Chondrocyte growth dynamics and spatial pattern formation

In this thesis we study the generation of patterns in cell culture refering to the important work of Elsdale where fibroblast
cultures were analyzed to investigate how densely packed cells organize.
A mathematical model was introduced in Edelstein-Keshet and Ermentrout (1990)
to prove that the pattern formation can be caused by the mere interactions of individual cells, although it is a
population phenomena. Until then the formation of structures was only attributed to other mechanisms as chemical gradients (chemotaxis)
or mechanical stresses. In this regard we refer in particular to Oster where a mathematical analysis was proposed
to understand how these mechanisms conspire to generate organized spatial aggregations. In Edelstein-Keshet and Ermentrout (1990)
indeed the authors showed that the self-organization of cells can actually be explained from contact-responses of the cells alone.
Their integro-differential equations considered the distribution of the cells as a variable of the time and the angle of orientation.
They presented two equations, one for cells that are bounded and one for free cells. Furthermore Mogilner extended the model to also take into account
the spatial distribution of the cells.
We could not find in the literature any similar works applied to the cells
considered in this dissertation. Because of the similarities between these cells and the fibroblasts we decided to start from this last model.
We worked in tight collaboration with the Tissue Engineering Group (TEG) at University Hospital in Basel.
Cartilage tissue engineering is a novel and promising approach to repair articular cartilage defects. This procedure requires that
cartilage cells (chondrocytes) are isolated from a small biopsy and expanded in vitro, generally on two-dimensional culture plates (monolayer),
to augment their original number. Post expanded cells are then cultured on specific biosynthetic materials and grafted in the
cartilage defects. One of the challenges that arise in this procedure is that the chondrocytes undergo only a limited number of divisions in vitro.
A possible way to overcome this limit consists in the supplantation of specific bioactive molecules (growth factors)
during the culture of chondrocytes. To investigate how these growth factors influence the cell expansion we were asked to seek an
appropriate mathematical model.
In a first step, we developed a model combining time-lag (delay) and logistic equations to capture the kinetic
parameters and to enable the description of the complete growth process of the cell culture.
However, this model only describes how the number of cells changes in time, without considering the spatial evolution of the cells
on a two-dimensional substrate. In previous experiments we observed that chondrocytes cultured with growth factors change not only their shapes, but also their main characteristics,
being then very similar to fibroblasts . This suggested that we
start from the model developed by Mogilner which, however, does not consider the cell duplication. We extended
this model in an innovative way, adding a logistic terms to follow the cell dynamics during the entire culture time. In particular, we used this model
to analyze the formation of patterns at confluence. Indeed it was observed in experiments that when the density of the cells reaches a critical
level there is a spontaneous tendency to align along some common axis of orientation. The selection of a preferred axis of
orientation can be explained by the fact that the uniform steady state
(one in which cells are uniformly distributed in orientation and space) could be unstable under particular conditions.
We used linear stability theory to test for the presence of such instability. Indeed, bifurcations can lead to loss of
stability of a uniform steady state in favor of patterned states, where cells are aligned in parallel arrays or aggregated
in clusters. We remark that we always tried not to loose
the link with the biological context by discussing constantly our results with the TEG. In particular, for the comparison
them with biological experiments it was essential to use sophisticated image analysis tools which also permit to analyze the orientation
of the cells.
One of the main problems the TEG is confronting consists in the variability of the behavior of chondrocytes isolated from different donors. In a study performed to
investigate age related changes in proliferation and post-expansion tissue-forming capacity \cite{Barbero04} an extreme variability in
these properties was unexpectedly observed among chondrocytes derived from donors within the same age range.
In this regard, the model we present could help biologists either in
defining conditions that improve chondrocyte properties or in identifying donor cells that have adequate characteristics for clinical application.