Thesis title: Mathematical modeling of cell migration in response to chemical and physical stimuli: from immune to endothelial cells
Mathematical and computational models are becoming increasingly important in
deciphering biomedical data and testing biological hypotheses. For this reason,
they can be considered as “Virtual Laboratories”, i.e. venues where a large number
of tests can be performed in order to accompany and sustain their experimental
In the present thesis, inspired by laboratory experiments conducted on organs-onchip and on micropatterned surfaces, we have developed two different mathematical
models to describe and improve the knowledge of two complex biological problems
related to cancer and vascular research.
Cancer is one of the leading causes of death in industrialized countries. In recent
years, immunotherapy has emerged as a leading therapy to combat malignant cells.
Specifically, experiments on immunocompetent organs-on-chips were conducted to
investigate how the immune system reacts after the treatment of cancer cells with
chemotherapy drugs. Experimentally, a migratory activity of immune cells was
observed towards the cancer, and this motivated us to develop a mathematical model
able to reproduce the observed behavior.
To this aim, we developed a hybrid model of equations using a discrete in continuous
approach to mimic immune cell migration towards dying cancer cells. With such
a formulation, we can turn our gaze within the chip and look at immune cells as
single entities whose movement is governed by specific forces acting on them, which
is made possible thanks to a micro approach. On the other hand, the presence of
cancer cells involves the release of chemical species that can be interpreted as an
average on the space, thus bringing up the macro component of the problem.
In a preliminary phase of the present work, an analysis of the experimental data was
carried out, characterized by the “fixed” positions of the tumor cells and the positions
occupied over time by the immune cells. This kind of data analysis, preparatory to
the development of the model, allowed the identification of the different positions
and velocities of the immune cells. However, the shortage of experimental data did
not allow to work directly with them. For this reason we have produced synthetic
data, i.e., artificially produced data with our model, in the form of chemotherapy
density and trajectories traveled by the cells.
It should be emphasized that currently it is not possible to carry out experimental
measurements to measure the quantity and type of chemoattractant in the microchip,
therefore the reconstruction of the chemical signal is completely entrusted to the
model. The parameters used to produce these data were manually tested in order to
obtain numerical solutions close to the experimental behaviors.
In the end, we performed a calibration of the model based on a multidimensional
interpolation technique. In particular, we generated a velocity field at each time
step by interpolating the velocities of the single cells taken from synthetic data. The
objective function, i.e. the functional to be minimized, was constructed using the
interpolated velocity fields as targets to be compared with the numerical velocities
obtained with the model at each iteration of the calibration algorithm.
As a result, we completed the functional minimization procedure with negligible
errors and it was possible to find the same parameter values used to produce the
artificial data of the immune cell trajectories. Furthermore, even if applied to
synthetic data, the success of the above strategy shows the validity of the calibration
algorithm of the model parameters proposed in the present work. In conclusion, we
believe our approach is promising for future real data applications.
“Why do endothelial cells tumble on line patterns?”. This is the question which
inspired the second part of our work. In the Hydrodynamics Laboratory of Ecole
Polytechnique, when looking at movies of experiments on endothelial cells on line
patterns, i.e. adhesive lines which confer physical confinement to the cells, three
different migration behaviors and phenotypes unexpectedly arose. Specifically, we
observed changes in cell length and shape correlated to modifications in cell migration.
In order to study this phenomenon, as a first approach we measured the different
cell lengths and made a categorization of them according to cell type. Successively,
to interpret this finding, we hypothesized that behind the three cellular phenotypes
there were changes in intracellular ATP levels and F-actin. ATP is the main source
of energy of the cell and F-actin is a component of the cytoskeleton, therefore they
are both important for cell shape and cell migration.
Based on the hypothesis that ATP and F-actin changes are related to the dinstict
cell types, we designed a system of stochastic differential equations that describes the
dynamics of cell length, intracellular ATP and F-actin. These equations are intended
to capture and therefore classify the different cell states. To achieve such a result,
after the data analysis, we applied a calibration procedure to the model in order to
determine the parameter values that best fit the different sets of experimental data.
The results demonstrate that the proposed model is capable of generating profiles
of cell length dynamics that closely match those observed experimentally, together
with different ATP levels and F-actin concentrations. Different ranges of model
parameters lead to behavior that resembles that of the three observed endothelial
cell phenotypes. Furthermore, a detailed sensitivity analysis revealed which model
parameters dictate which features of the observed dynamics for each of the different