# Lattice models in materials science diffusion, trabecular bone remodelling and linear elastic networks

Abstract (Summary)

This thesis presents the results of investigations on three independent research
topics of modern biophysical and materials science research: substitutional
diffusion in binary alloys, the remodelling process in trabecular
bone and the prediction of mechanical properties of self assembling, amphiphilic
bilayers. The basic description of all three projects is based on
lattice models, a highly successful class of models that are used in several
fields of modern physics to describe physical processes. For the diffusional
process in alloys, which on a microscopic scale manifests itself in a discrete
site exchange between one atom and a neighbouring vacancy, it was investigated
how this microscopic description can be reconciled with a macroscopic
continuum model. In a computer simulation exact microscopic averages were
used to determine macroscopic properties, like Onsager’s coefficients. These
were then compared to theoretical predictions of different accuracy. Following
the same strategy – comparing averaged results from microscopical
simulations with purely continuum mechanical calculations – interdiffusion
problems were investigated. It was shown that for obtaining an appropriate
macroscopic description it is essential to fully include the behaviour of the
vacancy in the description, which is – due its complexity – often omitted.
For the investigations on remodelling of trabecular bone, bone’s architecture
was mapped onto a lattice and the local mechanical state of each element
was determined by a simplified mechanical model. A local remodelling law
was then used to translate this mechanical information into a signal that
determined the rate of change of the architecture at that special point. This
rate of change was given by a stochastic description, i.e. the remodelling law
gave the probabilities for bone formation and resorption, respectively. The
development of the model was guided by the aim to give a good balance in
the accuracy of the description of the mechanical and biological part. The
simple, but fast, algorithm to assess the mechanical properties of the structure
gave the possibility to test a variety of biological hypotheses, concerning
the special form of the remodelling law. It was shown that a stochastic
description of the remodelling process demands the formulation of both, a
formation and a resorption probability, since – in contrast to conventional
simulations with deterministic rate equations – a pure net effect does not
suffice to describe the process. Furthermore it was shown that a non-linear
remodelling law is a better candidate to describe the remodelling process
in real bone than a linear one. Finally the model was used to describe osteoporosis,
a wide spread disease affecting trabecular architecture. It was
concluded that in the features attributed to osteoporosis one has to distinguish
between normal ageing of bone’s architecture and additional changes
that stem from pathological alterations in the regulatory system.
A simple concept was introduced to model the mechanical properties of
self-assembled membranes. The (amphiphilic) molecules forming the membrane
are assumed to occupy a regular lattice, nearest neighbours are connected
by linear, elastic springs. Different spring constants are assumed
for different atomic pairs. The full elastic matrix of a given structure was
solved and the elastic modulus, the Poisson ratio and the bending rigidity
of the system determined. It was shown that the bending rigidity exhibits
a pronounced concentration dependence, varying over orders of magnitude
in a small concentration regime, giving very flexible membranes at one end
(bending rigidities of the order of kT ), very stiff ones at the other (bending
rigidities up to three orders of magnitude larger than kT ).
Bibliographical Information:

Advisor:

School:Oberlin College

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:

ISBN:

Date of Publication: