Conference Agenda

There is growing interest in the application of mathematical and numerical models to gain deeper insight into physiologically interesting problems. For instance, cardiovascular disease of general interest to modellers and clinicians given the large global disease burden. Via mathematical models and simulation, we are able to gain a deeper insight into disease processes than is necessarily feasible or ethical to gain via physiological experimentation.

All mathematical models of physiological phenomena require at least two parts: the model equations to solve, and a computational domain in which to solve them.

Here, we discuss the development of a 1D flow model for arterial and venous perfusion including explicitly specified large blood vessels and vascular beds that are implicitly modelled via a structured tree approach. The model can be implemented using only physiologically meaningful parameters, or parameters that can be derived from physiological data. Further, as many blood vessels, such as the microvasculature of the coronary and pulmonary systems are embedded within moving tissues, we include the periodic external pressure to which vessels are subject to as a result of tissue motion.

As the model can be implemented with only physiologically meaningful parameters, it is reasonable that the large blood vessels are defined using physiologically realistic values, i.e. the computational domain in which we simulate blood flow resembles that of the organ system in which we wish to simulate blood flow. To do this, we require the length, radius, and hierarchical information of all blood vessels that are to be modelled explicitly. These data are difficult to find in the literature, and where they are reported, they are given in aggregate. However, imaged vascular networks are increasingly easy to obtain.

Given the need for realistic computational domains, and the increasing availability of medical images, we present the development of robust algorithms that are used to simplify large data sets that represent the vasculature of an organ system without loss of information, and pruning techniques to obtain a computational domain that conforms to given criteria required of the computational domain. In the specific example of the coronary circulation, we separate the arteries into those that lie within and outwith the myocardium. We are also able to divide the given organ into subregions that are perfused from a given large artery and compare these against the American Heart Associations division of the left ventricle as a sense-check.

Finally, we will simulate blood flow in the obtained computational domains to investigate the numerical model’s sensitivity to computational domain morphometry.

This research was supported by the UK Engineering and Physical Sciences Research Council (EP/N014642/1, EP/T017899/1) .

Human skin is a complex material to test and model as its physical and geometric properties depend on a host of parameters and conditions: thickness, location, age, ethnicity, hydration, etc. Skin tension plays a pivotal role in clinical settings, affecting scarring, wound healing and skin necrosis. Despite its importance, there is no widely accepted method for assessing in vivo skin tension. Destructive testing of skin samples only gives a partial picture, as harvesting skin dehydrates the sample and releases its residual stress, which is likely to alter its behaviour significantly. Knowledge of the in vivo tension in skin will aid in preoperative reconstructive surgery planning, e.g., by providing safe limits of skin stress or accurately estimating the skin area required for repairs. Here, we develop and validate two methods to quantify the in vivo tension in skin using acoustic measurements. We find that it is possible to determine the difference in residual stress along in-plane directions on a patient-specific basis, using a simple in vivo measurement technique. Additionally, we find that coupling elastic wave measurements with machine learning (ML) models is a viable non-invasive method to determine in vivo skin tension.

Firstly, we model the skin as an incompressible, anisotropic hyperelastic material with one family of fibres, where the principal pre-stress is aligned along the fibres. A detailed analysis of the theoretical model reveals that the in-plane stress difference is related to the surface wave speeds, via a simple formula with a known error of less than 9%. The proposed formula is universal and depends on neither a specific energy density function nor material properties. We validate the formula with finite element (FE) simulations. We replicate the in vivo stress by applying a pre-stretch and induce a wave using an instantaneous impulse. We measure the wave speeds parallel and perpendicular to the fibres at various levels of pre-stress. We find that the error between the actual stress difference and that calculated with the formula is less than 3% for the simulations, which is within the range determined by the analytic model.

Secondly, we train a ML model that uses surface wave speed measurements to predict the in vivo skin tension. We create a large dataset consisting of simulated wave propagation experiments using an FE model. We then train a Gaussian process regression model to solve the inverse problem of predicting stress and pre-stretch in skin using the wave speed measurements. The ML model demonstrated good predictive performance, highlighting the feasibility of the method.

In vivo stress is difficult to estimate in general, however, the methods proposed here will enable on-demand patient-specific measurement of the in vivo stress in skin in a non-invasive manner. They are based on easily accessible parameters and could replace existing qualitative techniques with more accurate quantitative measurements, aiding preoperative reconstructive surgery planning and ultimately improving surgical outcomes.

This is joint work with Matt Nagle, Christelle Vedel, Wenting Shu, Michel Destrade, Michael Fop and Aisling Ní Annaidh.

Age-related vision problems significantly impact the quality of life. Among these, macular holes are a leading cause of vision loss in the elderly, resulting from the formation of a hole in the macula, the central region of the retina responsible for high-acuity vision. Damage to the macula can lead to severe visual impairment. Although the precise mechanism of macular hole formation remains unclear, it is thought to be associated with mechanical forces exerted on the retina by the vitreous. The vitreous is a clear, viscoelastic gel composed of collagen fibres that occupies the space between the lens and the retina and is attached to the retina at its posterior end. With age, the vitreous undergoes structural changes and shrinks, increasing the traction forces on the retina. Simultaneously, the retinal tissue gets thinner and fragile with age. In combination, these processes compromise the integrity of the macula, cause swelling, and eventually lead to the formation of a hole in the retina. Prolonged traction forces can further result in the retina separating from the retinal pigment epithelium (RPE), a layer of epithelial cells located at the back of the retina, leading to retinal detachment. While the initiation of this pathology is qualitatively understood, a quantitative analysis of the mechanical forces contributing to macular hole formation is still lacking. It is observed that the size of the macular hole affects postoperative closure and visual acuity. Hence, a better understanding of macular hole formation and growth mechanics will help improve surgical techniques such as vitrectomy.
In this study, we introduce a poroelastic model of the retina to investigate the mechanical forces involved in macular hole formation and retinal detachment. Our model captures the age-related changes in vitreous traction forces and their impact on the retina. Using retinal OCT images as input data, we simulate the patient-specific retinal geometry with symmetry at the macula. The model incorporates a multiphasic contact framework between the two halves of the retina, ensuring continuity in fluid pressure, solute concentration, and displacement degrees of freedom. A traction-separation law-based damage criterion is employed to simulate the detachment of the retinal halves. Additionally, a multiphasic contact is defined between the retina and the RPE to simulate retinal detachment.
Our findings demonstrate that as the traction forces on the retina increase with age, the size of the macular hole also grows, consistent with clinical observations. The model further reveals how fluid and solute flows are redistributed due to macular hole formation. Moreover, it highlights the significant influence of retinal stiffness on macular hole formation and retinal detachment, suggesting a link between these conditions and other pathologies associated with increased retinal stiffness. This work provides a quantitative framework for understanding the mechanical forces driving macular hole formation and offers insights into the relationship between various retinal pathologies.

This study investigates the mechanics of avascular solid tumour growth, examining how chemical interactions in the microenvironment shape its development, with a particular focus on the delayed effects these interactions have on tissue evolution. We model the tumour's progression by using the multiplicative decomposition of the deformation gradient tensor [1], which separates the total deformation into two distinct parts, one capturing inelastic distortions caused by growth, and the other representing elastic deformations accommodating these changes.
Building on [2,3], we formulate a growth law to govern these inelastic distortions, based on a balance of generalised mechanical forces and the dissipation inequality, allowing for the inclusion of constitutive relations. This structure provides a more physically grounded framework and also permits the inclusion of mass source or sink terms from conventional phenomenological models. Furthermore, this growth law is determined a posteriori and governed by internal and external non-conventional forces. The internal force, associated with the Eshelby stress tensor, captures configurational mechanical stress arising from inhomogeneities within the tumour structure. The external force is conjugate to growth-related kinematic descriptors and models microscale biochemical interactions, such as nutrient consumption, drug treatment strategies, and other factors that may influence tumour growth and tissue behaviour over time.
A further key aspect of this study is the inclusion of time-delay effects within the growth model. In real biological systems, particularly in heterogeneous tumour microenvironments, chemical processes are not instantaneous. We account for this by introducing memory effects into the external non-conventional force through integral operators, drawing on concepts from fractional calculus, to reflect how earlier processes can continue to influence the current state of growth. Such delays become especially relevant when considering tumour response to temporally varying environmental inputs, including nutrient supply or therapeutic interventions. These time-dependent effects improve the model's ability to reflect tumour progression beyond time-local formulations and offer a mathematically consistent way to account for non-instantaneous biochemical effects.
Our findings show how time-delay effects, combined with the growth law, affect key tumour descriptors over time. This approach highlights the connection between delayed chemical influences and a mechanically grounded growth law, offering a new perspective on tumour mechanics and contributing to a deeper understanding of the complexity of the tumour microenvironment.
References
[1] Micunovic, M. (2009). Thermomechanics of Viscoplasticity. Springer. https://doi.org/10.1007/978-0-387-89490-4
[2] Grillo, A., & Di Stefano, S. (2023). Mathematics and Mechanics of Complex Systems, 11(1), 57–86. https://doi.org/10.2140/memocs.2023.11.57
[3] Almudarra, M., & Ramirez-Torres, A. (2025). Mathematics and Mechanics of Solids, 30(2), 501–526. https://doi.org/10.1177/10812865241230269

We study numerically the conformations of self-avoiding active ring polymers. They are modeled as closed bead-spring chains, driven by tangential forces, whose dynamics is implemented via the Brownian multiparticle collision scheme.
The mean-square intrachain distances, the mean-square gyration radius, and the bond correlation functions are calculated, analyzed using scaling assumptions, and comprared to the same quantities of the corresponding passive case.
We show that all these quantities can be described asymptotically by the same scaling forms of passive ring polymers but with a new active scaling exponent.
At high activity it is found that an effective persistence length characterizes the conformations of flexible active rings.