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. © 2021

The focus of this investigation lies on the development of a two-scale homogenization scheme for poro-elastic fluid-saturated porous media. For this purpose, the general concepts of the Theory of Porous Media (TPM) are combined with the FE2 method. After an introduction of the basics of TPM, the weak forms for the macroscopic and the microscopic scale will be formulated and the averaged macroscopic tangent moduli considering the microscale will be derived. Additionally, the formulation of lower level boundary conditions, which refer to the quantities that will be transmitted from the macro- to the microscale, in strict compliance with the Hill–Mandel homogeneity condition, is derived. Finally, a numerical example will be presented, pointing out the gained features of the methodology. © 2021

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. © Copyright © 2021 Seyedpour, Nabati, Lambers, Nafisi, Tautenhahn, Sack, Reichenbach and Ricken.

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. Copyright © 2021 Christ, Collatz, Dahmen, Herrmann, Höpfl, König, Lambers, Marz, Meyer, Radde, Reichenbach, Ricken and Tautenhahn.

We study fluid-saturated porous materials that undergo poro-elastic deformations in thin domains. The mechanics in such materials are described using a biphasic model based on the theory of porous media (TPM) and consisting of a system of differential equations for material’s displacement and fluid’s pressure. These equations are in general strongly coupled and nonlinear, such that exact solutions are hard to obtain and numerical solutions are computationally expensive. This paper reduces the complexity of the biphasic model in thin domains with a scale separation between domain’s width and length. Based on standard asymptotic analysis, we derive a reduced model that combines two sub-models. Firstly, a limit model consists of averaged equations that describe the fluid pore pressure and displacement in the longitudinal direction of the domain. Secondly, a corrector model re-captures the mechanics in the transverse direction. The validity of the reduced model is finally tested using a set of numerical examples. These demonstrate the computational efficiency of the reduced model, while maintaining reliable solutions in comparison with original biphasic TPM model in thin domain. © 2021, The Author(s).

In situ chemical oxidation using permanganate as an oxidant is a remediation technique often used to treat contaminated groundwater. In this paper, groundwater flow with a full hydraulic conductivity tensor and remediation process through in situ chemical oxidation are simulated. The numerical approach was verified with a physical sandbox experiment and analytical solution for 2D advection-diffusion with a first-order decay rate constant. The numerical results were in good agreement with the results of physical sandbox model and the analytical solution. The developed model was applied to two different studies, using multi-objective genetic algorithm to optimize remediation design. In order to reach the optimised design, three objectives considering three constraints were defined. The time to reach the desired concentration and remediation cost regarding the number of required oxidant sources in the optimised design was less than any arbitrary design. © 2021 by the authors.Licensee MDPI, Basel, Switzerland.

Rheumatic heart disease (RHD) is identified as a serious health concern in developing countries, specifically amongst young individuals, accounting for between 250 000 and 1.4 million deaths annually. As such, attention is initially placed on the importance of the development of a cardiac analysis toolbox with functionality for pathophysiological analysis of the disease. Subsequently, in order to develop a toolbox to further the understanding of the mechanisms of the disease as linked to changes in the cytoskeletal architecture and hypertrophy of cardiac myocytes, a continuum bi-phasic model applicable to cardiac tissue is formulated based on the theory of porous media (TPM). This makes it possible to account for interactions and contributions of multiple phases of constituent materials as well as concentrations of solved components, which in computational cardiac modelling are the solid phase – the cardiac tissue – and the liquid phase – blood and interstitial fluid. Therefore, subsequent attention is paid to the cardiac model development in order to implement a sound base on which to add strain- and nutrient-driven phase transition, in addition to a nutrient phase contained within the liquid phase. To this end, based on thermodynamical restrictions, constitutive relations are proposed for stress, permeability, seepage velocity and interaction forces. The approach is implemented in the in-house computational cardiac mechanics toolbox SESKA which supports finite element as well as Element-free Galerkin-based approximations. This paper considers the passive and active non-linear elastic material behaviour of the myocardium of the left ventricle coupled with porous media theory, along with an additional coupling to the haemodynamics of the circulatory system, facilitating modelling of the full cardiac cycle. In order to illustrate the potential and efficacy of the approach with qualitative results, a human heart affected by RHD is investigated, making use of cardiovascular magnetic resonance scans taken over a period of two years to generate realistic 3D computer models. © 2018 Elsevier Ltd

Cartilage has the task of creating a joint connection that is as frictionless and damping as possible. It is a complex multi-phase material consisting of a liquid saturated collagen fiber-reinforced porous solid. The mechanical properties of cartilage are highly dependent on the interaction of the isotropically distributed, electrically negative charged proteoglycans and the highly anisotropic and inhomogeneously distributed collagen fiber network. The negatively charged ions of the proteoglycans lead to an electrical gradient which in turn causes an osmotic negative pressure. This causes a swelling inside the cartilage whereby the collagen fibers are mechanically prestressed and thus reducing the total pressure acting on the cartilage matrix. A major cartilage disease that currently affects more and more people especially in developed countries is osteoarthritis. It changes the composition of the proteoglycans which results in a decrease in osmotic pressure, leading to increased abrasion. In addition, a defibration of the collagen fibers occurs which causes a decrease of tensile strength and viscoelasticity in fiber direction. In order to account for these age-related degeneration processes a model using the Theory of Porous Media as a homogenization approach is developed. The model describing the mechanical behavior of articular cartilage is based on the biphasic model presented in Pierce et al. 2013. It includes the incompressible poroelastic solid matrix reinforced with collagen fibers and the incompressible fluid in the pores. We aim to incorporate the influences of osmotic pressure (Wang et al. 2018) in order to determine the correct initial stress state under imaged configuration. © 2019 Taylor & Francis Group, London, UK.

Numerical simulations of biological systems have become more and more important in recent years. In order to understand and predict hepatic function in health and disease, we developed a multiscale and multiphase model for the description of function-perfusion processes in the liver. With respect to the different scales of the hierarchically structured liver, processes like glucose homeostasis or detoxification of paracetamol as well as the influence of hepatic disease like the non-alcoholic fatty liver disease (NAFLD) can be investigated. On the tissue scale the liver consists of hexagonal liver lobules, containing anisotropically oriented capillaries, called sinusoids, which lead macroscopically to an anisotropic blood flow as well as a nonlinear, anisotropic and poro-elastic response. This structure is described using a homogenization method based on the Theory of Porous Media (TPM). This leads to a coupled set of partial differential equations (PDE) describing the tissue deformation as well as the transport of blood and metabolites like nutrients or xenobiotics. The lobule scale is coupled to the cellular scale where hepatic metabolism takes place. With the use of embedded ODE equations we can simulate metabolic processes depending on various nutrients or substances. The performance of the developed theory is demonstrated by a numerical example. The results provide an overview of possible applications of our approach using the NAFLD as a showcase. During the development of fatty liver, fat is stored in the liver cells resulting in swelling of the hepatocytes. This accumulation of fat drives tissue growth, which has an impact on the blood perfusion in the liver lobules. The results also clarify the spatial distribution of flow and fat accumulation since many hepatic processes proceed zonated. The processes in one single lobule are then expanded to a group of lobules to investigate the mutual liver lobe interaction. © 2019 Taylor & Francis Group, London, UK.

It is evident that the sea ice cycle, from its formation to its melt, is governed by a complex interaction of the ocean, atmosphere and surrounding continents. Once sea water begins to freeze, physical, biological and chemical processes have implications on the evolution of the sea ice morphology (Thomas, 2017). The distinguishing factor between fresh and sea water ice is brine inclusions that get trapped within the ice pores during freezing. Salt inclusions within frozen ice influence the salinity of the seawater as well as the physical properties of the sea ice (Hunke et al., 2010). These brine inclusions form part of a dynamic process within the ice characterized by the movement of brine and phase transition which are the foundation of many of its physical properties (Hunke et al., 2010). Brine removal subsequently begins to occur due to vertical gravity drainage into the underlying ocean water. This review article provides an overview of models that have attempted to describe this complex system. It also introduces a biphasic model being developed based on the Theory of Porous Media (TPM) which considers a solid phase for the pore structure of the ice matrix as well as a liquid phase for the brine inclusions, respectively. The TPM framework is able to describe and study the various desalination mechanisms that are significant in aiding the salt flux into the Southern Ocean. This will foster understanding of brine rejection and how it is linked to the porous microstructure of Antarctic sea ice. © 2019 Taylor & Francis Group, London, UK.

In this contribution, several case studies with data uncertainties are presented which have been performed in individual projects as part of the DFG (German Research Foundation) Priority Programme SPP 1886 “Polymorphic uncertainty modelling for the numerical design of structures.” In all case studies numerical models with uncertainties are derived from engineering problems describing concepts for handling and incorporating measurement data, either of model input parameters or of the system response. The first case study deals with polymorphic uncertain data based on computer tomographic scans with respect to air voids which are acquired, simplified and integrated in numerical models of adhesive bonds. In the second case study, the variation sensitivity analysis is presented to provide suitable prior knowledge for numerical soil analyses, for example, in order to reduce required input data. The uncertainty in friction processes is treated in case study 3 whereby measurement data are used in data driven methods to improve the numerical predictions. In case study 4, the failure behavior of die-cast window hinges, which is affected by an uncertain initial pore distribution, is investigated by means of a Markov chain approach. In the last two case studies, mathematical methods of statistical inference and updating algorithms for uncertainty models are shown. Due to the heterogeneous spectrum of problems, a generalized strategy for data modeling, acquisition, and assimilation is developed and applied on each case study. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Modeling of mechanical systems with uncertainties is extremely challenging and requires a careful analysis of a huge amount of data. Both, probabilistic modeling and nonprobabilistic modeling require either an extremely large ensemble of samples or the introduction of additional dimensions to the problem, thus, resulting also in an enormous computational cost growth. No matter whether the Monte-Carlo sampling or Smolyak's sparse grids are used, which may theoretically overcome the curse of dimensionality, the system evaluation must be performed at least hundreds of times. This becomes possible only by using reduced order modeling and surrogate modeling. Moreover, special approximation techniques are needed to analyze the input data and to produce a parametric model of the system's uncertainties. In this paper, we describe the main challenges of approximation of uncertain data, order reduction, and surrogate modeling specifically for problems involving polymorphic uncertainty. Thereby some examples are presented to illustrate the challenges and solution methods. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The motion of sea ice in large scales of several thousand kilometers is modeled by the viscous-plastic (VP) sea ice rheology. The sea ice motion model is based on the findings of Hibler III (1979), who introduced a numerical model for the simulation of sea ice circulation and thickness evolution over a seasonal cycle. The velocity and stress fields, as well as the sea ice thickness and sea ice concentration, are included in the model. Recent research on a finite element implementation of the model is devoted to formulations based on the (mixed) Galerkin variational approach. Here, special treatments are necessary regarding the stabilization of the numerical complex scheme. It is therefore suggested to utilize a mixed least-squares formulation to overcome possible numerical drawbacks. The least-squares finite element method is well established, especially in the branch of fluid mechanics, see e.g. Jiang (1998), Cai et al. (2004) and Bochev & Gunzburger (2009). A significant advantage of the method is its applicability to first-order systems, such that it results in stable and robust formulations also for not self-adjoint operators like in the Navier-Stokes equations. The presented least-squares finite element formulations are based on the instationary sea ice equations including two tracer equations of transient convection type. Therefore, the mixed least-squares approach includes four primary fields, which are the stress tensor σ, the velocity field v and two scalar tracers Aice and Hice. Different time-integrators are investigated regarding accuracy and stability with a particular focus on the treatment of the tracer equations. The investigation of the formulation with the help of a boundary value problem is provided. © 2019 Taylor & Francis Group, London, UK.

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. © 2019

According to NASA (National Aeronautics and Space Administration), Antarctic sea ice reached its lowest extent ever recorded by satellites at the end of summer 2017 after decades of moderate sea ice expansion. Besides a strong influence on the global climate, linking the exchange of energy and gases between the atmosphere and the ocean, changes in sea ice also have a biological impact on the structure and function of the ecosystem. These effects are strongly related to the physical and mechanical properties of the sea ice structure. Seawater is trapped in so-called brine pockets during the growth of sea ice, providing a natural habitat for sea ice microorganisms. The microorganisms are supplied with nutrients they need for primary production from the seawater. A small-scale modeling of the porosity of the sea ice and its inclusions and the solid/brine multiphase microstructure, respectively, thermodynamics of air-sea interactions as well as sea ice-biological linkages is a necessary tool to better understand the heterogeneous nature of sea ice. Based on the extended Theory of Porous Media (eTPM), the development of a multiphasic, multi-component model which enables the continuum mechanical description of transport and phase transition phenomena in sea ice at a homogenized pore scale is developed. The foundation builds a biphasic (ice and brine) model, in which the brine is composed of freshwater and salt. © 2019 Taylor & Francis Group, London, UK.

Artificial neural networks (ANNs) and quantum computing (QC) have both received rapidly increasing interest in recent years for several reasons. Their roles in the mechanics community remain a subject of ongoing debate and research. ANNs have successfully been employed on problems previously considered unsolvable (Silver et al. 2016), as well as on practical and illustrative problems especially in the image recognition domain (Szegedy et al. 2013, among others). The most straightforward way to interpret the behavior of ANNs is focusing on their ability to approximate any continuous function to arbitrary precision (Kreinovich 1991). The adjustment to a given function happens automatically except for hyperparameters, such as the number of hidden layers or the learning rate, which remain constant during training. This makes them a perfect tool for reducing computational cost in otherwise time-consuming simulations. Quantum computing on the other hand has received some publicity in 2018 mainly because of a paper published by Google Quantum A.I. Lab scientists defining quantum supremacy (Boixo & Isakov 2018), where it is shown that a fully working universal quantum computer with only 50 qubits will outperform any near-term realizable classical supercomputer on certain tasks. This publication was followed up by Google revealing Bristlecone (Kelly 2018), a quantum processor consisting of 72 qubits. While many interesting quantum algorithms have proven to run polynomially or even exponentially faster than their classical analogons (Montanaro 2016), these results only hold true for a decoherence-free machine. Therefore, to actually achieve quantum supremacy, quantum error correction schemes need to be implemented, requiring several physical qubits to make up a single logical qubit suitable for computation. © 2019 Taylor & Francis Group, London, UK.

As a smart material, hydrogels react to different stimuli. These could be – for example – of chemical, electrical, mechanical or thermal nature. The hydrogels consist of a more or less crosslinked polyelectrolytic polymer network, depending on the conditions, components, and time spent for its fabrication. The reaction of these stimulated hydrogels is to uptake or deliver ions with hydration shells or solvent, followed by e.g. an elongation or bending deformation of the gels. Gels can reach enormous swelling ratios and are applicable as actuators, energy converters or sensors. In the present approach, the underlying physical phenomena of the chemical, electrical as well as of the mechanical field are incorporated in a homogenized way by using different partial differential equations. Local effects, e.g. the osmotic pressure differences in the mechanical field, are derived over reference and local concentrations. Depending on the type of the stimulus, the hydrogel reaction is more or less sensitive. For example, the mechanical reaction under chemical stimulation is far beyond the reaction under electrical stimulation. The applied coupled multi-field formulation is capable of providing local concentrations, electric potential distributions and displacements. The hydrogel model of a finger gripper is formulated by the use of finite elements. For the simulation the Newton-Raphson method and implicit Euler time integration are applied. Here only small volume changes, corresponding swelling ratios and deformations are considered. The results to different kinds of stimulation will be presented. © 2019 Taylor & Francis Group, London, UK.

To account for the natural variability of material parameters in multiphasic and hydro-mechanical coupled finite element analyses of soil and earth structure applications, the use of probabilistic methods may be effective. Here, selecting the appropriate soil auto-correlation functions for random field realizations plays an essential role. In a joint study, the general influence of auto-correlation lengths on the results of strongly coupled models is determined. Subsequently, a polymorphic approach using fuzzy probability based random fields is used to capture the solution space for fuzzy auto-correlation lengths. To adequately describe the behavior of the soil the theory of porous media is implemented, which uses a homogenization approach for the multiple phases on the soil microstructure. Its foundations and the differentiated methods used for the polymorphic uncertainty quantification are explained in this contribution. Based on two representative examples, the requirements and advantages of a polymorphic uncertainty model are worked out30. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In recent years computational models have become more important for simulating hepatic processes and investigating liver diseases in silico and so various liver models have been published. The complex behavior of biological tissue with its hierarchical structure as well as the blood perfusion through the organ have been described using different approaches and numerical techniques. This paper shows and compares numerical approaches for function and perfusion simulation recently published and compares them with a multiscale function-perfusion model using the extended theory of porous media. We focus on the description of blood perfusion and liver tissue, but also on the simulation of liver diseases or the zonation of processes in the liver. Furthermore, the selected geometry is taken into account. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The simulation-optimisation models of groundwater and contaminant transport can be a powerful tool in the management of groundwater resources and remediation design. In this study, using Multiquadratic Radial Basis Function (MRBF) a coupled groundwater flow and reactive transport of contaminant and oxidant was developed in the framework of the Meshfree method. The parameter analysis has determined the optimum shape parameter (0.97), and the results of the model were compared with a physical sandbox model which were in good agreement. The genetic algorithm approach was used to find the optimum design of the remediation using permanganate as an oxidant. To find the optimum design we considered two objectives and two constraints. The results revealed that the breakthrough of contaminant to the downstream area of interest and the concentration of the contaminant in this area is reduced significantly with optimisation. © 2019

Reliability assessment of soil structures and suitable numerical methods for computational simulations still are considerable challenges for industry and science. This is due to the natural scattering of material properties, the very high complexity of the soil structure on different length scales and the load bearing mechanisms as well as the residual stress state due to individual load histories. The design security concept of soil structures requires high safety factors unlike industrially manufactured materials like steel or concrete. A better understanding of the nature and influence of the uncertainties and errors could help to develop a more reliable risk assessment and to reduce inefficient safety factors. For this, we realize a new research project within the new priority program SPP 1886, installed by the DFG and focusing on polymorphic uncertainty quantification. In the present subproject (sp12), the focus is on quantification and assessment of polymorphic uncertainties in computational simulations of earth structures, especially for fluid-saturated soils. To describe the strongly coupled solid-fluid response behavior, the theory of porous media (TPM) is used and solved with the Finite Element Method (FEM) (De Boer 2012, Ehlers 2002). Motivated by structural optimization research, the Variational Sensitivity Analysis (VSA) provides detailed information about the current equilibrium state by means of just one single run of the simulation. In this work the methodology is transferred to quantify aleatoric uncertainties, see (Henning & Ricken 2017). The method allows an immediate decision support, e.g. for site investigators. Not only the computational effort but also the applicability to any arbitrary FE-Model are great advantages of this approach, which provides a lot of additional information for end-users as well as researchers. After discussing the basic framework, governing equations and algorithmic implementation of the proposed model, we demonstrate the applicability by a simple boundary value problem. © 2019 Taylor & Francis Group, London, UK.

The human liver is an important metabolic organ which regulates metabolism of the body in a complex time depending and non-linear coupled functionperfusion-mechanism. Harmful microstructure failure strongly affects the viability of the organ. The excessive accumulation of fat in the liver tissue, known as a fatty liver, is one of the most common liver micro structure failures, especially in western countries. The growing fat has a high impact on the blood perfusion and thus on the functionality of the organ. This interaction between perfusion, growth of fat and functionality on the hepatic microcirculation is poorly understood and many biological aspects of the liver are still subject of discussion. The presented computational model consists of a bi-scale, tri-phasic, multi-component approach based on the theory of porous media. The model includes the stress and strain state of the liver tissue, the transverse isotropic blood perfusion in the sinusoidal micro perfusion system. Furthermore, we describe the glucose metabolism in a two-scale PDE-ODE approach whereas the fat metabolism is included via phenomenological functions. Different inflow boundary conditions are tested against the influence on fat deposition and zonation in the liver lobules. With this example we can discuss biological assumptions and get a better understanding of the coupled function-perfusion ability of the liver. © Springer International Publishing AG 2018.

Biological methane oxidation may be regarded as a method of aftercare treatment for landfills to reduce climate relevant methane emissions. It is of social and economic interest to estimate the behavior of bacterial methane oxidation in aged landfill covers due to an adequate long-term treatment of the gas emissions. Different approaches assessing methane oxidation in laboratory column studies have been investigated by other authors recently. However, this work represents the first study in which three independent approaches, ((i) mass balance, (ii) stable isotope analysis, and (iii) stoichiometric balance of product (CO2) and reactant (CH4) by CO2/CH4-ratio) have been compared for the estimation of the biodegradation by a robust statistical validation on a rectangular, wide soil column. Additionally, an evaluation by thermal imaging as a potential technique for the localization of the active zone of bacterial methane oxidation has been addressed in connection with stable isotope analysis and CO2/CH4-ratios. Although landfills can be considered as open systems the results for stable isotope analysis based on a closed system correlated better with the mass balance than calculations based on an open system. CO2/CH4-ratios were also in good agreement with mass balance. In general, highest values for biodegradation were determined from mass balance, followed by CO2/CH4-ratio, and stable isotope analysis. The investigated topsoil proved to be very suitable as a potential cover layer by removing up to 99% of methane for CH4 loads of 35–65 g m–2 d–1 that are typical in the aftercare phase of landfills. Finally, data from stable isotope analysis and the CO2/CH4-ratios were used to trace microbial activity within the reactor system. It was shown that methane consumption and temperature increase, as a cause of high microbial activity, correlated very well. © 2017 Elsevier Ltd

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. © 2017 Christ, Dahmen, Herrmann, König, Reichenbach, Ricken, Schleicher, Schwen, Vlaic and Waschinsky.

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress. © 2015, Springer-Verlag Berlin Heidelberg.

The model depSIM is a dump simulation model, which allows a detailed and time-scaled focus into the complex processes of a landfill. Description of the mechanical model: The biological, chemical and physical processes in the waste body are closely connected with each other and can be described mechanically. Therefore, a number of differential equations are needed and implemented in the model. The porous media body is examined under the acceptance of a compressible gas phase, a materially incompressible solid state, an organic phase and a liquid phase. For the verification of the numerical model the long-time behaviour (100 years) was simulated. Further details about the model and the mechanical background are summarized in Robeck, Ricken et Widmann: A finite element simulation model of biological conversion processes in landfills [1]. Use potentials: The developed model allows a differentiated, time wise and locally calculation and representation of the temperature, the organic conversion rate, the local pressure ratios and the gas current speeds. There were several case studies with the depSIM model in Germany which show the correlation between the temperature, gas production and gas potential. Therefore three different landfills were evaluated. Here, in the correlation between measured temperature in the landfill body and the temperature in the model was shown. The average divergence between both was less than 2 degree. By the detailed calculation of the gas speeds in every point of the dump an essential improvement arises compared with conventional arithmetic models for gas forecast and gas capture. These forecast models are based on estimated initial parameters. This allows only forecasts for a complete dump or a dump segment, but allows no coupled calculation of the relevant parameters. The model depSIM offers a spatially differentiated consideration of the gas production. However, just a spatially exact, quantitative forecast of the gas production is necessary for dump operator and authorities. The right forecast is elementary for the right dimensioning of the gas collection system and gas treatment and the possible use in combined heat and power units. All gas streams can be shown with the simulation model along the dump surface spatially and time wise differentiated. This allows a locally differentiated dump gas management with a division in areas with active or passive gas collection or to estimate the feasibility of a methane oxidation layer. © 2016 WIT Press.

This study focuses on a two-scale, continuum multicomponent model for the description of blood perfusion and cell metabolism in the liver. The model accounts for a spatial and time depending hydro-diffusion–advection–reaction description. We consider a solid-phase (tissue) containing glycogen and a fluid-phase (blood) containing glucose as well as lactate. The five-component model is enhanced by a two-scale approach including a macroscale (sinusoidal level) and a microscale (cell level). The perfusion on the macroscale within the lobules is described by a homogenized multiphasic approach based on the theory of porous media (mixture theory combined with the concept of volume fraction). On macro level, we recall the basic mixture model, the governing equations as well as the constitutive framework including the solid (tissue) stress, blood pressure and solutes chemical potential. In view of the transport phenomena, we discuss the blood flow including transverse isotropic permeability, as well as the transport of solute concentrations including diffusion and advection. The continuum multicomponent model on the macroscale finally leads to a coupled system of partial differential equations (PDE). In contrast, the hepatic metabolism on the microscale (cell level) was modeled via a coupled system of ordinary differential equations (ODE). Again, we recall the constitutive relations for cell metabolism level. A finite element implementation of this framework is used to provide an illustrative example, describing the spatial and time-depending perfusion–metabolism processes in liver lobules that integrates perfusion and metabolism of the liver. © 2014, Springer-Verlag Berlin Heidelberg.

This study focuses on a formulation within the theory of porus media for continuum multicomponent modeling of bacterial driven methane oxidation in a porous landfill cover layer which consists of a porous solid matrix (soil and bacteria) saturated by a liquid (water) and gas phase. The solid, liquid, and gas phases are considered as immiscible constituents occupying spatially their individual volume fraction. However, the gas phase is composed of three components, namely methane (CH4), oxygen (O2), and carbon dioxide (CO2). A thermodynamically consistent constitutive framework is derived by evaluating the entropy inequality on the basis of Coleman and Noll [8], which results in constitutive relations for the constituent stress and pressure states, interaction forces, and mass exchanges. For the final set of process variables of the derived finite element calculation concept we consider the displacement of the solid matrix, the partial hydrostatic gas pressure and osmotic concentration pressures. For simplicity, we assume a constant water pressure and isothermal conditions. The theoretical formulations are implemented in the finite element code FEAP by Taylor [29]. A new set of experimental batch tests has been created that considers the model parameter dependencies on the process variables; these tests are used to evaluate the nonlinear model parameter set. After presenting the framework developed for the finite element calculation concept, including the representation of the governing weak formulations, we examine representative numerical examples. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In civil engineering, the frost durability of partly liquid saturated porous media under freezing and thawing conditions is a point of great discussion. Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase in space and time corresponds to the moving freezing front inside the porous solid. In this paper, a macroscopic model based on the Theory of Porous Media (TPM) is presented which describes energetic effects of freezing and thawing processes. For simplification a ternary model consisting of the phases solid, ice and liquid is used. Attention is paid to the description of the temperature development, the determination of energy, enthalpy and mass supply as well as volume deformations due to ice formation during a freezing and thawing cycle. For the detection of energetic effects regarding the characterization and control of phase transition of water and ice, a physically motivated evolution equation for the mass exchange between ice and liquid is presented. Comparing experimental data with numerical examples shows that the simplified model is indeed capable of simulating the temperature development and energetic effects during phase change. In civil engineering, the frost durability of partly liquid saturated porous media under freezing/thawing conditions is a point of great discussion. Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase corresponds to the moving freezing front inside the porous solid. A macroscopic model based on the Theory of Porous Media (TPM) is presented which describes energetic effects of freezing/thawing processes. To simplify a ternary model consisting of the phases solid, ice and liquid is used which describes the temperature development, the determination of energy, enthalpy and mass supply as well as volume deformations due to ice formation during a freezing and thawing cycle. Energetic effects regarding the phase transition are modelled by a physically motivated evolution equation for the mass exchange between ice and liquid. Comparing experimental data with numerical examples shows that the simplified model is capable of simulating the temperature development and energetic effects during phase change. Copyright © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Cartilage is a multi-phase material composed of fluid and electrolytes (68-85% by wet weight), proteoglycans (5-10% by wet weight), chondrocytes, collagen fibres and other glycoproteins. The solid phase constitutes an isotropic proteoglycan gel and a fibre network of predominantly type II collagen, which provides tensile strength and mechanical stiffness. The same two components control diffusion of the fluid phase, e.g. as visualised by diffusion tensor MRI: (i) the proteoglycan gel (giving a baseline isotropic diffusivity) and (ii) the highly anisotropic collagenous fibre network. We propose a new constitutive model and finite element implementation that focus on the essential load-bearing morphology: an incompressible, poroelastic solid matrix reinforced by an inhomogeneous, dispersed fibre fabric, which is saturated with an incompressible fluid residing in strain-dependent pores of the collagen-proteoglycan solid matrix. The inhomogeneous, dispersed fibre fabric of the solid further influences the fluid permeability, as well as an intrafibrillar portion that cannot be 'squeezed out' from the tissue. Using representative numerical examples on the mechanical response of cartilage, we reproduce several features that have been demonstrated experimentally in the cartilage mechanics literature. © 2013 © 2013 Taylor & Francis.

We propose a new 3D biphasic constitutive model designed to incorporate structural data on the sample/patient-specific collagen fiber network. The finite strain model focuses on the load-bearing morphology, that is, an incompressible, poroelastic solid matrix, reinforced by an inhomogeneous, dispersed fiber fabric, saturated with an incompressible fluid at constant electrolytic conditions residing in strain-dependent pores of the collagen-proteoglycan solid matrix. In addition, the fiber network of the solid influences the fluid permeability and an intrafibrillar portion that cannot be 'squeezed out' from the tissue. We implement the model into a finite element code. To demonstrate the utility of our proposed modeling approach, we test two hypotheses by simulating an indentation experiment for a human tissue sample. The simulations use ultra-high field diffusion tensor magnetic resonance imaging that was performed on the tissue sample. We test the following hypotheses: (i) the through-thickness structural arrangement of the collagen fiber network adjusts fluid permeation to maintain fluid pressure (Biomech. Model. Mechanobiol. 7:367-378, 2008); and (ii) the inhomogeneity of mechanical properties through the cartilage thickness acts to maintain fluid pressure at the articular surface (J. Biomech. Eng. 125:569-577, 2003). For the tissue sample investigated, both through-thickness inhomogeneities of the collagen fiber distribution and of the material properties serve to influence the interstitial fluid pressure distribution and maintain fluid pressure underneath the indenter at the cartilage surface. Tissue inhomogeneity appears to have a larger effect on fluid pressure retention in this tissue sample and on the advantageous pressure distribution. © 2012 John Wiley & Sons, Ltd.

Landfills are the most common way of waste disposal worldwide. Biological processes convert the organic material into an environmentally harmful landfill gas, which has an impact on the greenhouse effect. After the depositing of waste has been stopped, current conversion processes continue and emissions last for several decades and even up to 100. years and longer. A good prediction of these processes is of high importance for landfill operators as well as for authorities, but suitable models for a realistic description of landfill processes are rather poor. In order to take the strong coupled conversion processes into account, a constitutive three-dimensional model based on the multiphase Theory of Porous Media (TPM) has been developed at the University of Duisburg-Essen. The theoretical formulations are implemented in the finite element code FEAP. With the presented calculation concept we are able to simulate the coupled processes that occur in an actual landfill. The model's theoretical background and the results of the simulations as well as the meantime successfully performed simulation of a real landfill body will be shown in the following. © 2010 Elsevier Ltd.

Cartilage from the medial femoral condyle of chicken was sectioned and imaged using second harmonic generation microscopy. Using image analysis techniques based on the Fourier transform we derived quantitative threedimensional data of the fiber direction and dispersion of the collagen fiber network in the superficial layer. These data can be used directly in biomechanical models to enhance the fidelity of these models. © 2011 SPIE.

Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase in space and time corresponds to the moving freezing front inside the porous solid. In this contribution, a macroscopic model based on the Theory of Porous Media (TPM) is presented toward the description of freezing and thawing processes in saturated porous media. Therefore, a quadruple model consisting of the constituents solid, ice, liquid and gas is used. Attention is paid to the description of capillary suction, liquid- and gas pressure on the surrounding surfaces, volume deformations due to ice formation, temperature distribution as well as influence of heat of fusion under thermal loading. For detection of energetic effects regarding the control of phase transition of water and ice, a physically motivated evolution equation for the mass exchange based on the local divergence of the heat flux is used. Numerical examples are presented to the applications of the model. © 2011 Springer-Verlag Berlin Heidelberg.

The 3-D morphology of chicken articular cartilage was quantified using multiphoton microscopy (MPM) for use in continuum-mechanical modeling. To motivate this morphological study we propose aspects of a new, 3-D finite strain constitutive model for articular cartilage focusing on the essential load-bearing morphology: an inhomogeneous, poro-(visco)elastic solid matrix reinforced by an anisotropic, (visco)elastic dispersed fiber fabric which is saturated by an incompressible fluid residing in strain-dependent pores. Samples of fresh chicken cartilage were sectioned in three orthogonal planes and imaged using MPM, specifically imaging the collagen fibers using second harmonic generation. Employing image analysis techniques based on Fourier analysis, we derived the principal directionality and dispersion of the collagen fiber fabric in the superficial layer. In the middle layer, objective thresholding techniques were used to extract the volume fraction occupied by extracellular collagen matrix. In conjunction with information available in the literature, or additional experimental testing, we show how this data can be used to derive a 3-D map of the initial solid volume fraction and Darcy permeability. © 2011 IEEE.

Liver resection can lead to focal outflow obstruction due to transection of hepatic veins. Outflow obstruction may cause additional damage to the small remnant liver. Drainage of the obstructed territories is reestablished via dilatation of sinusoids. Subsequently, sinusoidal canals are formed draining the blood from the obstructed territory to the neighboring unobstructed territories. We raised the phenomenological hypothesis that the blood pressure gradient is the main driving force for the formation of sinusoidal vascular canals. We generated a biphasic mechanical model to describe this vascular remodeling process in relation to the variable pressure gradient. Therefore, we introduced a transverse isotropic permeability relation aswell as an evolutional optimization rule to describe the relationship between pressure gradient and the direction of the sinusoidal blood flow in the fluid phase. As a next step, we developed a framework for the calculation concept including the representation of the governing weak formulations. Then, we examined a representative numerical example with simulation of the blood flow under both conditions, the physiological situation as well as after outflow obstruction. Doing so, we were able to reproduce numerically the experimentally observed process of reestablishing hepatic venous drainage via redirection of blood flow and formation of new vascular structures in respect to the fluid flow. The calculated results support the hypothesis that the reorientation of blood flow mainly depends on the pressure gradient. Further investigations are needed to determine the micromechanical influences on the reorientation of the sinusoids. © Springer-Verlag 2010.

Freezing and thawing are important processes in civil engineering. On the one hand frost damage of porous building materials like road pavements and concrete in regions with periodical freezing is well known. On the other hand, artificial freezing techniques are widely used, e.g. for tunneling in non-cohesive soils and other underground constructions as well as for the protection of excavation and compartmentalization of contaminated tracts. Ice formation in porous media results from a coupled heat and mass transport and is accompanied by the ice expansion. The volume increase in space and time is assigned to the moving freezing front inside the porous solid. In this paper, a macroscopic ternary model is presented within the framework of the Theory of Porous Media (TPM) in view of the description of phase transition. For the mass exchange between ice and water an evolution equation based on the local balance of the heat flux vector is used. Examples illustrate the application of the model for saturated porous solids under thermal loading. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The 3D structure of collagen fibers in chicken cartilage was quantified using multiphoton microscopy. Samples of fresh chicken cartilage were sectioned in three orthogonal planes using a vibratome. The sections were imaged using multiphoton microscopy, specifically imaging the collagen fibers using the second harmonic signal. Employing image analysis techniques based on Fourier analysis, the primary direction and anisotropy of the collagen fibers were extracted for the superficial layer resulting in a 3D map of the collagen fiber fabric. In the middle layer, image analysis using objective thresholding techniques was employed to extract the volume fraction occupied by extracellular matrix, the rest being occupied by the lacunae and residing chondrocytes. To implement these imaging data in a computational setting, we propose a new, 3D large strain constitutive model for articular cartilage, focused on the essential load-bearing morphology: an inhomogeneous, visco-poroelastic solid matrix reinforced by an anisotropic, viscoelastic dispersed fiber fabric which is saturated by an incompressible fluid residing in strain-dependent pores of the collagen-proteoglycan solid matrix. High-fidelity models, combining advanced imaging and computational biomechanics, will allow us to consider complex problems in structure-function relationships and provide insight to microphysical (mechanobiological) cellular stimuli. © 2010 International Federation for Medical and Biological Engineering.

This study formulates a theory for multigenerational interstitial growth of biological tissues whereby each generation has a distinct reference configuration determined at the time of its deposition. In this model, the solid matrix of a growing tissue consists of a multiplicity of intermingled porous permeable bodies, each of which represents a generation, all of which are constrained to move together in the current configuration. Each generation's reference configuration has a one-to-one mapping with the master reference configuration, which is typically that of the first generation. This mapping is postulated based on a constitutive assumption with regard to that generations' state of stress at the time of its deposition. For example, the newly deposited generation may be assumed to be in a stress-free state, even though the underlying tissue is in a loaded configuration. The mass content of each generation may vary over time as a result of growth or degradation, thereby altering the material properties of the tissue. A finite element implementation of this framework is used to provide several illustrative examples, including interstitial growth by cell division followed by matrix turnover. © 2010 Springer-Verlag.

A continuum triphase model (i.e., a solid filled with fluid containing nutrients) based on the theory of porous media (TPM) is proposed for the phenomenological description of growth and remodeling phenomena in isotropic and transversely isotropic biological tissues. In this study, particular attention is paid on the description of the mass exchange during the stress-strain- and/or nutrient-driven phase transition of the nutrient phase to the solid phase. In order to define thermodynamically consistent constitutive relations, the entropy inequality of the mixture is evaluated in analogy to Coleman and Noll (Arch Ration Mech Anal 13:167-178, 1963). Thereby, the choose of independent process variables is motivated by the fact that the resulting phenomenological description derives both a physical interpretability and a comprehensive description of the coupled processes. Based on the developed thermodynamical restrictions constitutive relations for stress, mass supply and permeability are proposed. The resulting system of equation is implemented into a mixed finite element scheme. Thus, we obtain a coupled calculation concept to determine the solid motion, inner pressure as well as the solid, fluid and nutrient volume fractions. © 2009 Springer-Verlag.