Modeling function–perfusion behavior in liver lobules including tissue, blood, glucose, lactate and glycogen by use of a coupled two-scale PDE–ODE approach

Cross-species variability in lobular geometry and cytochrome P450 hepatic zonation: insights into CYP1A2, CYP2D6, CYP2E1 and CYP3A4

There is a lack of systematic research exploring cross-species variation in liver lobular geometry and zonation patterns of critical drug-metabolizing enzymes, a knowledge gap essential for translational studies. This study investigated the critical interplay between lobular geometry and key cytochrome P450 (CYP) zonation in four species: mouse, rat, pig, and human. We developed an automated pipeline based on whole slide images (WSI) of hematoxylin-eosin-stained liver sections and immunohistochemistry. This pipeline allows accurate quantification of both lobular geometry and zonation patterns of essential CYP proteins. Our analysis of CYP zonal expression shows that all CYP enzymes (besides CYP2D6 with panlobular expression) were observed in the pericentral region in all species, but with distinct differences. Comparison of normalized gradient intensity shows a high similarity between mice and humans, followed by rats. Specifically, CYP1A2 was expressed throughout the pericentral region in mice and humans, whereas it was restricted to a narrow pericentral rim in rats and showed a panlobular pattern in pigs. Similarly, CYP3A4 is present in the pericentral region, but its extent varies considerably in rats and appears panlobular in pigs. CYP2D6 zonal expression consistently shows a panlobular pattern in all species, although the intensity varies. CYP2E1 zonal expression covered the entire pericentral region with extension into the midzone in all four species, suggesting its potential for further cross-species analysis. Analysis of lobular geometry revealed an increase in lobular size with increasing species size, whereas lobular compactness was similar. Based on our results, zonated CYP expression in mice is most similar to humans. Therefore, mice appear to be the most appropriate species for drug metabolism studies unless larger species are required for other purposes, e.g., surgical reasons. CYP selection should be based on species, with CYP2E1 and CYP2D6 being the most preferable to compare four species. CYP1A2 could be considered as an additional CYP for rodent versus human comparisons, and CYP3A4 for mouse/human comparisons. In conclusion, our image analysis pipeline together with suggestions for species and CYP selection can serve to improve future cross-species and translational drug metabolism studies.

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.

Mathematical models of cystic fibrosis as a systemic disease

Cystic fibrosis (CF) is widely known as a disease of the lung, even though it is in truth a systemic disease, whose symptoms typically manifest in gastrointestinal dysfunction first. CF ultimately impairs not only the pancreas and intestine but also the lungs, gonads, liver, kidneys, bones, and the cardiovascular system. It is caused by one of several mutations in the gene of the epithelial ion channel protein CFTR. Intense research and improved antimicrobial treatments during the past eight decades have steadily increased the predicted life expectancy of a person with CF (pwCF) from a few weeks to over 50 years. Moreover, several drugs ameliorating the sequelae of the disease have become available in recent years, and notable treatments of the root cause of the disease have recently generated substantial improvements in health for some but not all pwCF. Yet, numerous fundamental questions remain unanswered. Complicating CF, for instance in the lung, is the fact that the associated insufficient chloride secretion typically perturbs the electrochemical balance across epithelia and, in the airways, leads to the accumulation of thick, viscous mucus and mucus plaques that cannot be cleared effectively and provide a rich breeding ground for a spectrum of bacterial and fungal communities. The subsequent infections often become chronic and respond poorly to antibiotic treatments, with outcomes sometimes only weakly correlated with the drug susceptibility of the target pathogen. Furthermore, in contrast to rapidly resolved acute infections with a single target pathogen, chronic infections commonly involve multi-species bacterial communities, called "infection microbiomes," that develop their own ecological and evolutionary dynamics. It is presently impossible to devise mathematical models of CF in its entirety, but it is feasible to design models for many of the distinct drivers of the disease. Building upon these growing yet isolated modeling efforts, we discuss in the following the feasibility of a multi-scale modeling framework, known as template-and-anchor modeling, that allows the gradual integration of refined sub-models with different granularity. The article first reviews the most important biomedical aspects of CF and subsequently describes mathematical modeling approaches that already exist or have the potential to deepen our understanding of the multitude aspects of the disease and their interrelationships. The conceptual ideas behind the approaches proposed here do not only pertain to CF but are translatable to other systemic diseases. This article is categorized under: Congenital Diseases > Computational Models.

The influence of static portal pressure on liver biophysical properties

The liver is a highly vascularized organ where fluid properties, including vascular pressure, vessel integrity and fluid viscosity, play a critical role in gross mechanical properties. To study the effects of portal pressure, liver confinement, fluid viscosity, and tissue crosslinking on liver stiffness, water diffusion, and vessel size, we applied multiparametric magnetic resonance imaging (mpMRI), including multifrequency magnetic resonance elastography (MRE) and apparent diffusion coefficient (ADC) measurements, to ex vivo livers from healthy male rats (13.6±1.6 weeks) at room temperature. Four scenarios including altered liver confinement, tissue crosslinking, and vascular fluid viscosity were investigated with mpMRI at different portal pressure levels (0-17.5 cmH2O). Our experiments demonstrated that, with increasing portal pressure, rat livers showed higher water content, water diffusivity, and increased vessel sizes quantified by the vessel tissue volume fraction (VTVF). These effects were most pronounced in native, unconfined livers (VTVF: 300±120%, p<0.05, ADC: 88±29%, p<0.01), while still significant under confinement (confined: VTVF: 53±32%, p<0.01, ADC: 28±19%, p<0.05; confined-fixed: VTVF: 52±20%, p<0.001, ADC: 11±2%, p<0.01; confined-viscous: VTVF: 210±110%, p<0.01, ADC: 26±9%, p<0.001). Softening with elevated portal pressure (-12±5, p<0.05) occurred regardless of confinement and fixation. However, the liver stiffened when exposed to a more viscous inflow fluid (11±4%, p<0.001). Taken together, our results elucidate the complex relationship between macroscopic-biophysical parameters of liver tissue measured by mpMRI and vascular-fluid properties. Influenced by portal pressure, vascular permeability, and matrix crosslinking, liver stiffness is sensitive to intrinsic poroelastic properties, which, alongside vascular architecture and water diffusivity, may aid in the differential diagnosis of liver disease. STATEMENT OF SIGNIFICANCE: Using highly controllable ex vivo rat liver phantoms, hepatic biophysical properties such as tissue-vascular structure, stiffness, and water diffusivity were investigated using multiparametric MRI including multifrequency magnetic resonance elastography (MRE) and diffusion-weighted imaging (DWI). Through elaborate tuning of the experimental conditions such as the static portal pressure, flow viscosity, amount and distribution of fluid content in the liver, we identified the contributions of the fluid component to the overall imaging-based biophysical properties of the liver. Our finding demonstrated the sensitivity of liver stiffness to the hepatic poroelastic properties, which may aid in the differential diagnosis of liver diseases.

3D Bioprinting of an Endothelialized Liver Lobule-like Construct as a Tumor-Scale Drug Screening Platform

3D cell culture models replicating the complexity of cell-cell interactions and biomimetic extracellular matrix (ECM) are novel approaches for studying liver cancer, including in vitro drug screening or disease mechanism investigation. Although there have been advancements in the production of 3D liver cancer models to serve as drug screening platforms, recreating the structural architecture and tumor-scale microenvironment of native liver tumors remains a challenge. Here, using the dot extrusion printing (DEP) technology reported in our previous work, we fabricated an endothelialized liver lobule-like construct by printing hepatocyte-laden methacryloyl gelatin (GelMA) hydrogel microbeads and HUVEC-laden gelatin microbeads. DEP technology enables hydrogel microbeads to be produced with precise positioning and adjustable scale, facilitating the construction of liver lobule-like structures. The vascular network was achieved by sacrificing the gelatin microbeads at 37 °C to allow HUVEC proliferation on the surface of the hepatocyte layer. Finally, we used the endothelialized liver lobule-like constructs for anti-cancer drug (Sorafenib) screening, and stronger drug resistance results were obtained when compared to either mono-cultured constructs or hepatocyte spheroids alone. The 3D liver cancer models presented here successfully recreate liver lobule-like morphology, and may have the potential to serve as a liver tumor-scale drug screening platform.

Digital twin demonstrates significance of biomechanical growth control in liver regeneration after partial hepatectomy

Partial liver removal is an important therapy option for liver cancer. In most patients within a few weeks, the liver is able to fully regenerate. In some patients, however, regeneration fails with often severe consequences. To better understand the control mechanisms of liver regeneration, experiments in mice were performed, guiding the creation of a spatiotemporal 3D model of the regenerating liver. The model represents cells and blood vessels within an entire liver lobe, a macroscopic liver subunit. The model could reproduce the experimental data only if a biomechanical growth control (BGC)-mechanism, inhibiting cell cycle entrance at high compression, was taken into account and predicted that BGC may act as a short-range growth inhibitor minimizing the number of proliferating neighbor cells of a proliferating cell, generating a checkerboard-like proliferation pattern. Model-predicted cell proliferation patterns in pigs and mice were found experimentally. The results underpin the importance of biomechanical aspects in liver growth control.

An in vitro fibrotic liver lobule model through sequential cell-seeding of HSCs and HepG2 on 3D-printed poly(glycerol sebacate) acrylate scaffolds

Cirrhosis is a major cause of global morbidity and mortality, and significantly leads to a heightened risk of liver cancer. Despite decades of efforts in seeking for cures for cirrhosis, this disease remains irreversible. To assist in the advancement of understanding toward cirrhosis as well as therapeutic options, various disease models, each with different strengths, are developed. With the development of three-dimensional (3D) cell culture in recent years, more realistic biochemical properties are observed in 3D cell models, which have gradually taken over the responsibilities of traditional 2D cell culture, and are expected to replace some of the animal models in the near future. Here, we propose a 3D fibrotic liver model inspired by liver lobules. In the model, 3D-printed poly(glycerol sebacate) acrylate (PGSA) scaffolds facilitated the formation of 3D tissues and guided the deposition of fibrotic structures. Through the sequential seeding of hepatic stellate cells (HSCs), HepG2 and HSCs, fibrotic septum-like tissues were created on PGSA scaffolds. As albumin secretion is considered a rather important function of the liver and is found only among hepatic cells, the detection of albumin secretion up to 30 days indicates the mimicking of basic liver functions. Moreover, the in vivo fibrotic tissue shows a high similarity to fibrotic septa. Finally, via complete encapsulation of HSCs, a down-regulated albumin secretion profile was observed in the capped model, which is a metabolic indicator that is important for the prognosis for liver cirrhosis. Looking forward, the incorporation of the vasculature will further upgrade the model into a sound tool for liver research and associated treatments.

Multiscale biomechanics and mechanotransduction from liver fibrosis to cancer

A growing body of multiscale biomechanical studies has been proposed to highlight the mechanical cues in the development of hepatic fibrosis and cancer. At the cellular level, changes in mechanical microenvironment induce phenotypic and functional alterations of hepatic cells, initiating a positive feedback loop that promotes liver fibrogenesis and hepatocarcinogenesis. Tumor mechanical microenvironment of hepatocellular carcinoma facilitates tumor cell growth and metastasis, and hinders the drug delivery and immunotherapy. At the molecular level, mechanical forces are sensed and transmitted into hepatic cells via allosteric activation of mechanoreceptors on the cell membrane, leading to the activation of various mechanotransduction pathways including integrin and YAP signaling and then regulating cell function. Thus, the application of mechanomedicine concept in the treatment of liver diseases is promising for rational design and cell-specific delivery of therapeutic drugs. This review mainly discusses the correlation between biomechanical cues and liver diseases from the viewpoint of mechanobiology.

Rigorous mathematical optimization of synthetic hepatic vascular trees

In this paper, we introduce a new framework for generating synthetic vascular trees, based on rigorous model-based mathematical optimization. Our main contribution is the reformulation of finding the optimal global tree geometry into a nonlinear optimization problem (NLP). This rigorous mathematical formulation accommodates efficient solution algorithms such as the interior point method and allows us to easily change boundary conditions and constraints applied to the tree. Moreover, it creates trifurcations in addition to bifurcations. A second contribution is the addition of an optimization stage for the tree topology. Here, we combine constrained constructive optimization (CCO) with a heuristic approach to search among possible tree topologies. We combine the NLP formulation and the topology optimization into a single algorithmic approach. Finally, we attempt the validation of our new model-based optimization framework using a detailed corrosion cast of a human liver, which allows a quantitative comparison of the synthetic tree structure with the tree structure determined experimentally down to the fifth generation. The results show that our new framework is capable of generating asymmetric synthetic trees that match the available physiological corrosion cast data better than trees generated by the standard CCO approach.

Automated Detection of Portal Fields and Central Veins in Whole-Slide Images of Liver Tissue

Many physiological processes and pathological phenomena in the liver tissue are spatially heterogeneous. At a local scale, biomarkers can be quantified along the axis of the blood flow, from portal fields (PFs) to central veins (CVs), i.e., in zonated form. This requires detecting PFs and CVs. However, manually annotating these structures in multiple whole-slide images is a tedious task. We describe and evaluate a fully automated method, based on a convolutional neural network, for simultaneously detecting PFs and CVs in a single stained section. Trained on scans of hematoxylin and eosin-stained liver tissue, the detector performed well with an F1 score of 0.81 compared to annotation by a human expert. It does, however, not generalize well to previously unseen scans of steatotic liver tissue with an F1 score of 0.59. Automated PF and CV detection eliminates the bottleneck of manual annotation for subsequent automated analyses, as illustrated by two proof-of-concept applications: We computed lobulus sizes based on the detected PF and CV positions, where results agreed with published lobulus sizes. Moreover, we demonstrate the feasibility of zonated quantification of biomarkers detected in different stainings based on lobuli and zones obtained from the detected PF and CV positions. A negative control (hematoxylin and eosin) showed the expected homogeneity, a positive control (glutamine synthetase) was quantified to be strictly pericentral, and a plausible zonation for a heterogeneous F4/80 staining was obtained. Automated detection of PFs and CVs is one building block for automatically quantifying physiologically relevant heterogeneity of liver tissue biomarkers. Perspectively, a more robust and automated assessment of zonation from whole-slide images will be valuable for parameterizing spatially resolved models of liver metabolism and to provide diagnostic information.

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.

Magnetic Resonance Imaging-based biomechanical simulation of cartilage: A systematic review

MRI-based mathematical and computational modeling studies can contribute to a better understanding of the mechanisms governing cartilage's mechanical performance and cartilage disease. In addition, distinct modeling of cartilage is needed to optimize artificial cartilage production. These studies have opened up the prospect of further deepening our understanding of cartilage function. Furthermore, these studies reveal the initiation of an engineering-level approach to how cartilage disease affects material properties and cartilage function. Aimed at researchers in the field of MRI-based cartilage simulation, research articles pertinent to MRI-based cartilage modeling were identified, reviewed, and summarized systematically. Various MRI applications for cartilage modeling are highlighted, and the limitations of different constitutive models used are addressed. In addition, the clinical application of simulations and studied diseases are discussed. The paper's quality, based on the developed questionnaire, was assessed, and out of 79 reviewed papers, 34 papers were determined as high-quality. Due to the lack of the best constitutive models for various clinical conditions, researchers may consider the effect of constitutive material models on the cartilage disease simulation. In the future, research groups may incorporate various aspects of machine learning into constitutive models and MRI data extraction to further refine the study methodology. Moreover, researchers should strive for further reproducibility and rigorous model validation and verification, such as gait analysis.

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.

Influence of microflow on hepatic sinusoid blood flow and red blood cell deformation

Hepatic sinusoids present complex anatomical structures such as the endothelial sieve pores and the Disse space, which govern the microscopic blood flow in the sinusoids and are associated with structural variations in liver fibrosis and cirrhosis. However, the contributions of the permeability of endothelial and collagen layers and the roughness of hepatocyte microvilli to the features of this microflow remain largely unknown. Here, an immersed boundary method coupled with a lattice Boltzmann method was adopted in an in vitro hepatic sinusoidal model, and flow field and erythrocyte deformation analyses were conducted by introducing three new source terms including permeability of the endothelial layer, resistance of hepatocyte microvilli and collagen layers, and deformation of red blood cells (RBCs). Numerical calculations indicated that alterations in endothelial permeability could significantly affect the flow velocity and flow rate distributions in hepatic sinusoids. Interestingly, a biphasic regulating pattern of shear stress occurred simultaneously on the surface of hepatocytes and the lower side of endothelium, i.e., the shear stress increased with increased thickness of hepatocyte microvilli and collagen layer when the endothelial permeability was high but decreased with the increase of the thickness at low endothelial permeability. Additionally, this specified microflow manipulates typical RBC deformation inside the sinusoid, yielding one-third of the variation of deformable index with varied endothelial permeability. These simulations not only are consistent with experimental measurements using in vitro liver sinusoidal chip but also elaborate the contributions of endothelial and collagen layer permeability and wall roughness. Thus, our results provide a basis for further characterizing this microflow and understanding its effects on cellular migration and deformation in the hepatic sinusoids.

Application of Magnetic Resonance Imaging in Liver Biomechanics: A Systematic Review

MRI-based biomechanical studies can provide a deep understanding of the mechanisms governing liver function, its mechanical performance but also liver diseases. In addition, comprehensive modeling of the liver can help improve liver disease treatment. Furthermore, such studies demonstrate the beginning of an engineering-level approach to how the liver disease affects material properties and liver function. Aimed at researchers in the field of MRI-based liver simulation, research articles pertinent to MRI-based liver modeling were identified, reviewed, and summarized systematically. Various MRI applications for liver biomechanics are highlighted, and the limitations of different viscoelastic models used in magnetic resonance elastography are addressed. The clinical application of the simulations and the diseases studied are also discussed. Based on the developed questionnaire, the papers' quality was assessed, and of the 46 reviewed papers, 32 papers were determined to be of high-quality. Due to the lack of the suitable material models for different liver diseases studied by magnetic resonance elastography, researchers may consider the effect of liver diseases on constitutive models. In the future, research groups may incorporate various aspects of machine learning (ML) into constitutive models and MRI data extraction to further refine the study methodology. Moreover, researchers should strive for further reproducibility and rigorous model validation and verification.

Intravital Microscopy for the Study of Hepatic Glucose Uptake

The liver is central in maintaining glucose homeostasis. Indeed, impaired hepatic glucose uptake has been implicated in the development of hyperglycemia in type II diabetes (T2D) and non-alcoholic fatty liver disease (NAFLD). However, current approaches to evaluate glucose mobilization rely on indirect measurements that do not provide spatial and temporal information. Here, we describe confocal-based intravital microscopy (IVM) of the liver that allows the identification of hepatocyte spatial organization and glucose transport. Specifically, we describe a method to fluorescently label hepatic landmarks to identify different compartments within the liver. In addition, we outline an in vivo fluorescent glucose uptake assay to quantitatively measure glucose mobilization in space and time. These protocols allow direct investigation of hepatic glycemic control and can be further applied to murine models of liver disease. © Published 2021. This article is a U.S. Government work and is in the public domain in the USA. Basic Protocol 1: Mouse surgical procedure and positioning for liver intravital imaging Basic Protocol 2: Fluorescent labeling and intravital imaging of mouse hepatic compartments Basic Protocol 3: Mouse hepatic glucose uptake assay and intravital imaging analysis.

Impact of tumor-parenchyma biomechanics on liver metastatic progression: a multi-model approach

Colorectal cancer and other cancers often metastasize to the liver in later stages of the disease, contributing significantly to patient death. While the biomechanical properties of the liver parenchyma (normal liver tissue) are known to affect tumor cell behavior in primary and metastatic tumors, the role of these properties in driving or inhibiting metastatic inception remains poorly understood, as are the longer-term multicellular dynamics. This study adopts a multi-model approach to study the dynamics of tumor-parenchyma biomechanical interactions during metastatic seeding and growth. We employ a detailed poroviscoelastic model of a liver lobule to study how micrometastases disrupt flow and pressure on short time scales. Results from short-time simulations in detailed single hepatic lobules motivate constitutive relations and biological hypotheses for a minimal agent-based model of metastatic growth in centimeter-scale tissue over months-long time scales. After a parameter space investigation, we find that the balance of basic tumor-parenchyma biomechanical interactions on shorter time scales (adhesion, repulsion, and elastic tissue deformation over minutes) and longer time scales (plastic tissue relaxation over hours) can explain a broad range of behaviors of micrometastases, without the need for complex molecular-scale signaling. These interactions may arrest the growth of micrometastases in a dormant state and prevent newly arriving cancer cells from establishing successful metastatic foci. Moreover, the simulations indicate ways in which dormant tumors could "reawaken" after changes in parenchymal tissue mechanical properties, as may arise during aging or following acute liver illness or injury. We conclude that the proposed modeling approach yields insight into the role of tumor-parenchyma biomechanics in promoting liver metastatic growth, and advances the longer term goal of identifying conditions to clinically arrest and reverse the course of late-stage cancer.

Modeling of Nanotherapy Response as a Function of the Tumor Microenvironment: Focus on Liver Metastasis

The tumor microenvironment (TME) presents a challenging barrier for effective nanotherapy-mediated drug delivery to solid tumors. In particular for tumors less vascularized than the surrounding normal tissue, as in liver metastases, the structure of the organ itself conjures with cancer-specific behavior to impair drug transport and uptake by cancer cells. Cells and elements in the TME of hypovascularized tumors play a key role in the process of delivery and retention of anti-cancer therapeutics by nanocarriers. This brief review describes the drug transport challenges and how they are being addressed with advanced in vitro 3D tissue models as well as with in silico mathematical modeling. This modeling complements network-oriented techniques, which seek to interpret intra-cellular relevant pathways and signal transduction within cells and with their surrounding microenvironment. With a concerted effort integrating experimental observations with computational analyses spanning from the molecular- to the tissue-scale, the goal of effective nanotherapy customized to patient tumor-specific conditions may be finally realized.

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.

Resilience of three-dimensional sinusoidal networks in liver tissue

Can three-dimensional, microvasculature networks still ensure blood supply if individual links fail? We address this question in the sinusoidal network, a plexus-like microvasculature network, which transports nutrient-rich blood to every hepatocyte in liver tissue, by building on recent advances in high-resolution imaging and digital reconstruction of adult mice liver tissue. We find that the topology of the three-dimensional sinusoidal network reflects its two design requirements of a space-filling network that connects all hepatocytes, while using shortest transport routes: sinusoidal networks are sub-graphs of the Delaunay graph of their set of branching points, and also contain the corresponding minimum spanning tree, both to good approximation. To overcome the spatial limitations of experimental samples and generate arbitrarily-sized networks, we developed a network generation algorithm that reproduces the statistical features of 0.3-mm-sized samples of sinusoidal networks, using multi-objective optimization for node degree and edge length distribution. Nematic order in these simulated networks implies anisotropic transport properties, characterized by an empirical linear relation between a nematic order parameter and the anisotropy of the permeability tensor. Under the assumption that all sinusoid tubes have a constant and equal flow resistance, we predict that the distribution of currents in the network is very inhomogeneous, with a small number of edges carrying a substantial part of the flow-a feature known for hierarchical networks, but unexpected for plexus-like networks. We quantify network resilience in terms of a permeability-at-risk, i.e., permeability as function of the fraction of removed edges. We find that sinusoidal networks are resilient to random removal of edges, but vulnerable to the removal of high-current edges. Our findings suggest the existence of a mechanism counteracting flow inhomogeneity to balance metabolic load on the liver.

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.

Contaminant transport in soil: A comparison of the Theory of Porous Media approach with the microfluidic visualisation

Visualisation of the groundwater flow and contaminant transport can play a significant role for a better understanding of contaminant fate, which helps decision-makers and contaminated site planners to choose and implement the best remediation strategies. In this paper, a microfluidic chip coated with nanoclay was developed to mimic soil behaviour. Scanning electron microscopy (SEM) images and Fourier-transform infrared spectroscopy (FTIR) analysis confirmed that all the features and surfaces are coated with nanoclay. The change of contact angle for the native polydimethylsiloxane (PDMS) from 151^° ± 4^° to 73^° ± 6^° for modified ones is indicative of a considerable shift to hydrophilic behaviour. Moreover, the transport process in the developed chip was simulated utilising the Theory of Porous Media (TPM) and computational fluid dynamic (CFD) approaches. Although the results of both numerical approaches are in good agreement with experiments, the Root Mean Square Error (RMSE) of the predicted contaminant concentration by TPM at two observation points is less than that of CFD.

Spatial heterogeneity in the mammalian liver

Hepatocytes operate in highly structured repeating anatomical units termed liver lobules. Blood flow along the lobule radial axis creates gradients of oxygen, nutrients and hormones, which, together with morphogenetic fields, give rise to a highly variable microenvironment. In line with this spatial variability, key liver functions are expressed non-uniformly across the lobules, a phenomenon termed zonation. Technologies based on single-cell transcriptomics have constructed a global spatial map of hepatocyte gene expression in mice revealing that ~50% of hepatocyte genes are expressed in a zonated manner. This broad spatial heterogeneity suggests that hepatocytes in different lobule zones might have not only different gene expression profiles but also distinct epigenetic features, regenerative capacities, susceptibilities to damage and other functional aspects. Here, we present genomic approaches for studying liver zonation, describe the principles of liver zonation and discuss the intrinsic and extrinsic factors that dictate zonation patterns. We also explore the challenges and solutions for obtaining zonation maps of liver non-parenchymal cells. These approaches facilitate global characterization of liver function with high spatial resolution along physiological and pathological timescales.

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.

Zonation of hepatic fat accumulation: insights from mathematical modelling of nutrient gradients and fatty acid uptake

Intrinsic of non-alcoholic fatty liver diseases is an aberrant accumulation of triglycerides (steatosis), which occurs inhomogeneously within lobules. To improve our understanding of the mechanisms involved in this zonation patterning, we developed a mathematical multicompartment model of hepatic fatty acid metabolism accompanied by blood flow simulations. A model analysis determines the influence of the uptake process of fatty acids, the porto-central gradient of plasma fatty acid concentration, and the oxygen supply via blood on the zonation of triglyceride accumulation. From this theoretical perspective, the plasma oxygen gradient, but not the fatty acid gradient, leads the way to a zonated triglyceride accumulation by its decisive role in oxidative processes. In addition, the uptake mechanism of fatty acids seems to be fundamental for a pericentral dominance of steatosis. However, the mechanism of cellular fatty acid uptake from the blood is still under debate. Our theoretical approach supports the transporter-mediated uptake mechanism and reveals that the maximal velocity of fatty acid uptake affects the switching between a periportal and a pericentral triglyceride accumulation. Further research on hepatic fatty acid uptake is needed to push forward our understanding of aberrant triglyceride accumulation in diet-induced steatosis.

Model Prediction and Validation of an Order Mechanism Controlling the Spatiotemporal Phenotype of Early Hepatocellular Carcinoma

Recently, hepatocyte-sinusoid alignment (HSA) has been identified as a mechanism that supports the coordination of hepatocytes during liver regeneration to reestablish a functional micro-architecture (Hoehme et al. in Proc Natl Acad Sci 107(23):10371-10376, 2010). HSA means that hepatocytes preferentially align along the closest micro-vessels. Here, we studied whether this mechanism is still active in early hepatocellular tumors. The same agent-based spatiotemporal model that previously correctly predicted HSA in liver regeneration was further developed to simulate scenarios in early tumor development, when individual initiated hepatocytes gain increased proliferation capacity. The model simulations were performed under conditions of realistic liver micro-architectures obtained from 3D reconstructions of confocal laser scanning micrographs. Interestingly, the established model predicted that initiated hepatocytes at first arrange in elongated patterns. Only when the tumor progresses to cell numbers of approximately 4000, does it adopt spherical structures. This prediction may have relevant consequences, since elongated tumors may reach critical structures faster, such as larger vessels, compared to a spherical tumor of similar cell number. Interestingly, this model prediction was confirmed by analysis of the spatial organization of initiated hepatocytes in a rat liver tumor initiation study using single doses of 250 mg/kg of the genotoxic carcinogen N-nitrosomorpholine (NNM). Indeed, small clusters of GST-P positive cells induced by NNM were elongated, almost columnar, while larger GDT-P positive foci of approximately the size of liver lobuli adopted spherical shapes. From simulations testing numerous possible mechanisms, only HSA could explain the experimentally observed initial deviation from spherical shape. The present study demonstrates that the architecture of small cell clusters of hepatocytes early after initiation is still controlled by physiological mechanisms. However, this coordinating influence is lost when the tumor grows to approximately 4000 cells, leading to further growth in spherical shape. Our findings stress the potential importance of organ micro-architecture in understanding tumor phenotypes.

Modeling of the contrast-enhanced perfusion test in liver based on the multi-compartment flow in porous media

The paper deals with modeling the liver perfusion intended to improve quantitative analysis of the tissue scans provided by the contrast-enhanced computed tomography (CT). For this purpose, we developed a model of dynamic transport of the contrast fluid through the hierarchies of the perfusion trees. Conceptually, computed time-space distributions of the so-called tissue density can be compared with the measured data obtained from CT; such a modeling feedback can be used for model parameter identification. The blood flow is characterized at several scales for which different models are used. Flows in upper hierarchies represented by larger branching vessels are described using simple 1D models based on the Bernoulli equation extended by correction terms to respect the local pressure losses. To describe flows in smaller vessels and in the tissue parenchyma, we propose a 3D continuum model of porous medium defined in terms of hierarchically matched compartments characterized by hydraulic permeabilities. The 1D models corresponding to the portal and hepatic veins are coupled with the 3D model through point sources, or sinks. The contrast fluid saturation is governed by transport equations adapted for the 1D and 3D flow models. The complex perfusion model has been implemented using the finite element and finite volume methods. We report numerical examples computed for anatomically relevant geometries of the liver organ and of the principal vascular trees. The simulated tissue density corresponding to the CT examination output reflects a pathology modeled as a localized permeability deficiency.

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.

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.

Zonated quantification of steatosis in an entire mouse liver

Many physiological processes and pathological conditions in livers are spatially heterogeneous, forming patterns at the lobular length scale or varying across the organ. Steatosis, a common liver disease characterized by lipids accumulating in hepatocytes, exhibits heterogeneity at both these spatial scales. The main goal of the present study was to provide a method for zonated quantification of the steatosis patterns found in an entire mouse liver. As an example application, the results were employed in a pharmacokinetics simulation. For the analysis, an automatic detection of the lipid vacuoles was used in multiple slides of histological serial sections covering an entire mouse liver. Lobuli were determined semi-automatically and zones were defined within the lobuli. Subsequently, the lipid content of each zone was computed. The steatosis patterns were found to be predominantly periportal, with a notable organ-scale heterogeneity. The analysis provides a quantitative description of the extent of steatosis in unprecedented detail. The resulting steatosis patterns were successfully used as a perturbation to the liver as part of an exemplary whole-body pharmacokinetics simulation for the antitussive drug dextromethorphan. The zonated quantification is also applicable to other pathological conditions that can be detected in histological images. Besides being a descriptive research tool, this quantification could perspectively complement diagnosis based on visual assessment of histological images.

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.

Constraint Based Modeling Going Multicellular

Constraint based modeling has seen applications in many microorganisms. For example, there are now established methods to determine potential genetic modifications and external interventions to increase the efficiency of microbial strains in chemical production pipelines. In addition, multiple models of multicellular organisms have been created including plants and humans. While initially the focus here was on modeling individual cell types of the multicellular organism, this focus recently started to switch. Models of microbial communities, as well as multi-tissue models of higher organisms have been constructed. These models thereby can include different parts of a plant, like root, stem, or different tissue types in the same organ. Such models can elucidate details of the interplay between symbiotic organisms, as well as the concerted efforts of multiple tissues and can be applied to analyse the effects of drugs or mutations on a more systemic level. In this review we give an overview of the recent development of multi-tissue models using constraint based techniques and the methods employed when investigating these models. We further highlight advances in combining constraint based models with dynamic and regulatory information and give an overview of these types of hybrid or multi-level approaches.

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.