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.