A 3D porous media liver lobule model: the importance of vascular septa and anisotropic permeability for homogeneous perfusion

Quantifying fat zonation in liver lobules: an integrated multiscale in silico model combining disturbed microperfusion and fat metabolism via a continuum biomechanical bi-scale, tri-phasic approach

Metabolic zonation refers to the spatial separation of metabolic functions along the sinusoidal axes of the liver. This phenomenon forms the foundation for adjusting hepatic metabolism to physiological requirements in health and disease (e.g., metabolic dysfunction-associated steatotic liver disease/MASLD). Zonated metabolic functions are influenced by zonal morphological abnormalities in the liver, such as periportal fibrosis and pericentral steatosis. We aim to analyze the interplay between microperfusion, oxygen gradient, fat metabolism and resulting zonated fat accumulation in a liver lobule. Therefore we developed a continuum biomechanical, tri-phasic, bi-scale, and multicomponent in silico model, which allows to numerically simulate coupled perfusion-function-growth interactions two-dimensionally in liver lobules. The developed homogenized model has the following specifications: (i) thermodynamically consistent, (ii) tri-phase model (tissue, fat, blood), (iii) penta-substances (glycogen, glucose, lactate, FFA, and oxygen), and (iv) bi-scale approach (lobule, cell). Our presented in silico model accounts for the mutual coupling between spatial and time-dependent liver perfusion, metabolic pathways and fat accumulation. The model thus allows the prediction of fat development in the liver lobule, depending on perfusion, oxygen and plasma concentration of free fatty acids (FFA), oxidative processes, the synthesis and the secretion of triglycerides (TGs). The use of a bi-scale approach allows in addition to focus on scale bridging processes. Thus, we will investigate how changes at the cellular scale affect perfusion at the lobular scale and vice versa. This allows to predict the zonation of fat distribution (periportal or pericentral) depending on initial conditions, as well as external and internal boundary value conditions.

Assessing Enhanced Acoustic Absorption From Sonosensitive Perfluorocarbon Emulsion With Magnetic Resonance-Guided High-Intensity Focused Ultrasound and a Percolated Tissue-Mimicking Flow Phantom

OBJECTIVE

Sonosensitive high-boiling point perfluorocarbon F₈TAC₁₈-PFOB emulsions previously exhibited thermal enhancement during focused ultrasound heating in ex vivo pig livers, kidneys and a laminar flow phantom. The main objectives of this study were to evaluate heating under turbulent conditions, observe perfusion effects, quantify heating in terms of acoustic absorption and model the experimental data.

METHODS

In this study, similar perfluorocarbon emulsions were circulated at incremental concentrations of 0.07, 0.13, 0.19 and 0.25% v:v through a percolated turbulent flow phantom, more representative of the biological tissue than a laminar flow phantom. The concentrations represent the droplet content in only the perfused fluid, rather than the droplet concentration throughout the entire cross-section. The temperature was measured with magnetic resonance thermometry, during focused ultrasound sonications of 67 W, 95% duty cycle and 33 s duration. These were used in Bioheat equation simulations to investigate in silico the thermal phenomena. The temperature change was compared with the control condition by circulating de-gassed and de-ionized water through the flow phantom without droplets.

RESULTS

With these 1.24 µm diameter droplets at 0.25% v:v, the acoustic absorption coefficient increased from 0.93 ± 0.05 at 0.0% v:v to 1.82 ± 0.22 m^-1 at 0.25% v:v using a 0.1 mL s^-1 flow rate. Without perfusion at 0.25% v:v, an increase was observed from 1.23 ± 0.07 m^-1 at 0.0% v:v to 1.65 ± 0.17 m^-1.

CONCLUSION

The results further support previously reported thermal enhancement with F₈TAC₁₈-PFOB emulsion, quantified the increased absorption at small concentration intervals, illustrated that the effects can be observed in a variety of visceral tissue models and provided a method to simulate untested scenarios.

Numerical prediction of portal hypertension by a hydrodynamic blood flow model combing with the fractal theory

Portal hypertension (PH) can cause a series of complications, therefore, early prediction of PH is important. Traditional diagnostic methods are harmful to the human body, while other non-invasive methods are inaccurate and lack physical meaning. Combining various fractal theories and flow laws, we establish a complete portal system blood flow model from the Computed Tomography (CT) and angiography images. The portal vein pressure (PP) is obtained by the flow rate data from the Doppler ultrasound and the pressure-velocity relationship is established by the model. Three normal participants and 12 patients with portal hypertension were divided into three groups. For the three normal participants (Group A), their mean PP calculated by the model is 1752 Pa, falling into the normal range of PP. The mean PP of three patients with portal vein thrombosis (Group B) is 2357 Pa; and for the 9 patients with cirrhosis (Group C), their mean PP is 2915 Pa. These results validate the classification performance of the model. Moreover, the blood flow model can give early warning parameters of the corresponding portal vein trunk and portal vein microtubules for thrombosis and liver cirrhosis. This model presents the complete process of blood flow from sinusoids to the portal vein, adapts to the diagnosis of portal hypertension of thrombosis and liver cirrhosis, and provides a new method for noninvasive portal vein pressure detection from the perspective of biomechanics.

In silico modeling for the hepatic circulation and transport: From the liver organ to lobules

The function of the liver depends critically on its blood supply. Numerous in silico models have been developed to study various aspects of the hepatic circulation, including not only the macro-hemodynamics at the organ level, but also the microcirculation at the lobular level. In addition, computational models of blood flow and bile flow have been used to study the transport, metabolism, and clearance of drugs in pharmacokinetic studies. These in silico models aim to provide insights into the liver organ function under both healthy and diseased states, and to assist quantitative analysis for surgical planning and postsurgery treatment. The purpose of this review is to provide an update on state-of-the-art in silico models of the hepatic circulation and transport processes. We introduce the numerical methods and the physiological background of these models. We also discuss multiscale frameworks that have been proposed for the liver, and their linkage with the large context of systems biology, systems pharmacology, and the Physiome project. This article is categorized under: Metabolic Diseases > Computational Models Metabolic Diseases > Biomedical Engineering Cardiovascular Diseases > Computational Models.

An ultrasonically actuated needle promotes the transport of nanoparticles and fluids

Non-invasive therapeutic ultrasound (US) methods, such as high-intensity focused ultrasound (HIFU), have limited access to tissue targets shadowed by bones or presence of gas. This study demonstrates that an ultrasonically actuated medical needle can be used to translate nanoparticles and fluids under the action of nonlinear phenomena, potentially overcoming some limitations of HIFU. A simulation study was first conducted to study the delivery of a tracer with an ultrasonically actuated needle (33 kHz) inside a porous medium acting as a model for soft tissue. The model was then validated experimentally in different concentrations of agarose gel showing a close match with the experimental results, when diluted soot nanoparticles (diameter < 150 nm) were employed as delivered entity. An additional simulation study demonstrated a threefold increase in the volume covered by the delivered agent in liver under a constant injection rate, when compared to without US. This method, if developed to its full potential, could serve as a cost effective way to improve safety and efficacy of drug therapies by maximizing the concentration of delivered entities within, e.g., a small lesion, while minimizing exposure outside the lesion.

Geometrical model of lobular structure and its importance for the liver perfusion analysis

A convenient geometrical description of the microvascular network is necessary for computationally efficient mathematical modelling of liver perfusion, metabolic and other physiological processes. The tissue models currently used are based on the generally accepted schematic structure of the parenchyma at the lobular level, assuming its perfect regular structure and geometrical symmetries. Hepatic lobule, portal lobule, or liver acinus are considered usually as autonomous functional units on which particular physiological problems are studied. We propose a new periodic unit-the liver representative periodic cell (LRPC) and establish its geometrical parametrization. The LRPC is constituted by two portal lobulae, such that it contains the liver acinus as a substructure. As a remarkable advantage over the classical phenomenological modelling approaches, the LRPC enables for multiscale modelling based on the periodic homogenization method. Derived macroscopic equations involve so called effective medium parameters, such as the tissue permeability, which reflect the LRPC geometry. In this way, mutual influences between the macroscopic phenomena, such as inhomogeneous perfusion, and the local processes relevant to the lobular (mesoscopic) level are respected. The LRPC based model is intended for its use within a complete hierarchical model of the whole liver. Using the Double-permeability Darcy model obtained by the homogenization, we illustrate the usefulness of the LRPC based modelling to describe the blood perfusion in the parenchyma.

Hepatectomy-Induced Alterations in Hepatic Perfusion and Function - Toward Multi-Scale Computational Modeling for a Better Prediction of Post-hepatectomy Liver Function

Liver resection causes marked perfusion alterations in the liver remnant both on the organ scale (vascular anatomy) and on the microscale (sinusoidal blood flow on tissue level). These changes in perfusion affect hepatic functions via direct alterations in blood supply and drainage, followed by indirect changes of biomechanical tissue properties and cellular function. Changes in blood flow impose compression, tension and shear forces on the liver tissue. These forces are perceived by mechanosensors on parenchymal and non-parenchymal cells of the liver and regulate cell-cell and cell-matrix interactions as well as cellular signaling and metabolism. These interactions are key players in tissue growth and remodeling, a prerequisite to restore tissue function after PHx. Their dysregulation is associated with metabolic impairment of the liver eventually leading to liver failure, a serious post-hepatectomy complication with high morbidity and mortality. Though certain links are known, the overall functional change after liver surgery is not understood due to complex feedback loops, non-linearities, spatial heterogeneities and different time-scales of events. Computational modeling is a unique approach to gain a better understanding of complex biomedical systems. This approach allows (i) integration of heterogeneous data and knowledge on multiple scales into a consistent view of how perfusion is related to hepatic function; (ii) testing and generating hypotheses based on predictive models, which must be validated experimentally and clinically. In the long term, computational modeling will (iii) support surgical planning by predicting surgery-induced perfusion perturbations and their functional (metabolic) consequences; and thereby (iv) allow minimizing surgical risks for the individual patient. Here, we review the alterations of hepatic perfusion, biomechanical properties and function associated with hepatectomy. Specifically, we provide an overview over the clinical problem, preoperative diagnostics, functional imaging approaches, experimental approaches in animal models, mechanoperception in the liver and impact on cellular metabolism, omics approaches with a focus on transcriptomics, data integration and uncertainty analysis, and computational modeling on multiple scales. Finally, we provide a perspective on how multi-scale computational models, which couple perfusion changes to hepatic function, could become part of clinical workflows to predict and optimize patient outcome after complex liver surgery.

3DICE coding matrix multidirectional macro-architecture modulates cell organization, shape, and co-cultures endothelization network

Natural extracellular matrix governs cells providing biomechanical and biofunctional outstanding properties, despite being porous and mostly made of soft materials. Among organs, specific tissues present specialized macro-architectures. For instance, hepatic lobules present radial organization, while vascular sinusoids are branched from vertical veins, providing specific biofunctional features. Therefore, it is imperative to mimic such structures while modeling tissues. So far, there is limited capability of coupling oriented macro-structures with interconnected micro-channels in programmable long-range vertical and radial sequential orientations. Herein, a three-directional ice crystal elongation (3DICE) system is presented to code geometries in cryogels. Using 3DICE, guided ice crystals growth templates vertical and radial pores through bulky cryogels. Translucent isotropic and anisotropic architectures of radial or vertical pores are fabricated with tunable mechanical response. Furthermore, 3D combinations of vertical and radial pore orientations are coded at the centimeter scale. Cell morphological response to macro-architectures is demonstrated. The formation of endothelial segments, CYP450 activity, and osteopontin expression, as liver fibrosis biomarkers, present direct response and specific cellular organization within radial, linear, and random architectures. These results unlock the potential of ice-templating demonstrating the relevance of macro-architectures to model tissues, and broad possibilities for drug testing, tissue engineering, and regenerative medicine.

Numerical Simulation of Insulin Depot Formation in Subcutaneous Tissue Modeled as a Homogeneous Anisotropic Porous Media

In this study, a numerical model of insulin depot formation in the subcutaneous adipose tissue of humans has been developed using the commercial computational fluid dynamics software. A better understanding of the underlying mechanisms can be helpful in the development of novel insulin administration devices and cannula geometries. Developing a model of insulin depot formation can provide faster results compared to extensive experimental studies which are typically done on porcine tissues. The injection method considered in this simulation involves an insulin pump that uses a rapid acting U100 insulin analogue. The depot formation has been studied by simulating Bolus injections ranging from 5 to 15 units of insulin, which corresponds to volumes of 50-150 μL. The insulin is injected into modeled subcutaneous tissues typically present in human abdominal regions. The subcutaneous tissue has been modeled as a fluid-saturated porous media. An anisotropic approach has been used to define the tissue permeability. The value of the porosity in parallel and perpendicular directions has been varied to modify the viscous resistance to the flow in these directions. The developed model has been validated by comparing with published experimental results, which show qualitative similarities in disk-shaped insulin depot formation. The validated model is then used to study formation of insulin depot inside the subcutaneous tissue at varying insulin flow rates involving different cannula geometries and arrays. The numerical model has been found to be an effective option to evaluate new cannula designs prior to the manufacturing and testing of prototypes, which can be rather time consuming and expensive.

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.

Modeling the initial-volume dependent approximate compressibility of porcine liver tissues using a novel volumetric strain energy model

Experimental observations in the open literature indicate that soft tissues are slightly compressible, and this characteristic affects not only their overall elastic response but also their damage evolution and failure mechanism. In this study, we find that the compressibility of liver tissues is also closely related to the initial specimen volume according to the confined compression tests: the samples with smaller initial volume exhibit more compressible behavior compared to the larger ones. To include this initial-volume dependent effect, we developed a novel volumetric strain energy model with two variables, i.e., the bulk modulus and the compressibility factor. A detailed scheme was proposed as well to identify these two parameters, and the relationship between the bulk modulus and the initial volume was clarified. Findings from this study will help to deepen the understanding of the biomechanical properties of soft tissues. STATEMENT OF SIGNIFICANCE: Liver is a highly vascular organ and traditionally assumed to be an incompressible medium. However, through the confined compression tests, we found that the samples with smaller initial volumes exhibit more compressible behavior. Hence, we developed a novel strain energy density model to characterize the initial-volume dependent hyperelastic response, and found that the bulk modulus of liver tissues is positively related to the initial volume. Our results suggest that the compressibility of liver tissues should be considered in the future study of liver biomechanics.

An image-based geometric model for numerical simulation of blood perfusion within the liver lobules

An image-based numerical algorithm is presented for simulating blood flow through the liver tissue. First, a geometric model is constructed by applying image processing techniques on a real microscopic image of a liver tissue. Then, incompressible blood flow through liver lobules is simulated. Effects of tissue heterogeneity and deformity, presence/absence of the second central vein in a particular lobule, and apparent sinusoids density in the liver cross section on the blood flow are investigated. Numerical results indicate that the existence of thick low permeability vascular septum, high permeability sinusoids, and lobule tissue heterogeneity can considerably affect interlobular and intralobular blood flow.

Hydrostatic pressure regulates CYP1A2 expression in human hepatocytes via a mechanosensitive aryl hydrocarbon receptor-dependent pathway

Approximately 75% of xenobiotics are primarily eliminated through metabolism; thus the accurate scaling of metabolic clearance is vital to successful drug development. Yet, when data is scaled from in vitro to in vivo, hepatic metabolic clearance, the primary source of metabolism, is still commonly underpredicted. Over the past decades, with biophysics used as a key component to restore aspects of the in vivo environment, several new cell culture settings have been investigated to improve hepatocyte functionalities. Most of these studies have focused on shear stress, i.e., flow mediated by a pressure gradient. One potential conclusion of these studies is that hepatocytes are naturally "mechanosensitive," i.e., they respond to a change in their biophysical environment. We demonstrate that hepatocytes also respond to an increase in hydrostatic pressure that, we suggest, is directly linked to the lobule geometry and vessel density. Furthermore, we demonstrate that hydrostatic pressure improves albumin production and increases cytochrome P-450 (CYP) 1A2 expression levels in an aryl hydrocarbon-dependent manner in human hepatocytes. Increased albumin production and CYP function are commonly attributed to the impacts of shear stress in microfluidic experiments. Therefore, our results highlight evidence of a novel link between hydrostatic pressure and CYP metabolism and demonstrate that the spectrum of hepatocyte mechanosensitivity might be larger than previously thought.

Liver Bioreactor Design Issues of Fluid Flow and Zonation, Fibrosis, and Mechanics: A Computational Perspective

Tissue engineering, with the goal of repairing or replacing damaged tissue and organs, has continued to make dramatic science-based advances since its origins in the late 1980’s and early 1990’s. Such advances are always multi-disciplinary in nature, from basic biology and chemistry through physics and mathematics to various engineering and computer fields. This review will focus its attention on two topics critical for tissue engineering liver development: (a) fluid flow, zonation, and drug screening, and (b) biomechanics, tissue stiffness, and fibrosis, all within the context of 3D structures. First, a general overview of various bioreactor designs developed to investigate fluid transport and tissue biomechanics is given. This includes a mention of computational fluid dynamic methods used to optimize and validate these designs. Thereafter, the perspective provided by computer simulations of flow, reactive transport, and biomechanics responses at the scale of the liver lobule and liver tissue is outlined, in addition to how bioreactor-measured properties can be utilized in these models. Here, the fundamental issues of tortuosity and upscaling are highlighted, as well as the role of disease and fibrosis in these issues. Some idealized simulations of the effects of fibrosis on lobule drug transport and mechanics responses are provided to further illustrate these concepts. This review concludes with an outline of some practical applications of tissue engineering advances and how efficient computational upscaling techniques, such as dual continuum modeling, might be used to quantify the transition of bioreactor results to the full liver scale.

Three-dimensional spatially resolved geometrical and functional models of human liver tissue reveal new aspects of NAFLD progression

Early disease diagnosis is key to the effective treatment of diseases. Histopathological analysis of human biopsies is the gold standard to diagnose tissue alterations. However, this approach has low resolution and overlooks 3D (three-dimensional) structural changes resulting from functional alterations. Here, we applied multiphoton imaging, 3D digital reconstructions and computational simulations to generate spatially resolved geometrical and functional models of human liver tissue at different stages of non-alcoholic fatty liver disease (NAFLD). We identified a set of morphometric cellular and tissue parameters correlated with disease progression, and discover profound topological defects in the 3D bile canalicular (BC) network. Personalized biliary fluid dynamic simulations predicted an increased pericentral biliary pressure and micro-cholestasis, consistent with elevated cholestatic biomarkers in patients' sera. Our spatially resolved models of human liver tissue can contribute to high-definition medicine by identifying quantitative multiparametric cellular and tissue signatures to define disease progression and provide new insights into NAFLD pathophysiology.

A fully coupled porous media and channels flow approach for simulation of blood and bile flow through the liver lobules

Two dimensional, steady state, and incompressible blood and bile flows through the liver lobules are numerically simulated. Two different geometric models A and B are proposed to study the effects of lobule structure on the fluid flow behaviour. In Model A, the lobule tissue is represented as a hexagonal shape porous medium with a set of flow channels at its vertices accounting for the hepatic artery, portal and central veins along with bile ductules. Model B is a channelized porous medium constructed by adding a set of flow channels, representing the bile canaliculies and lobule sinusoids, to Model A. The bile and blood flow through the lobule is simulated by the finite element approach, based on the Darcy/Brinkman equations in the lobule tissue and the Navier-Stokes (or Stokes) equations in the flow channels. In Model B, a transmission factor on the boundaries of the bile canaliculies is introduced to connect the bile and blood flows. First, a single regular lobule is utilized to exhibit the fluid flow pattern through the liver lobule represented by proposed geometric models. Then, the model is extended to a group of liver lobules to demonstrate the flow through a liver slice represented by irregular lobules. Numerical results indicate that the Darcy and Brinkman equations provide nearly the same solutions for Model A and similar solutions with a little difference for Model B. It is shown that the existence of sinusoids and bile canaliculies inside the liver lobules has noticeable effects on its fluid flow pattern, in terms of pressure and velocity fields.

Modeling of xenobiotic transport and metabolism in virtual hepatic lobule models

Computational models of normal liver function and xenobiotic induced liver damage are increasingly being used to interpret in vitro and in vivo data and as an approach to the de novo prediction of the liver’s response to xenobiotics. The microdosimetry (dose at the level of individual cells) of xenobiotics vary spatially within the liver because of both compound-independent and compound-dependent factors. In this paper, we build model liver lobules to investigate the interplay between vascular structure, blood flow and cellular transport that lead to regional variations in microdosimetry. We then compared simulation results obtained using this complex spatial model with a simpler linear pipe model of a sinusoid and a very simple single box model. We found that variations in diffusive transport, transporter-mediated transport and metabolism, coupled with complex liver sinusoid architecture and blood flow distribution, led to three essential patterns of xenobiotic exposure within the virtual liver lobule: (1) lobular-wise uniform, (2) radially varying and (3) both radially and azimuthally varying. We propose to use these essential patterns of exposure as a reference for selection of model representations when a computational study involves modeling detailed hepatic responses to xenobiotics.

Computational Modeling in Liver Surgery

The need for extended liver resection is increasing due to the growing incidence of liver tumors in aging societies. Individualized surgical planning is the key for identifying the optimal resection strategy and to minimize the risk of postoperative liver failure and tumor recurrence. Current computational tools provide virtual planning of liver resection by taking into account the spatial relationship between the tumor and the hepatic vascular trees, as well as the size of the future liver remnant. However, size and function of the liver are not necessarily equivalent. Hence, determining the future liver volume might misestimate the future liver function, especially in cases of hepatic comorbidities such as hepatic steatosis. A systems medicine approach could be applied, including biological, medical, and surgical aspects, by integrating all available anatomical and functional information of the individual patient. Such an approach holds promise for better prediction of postoperative liver function and hence improved risk assessment. This review provides an overview of mathematical models related to the liver and its function and explores their potential relevance for computational liver surgery. We first summarize key facts of hepatic anatomy, physiology, and pathology relevant for hepatic surgery, followed by a description of the computational tools currently used in liver surgical planning. Then we present selected state-of-the-art computational liver models potentially useful to support liver surgery. Finally, we discuss the main challenges that will need to be addressed when developing advanced computational planning tools in the context of liver surgery.

Microcirculation in the murine liver: a computational fluid dynamic model based on 3D reconstruction from in vivo microscopy

The liver is organized in hexagonal functional units - termed lobules - characterized by a rather peculiar blood microcirculation, due to the presence of a tangled network of capillaries - termed sinusoids. A better understanding of the hemodynamics that governs liver microcirculation is relevant to clinical and biological studies aimed at improving our management of liver diseases and transplantation. Herein, we built a CFD model of a 3D sinusoidal network, based on in vivo images of a physiological mouse liver obtained with a 2-photon microscope. The CFD model was developed with Fluent 16.0 (ANSYS Inc., Canonsburg, PA), particular care was taken in imposing the correct boundary conditions representing a physiological state. To account for the remaining branches of the sinusoids, a lumped parameter model was used to prescribe the correct pressure at each outlet. The effect of an adhered cell on local hemodynamics is also investigated for different occlusion degrees. The model here proposed accurately reproduces the fluid dynamics in a portion of the sinusoidal network in mouse liver. Mean velocities and mass flow rates are in agreement with literature values from in vivo measurements. Our approach provides details on local phenomena, hardly described by other computational studies, either focused on the macroscopic hepatic vasculature or based on homogeneous porous medium model.

Partial hepatectomy hemodynamics changes: Experimental data explained by closed-loop lumped modeling

The liver function may be degraded after partial liver ablation surgery. Adverse liver hemodynamics have been shown to be associated to liver failure. The link between these hemodynamics changes and ablation size is however poorly understood. This article proposes to explain with a closed-loop lumped model the hemodynamics changes observed during twelve surgeries in pigs. The portal venous tree is modeled with a pressure-dependent variable resistor. The variables measured, before liver ablation, are used to tune the model parameters. Then, the liver partial ablation is simulated with the model and the simulated pressures and flows are compared with post-operative measurements. Fluid infusion and blood losses occur during the surgery. The closed-loop model presented accounts for these blood volume changes. Moreover, the impact of blood volume changes and the liver lobe mass estimations on the simulated variables is studied. The typical increase of portal pressure, increase of liver pressure loss, slight decrease of portal flow and major decrease in arterial flow are quantitatively captured by the model for a 75% hepatectomy. It appears that the 75% decrease in hepatic arterial flow can be explained by the resistance increase induced by the surgery, and that no hepatic arterial buffer response (HABR) mechanism is needed to account for this change. The different post-operative states, observed in experiments, are reproduced with the proposed model. Thus, an explanation for inter-subjects post-operative variability is proposed. The presented framework can easily be adapted to other species circulations and to different pathologies for clinical hepatic applications.

A multilevel framework to reconstruct anatomical 3D models of the hepatic vasculature in rat livers

The intricate (micro)vascular architecture of the liver has not yet been fully unravelled. Although current models are often idealized simplifications of the complex anatomical reality, correct morphological information is instrumental for scientific and clinical purposes. Previously, both vascular corrosion casting (VCC) and immunohistochemistry (IHC) have been separately used to study the hepatic vasculature. Nevertheless, these techniques still face a number of challenges such as dual casting in VCC and limited imaging depths for IHC. We have optimized both techniques and combined their complementary strengths to develop a framework for multilevel reconstruction of the hepatic circulation in the rat. The VCC and micro-CT scanning protocol was improved by enabling dual casting, optimizing the contrast agent concentration, and adjusting the viscosity of the resin (PU4ii). IHC was improved with an optimized clearing technique (CUBIC) that extended the imaging depth for confocal microscopy more than five-fold. Using in-house developed software (DeLiver), the vascular network - in both VCC and IHC datasets - was automatically segmented and/or morphologically analysed. Our methodological framework allows 3D reconstruction and quantification of the hepatic circulation, ranging from the major blood vessels down to the intertwined and interconnected sinusoids. We believe that the presented framework will have value beyond studies of the liver, and will facilitate a better understanding of various parenchymal organs in general, in physiological and pathological circumstances.

Lactation in the Human Breast From a Fluid Dynamics Point of View

This study is a collaborative effort among lactation specialists and fluid dynamic engineers. The paper presents clinical results for suckling pressure pattern in lactating human breast as well as a 3D computational fluid dynamics (CFD) modeling of milk flow using these clinical inputs. The investigation starts with a careful, statistically representative measurement of suckling vacuum pressure, milk flow rate, and milk intake in a group of infants. The results from clinical data show that suckling action does not occur with constant suckling rate but changes in a rhythmic manner for infants. These pressure profiles are then used as the boundary condition for the CFD study using commercial ansys fluent software. For the geometric model of the ductal system of the human breast, this work takes advantage of a recent advance in the development of a validated phantom that has been produced as a ground truth for the imaging applications for the breast. The geometric model is introduced into CFD simulations with the aforementioned boundary conditions. The results for milk intake from the CFD simulation and clinical data were compared and cross validated. Also, the variation of milk intake versus suckling pressure are presented and analyzed. Both the clinical and CFD simulation show that the maximum milk flow rate is not related to the largest vacuum pressure or longest feeding duration indicating other factors influence the milk intake by infants.

Multiscale computational model of fluid flow and matrix deformation in decellularized liver

Currently little is known about the biomechanical environment in decellularized tissue. The goal of this research is to quantify the mechanical microenvironment in decellularized liver, for varying organ-scale perfusion conditions, using a combined experimental/computational approach. Needle-guided ultra-miniature pressure sensors were inserted into liver tissue to measure parenchymal fluid pressure ex-situ in portal vein-perfused native (n=5) and decellularized (n=7) ferret liver, for flow rates from 3-12mL/min. Pressures were also recorded at the inlet near the portal vein cannula to estimate total vascular resistance of the specimens. Experimental results were fit to a multiscale computational model to simulate perfusion conditions inside native versus decellularized livers for four experimental flow rates. The multiscale model consists of two parts: an organ-scale electrical analog model of liver hemodynamics and a tissue-scale model that predicts pore fluid pressure, pore fluid velocity, and solid matrix stress and deformation throughout the 3D hepatic lobule. Distinct models were created for native versus decellularized liver. Results show that vascular resistance decreases by 82% as a result of decellularization. The hydraulic conductivity of the decellularized liver lobule, a measure of tissue permeability, was 5.6 times that of native liver. For the four flow rates studied, mean fluid pressures in the decellularized lobule were 0.6-2.4mmHg, mean fluid velocities were 211-767μm/s, and average solid matrix principal strains were 1.7-6.1%. In the future this modeling platform can be used to guide the optimization of perfusion seeding and conditioning strategies for decellularized scaffolds in liver bioengineering.

Engineering point of view on liver transplantation strategies: multi-level modeling of hepatic perfusion

BACKGROUND

Hepatic perfusion plays a crucial role in liver transplantation strategies, for example, when preserving procured organs with the use of machine perfusion preservation (MP) and in the case of living donor liver transplantation (LDLT). Liver hemodynamics are not yet fully understood because of insufficient knowledge on the hepatic vascular morphology and its perfusion characteristics, hampering the optimization of liver transplantation procedures. To this end, we developed computer models to simulate the complex blood circulation through the liver from the macro-scale down to the terminal micro-scale level.

METHODS

A combination of state-of-the-art techniques (vascular corrosion casting, micro-CT scanning up to a 2.6-μm resolution, and image processing) led to 3D visualizations and detailed geometrical analyses of the complex architecture of the liver's 3 vascular trees, ranging from the largest vessels (macrocirculation) down to the sinusoids (microcirculation).

RESULTS

On the basis of these data, we developed various computational models (electrical analog models and 3D computational fluid dynamics models) to study the blood flow-induced forces acting on the hepatic blood vessels. The latter was done for physiological blood flow through the liver as well as for livers undergoing MP or LDLT procedures. Hereby, several scenarios were simulated to study the behavior of livers in different hemodynamic circumstances.

CONCLUSIONS

A novel, multi-level modeling framework was developed to simulate hepatic perfusion in support of liver transplantation strategies. We obtained unique anatomical data on the vascular architecture of both human and rat livers. These data formed the building blocks of electrical analog models of hepatic perfusion and numerical models of the liver microcirculation. The results revealed novel insights into the hemodynamic impact of liver MP and LDLT procedures as well as into the microcirculatory perfusion characteristics. The presented methodology is also applicable to other tree-like structures (eg, the biliary tree) or organs (eg, kidneys, lungs).

Representative Sinusoids for Hepatic Four-Scale Pharmacokinetics Simulations

The mammalian liver plays a key role for metabolism and detoxification of xenobiotics in the body. The corresponding biochemical processes are typically subject to spatial variations at different length scales. Zonal enzyme expression along sinusoids leads to zonated metabolization already in the healthy state. Pathological states of the liver may involve liver cells affected in a zonated manner or heterogeneously across the whole organ. This spatial heterogeneity, however, cannot be described by most computational models which usually consider the liver as a homogeneous, well-stirred organ.

The goal of this article is to present a methodology to extend whole-body pharmacokinetics models by a detailed liver model, combining different modeling approaches from the literature. This approach results in an integrated four-scale model, from single cells via sinusoids and the organ to the whole organism, capable of mechanistically representing metabolization inhomogeneity in livers at different spatial scales. Moreover, the model shows circulatory mixing effects due to a delayed recirculation through the surrounding organism.

To show that this approach is generally applicable for different physiological processes, we show three applications as proofs of concept, covering a range of species, compounds, and diseased states: clearance of midazolam in steatotic human livers, clearance of caffeine in mouse livers regenerating from necrosis, and a parameter study on the impact of different cell entities on insulin uptake in mouse livers.

The examples illustrate how variations only discernible at the local scale influence substance distribution in the plasma at the whole-body level. In particular, our results show that simultaneously considering variations at all relevant spatial scales may be necessary to understand their impact on observations at the organism scale.

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

Liver cirrhosis represents the end-stage of different liver disorders, progressively affecting hepatic architecture, hemodynamics, and function. Morphologically, cirrhosis is characterized by diffuse fibrosis, the conversion of normal liver architecture into structurally abnormal regenerative nodules and the formation of an abundant vascular network. To date, the vascular remodeling and altered hemodynamics due to cirrhosis are still poorly understood, even though they seem to play a pivotal role in cirrhogenesis. This study aims to determine the perfusion characteristics of the cirrhotic circulation using a multilevel modeling approach including computational fluid dynamics (CFD) simulations. Vascular corrosion casting and multilevel micro-CT imaging of a single human cirrhotic liver generated detailed datasets of the hepatic circulation, including typical pathological characteristics of cirrhosis such as shunt vessels and dilated sinusoids. Image processing resulted in anatomically correct 3D reconstructions of the microvasculature up to a diameter of about 500 μm. Subsequently, two cubic samples (150 × 150 × 150 μm3) were virtually dissected from vascularized zones in between regenerative nodules and applied for CFD simulations to study the altered cirrhotic microperfusion and permeability. Additionally, a conceptual 3D model of the cirrhotic macrocirculation was developed to reveal the hemodynamic impact of regenerative nodules. Our results illustrate that the cirrhotic microcirculation is characterized by an anisotropic permeability showing the highest value in the direction parallel to the central vein (kd,zz = 1.68 × 10−13 m2 and kd,zz = 7.79 × 10−13 m2 for sample 1 and 2, respectively) and lower values in the circumferential (kd,ϑϑ = 5.78 × 10−14 m2 and kd,ϑϑ = 5.65 × 10−13 m2 for sample 1 and 2, respectively) and radial (kd,rr = 9.87 × 10−14 m2 and kd,rr = 5.13 × 10−13 m2 for sample 1 and 2, respectively) direction. Overall, the observed permeabilities are markedly higher compared to a normal liver, implying a locally decreased intrahepatic vascular resistance (IVR) probably due to local compensation mechanisms (dilated sinusoids and shunt vessels). These counteract the IVR increase caused by the presence of regenerative nodules and dynamic contraction mechanisms (e.g., stellate cells, NO-concentration, etc.). Our conceptual 3D model of the cirrhotic macrocirculation indicates that regenerative nodules severely increase the IVR beyond about 65 vol. % of regenerative nodules. Numerical modeling allows quantifying perfusion characteristics of the cirrhotic macro- and microcirculation, i.e., the effect of regenerative nodules and compensation mechanisms such as dilated sinusoids and shunt vessels. Future research will focus on the development of models to study time-dependent degenerative adaptation of the cirrhotic macro- and microcirculation.

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

Biological responses are determined by information processing at multiple and highly interconnected scales. Within a tissue the individual cells respond to extracellular stimuli by regulating intracellular signaling pathways that in turn determine cell fate decisions and influence the behavior of neighboring cells. As a consequence the cellular responses critically impact tissue composition and architecture. Understanding the regulation of these mechanisms at different scales is key to unravel the emergent properties of biological systems. In this perspective, a multidisciplinary approach combining experimental data with mathematical modeling is introduced. We report the approach applied within the Virtual Liver Network to analyze processes that regulate liver functions from single cell responses to the organ level using a number of examples. By facilitating interdisciplinary collaborations, the Virtual Liver Network studies liver regeneration and inflammatory processes as well as liver metabolic functions at multiple scales, and thus provides a suitable example to identify challenges and point out potential future application of multi-scale systems biology.

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

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.