As the key organ for metabolic processes in the human body, the liver is responsible for essential processes like fat storage or detoxification. Liver diseases can trigger growth processes in the liver, disrupting important hepatic function-perfusion processes[1]. To better understand the interplay between hepatic perfusion, metabolism and tissue in the hierarchically organized liver structure, we developed a multicomponent, poro-elastic multiphasic and multiscale function-perfusion model [2,3], using a multicomponent mixture theory based on the Theory of Porous Media (TPM). The multiscale approach considers the different functional units of the liver, so-called liver lobules, with an anisotropic blood flow via the sinusoids (slender capillaries between periportal field and central vein), and the hepatocytes, where the biochemical metabolic reactions take place. On the lobular scale, we consider a tetra-phasic body, composed of a porous solid structure representing healthy tissue, a liquid phase describing the blood, and two solid phases with the ability of growth and depletion representing the fat tissue and the tumor tissue. The phases consist of a carrier phase, called solvent, and solutes, representing microscopic components, e.g. nutrients, dissolved in the solvent. To describe the influences of the resulting tissue growth, the model is enhanced by a kinematic growth approach using a multiplicative split of the deformation gradient into an elastic and a growth part, dependent on the fat accumulation and tumor development. To describe the metabolic processes as well as the production, utilization and storage of the metabolites on the cellular scale, a bi-scale PDE-ODE approach with embedded coupled ordinary differential equations is used. In order to represent realistic conditions of the liver, experimentally or clinically obtained data such as changes in perfusion, material parameters or tissue morphology and geometry are integrated as initial boundary conditions or used for parametrization and validation [4]. Data integration approaches like machine learning are developed for the identification, processing and integration of data. A workflow is designed that directly prepares the model for clinical application by (semi-)automatically processing the data, considering uncertainties, and reducing computation time.

Acknowledgements: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the following projects: 312860381 (Priority Programme SPP 1886, Subproject 12); 390740016 (Germany’s Excellence StrategyEXC 2075/1, Project PN 2-2A); 436883643 (Research Unit Programme FOR5151, Project P7); 465194077 (Priority Programme SPP 2311, Project Sim-LivA); 463296570 (Priority Programme SPP 1158, Antarctica)

REFERENCES
[1] Christ, B., ..., Ricken, T. et al. [2017], Front. Physiol., doi: 10.3389/fphys.2017.00906 .
[2] Ricken, T., et al. [2015], Biomech. Model. Mechanobiol., doi: 10.1007/s10237-014-0619-z .
[3] Ricken, T., and Lambers, L. [2019], GAMM-Mitteilungen, doi: 10.1002/gamm.201900016 .
[4] Seyedpour, S. M, ..., and Ricken, T. [2021], Front. Physiol., doi: 10.3389/fphys.2021.733393

Physiologically-based pharmacokinetic (PBPK) models are powerful tools for studying drug metabolism and its effect on the human body. Here, we present our work on PBPK models for metabolic phenotyping and liver function evaluation [1-5]. To develop and validate our models, we established the first open pharmacokinetics database, PK-DB, which contains curated data from over 600 clinical studies. Our models can be individualized and stratified, allowing us to simulate the effect of lifestyle factors and co-administration on drug metabolism.

We have applied our models to various clinical questions, such as simulating individual outcomes after hepatectomy using an indocyanine green model and studying the effect of CYP2D6 gene variants using a model of dextromethorphan coupled with drug-gene interactions. These models are constructed hierarchically, describing metabolism and other biological processes of organs such as liver and kidney coupled to whole-body physiology. All models and data are available for reuse in a reproducible manner encoded in the Systems Biology Markup Language (SBML).

One of the major challenges in PBPK modeling is the integration of these models with cellular and continuum-biomechanical models. By coupling PBPK models with these models, it becomes possible to provide drug concentrations at the site of action as boundary conditions and time-varying inputs to continuum-biomechanical models. This approach enables us to study the impact of systemic chemotherapy on tumor growth or to investigate changes in perfusion and pressure at the whole-body level on transport and biomechanical properties at the tissue scale.

To demonstrate the effectiveness of this approach, we present an example of PBPK modeling coupled with a multiscale and multiphase continuum-biomechanical model for the liver. Our model integrates partial differential equations (PDE) on the lobular scale with ordinary differential equations (ODE) on the cellular and whole-body scale, resulting in a PDE-ODE coupling. In our showcase, we investigate the drug metabolization process in the liver, considering the possible damage to the liver tissue due to necrosis caused by a toxic side product. The substrates and products of this process can be transported via the systemic circulation and be removed by the kidneys. By coupling PBPK models with continuum-biomechanical models, we can study the transport and biomechanical properties of these substances at both the tissue and whole-body levels. This coupling approach provides a promising method for studying drug metabolism and its effect on the human body at multiple scales, ranging from cellular to whole-body.

Our approach allows us to investigate the impact of this drug metabolization process on the liver tissue and how it affects the overall functioning of the body. Additionally, we can study the impact of various factors, such as different drug dosages, on this process and determine the optimal drug dosage for a given patient. This integrated approach provides a comprehensive understanding of the drug metabolization process and its impact on the human body, paving the way for more effective drug development and personalized medicine.

[1] https://doi.org/10.1093/nar/gkaa990
[2] https://doi.org/10.3389/fphar.2021.752826
[3] https://doi.org/10.3389/fphys.2021.730418
[4] https://doi.org/10.3389/fphys.2021.757293
[5] https://doi.org/10.1101/2022.08.23.504981

Liver transplantation is the only curative treatment option for acute and chronic end-stage liver disease. As a result of demographic change and a more western way of life, the number of elderly multi-morbid prospective recipients and donors grows. Liver grafts from such donors, so-called marginal liver grafts, are often affected by hepatic steatosis compromising the quality of the donor organ. One reason for this is the alteration of the tissue structure, resulting in an impaired perfusion, in turn affecting hepatic metabolism and organ function. In case of a marginal graft, the surgeon is faced with the clinical decision to accept or reject the organ, increasing either the recipient’s postoperative risk or the risk of death on the waiting list significantly. Two major challenges for marginal grafts are the storage between procurement of the organ and transplantation (cold ischemia time) alongside damage after reperfusion, the so-called ischemia reperfusion injury (IRI).

For this purpose, we use computational multiphase and multiscale continuum-biomechanical modeling of biological tissue to simulate the hepatic deformation-perfusion-function relationship (cf. [1, 2]). Based on the theory of porous media (TPM, cf. [3]), we describe the functional liver units, the liver lobules, as a homogenized porous medium. A poroelastic multiphase and multiscale function-perfusion model is obtained by considering the porous liver tissue saturated with an anisotropic blood flow and coupling the metabolic processes at the cellular level within a bi-scale approach (cf. [4]). This approach combines partial differential equations (PDE) on the lobular scale with ordinary differential equations (ODE) on the cellular scale resulting in a PDE-ODE coupling. At the lobular scale, we consider healthy liver, necrotic, and fat tissue as solid phases, while blood is given as a fluid phase. Based on this, systems biology models can be used to model the energy balance, cell death, and functionality of each hepatocyte at the cellular level. Thus, it is possible to describe the ischemic damage caused by nutrient depletion, in turn influencing the perfusion at the lobular scale.

To make patient-specific predictions, we enhance the model through the integration of experimental and clinical data for validation and parameterization. This involves using machine-learning-based image analysis to read the geometry of liver lobules and zonation patterns of steatosis from histopathological images, in addition to laboratory data to provide initial and boundary values for solutes and the ODE models as well as information on the transplantation process, e.g., cold ischemia time. The simulation yields spatial evolutions of the considered phases and solutes within the liver lobules over time based on the given data. Such an in-silico model, which connects structure, perfusion, and function of the liver via the interaction between mechanical properties of the graft, hepatic perfusion, and the affected metabolism could facilitate clinical decision-making cf. [2].

[1] B. Christ et al. Frontiers in Physiology 8:00906 (2017)

[2] B. Christ et al. Frontiers in Physiology 12:733868 (2021)

[3] W. Ehlers. „Foundations of multiphasic and porous materials” in Porous Media: Theory, Experiments and Numerical Applications (2002).

[4] T. Ricken et al. BMMB 14:515-536 (2015)

Various research works from the biomechanics community focus on the inflation of pressurized tubular materials as this problem can be directly linked to the modeling of arterial tissue (see, e.g. Merodio & Ogden). In the modeling, the material is often taken to be hyperelastic, and the relation between loading deformation is given in terms of a strain energy density function. This function is often decomposed to account for the contribution of the different types of constitutes. The pressurization of the tubular structure is accompanied by various types of responses in both the stable and unstable parts of the inflation (see, e.g. Topol et al. 2022).

Biological soft tissue consists of cellular and non-cellular components. The relationship between stress and strain in the material evolves, and is determined by viscoelastic and tissue remodeling processes (see, e.g. Topol et al. (2021), Wineman & Pence (2021)). Due to the interplay of countless biological and chemical processes, the mechanical properties of tissue will be time-dependent. These processes may be initiated, stimulated, and mediated by different physical processes, for example in the form of mechanical stimuli in static or dynamic form (Stoffel et al., 2017).

This work on the inflation behavior of pressurized tubes under a time-dependent behavior of the material. It can be shown that the inflation of the material shows a large variety of responses that depend on various factors, that include the geometry of the considered problem (H. Topol et al. 2019)). A certain focus is given to the influence of the fiber natural configuration of the inflation behavior, which may differ from that of the embedding ground substance.

References
J. Merodio and R. W. Ogden. Extension, inflation and torsion of a residually stressed circular cylindrical tube. Continuum Mech. Thermodyn. 28:157–174 (2016)

M. Stoffel, W. Willenber, M. Azarnoosh, N. Fuhrmann-Nelles, B. Zhou, B. Markert. Towards bioreactor development with physiological motion control and its applications. Med. Eng. Phys 39: 106–112 (2017)

H. Topol, H. Demirkoparan, T. J. Pence. Morphoelastic fiber remodeling in pressurized thick-walled cylinders with application to soft tissue collagenous tubes. Eur. J. Mech. A/Solids 77: 103800 (2019)

H. Topol, H. Demirkopararn, T. J. Pence. Fibrillar Collagen: A Review of the Mechanical Modeling of Strain Mediated Enzymatic Turnover. Appl. Mech. Rev. 73: 050802, 2021

H. Topol, N. K. Jha, H. Demirkoparan, M. Stoffel, and J. Merodio. Bulging of inflated membranes made of fiber reinforced materials with different natural configurations. Eur. J. Mech. A/Solids 94: 104670 (2022)

A. Wineman and T. J. Pence. Fiber-reinforced composites: nonlinear elasticity and beyond. J. Eng. Math. 127: 30 (2021)