Liver fibrosis emulation: Impact of the vascular fibrotic alterations on hemodynamics.

The liver circulatory system comprises two blood supply vascular trees (the hepatic artery and portal venous networks), microcirculation through the hepatic capillaries (the sinusoids), and a blood drainage vascular tree (the hepatic vein network). Vasculature changes due to fibrosis -located predominantly at the microcirculation level- lead to a marked increase in resistance to flow causing an increase in portal pressure (portal hypertension). Here, we present a liver fibrosis/cirrhosis model. We build on our 1D model of the healthy hepatic circulation, which considers the elasticity of the vessels walls and the pulsatile character of blood flow and pressure, and recreate the deteriorated liver vasculature due to fibrosis. We emulate altered sinusoids by fibrous tissue (stiffened, compressed and splitting) and propose boundary conditions to investigate the impact of fibrosis on hemodynamic variables within the organ. We obtain that the sinusoids stiffness leads to changes in the amplitude and shape of the blood flow and pressure waveforms but not in their mean value. For the compressed and splitting sinusoids, we observe significant increases in the mean value and amplitude of the pressure waveform in the altered sinusoids and in the portal venous network. In other words, we obtain the portal hypertension clinically observed in fibrotic/cirrhotic patients. We also study the extent of the spreading fibrosis by performing the structural fibrotic changes in an increasingly number of sinusoids. Finally, we calculate the portal pressure gradient (PPG) in the model and obtain values in agreement with those reported in the literature for fibrotic/cirrhotic patients.

The growth of capillary networks by branching for maximum fluid access.

Here we document the deterministic evolution of capillary networks that morph by connecting more and more branches to water sources. The network grows with the objective of extracting in steady state higher and higher liquid flow rates. Growth happens through the generation of tree-shaped structures and the geometrical configuration of the dendritic network evolves as the number of connected sources increases. We present a novel methodology to generate capillary architectures and show how the evolution of the network leads to pump higher volumetric flow rates by capillary suction. The results suggest that networks generated within a plane lead to higher flow rates than networks generated within a three-dimensional domain, for the same volume of fluid.

1D-model of the human liver circulatory system.

BACKGROUND AND OBJECTIVE: Blood flow rate and pressure can be measured in vivo by invasive and non-invasive techniques in the large vessels of the hepatic vasculature, but it is not possible to do so along the entire liver circulatory system. Here, we develop a novel 1D model of the liver circulatory system to obtain the hemodynamic signals from macrocirculation to microcirculation with a very low computational cost. METHODS: The model considers structurally well-defined elements that constitute the entire hepatic circulatory system, the hemodynamics (the temporal-dependence of the blood flow rate and pressure), and the elasticity of the vessel walls. RESULTS: Using flow rate signals from in vivo measurements as inputs in the model, we obtain pressure signals within their physiological range of values. Furthermore, the model allows to get and analyze the blood flow rate and pressure signals along any vessel of the hepatic vasculature. The impact of the elasticity of the different model components on the inlet pressures is also tested. CONCLUSIONS: A 1D model of the entire blood vascular system of the human liver is presented for the first time. The model allows to obtain the hemodynamic signals along the hepatic vasculature at a low computational cost. The amplitude and shape of the flow and pressure signals has hardly been studied in the small liver vessels. In this sense, the proposed model is a useful non-invasive exploration tool of the characteristics of the hemodynamic signals. In contrast to models that partially address the hepatic vasculature or those using an electrical analogy, the model presented here is made entirely of structurally well-defined elements. Future works will allow to directly emulate structural vascular alterations due to hepatic diseases and studying their impact on pressure and blood flow signals at key locations of the vasculature.

Hierarchical Modeling of the Liver Vascular System.

The liver plays a key role in the metabolic homeostasis of the whole organism. To carry out its functions, it is endowed with a peculiar circulatory system, made of three main dendritic flow structures and lobules. Understanding the vascular anatomy of the liver is clinically relevant since various liver pathologies are related to vascular disorders. Here, we develop a novel liver circulation model with a deterministic architecture based on the constructal law of design over the entire scale range (from macrocirculation to microcirculation). In this framework, the liver vascular structure is a combination of superimposed tree-shaped networks and porous system, where the main geometrical features of the dendritic fluid networks and the permeability of the porous medium, are defined from the constructal viewpoint. With this model, we are able to emulate physiological scenarios and to predict changes in blood pressure and flow rates throughout the hepatic vasculature due to resection or thrombosis in certain portions of the organ, simulated as deliberate blockages in the blood supply to these sections. This work sheds light on the critical impact of the vascular network on mechanics-related processes occurring in hepatic diseases, healing and regeneration that involve blood flow redistribution and are at the core of liver resilience.

The liver, a functionalized vascular structure.

The liver is not only the largest organ in the body but also the one playing one of the most important role in the human metabolism as it is in charge of transforming toxic substances in the body. Understanding the way its blood vasculature works is key. In this work we show that the challenge of predicting the hepatic multi-scale vascular network can be met thanks to the constructal law of design evolution. The work unveils the structure of the liver blood flow architecture as a combination of superimposed tree-shaped networks and porous system. We demonstrate that the dendritic nature of the hepatic artery, portal vein and hepatic vein can be predicted, together with their geometrical features (diameter ratio, duct length ratio) as the entire blood flow architectures follow the principle of equipartition of imperfections. At the smallest scale, the shape of the liver elemental systems-the lobules-is discovered, while their permeability is also predicted. The theory is compared with good agreement to anatomical data from the literature.

Analysis of thermochemical energy storage in an elemental configuration.

Here we show theoretically that the design of a thermochemical energy storage system for fast response and high thermal power can be predicted in accord with the constructal law of design. In this fundamental configuration, the walls of the elemental cylinder are impregnated with salt, while humid air is blown through the tube. Cases with constant salt volume or constant fluid volume or both are considered. It is shown that the best design in each case meets the equipartition of imperfections principle. The predictions are confirmed by full numerical experiments, allowing to consider various shape ratios and study their impact on the overall performance.

The constructal law and the evolution of design in nature.

The constructal law accounts for the universal phenomenon of generation and evolution of design (configuration, shape, structure, pattern, rhythm). This phenomenon is observed across the board, in animate, inanimate and human systems. The constructal law states the time direction of the evolutionary design phenomenon. It defines the concept of design evolution in physics. Along with the first and second law, the constructal law elevates thermodynamics to a science of systems with configuration. In this article we review the more recent work of our group, with emphasis on the advances made with the constructal law in the natural sciences. Highlighted are the oneness of animate and inanimate designs, the origin of finite-size organs on animals and vehicles, the flow of stresses as the generator of design in solid structures (skeletons, vegetation), the universality and rigidity of hierarchy in all flow systems, and the global design of human flows. Noteworthy is the tapestry of distributed energy systems, which balances nodes of production with networks of distribution on the landscape, and serves as key to energy sustainability and empowerment. At the global level, the constructal law accounts for the geography and design of human movement, wealth and communications.

The constructal law of design and evolution in nature.

Constructal theory is the view that (i) the generation of images of design (pattern, rhythm) in nature is a phenomenon of physics and (ii) this phenomenon is covered by a principle (the constructal law): 'for a finite-size flow system to persist in time (to live) it must evolve such that it provides greater and greater access to the currents that flow through it'. This law is about the necessity of design to occur, and about the time direction of the phenomenon: the tape of the design evolution 'movie' runs such that existing configurations are replaced by globally easier flowing configurations. The constructal law has two useful sides: the prediction of natural phenomena and the strategic engineering of novel architectures, based on the constructal law, i.e. not by mimicking nature. We show that the emergence of scaling laws in inanimate (geophysical) flow systems is the same phenomenon as the emergence of allometric laws in animate (biological) flow systems. Examples are lung design, animal locomotion, vegetation, river basins, turbulent flow structure, self-lubrication and natural multi-scale porous media. This article outlines the place of the constructal law as a self-standing law in physics, which covers all the ad hoc (and contradictory) statements of optimality such as minimum entropy generation, maximum entropy generation, minimum flow resistance, maximum flow resistance, minimum time, minimum weight, uniform maximum stresses and characteristic organ sizes. Nature is configured to flow and move as a conglomerate of 'engine and brake' designs.