Multiscale Models in Mechano and Tumor Biology

We present an introduction to the asymptotic homogenization technique as a follow up of the four hours course held at the International Workshop on Multiscale Models in Mechano and Tumor Biology: Modeling, Homogenization, and Applications (M3TB2015) at TU Darmstadt (Germany) on 28 September 2015. The content is well-known, although revisited to provide a first insight to scientists and (especially) students who are unaware of the topic. We present the technique via three simple and instructive examples (one-dimensional and multi-dimensional diffusion and the Stokes’s problem for porous media) remarking on the role of regularity assumptions (periodicity vs. local boundedness) and non-dimensionalization, which are sometimes not sufficiently clarified in the existing literature.

Asymptotic homogenization is a useful mathematical tool that can be used to reduce the complexity of a problem with a periodic geometry. Generally, for asymptotic homogenization to be applicable, the full problem must have: (i) a periodic microstructure and (ii) a small ratio between the typical lengths of the periodic cell and the macroscale variation. In this chapter we consider a model for drug delivery. Namely, we discuss an asymptotic homogenization for the concentration field of a drug diffusing within a domain that contains a near-periodic array of circular obstructions whose boundaries can absorb the drug. In particular, the radii of these circular obstructions can slowly vary in space, and thus the microscale geometry varies in the macroscale. Constraining the shape of the obstacles to a one-parameter family, where the only variation is circle radius, allows us to homogenize this problem in a computationally efficient manner. Moreover, the method we present allows us to determine the homogenized equation for any arrangement of the microstructure within the one-parameter constraint.

A common feature of human cancers is the acidity of the tumor microenvironment, which is essentially influencing the development and spread of the neoplastic tissue, tumor invasion and formation of metastases, degradation of normal tissue, and response to therapies. In this chapter we review some of the continuum mathematical modeling approaches used so far to describe these phenomena, with a focus on the invasive behavior of cancer mediated by acidic pH of the surrounding tissue. We also pay particular attention to the issues of multiscaledness and randomness and compare the qualitative behavior of a class of deterministic models with their stochastic counterparts.

We present a problem-suited numerical method for a particularly challenging cancer invasion model. This model is a multiscale haptotaxis advection-reaction-diffusion system that describes the macroscopic dynamics of two types of cancer cells coupled with microscopic dynamics of the cells adhesion on the extracellular matrix. The difficulties to overcome arise from the non-constant advection and diffusion coefficients, a time delay term, as well as stiff reaction terms.

Our numerical method is a second order finite volume implicit-explicit scheme adjusted to include (a) non-constant diffusion coefficients in the implicit part, (b) an interpolation technique for the time delay, and (c) a restriction on the time increment for the stiff reaction terms.

In this chapter, we present recent works concerned with the derivation of a macroscopic model for complex interconnected fiber networks from an agent-based model, with applications to, but not limited to, adipose tissue self-organization. Starting from an agent-based model for interconnected fibers interacting through alignment interactions and having the ability to create and suppress cross-links, the formal limit of large number of individuals is first investigated. It leads to a kinetic system of two equations: one for the individual fiber distribution function and one for the distribution function of connected fiber pairs. The hydrodynamic limit, in a regime of instantaneous fiber linking/unlinking then leads to a macroscopic model describing the evolution of the fiber local density and mean orientation. These works are the first attempt to derive a macroscopic model for interconnected fibers from an agent-based formulation and represent a first step towards the formulation of a large scale synthetic tissue model which will serve for the investigation of large scale effects in tissue homeostasis.

Nanomedicine is the emerging medical research branch which employs nanotechnological devices to improve clinical diagnosis and to design more effective therapeutic methodologies. In particular, functionalized nanoparticles have proved their clinical usefulness for cancer therapy, either as vectors for targeted drug delivery or for hyperthermia treatment. The effectiveness of such novel therapeutic strategies in nanomedicine exploits the capability of the nanoparticles to penetrate into the living tissue through the vascular network and to reach the targeted site. Accordingly, their success is tightly related to the control of the the multi-physics and multiscale phenomena governing the diffusion and transport properties of the nanoparticles, together with the geometrical and chemo-mechanical factors regulating the nanoparticles-tissue interactions. Indeed, the therapeutic effectiveness of earlier approaches was hindered by a limited ability in penetrating within the tumor tissue essentially due to microfluidic effects. Mathematical modeling is often employed in nanomedicine to analyze in silico the key biophysical mechanisms acting at different scales of investigations, providing useful guidelines to foresee and possibly optimize novel experimental techniques. Since these phenomena involve different characteristic time- and length-scales, a multiscale modeling approach is mandatory. In this work we outline how a multiscale analysis starts at the smallest scale, and its results are injected in large-scale models. At the microscale, the transport of nanoparticles is modeled either by the stochastic Langevin equation or by its continuous limit; in both cases short distance interaction forces between particles are considered, such as Coulomb and van der Waals interactions, and small disturbances of the fluid velocity field induced by the presence of nanoparticles are assumed. At the macroscopic scale, the living tissue is typically modeled as a homogeneous (homogenized) porous material of varying permeability, where the fluid flow is modeled by Darcy’s equation and nanoparticle transport is described by a continuum Diffusion-Reaction-Advection equation. One of the most significant features of the model is the ability to incorporate information on the microvascular network based on physiological data. The exploitation of the large aspect ratio between the diameter of a capillary and the intercapillary distance makes it possible to adopt an advanced computational scheme as the embedded multiscale method: with this approach the capillaries are represented as one-dimensional (1D) channels embedded and exchanging mass in a porous medium. Special mathematical operators are used to model the interaction of capillaries with the surrounding tissue. In this general context, we illustrate a bottom-up approach to study the transport and the diffusion of nanoparticles in living materials. We determine the permeability as well as the lumped parameters appearing in the nanoparticle transport equation at the tissue level by means of simulations at the microscale, while the macroscale tissue deposition rate is derived from the results of microscale simulations by means of a suitable upscaling technique.

We present a continuum model for the mechanical behavior of the skeletal muscle tissue when its functionality is reduced due to aging. The loss of ability of activating is typical of the geriatric syndrome called sarcopenia. The material is described by a hyperelastic, polyconvex, transversely isotropic strain energy function. The three material parameters appearing in the energy are fitted by least square optimization on experimental data, while incompressibility is assumed through a Lagrange multiplier representing the hydrostatic pressure. The activation of the muscle fibers, which is related to the contraction of the sarcomere, is modeled by the so called active strain approach. The loss of performance of an elder muscle is then obtained by lowering of some percentage the active part of the stress. The model is implemented numerically and the obtained results are discussed and graphically represented.

We discuss a reduced model to compute the motion of slender swimmers which propel themselves by changing the curvature of their body. Our approach is based on the use of Resistive Force Theory for the evaluation of the viscous forces and torques exerted by the surrounding fluid, and on discretizing the kinematics of the swimmer by representing its body through an articulated chain of N rigid links capable of planar deformations. The resulting system of ODEs, governing the motion of the swimmer, is easy to assemble and to solve, making our reduced model a valuable tool in the design and optimization of bio-inspired artificial microdevices. We prove that the swimmer is controllable in the whole plane, for N ≥ 3 and for almost every set of stick lengths. As a direct result, there exists an optimal swimming strategy to reach a desired configuration in minimum time. Numerical experiments for N = 3 (Purcell swimmer) suggest that the optimal strategy is periodic, namely a sequence of identical strokes. Our results indicate that this candidate for an optimal stroke, indeed gives a better displacement speed than the classical Purcell stroke.