Unlike standard applications of transport theory, the transport of molecules and cells during embryonic development often takes place within growing multidimensional tissues. In this work, we consider a model of diffusion on uniformly growing lines, disks, and spheres. An exact solution of the partial differential equation governing the diffusion of a population of individuals on the growing domain is derived. Using this solution, we study the survival probability, S ( t ). For the standard non-growing case with an absorbing boundary, we observe that S ( t ) decays to zero in the long time limit. In contrast, when the domain grows linearly or exponentially with time, we show that S ( t ) decays to a constant, positive value, indicating that a proportion of the diffusing substance remains on the growing domain indefinitely. Comparing S ( t ) for diffusion on lines, disks, and spheres indicates that there are minimal differences in S ( t ) in the limit of zero growth and minimal differences in S ( t ) in the limit of fast growth. In contrast, for intermediate growth rates, we observe modest differences in S ( t ) between different geometries. These differences can be quantified by evaluating the exact expressions derived and presented here.
Chemistry and Pharmacology