Mathematical model of blood and interstitial flow and lymph production in the liver

Abstract

We present a mathematical model of blood and interstitial flow in the liver. The liver is treated as a lattice of hexagonal ‘classic’ lobules, which are assumed to be long enough that end effects may be neglected and a two-dimensional problem considered. Since sinusoids and lymphatic vessels are numerous and small compared to the lobule, we use a homogenized approach, describing the sinusoidal and interstitial spaces as porous media. We model plasma filtration from sinusoids to the interstitium, lymph uptake by lymphatic ducts, and lymph outflow from the liver surface. Our results show that the effect of the liver surface only penetrates a depth of a few lobules’ thickness into the tissue. Thus, we separately consider a single lobule lying sufficiently far from all external boundaries that we may regard it as being in an infinite lattice, and also a model of the region near the liver surface. The model predicts that slightly more lymph is produced by interstitial fluid flowing through the liver surface than that taken up by the lymphatic vessels in the liver and that the non-peritonealized region of the surface of the liver results in the total lymph production (uptake by lymphatics plus fluid crossing surface) being about 5 % more than if the entire surface were covered by the Glisson–peritoneal membrane. Estimates of lymph outflow through the surface of the liver are in good agreement with experimental data. We also study the effect of non-physiological values of the controlling parameters, particularly focusing on the conditions of portal hypertension and ascites. To our knowledge, this is the first attempt to model lymph production in the liver. The model provides clinically relevant information about lymph outflow pathways and predicts the systemic response to pathological variations.

Generalization of intravoxel incoherent motion model by introducing the notion of continuous pseudodiffusion variable

“…In the liver, the sinusoidal capillaries constitute the main part of perfusion system (39). The portal veins, hepatic arteries, and hepatic veins have higher perfusion ability than sinusoidal capillaries (19,20), while the bile ducts and rich lymphatic vessels have lower perfusion ability than sinusoidal capillaries (40,41). In other words, this physiological property of hepatic vascular system results in a peak-shaped distribution function, q(D 0 ).…”

A Multilevel Modeling Framework to Study Hepatic Perfusion Characteristics in Case of Liver Cirrhosis

“…Contrary to the macrolevel, only little research has focused on mathematically modeling the microcirculation, often based on idealized geometries (e.g., mathematical 2D porous medium mod els of hexagonal lobules [31][32][33] or a 3D segment of an idealized lobule [34]). Nevertheless, a few studies have investigated the hepatic microcirculation based on anatomically realistic structures using vascular corrosion casts [9,35], Van Steenkiste et al [35] studied microvascular morphological changes and altered macrovascular hemodynamics in portal hypertensive and cirrhotic rodents.…”

Unraveling liver complexity from molecular to organ level: Challenges and perspectives

“…This requires suitable models of perfusion and flow (Debbaut et al, 2014;Ricken et al, 2010;Schwen et al, 2014;Siggers et al, 2014) that will be integrated with the spatial-temporal models, and at the body scale, the contribution of external influences, for example transport of material from extra-hepatic sources into the liver, that is usually addressed by compartment models (Krauss et al, 2012).…”