  As motivated by studies of cellular motility driven by spatiotemporal
chemotactic gradients in microdevices, we develop a framework for constructing
approximate analytical solutions for the location, speed and cellular densities
for cell chemotaxis waves in heterogeneous fields of chemoattractant from the
underlying partial differential equation models. In particular, such
chemotactic waves are not in general translationally invariant travelling
waves, but possess a spatial variation that evolves in time, and may even may
oscillate back and forth in time, according to the details of the chemotactic
gradients. The analytical framework exploits the observation that unbiased
cellular diffusive flux is typically small compared to chemotactic fluxes and
is first developed and validated for a range of exemplar scenarios. The
framework is subsequently applied to more complex models considering the full
dynamics of the chemoattractant and how this may be driven and controlled
within a microdevice by considering a range of boundary conditions. In
particular, even though solutions cannot be constructed in all cases, a wide
variety of scenarios can be considered analytically, firstly providing global
insight into the important mechanisms and features of cell motility in complex
spatiotemporal fields of chemoattractant. Such analytical solutions also
provide a means of rapid evaluation of model predictions, with the prospect of
application in computationally demanding investigations relating theoretical
models and experimental observation, such as Bayesian parameter estimation.
