A multi-compartment homogenized perfusion model for deforming hierarchical vasculature

The simulation of tissue perfusion based on highly detailed synthetic vasculature that often consists of multiple supplying and draining trees with millions of vascular segments is computationally expensive. Converting highly detailed synthetic vasculature into a homogenized continuum flow representation offers a computationally efficient alternative. In this paper, we investigate such a modeling approach that retains the essential features of potentially deforming hierarchical vascular networks. It is based on multi-compartment homogenization, where each compartment represents homogenized perfusion via a Darcy-type flow model associated with vascular segments at a specific spatial resolution in one individual tree of the network. The compartments are coupled through a pressure-dependent mass exchange, applied in a smeared manner everywhere within the perfusion domain. Key parameters, namely the permeability tensors of each compartment and the intercompartmental perfusion coefficients, are estimated directly from the vascular segments of the synthetic vasculature using averaging techniques. Our approach leverages spectral decomposition and a reduced set of representative vessel segments to balance computational efficiency with physical fidelity. For scenarios involving deformation, such as in a pumping heart or a regenerating liver, we introduce a computationally efficient parameter update based on geometric mapping, avoiding full re-homogenization in nonlinear simulations. We demonstrate the effectiveness and accuracy of the approach for several benchmark examples, including a full-scale multi-compartment liver perfusion simulation that explicitly incorporates three non-intersecting vascular trees, reflecting the hepatic artery, portal vein, and hepatic vein.