathematical and computational modelling can offer an additional perspective and enhance understanding. The majority of mathematical models for the within-host spread of SARS-CoV-2 infection are ordinary differential equations, which neglect spatial variation. In this article, we present a hybrid, multiscale, individual-based model to study the within-host spread of SARS-CoV-2 infection. The model incorporates epithelial cells (each containing a dynamical model for viral entry and replication), macrophages and a subset of cytokines. We investigate the role of increasing initial viral deposition, increasing delay in type I interferon secretion from epithelial cells (as well as the magnitude of secretion), increasing macrophage virus internalisation rate and macrophage activation, on the spread of infection.

The contents of this editorial is proposed in three parts devoted to the modeling and simulation of mutating virus pandemics in a globally connected world. The presentation is as follows: first, a general framework is presented according to the idea that mathematical models should go beyond deterministic population dynamics by considering the multiscale, heterogeneous features of the complex system under consideration; second, the contents of the papers in this issue are presented; finally, there is a critical analysis looking ahead to research perspectives.

Rheumatoid arthritis (RA) is a chronic inflammatory disorder that causes pain, swelling and stiffness in the joints, and negatively impacts the life of affected patients. The disease does not have a cure yet, as there are still many aspects of this complex disorder that are not fully understood. While mathematical models can shed light on some of these aspects, to date there are few such models that can be used to better understand the disease. As a first step in the mechanistic understanding of RA, in this study we introduce a new hybrid mathematical modelling framework that describes pannus formation in a small proximal interphalangeal (PIP) joint. We perform numerical simulations with this new model, to investigate the impact of different levels of immune cells (macrophages and fibroblasts) on the degradation of bone and cartilage. Since many model parameters are unknown and cannot be estimated due to a lack of experiments, we also perform a sensitivity analysis of model outputs to various model parameters (single parameters or combinations of parameters). Finally, we discuss how our model could be applied to investigate current treatments for RA, for example, methotrexate, TNF-inhibitors or tocilizumab, which can impact different model parameters.

Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin–Voigt model of linear viscoelasticity to represent the stress–strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this paper, we systematically assess the pattern formation potential of different stress–strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin–Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress–strain relation of the ECM, while no patterns are observed when the Kelvin–Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress–strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.

The formation of new vascular networks is essential for tissue development and regeneration, in addition to playing a key role in pathological settings such as ischemia and tumour development. Experimental findings in the past two decades have led to the identification of a new mechanism of neovascularisation, known as cluster-based vasculogenesis, during which endothelial progenitor cells (EPCs) mobilised from the bone marrow are capable of bridging distant vascular beds in a variety of hypoxic settings in vivo. This process is characterised by the formation of EPC clusters during its early stages and, while much progress has been made in identifying various mechanisms underlying cluster formation, we are still far from a comprehensive description of such spatio-temporal dynamics. In order to achieve this, we propose a novel mathematical model of the early stages of cluster-based vasculogenesis, comprising of a system of nonlocal partial differential equations including key mechanisms such as endogenous chemotaxis, matrix degradation, cell proliferation and cell-to-cell adhesion. We conduct a linear stability analysis on the system and solve the equations numerically. We then conduct a parametric analysis of the numerical solutions of the one-dimensional problem to investigate the role of underlying dynamics on the speed of cluster formation and the size of clusters, measured via appropriate metrics for the cluster width and compactness. We verify the key results of the parametric analysis with simulations of the two-dimensional problem. Our results, which qualitatively compare with data from in vitro experiments, elucidate the complementary role played by endogenous chemotaxis and matrix degradation in the formation of clusters, suggesting chemotaxis is responsible for the cluster topology while matrix degradation is responsible for the speed of cluster formation. Our results also indicate that the nonlocal cell-to-cell adhesion term in our model, even though it initially causes cells to aggregate, is not sufficient to ensure clusters are stable over long time periods. Consequently, new modelling strategies for cell-to-cell adhesion are required to stabilise in silico clusters. We end the paper with a thorough discussion of promising, fruitful future modelling and experimental research perspectives.

Background Simultaneous inhibition of multiple components of the BRAF-MEK-ERK cascade (vertical inhibition) has become a standard of care for treating BRAF-mutant melanoma. However, the molecular mechanism of how vertical inhibition synergistically suppresses intracellular ERK activity, and consequently cell proliferation, are yet to be fully elucidated. Methods We develop a mechanistic mathematical model that describes how the mutant BRAF inhibitor, dabrafenib, and the MEK inhibitor, trametinib, affect BRAFV600E-MEK-ERK signalling. The model is based on a system of chemical reactions that describes cascade signalling dynamics. Using mass action kinetics, the chemical reactions are re-expressed as ordinary differential equations that are parameterised by in vitro data and solved numerically to obtain the temporal evolution of cascade component concentrations. Results The model provides a quantitative method to compute how dabrafenib and trametinib can be used in combination to synergistically inhibit ERK activity in BRAFV600E-mutant melanoma cells. The model elucidates molecular mechanisms of vertical inhibition of the BRAFV600E-MEK-ERK cascade and delineates how elevated BRAF concentrations generate drug resistance to dabrafenib and trametinib. The computational simulations further suggest that elevated ATP levels could be a factor in drug resistance to dabrafenib. Conclusions The model can be used to systematically motivate which dabrafenib–trametinib dose combinations, for treating BRAFV600E-mutated melanoma, warrant experimental investigation.