The next seminar “Modelling of materials – theory, model reduction and efficient numerical methods” will take place next **Monday** (February 18, 2019) in **K2** from **9:00** till **10:30**. The talk will be given by Václav Klika (FJFI ČVUT). Please see the details below.

Speaker: **Václav Klika**

Title: **Implications of tissue heterogeneity in cartilage modelling**

Abstract: Current models targeting mechanobiology of cartilage are

becoming increasingly refined and complex by the inclusion of ever more

details such as heterogeneous distribution of solid matrix within

cartilage, fixed charged density, heterogeneous Darcy’s law with

preferential directions of flow determined by deformation, fibres,

compaction effects (the closing of pores), and complex 3D geometries.

Despite the undoubted benefit of having a finer description of the

cartilage tissue and hence the prospect of capturing its behaviour in a

wider context there is at the same time the issue of model verification as

the amount of data necessary for parameter estimation and subsequent

independent model validation rapidly increases.

In this talk we follow a different path to minimise the problem of

overestimation by revisiting the 1D experimentally relevant (confined

compression with rotational symmetry) biphasic model which allows for

qualitative insight and more reliable parameter estimation. Particularly

we shall see that the inclusion of heterogeneity in the initial solid

volume fraction corresponding to the presence of proteoglycans in

cartilage matrix has profound implications on both bulk equations, and

initial and boundary conditions. This influence is mediated by swelling

pressure being a consequence of achieving electroneutrality in the

system.

We shall rederive the 1D biphasic model carefully, as the linear

bihpasic model previously used that allows for an analytical solution has

some limitations in its presentation and derivation and is a special case

of the model formulated here with the swelling pressure contribution. Then

we continue with exploring the fundamental consequences of heterogeneous

distribution of initial volume fraction via the swelling pressure term

noting that compactification (pores closing) is naturally reflected in the

swelling pressure term.

If time permits, we shall discuss possible replacement of the

classically used Donnan theory for swelling pressure by a model reflecting

some of the microscopic natures of the cartilage tissue: a macroscale

model for swelling pressure that is an upscaled version of

Poisson-Nernst-Planck microscopic description. To this end we use the

method of multiple scales which is suitable even for systems with slowly

varying periodicity.