Derivation and Validation of Compartment Models: Implications for Dynamic Imaging

Compartment models are mathematical representations of the transport of a chemical substance in the human body, assuming uniform concentration within each compartment, which corresponds to a distinct body part. Widely used in pharmaceutical and medical imaging through pharmacokinetic and tracer kinetic (TK) models, they are formulated as systems of time-dependent ordinary differential equations. However, these models do not account for the spatial dependence of physiological processes within individual compartments. This work aims to explore the physical interpretation of these models through mathematical derivation and to analyze their limitations. Three TK models are derived from more complex models regarding the physiological processes represented. The derivation introduces the hypotheses relevant to these processes. The most notable is the well-mixed hypothesis, resembling concentration homogeneity in the region of interest. The hypotheses are numerically tested by simulating the more complex model and evaluating the reduced models' residuals. For a given tracer molecule, the validity of TK models is found to decrease as voxel size increases. This work proposes an algorithm to compute the maximum voxel size, above which the reduced models' residuals exceed a chosen error threshold. For a 3 % error, voxel sizes larger than 250 $\upmu $ m are found critical for highly permeable vessels, while sizes smaller than 4 mm induce less than 3 % error in non-permeable vessels. Tested on literature-derived datasets, these findings identify diffusion as the underlying phenomenon leading to well-mixed compartments, and highlight the need for high spatial resolution imaging to improve TK parameters ground truth estimation.