On the Influence of Growth in Perfusion Dependent Biological Systems – at the Example of the Human Liver

Abstract

Biochemical reactions in the liver are a complex non-linear and time depending coupled function-perfusion-mechanisms. The metabolism of the liver, e.g. nutrient storage, is directly coupled to its blood perfusion. Growth phenomena, such as accumulation of fat, have a strong impact onto micro-perfusion within the liver. Here, we present the results of a computational model that describes selected coupled functionalities. The metabolic processes take place in the liver cells, the hepatocytes, which are arranged in a delicate system of capillaries, so called sinusoids. Nutrients, oxygen, and other components of the blood are transported through the sinusoids. Due to this highly complex inner structure of the lobules it is impracticable to give a microscopic geometrical description in a continuum mechanical manner. Therefore, a homogenization scheme based on the theory of porous media (TPM) is used; [1]. In case of the presented liver model we consider a porous solid body, that consists of tissue and fat, and a fluid representing blood. In each phase several miscible components are considered. For the microscopic cell level use has been made of an embedded set of ordinary differential equations (ODE) that mimics a simplified metabolism, e.g. glucose utilization and production; [2]. In case of growth of the fat phase we follow a phenomenological approach. Numerical examples for this coupled process are presented and discussed. MotivationIn the European Union, approximately 29 million people are affected by a chronic liver disease. This high number is caused by the sensitivity of the liver due to negative environmental influences such as individual personal habits/addictions (alcohol and drugs) and social life style (western diet). Especially, the latter risk factor has drastically increased during the last decades as indicated by the rising rate of obesity. Obesity is always connected with accumulations of fat in the liver, called hepatic steatosis in its chronic form. Here, fatty droplets are long-term stored within the hepatocytes. Due to the additional space required by the stored fat, the delicate system of sinusoids, which guide the blood from the corners of the lobules to the draining central vein, narrows. To validate open questions, such as how the liver adepts to growth, by experimental observations and measurements is limited due to small dimension of sinusoids. Numerical simulations help to understand and investigate the complex, coupled interplay of micro-perfusion and metabolism in the liver. Triphasic Liver ModelThe here considered liver model consists of three immiscible, heterogeneously composed carrier phases, namely a porous solid body (tissue) ϕ S , a fat phase ϕ F and a fluid phase (blood) ϕ L . A set of four miscible substances is included: three external substances, namely glycogen, lactate, and free fatty acids (FFA), carried by the blood and one internal substance, glycogen, stored within the liver cells. Following the standard formulation within the framework of TPM as given in [1], [3] and ...

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“…To encapsulate the intricate relationship between structure, perfusion, and function, a coupled model of partial differential equations (PDEs) and ODEs was proposed as discussed in [21,32,39]. This model leverages a homogenized multi-scale, multi-component methodology within the TPM framework.…”