Conference Agenda

In the context of cell mechanics, we depict a multi-scale continuum approach to study the exchange of mechnanical actions between focal adhesions (FAs), the extra-cellular matrix (ECM) and living cells [1]. In particular, we investigate the role of force- and stress-type stimuli in determining the remodelling, or structural time evolution, of both FAs and ECM, and we investigate how possible heterogeneities attainable at the micro-scale, i.e., the scale characterising the internal structure of each component of focal adhesions, may influence their behaviour [2, 3, 5]. With reference to some structural models available in the literature, we describe a focal adhesion as a sandwich structure, accounting for three main components: the adhesion plaque, the integrins receptors and the ECM [1, 3]. Following previous works [1, 2, 3], we employ a mono-dimensional shear lag model [2, 3], so that, both the adhesion plaque and the ECM are modelled as linear elastic straight fibres subjected to axial deformation only, while the family of integrins receptors is represented by a system of elastic and non-elastic forces. Furthermore, we consider the description of focal adhesions’ dynamics by accounting for their remodelling and micro-scale inhomogeneities. To achieve this, we follow and adapt to our scopes some tools of non-linear elastoplasticity and we adhere to techniques of Asymptotic Homogenization to elucidate how the micro-structure of focal adhesions influences their overall behaviour [3, 4, 5]. In this regard, we obtain closed form of the effective coefficients characterising the FA-ECM complex, with the latter containing both constitutive and geometric information available at the micro-structure.

The obtained multi-scale model, and the related field equations, are solved and the forces exchanged by focal adhesions with the ECM and cells are studied. Detailed findings from this investigation are summarized in [5].

References

[1] Cao X., et al., “A chemomechanical model of matrix and nuclear rigidity regulation of focal

adhesion size,” Biophys. J., 109.9, 1807-1817 (2015).

[2] Di Stefano S., et al., “On the role of elasticity in focal adhesion stability within the passive

regime,” Int J Non Linear Mech, 146, 104157 (2022).

[3] Di Stefano S., et al., “On the role of friction and remodelling in cell–matrix interactions: a

continuum mechanical model,” Int J Non Linear Mech, 142, 103966 (2022).

[4] Ramírez-Torres-Torres A., et al., “An asymptotic homogenization approach to the microstructural evolution of heterogeneous media,” Int J Non Linear Mech, 106, 245-257 (2018).

[5] Di Stefano S., et al., ''Homogenised structural behaviour of remodelling cell-matrix systems: the case of focal adhesions''. In preparation.

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

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) .