Spatial processes are attacked on two fronts. On the one hand, tools from theoretical and
statistical physics can be used to understand behaviour in complex, spatially-extended
multi-body systems. On the other hand, computer vision and statistical analysis can be
used to study 4D microscopy data to observe and understand real spatial processes in
vivo.
On the rst of these fronts, analytical models are developed for abstract processes, which
can be simulated on graphs and lattices before considering real-world applications in elds
such as biology, epidemiology or ecology. In the eld theoretic formulation of spatial processes,
techniques originating in quantum eld theory such as canonical quantisation and
the renormalization group are applied to reaction-di usion processes by analogy. These
techniques are combined in the study of critical phenomena or critical dynamics. At this
level, one is often interested in the scaling behaviour; how the correlation functions scale
for di erent dimensions in geometric space. This can lead to a better understanding of how
macroscopic patterns relate to microscopic interactions. In this vein, the trace of a branching
random walk on various graphs is studied. In the thesis, a distinctly abstract approach
is emphasised in order to support an algorithmic approach to parts of the formalism.
A model of self-organised criticality, the Abelian sandpile model, is also considered. By
exploiting a bijection between recurrent con gurations and spanning trees, an e cient
Monte Carlo algorithm is developed to simulate sandpile processes on large lattices.
On the second front, two case studies are considered; migratory patterns of leukaemia cells
and mitotic events in Arabidopsis roots. In the rst case, tools from statistical physics
are used to study the spatial dynamics of di erent leukaemia cell lineages before and after
a treatment. One key result is that we can discriminate between migratory patterns in
response to treatment, classifying cell motility in terms of sup/super/di usive regimes.
For the second case study, a novel algorithm is developed to processes a 4D light-sheet
microscopy dataset. The combination of transient uorescent markers and a poorly localised
specimen in the eld of view leads to a challenging tracking problem. A fuzzy
registration-tracking algorithm is developed to track mitotic events so as to understand
their spatiotemporal dynamics under normal conditions and after tissue damage.Open Acces
