peter wriggers thomas lenarz editors biomedical technology

Lecture Notes in Applied and Computational Mechanics 84

Peter WriggersThomas Lenarz Editors

Biomedical TechnologyModeling, Experiments and Simulation

Lecture Notes in Applied and ComputationalMechanics

Volume 84

Series editors

Peter Wriggers, Leibniz Universität Hannover, Hannover, Germanye-mail: [email protected] Eberhard, University of Stuttgart, Stuttgart, Germanye-mail: [email protected]

About this Series

This series aims to report new developments in applied and computationalmechanics—quickly, informally and at a high level. This includes the fields of fluid,solid and structural mechanics, dynamics and control, and related disciplines. Theapplied methods can be of analytical, numerical and computational nature.

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Peter Wriggers ⋅ Thomas LenarzEditors

Biomedical TechnologyModeling, Experiments and Simulation

123

EditorsPeter WriggersInstitute of Continuum MechanicsLeibniz Universität HannoverHannoverGermany

Thomas LenarzHals-Nasen-OhrenklinikMedical School HannoverHannoverGermany

ISSN 1613-7736 ISSN 1860-0816 (electronic)Lecture Notes in Applied and Computational MechanicsISBN 978-3-319-59547-4 ISBN 978-3-319-59548-1 (eBook)DOI 10.1007/978-3-319-59548-1

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Preface

A major objective of medical research is related to the development of improvedmedication and implants. Due to the individual anatomy of each human being, theresearch direction points more and more towards a patient specific medicine. This inturn requires a better understanding of biological systems and of the performance ofimplants in humans. In engineering disciplines the application of virtual processdesign has originated many important innovations. Virtual modelling helpsunderstand and control processes. Furthermore, virtual testing is fast and flexible.Hence, many new products can be efficiently designed and verified by numericalapproaches.

In recent years these concepts were successfully applied in the field ofbiomedical technology. Based on the tremendous advances in medical imaging,modern CAD systems, high-performance computing and new experimental testdevices, engineering can provide a refinement of implant design and lead to saferproducts. Computational tools and methods can be applied to predicting the per-formance of medical devices in virtual patients. Physical and animal testing pro-cedures can be reduced by use of virtual prototyping of medical devices. Theseadvancements enhance medical decision processes in many areas of clinicalmedicine.

In this book, scientists from different areas of medicine, engineering and naturalsciences are contributing to the above research areas and ideas. The book provides agood overview of new mathematical models and computational simulations as wellas new experimental tests in the field of biomedical technology.

In the first part of the book the virtual environment is used in studying biologicalsystems at different scales and under multiphysics conditions. Modelling schemesare applied to human brain tissue, blood perfusion and metabolism in the livinghuman, investigation on the effect of mutations on the spectrin molecules in redblood cells and numerical strategies to model transdermal drug delivery systems.

The second part is devoted to modelling and computational approaches in thefield of cardiovascular medicine. The contributions start with an overview of cur-rent methods and challenges in the field of vascular haemodynamics. This is fol-lowed by new methods to accurately predict heart flow with contact between

v

the leaflets, estimation of a suitable zero stress state in arterial fluid structureinteraction, solution strategies for stable partitioned fluid-structure interactionsimulations, methods for stable large eddy simulation of turbulence in cardiovas-cular flow, a demonstration of the importance using non-Newtonian models inspecific hemodynamic cases, a multiscale modelling of artificial textile reinforcedheart valves, and new strategies to reduce the computational cost in fluid-structureinteraction modelling of haemodynamics. The part closes with a method to com-putationally assess the rupture risk of abdominal aortic aneurysm.

A parameter study of biofilm growth based on experimental observations andnumerical test as well as a multiscale modelling approach to dental enamel arecontributions that face current challenges in dentistry.

The part related to orthopaedics starts with an overview of challenges in total hiparthroplasty and is followed by a concept for a personalized orthopaedic traumasurgery based on computational simulations.

The last part addresses otology and shows that an off-the-shelf pressure mea-surement system can be successfully used for intrachochlear sound pressuremeasurements. The second contribution is a user-specific method for the auditorynerve activity, leading to a better understanding of the electrode nerve interface inthe case of cochlear implants.

All contributions highlight the state-of-the-art in biotechnology research andthus provide an extensive overview of this subject.

Hannover, Germany Peter WriggersJanuary 2017 Thomas Lenarz

vi Preface

Contents

Part I Biological Systems

Multiscale Aspects in the Multiphasic Modelling of Human BrainTissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Wolfgang Ehlers and Arndt Wagner

Simulation of Steatosis Zonation in Liver Lobule—AContinuummechanical Bi-Scale, Tri-Phasic, Multi-ComponentApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Tim Ricken, Navina Waschinsky and Daniel Werner

Nano-Mechanical Tensile Behavior of the SPTA1 Gene in the Presenceof Hereditary Hemolytic Anemia-Related Point Mutations . . . . . . . . . . . 35Melis Hunt

The Choice of a Performance Indicator of Release in TransdermalDrug Delivery Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Giuseppe Pontrelli and Laurent Simon

Part II Cardiovascular Medicine

Multiscale Multiphysic Approaches in Vascular Hemodynamics . . . . . . . 67Michael Neidlin, Tim A.S. Kaufmann, Ulrich Steinseiferand Thomas Schmitz-Rode

Heart Valve Flow Computation with the Space–Time Slip InterfaceTopology Change (ST-SI-TC) Method and Isogeometric Analysis(IGA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Kenji Takizawa, Tayfun E. Tezduyar, Takuya Teraharaand Takafumi Sasaki

Estimation of Element-Based Zero-Stress State in Arterial FSIComputations with Isogeometric Wall Discretization . . . . . . . . . . . . . . . . 101Kenji Takizawa, Tayfun E. Tezduyar and Takafumi Sasaki

vii

Fluid-Structure Interaction Modeling in 3D Cerebral Arteriesand Aneurysms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Yue Yu

Large-Eddy Simulation of Turbulence in Cardiovascular Flows . . . . . . . 147F. Nicoud, C. Chnafa, J. Siguenza, V. Zmijanovic and S. Mendez

Computational Comparison Between Newtonian and Non-NewtonianBlood Rheologies in Stenotic Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Bruno Guerciotti and Christian Vergara

Artificial Textile Reinforced Tubular Aortic Heart Valves—Multi-scale Modelling and Experimental Validation . . . . . . . . . . . . . . . . . . . . . . 185Deepanshu Sodhani, R. Varun Raj, Jaan Simon, Stefanie Reese,Ricardo Moreira, Valentine Gesché, Stefan Jockenhoevel, Petra Mela,Bertram Stier and Scott E. Stapleton

Preliminary Monolithic Fluid Structure Interaction Model forVentricle Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217D. Cerroni, D. Giommi, S. Manservisi and F. Mengini

The Biomechanical Rupture Risk Assessment of Abdominal AorticAneurysms—Method and Clinical Relevance . . . . . . . . . . . . . . . . . . . . . . 233T. Christian Gasser

Part III Dentistry

A Deeper Insight of a Multi-dimensional Continuum BiofilmGrowth Model: Experimental Observation and ParameterStudies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257Dianlei Feng, Henryke Rath, Insa Neuweiler, Nico Stumpp,Udo Nackenhorst and Meike Stiesch

Multiscale Experimental and Computational Investigation of Nature’sDesign Principle of Hierarchies in Dental Enamel . . . . . . . . . . . . . . . . . . 273Songyun Ma, Ingo Scheider, Ezgi D. Yilmaz, Gerold A. Schneiderand Swantje Bargmann

Part IV Orthopaedics

Challenges in Total Hip Arthroplasty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295Gabriela von Lewinski and Thilo Floerkemeier

Personalized Orthopedic Trauma Surgery by Applied ClinicalMechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313M. Roland, T. Tjardes, T. Dahmen, P. Slusallek, B. Bouillonand S. Diebels

viii Contents

Part V Otology

Measurement of Intracochlear Pressure Differences in HumanTemporal Bones Using an Off-the-Shelf Pressure Sensor . . . . . . . . . . . . . 335Martin Grossöhmichen, Rolf Salcher, Thomas Lenarz and Hannes Maier

Development of a Parametric Model of the Electrically StimulatedAuditory Nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349Waldo Nogueira and Go Ashida

Contents ix

Part IBiological Systems

Multiscale Aspects in the MultiphasicModelling of Human Brain Tissue

Wolfgang Ehlers and Arndt Wagner

Abstract The enormous microscopic complexity of the multicomponent brain-

tissue aggregate motivates the application of the well-known Theory of Porous

Media (TPM). Basically, a quaternary TPM-model is applied to brain tissue, cf.

Ehlers and Wagner, Comput Method Biomech Biomed Eng 18:861–879 (2015), [1].

Besides the model’s broad range of application, such as the simulation of brain-

tumour treatment, we focus in this article on a specific anatomical property of the

brain-tissue aggregate. Namely, its separated pore space which is concurrently per-

fused by two pore liquids. These are the blood in the blood-vessel system and the

interstitial fluid in the interstitial-fluid space. In this regard, the constitutive formu-

lation of evolving liquid saturations under certain loading conditions needs to be

found. In order to microscopically underlay and motivate such a macroscopic consti-

tutive relation within a thermodynamically consistent TPM approach, a microscopic

study of the interaction between the pore compartments is performed and discussed

in terms of scale-bridging aspects.

1 Introduction and Motivation

For various medical purposes regarding brain tissues, it could be of immense benefit

to use numerical simulations. The aim to simulate real applications on the length

scale of centimetres to metres requests a macroscopic modelling, for example, in

case of the simulation of a therapeutic infusion process within brain tissue during

brain-tumour treatment as it is shown in Fig. 1. Obviously, processes occurring on

the complex microscale have a certain impact on the overall behaviour and need

to be included. One example, on which we will focus in the present article, is the

stiffness of the blood vessel system in comparison to the surrounding tissue aggre-

gate (under physiological pressure conditions). This issue is still under controversial

W. Ehlers (✉) ⋅ A. Wagner

Institute of Applied Mechanics (CE), University of Stuttgart,

Pfaffenwaldring 7, 70569 Stuttgart, Germany

e-mail: [email protected]

A. Wagner


© Springer International Publishing AG 2018

P. Wriggers and T. Lenarz (eds.), Biomedical Technology, Lecture Notes in Applied

and Computational Mechanics 84, DOI 10.1007/978-3-319-59548-1_1

3

4 W. Ehlers and A. Wagner

Fig. 1 Simulation of a therapeutic infusion process within brain tissue during brain-tumour treat-

ment. The supplied domain of the brain hemisphere is displayed in green and the second-order

permeability tensors, governing the patient-specific anisotropic perfusion behaviour, are visualised

via ellipsoids in light blue on an exemplary cross section, cf. [3] for a comprehensive overview

discussion. Therefore, the purpose of this contribution is to perform within the

commercial finite-element (FE) software tool Abaqus®

(Dassault Systèmes, Vélizy-

Villacoublay, France, cf. http://www.3ds.com) microscopic and locally single-phasic

computations of a representative elementary volume of the brain-tissue aggregate.

This allows for the assignment of microscopic material parameters for the individ-

ual components. In this regard, elastically deformable blood vessels, separating the

pore compartments, are assumed. Thus, the liquid with the higher pressure displaces

the liquid with the lower pressure. The subsequent geometrical evaluation of certain

adjusted pressure situations (applied as boundary conditions) within such a micro-

scopic model allow for a meaningful postulation of a macroscopic constitutive equa-

tion. Note in passing that the presented study of scale-bridging aspects differs from

a complete two-scale approach using the FE2-Method (where both the macroscopic

as well as the underlying microscopic structure are analysed with the finite-element

method, cf. [2] in terms of the TPM). The scale-bridging approach proposed here

rather proceeds from an one-time analysis of a specific process on the microstruc-

ture and a subsequent transfer to the macroscopic model obtaining a microscopically

underlaid constitutive equation.

http://www.3ds.com

Multiscale Aspects in the Multiphasic Modelling of Human Brain Tissue 5

tissue cell

cell body interstitial fluid space

intracellular spaceintravascular space

extravascular space

blood plasma

blood vessel(capillary)

capillary membrane

blood cellextracellular space

Fig. 2 Exemplary sketch of microscopic fluid spaces in brain tissues, according to [3]

2 Anatomic Elements of Human Brain Tissue

Amongst other things, brain tissue includes the components interstitial fluid and

blood, which are mobile in two basically separated pore compartments. In this

regard, the interstitial fluid is situated within the interstitial fluid space, whereas

the intravascular space contains the blood. Moreover, further tissue spaces are com-

monly distinguished, such as the intra- and extracellular space or the extravascular

space, cf. Fig. 2 for a qualitative sketch. In general, the capillary membrane (wall) is

highly selective and governs the passing of molecules from the intravascular to the

extravascular space. Thus, the so-called blood-brain barrier separates the individual

pore compartments.

The interstitial fluid typically occupies a volume fraction of 20% of the brain’s

bulk volume, cf. [4]. However, a variation between 15% and 30% is normal and even

may fall to 5% during global ischaemia. Typical values of the blood volume fraction

are in the range of approximately 3% for healthy tissues, cf. [4] based on, e.g., [5]. In

contrast, in tumour-affected domains, the blood amount may vary between 1% and

20%.

3 TPMModel of the Overall Brain-Tissue Aggregate

To provide a framework for the considerations in this paper, the basis of the macro-

scopic thermodynamically consistent drug-infusion model is briefly described in this

section. Using the TPM, a volumetrical averaging procedure (smearing) of the under-

lying microscopic structure over a representative elementary volume (REV) leads

to an idealised macroscopic model of superimposed and mutually interacting con-

stituents. For a detailed discussion of the model, the interested reader is referred to

the underlying articles of [1, 6, 7] and citations therein. Basically, the quaternary

TPM model includes four constituents. In particular, three immiscible constituents

are given by the solid skeleton 𝜑S, the blood 𝜑

Band the overall interstitial fluid

𝜑I, where 𝜑

Icontains a dissolved miscible therapeutic solute 𝜑

D. To account for the

local composition of the aggregate, scalar structure parameters, the volume fractions


n𝛼 = dv𝛼dv

, (1)

are introduced defined as the local ratios of the partial volume elements dv𝛼 with

respect to the volume element dv of the overall aggregate. Assuming fully saturated

conditions (no vacant space within the domain), the well-known saturation condition∑𝛼n𝛼 = nS + nB + nI = 1 is obtained. As was already mentioned, the pore liquids

are mobile in their individual pore compartments. Therefore, the volumetric amount

(saturation) of the blood

sB = 1 − sI = nB1 − nS

(2)

is specified in relation to the overall pore space (porosity). The set of model equations

proceeds from the specific balance relations for multiphasic materials, cf., e.g., [8] or

[9], where preliminary assumptions are initially included, i. e. isothermal conditions,

materially incompressible constituents, negligible gravitational forces and accelera-

tion terms. Therefore, the governing balance relations are given by the momentum

balance (3)1 of the overall aggregate 𝜑, the volume balance (3)2 of 𝜑B, the volume

balance (3)3 of 𝜑I

and the concentration balance (3)4 of 𝜑D

, viz.:

div𝐓 = 𝟎 ,(nB)′S + div(nB𝐰B) + nB div(𝐮S)′S = 0 ,(nI)′S + div(nI𝐰I) + nI div(𝐮S)′S = 0 ,nI(cDm)

′S + cDm div(𝐮S)′S + div(nIcDm 𝐰D) + cDm div(nB𝐰B) = 0 .

(3)

Therein, the overall Cauchy stress 𝐓 in (3)1 can be derived for, e.g., finite and

anisotropic elastic deformation processes of the solid in combination with the overall

pore pressure as was described in [1]. The notion ( ⋅ )′S indicates the material time

derivative with respect to the solid motion. In terms of the volume fractions and

their temporal changes

nS = nS0S (det 𝐅S)−1 and (nS)′S = − nS div (𝐮S)′S ,

nB = sB (1 − nS) and (nB)′S = (sB)′S (1 − nS) − sB(nS)′S ,nI = 1 − nS − nB and (nI)′S = − (nS)′S − (nB)′S ,

(4)

it turns out that a constitutive relation for the blood’s saturation sB is required, since

the saturation condition is not sufficient to determine all volume fractions. Note in

passing that the temporal change (sB)′S is based on the chosen constitutive ansatz for

sB, cf. Sect. 4.2. In Eq. (4), the quantity nS0S denotes the initial solid volume fraction

and det 𝐅S is the determinant of the solid’s deformation gradient 𝐅S = 𝐈 + GradS𝐮S.

Therein, the operator GradS denotes the material gradient of 𝜑S

with respect to its

reference position. Finally, the set of equations in (3) contains the seepage velocities

𝐰B, 𝐰I and 𝐰D which are described via


nB 𝐰B = − 𝐊SB

𝜇BR ( grad pBR +pdif .sB

grad sB ) ,

nI 𝐰I = − 𝐊SI

𝜇IR grad pIR ,

nIcDm 𝐰D = −𝐃D grad cDm + nIcDm 𝐰I .

(5)

The perfusion of the solid skeleton by the mobile but separated pore liquids is con-

sidered by the permeability tensors 𝐊SBand 𝐊SI

, whereas 𝐃Dis the diffusivity of the

therapeutic agent. Moreover, 𝜇BR

and 𝜇IR

are the effective shear viscosities of the

liquids. The pressure difference pdif . = pBR − pIR indicates the difference between

the individual pore-liquid pressures. Finally, the TPM model is solved for the pri-

mary variables solid displacement 𝐮S, effective pore-liquid pressures pBR, pIR and

therapeutic concentration cDm after inserting the constitutive relations into the set

of equations given in (3). With regard to the corresponding numerical solution of

the coupled system of partial differential equations using the research code Pandas

(Porous media Adaptive Nonlinear finite element solver based on Differential Alge-

braic Systems, cf. http://www.get-pandas.com), the Bubnov-Galerkin mixed FEM is

applied as shown in [3] for several numerical examples.

4 Microscopically Underlaid Macroscopic ConstitutiveRelation

As was already mentioned, the TPM model includes two pore liquids. Therefore, a

constitutive equation to describe the division for the volume fractions is required,

since there is still no general agreement concerning the stiffness of the blood-vessel

system in comparison to the surrounding tissue aggregate. Therefore, the purpose

of this section is to perform a microscopic and locally singlephasic computation of

a representative elementary volume of the brain tissue aggregate. This is realised

using Abaqus and consequently allows for the assignment of microscopic material

parameters for the individual components. The evaluation of adjusted pressure sit-

uations within such a microscopic model allows for a meaningful foundation of the

developed macroscopic constitutive equation.

4.1 Microscopic Model Settings

In terms of a microscopic description, a tube-like geometry is considered for the

description of a single vascular vessel, cf. Fig. 3. In order to apply a pressure dif-

ference between the inside and the outside of a blood vessel, pdif . = pBR − pIR is

subjected at the inside of the blood-vessel wall, cf. Fig. 3a. For the sake of simplic-

ity, a linear elastic material behaviour is chosen. The corresponding parameters are

collected in Table 1. Due to the elastically deformable blood vessels, the liquid with


vessel

Rin

Rout

pdif.

1

2

(a) (b)

Fig. 3 Boundary conditions on a single blood vessel (a) and meshed geometry (b) with an exem-

plary point at the inside (1) and the outside (2) of the vessel, respectively

Table 1 Collection of parameters for a single blood vessel and a REV

Quantity Value Reference/Remark

Blood vessel inner radius Rin 1.6 × 10−4 m [10]

Blood vessel outer radius Rout 2.6 × 10−4 m [10]

Edge length lREV of a single

REV

7.3 × 10−4 m Assumption to satisfy the initial

(physiologic) condition sB = 0.2Poisson’s ratio 𝜇 0.48 For the cases (i) − (iii), according to [11]

Young’s modulus E 2792N/m2

For the cases (i) and (ii), assumption of

soft vessel walls

Young’s modulus E 1.2 × 105 N/m2

For case (iii), according to [12]

the higher pressure displaces the liquid with the lower pressure. In particular, three

different cases are studied:

∙ case (i): “soft” vessel walls under unconstrained conditions,

∙ case (ii): “soft” vessel walls under constrained conditions (cf. Sect. 4.1.1) and

∙ case (iii): “stiff” vessel walls under unconstrained conditions.

In these microscopic computations, the pressure difference is applied as a boundary

condition and the arising volume changes are geometrically evaluated in a subse-

quent step. To bridge from the microscopic to the macroscopic scale, many single

REV are arranged in a regular manner, cf. Fig. 4. In this regard, the edge length of

a single REV is chosen such that the initial saturation under zero pressure differ-

ence constitutes to sB = 0.2. Physically, a variation of the volume fractions within

a single REV requires an inflow (of the blood for a positive pressure difference) and

an outflow (of the interstitial fluid for a positive pressure difference) of the liquids.

This is provided by free-flow boundary conditions for both liquids in longitudinal

direction of the chosen blood-vessel orientation.


Fig. 4 Exemplary sketch of the macroscopic overall aggregate composed of multiple REVs

4.1.1 Geometrical Evaluation

Within Abaqus, the displacement of each node of the FE grid is obtained by the

solution of the considered initial-boundary-value problems. Note in passing that an

analytical solution for the widening of a linear-elastic cylindrical shell under con-

stant internal pressure does exist, cf., e.g., [13], and corresponds to an unconstrained

microscopic computation. In a geometric evaluation, the deformation of the solid

vessel wall can be captured by the displacements of the nodal points under load-

ing conditions. However, the focus of this contribution lies on the evaluation of the

liquid’s saturations. In this regard, the geometrical evaluation of the arising volume

fractions in the unconstrained cases (i) and (iii), cf. Fig. 5a, is trivial, since the dis-

placements of the nodes 1 (located at the inside of the vessel wall) and 2 (cor-

responding point in radial direction at the outside, cf. Fig. 3) are known from the

computation and allow for the determination using basic geometrical relations. In

addition, the widening of a single vessel is at a certain state somehow constrained by

the surrounding vessels, which themselves wide in a similar manner, as it is sketched

in Fig. 5b. This is approximated by a linear pathway in the contact domain and a cir-

cular shape in the edges. For the sake of simplicity, a constant thickness t of the

blood vessel is assumed. Obviously, this represents a simplification, since (at least

in the case of mutual contact) a deformation of the vessel wall may occur. However,

this assumption allows a relatively simple evaluation of the arising volume fractions.

Then, the volume fractions can be computed for the constrained case (ii), viz.:

VB =[𝜋 (d − t)2 + 4 (d − t) (lREV − 2 d) + (lREV − 2 d)2

]hREV ,

VI = (4 d2 − 𝜋 d2) hREV .(6)

Therein, hREV is the height of the REV required to compute a volume. However,

hREV vanishes within the determination of the liquid’s volume fractions via

nB = VB

VREVand nI = VI

VREV(7)


(a) (b)

Fig. 5 a Unconstrained vessel widening within the REV and b constrained vessel deformation

caused by interacting vessels

and, thus, has no influence. In conclusion, this allows for the evaluation of the liquid’s

saturations under applied pressure conditions. In particular, this is carried out for

typical pressure states and collected in Sect. 4.3.

4.2 Macroscopic Constitutive Relation

The aim to simulate bio-technical/engineering applications of human brain tissues

requires a thermodynamically consistent setting of the blood-saturation function in

the context of the TPM model. This is somehow comparable to the procedure within

unsaturated soil mechanics. Therein, the capillary pressure, defined as the difference

between the pressures of the non-wetting and the wetting fluids in a common pore

space is used to evaluate the saturations via capillary-pressure-saturation conditions,

cf., e.g., [14]. However, in the specific case of human brain tissue, the blood and

the interstitial fluid are not situated in the same pore compartment such that these

relations cannot be applied. Instead, further considerations are made here in order to

specify the blood saturation, while bearing the microscopic considerations in mind.

In general, the macroscopic constitutive equation should provide the flexibility to

capture several circumstances. For the case of sufficiently soft elastic arterial walls,

a mutually volumetrical interaction is induced by an upcoming pressure difference.

Therefore, a constitutively chosen ansatz for the Helmholtz free energy 𝜓B

of the

blood constituent is postulated according to [1], viz.:

𝜓B(sB) = �̃�

B

𝜌BR

((𝛽B + 1) ln (sB) + 1

sB− ln (1 − sB)

)+ �̃�

B0 . (8)

Therein, �̃�B

and 𝛽B

denote material parameters, which allow via �̃�B

for the adaption

of the pore-pressure difference to typical pressure values as they exist in the skull

and for the initial blood saturation via 𝛽B. Furthermore, �̃�

B0 denotes the constant

reference potential (standard state potential). This approach satisfies the thermody-

namical restrictions and consequently yields a relation for the pressure difference


Fig. 6 Thermodynamically

consistent constitutive

relation for the

blood-saturation function sB

pdif .(sB) = sB 𝜌BR𝜕𝜓

B

𝜕sB= �̃�

B(

1 − 2 sBsB (sB − 1)

+ 𝛽B)

,

where 𝜕𝜓B

𝜕sB= �̃�

B

𝜌BR

(𝛽B

sB− 1

sB (sB − 1)− 1

(sB)2

)

.

(9)

The inversion of (9) finally leads to the blood saturation function

sB(pdif .) = 1

2( pdif .

�̃�B − 𝛽B)

(( pdif .

�̃�B − 𝛽B − 2

)+√

4 +(pdif .�̃�B − 𝛽

B)2)

.(10)

It should be noted that Eq. (10) is found by rational investigations of a meaningful

inversion of (9), which is a second-order function in sB. The derived relation for the

blood saturation (10) allows for a proper determination of the volume fractions using

the relations in (4). To give an example, an equation satisfying typical initial volume

fractions of the brain tissue as well as typical pressure values as they exist in the skull

is adapted by 𝛽B = 3.75 and �̃�

B = 200.0 N/m2, cf. Fig. 6. Physically, this leads to a

replacement of the interstitial fluid if the pressure difference pdif . is positive. Note in

passing that a constant value of nB0S = 0.05 yields an initial value of sB0S = 0.2 and,

as a result, pdif . = 0.

4.3 Results and Discussion

In this section, the evaluated results of the microscopic computation are compared

with the macroscopic constitutive approach. For the pressure in the vascular sys-

tem (cerebral blood pressure depending on its hierarchical position), the range in

the capillary bed varies between 10 mm of mercury (mmHg) (corresponds to 1333

N/m2) at the venous end and 30 mmHg (4000 N/m

2) at the arterial end. Whereas

the interstitial-fluid pressure (tissue pressure) has a typical value of 6 mmHg (800


Fig. 7 Comparison of different approaches for the blood-saturation function. Evaluated pres-

sure states from microscopic computations are displayed with diamonds which are connected with

dashed lines

N/m2). To capture such typical conditions, the adjusting blood saturation is evalu-

ated for pressure differences in the range of −500 N/m2 ≤ pdif . ≤ 2000 N/m

2. In

particular, the pressure difference is applied in steps of 500 N/m2

with a subsequent

evaluation of the arising volume fractions according to Sect. 4.1.1. The evaluated

pressure states from the microscopic computations are then displayed with diamonds

and interpolated with dashed lines, cf. Fig. 7. The evaluation of case (i) (soft ves-

sel, unconstrained) yields to reasonable results in the pressure range of −500 N/m2

≤ pdif . ≤ 1500 N/m2. However, for a further increase of the pressure difference, non-

physical results (i. e. a blood saturation greater than one) occur. This is caused by the

possibility of an unconstrained widening of the blood vessel. In contrast, in the case

(ii) (soft vessel, constrained) it is nicely identified that the microstructural compu-

tations yield comparable results in relation to the constitutively chosen pressure-

saturation relation (Sect. 4.2) in the domain of nearly fully blood saturated condi-

tions. Again, the constitutive formulation of the blood saturation function is not arbi-

trary but has to fulfil the thermodynamic restrictions discussed in Sect. 4.2. There-

fore, the pathway vary slightly from the graph of case (ii) obtained by microscopic

computations. However, this allows for a scale-bridging in terms of microscopic and

macroscopic material parameters. In the specific case (ii), the microscopic material

parameters, Young’s modulus E = 2792 N/m2

and Poisson’s ratio 𝜇 = 0.48, of the

blood vessel wall correspond to the macroscopic material parameters �̃�B = 200.0

N/m2

and 𝛽B = 3.75 of the constitutive function. Assuming a higher stiffness of

the vessel (corresponding to somehow realistic experimental values of the human

intracranial artery) in case (iii) yields within the considered pressure regime a nearly

constant blood saturation. This would justify to assume a constant blood-volume

fraction within macroscopic simulations.

In conclusion, a constitutive approach for the relation between the pressure differ-

ence pdif . of the liquids and the blood saturation sB was presented which satisfies both


the thermodynamic consistence as well as physical conditions within human brain

tissue. Moreover, this macroscopic saturation condition was motivated and studied

by a microscopic computation of a representative microstructure using Abaqus and

a subsequent geometrical evaluation of the arising saturations under applied liquid

pressure conditions. With reliable experimental results, this would provide the pos-

sibility to identify the macroscopic material parameter of the constitutive function

based on the microscopic material parameter in terms of scale-bridging aspects.

Acknowledgements The authors would like to thank the German Research Foundation (DFG) for

the financial support of the project within the Cluster of Excellence in Simulation Technology (EXC

310/2) at the University of Stuttgart.

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(Springer, Netherlands, Dordrecht, 2013), pp. 263–276

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(2009)

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(1968)

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of human intracranial and extracranial arteries. J. Biomech. 13, 175–184 (1980)

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704–713 (2003)

Simulation of Steatosis Zonation in LiverLobule—A Continuummechanical Bi-Scale,Tri-Phasic, Multi-Component Approach

Tim Ricken, Navina Waschinsky and Daniel Werner

Abstract The human liver is an important metabolic organ which regulates metabo-

lism of the body in a complex time depending and non-linear coupled function-

perfusion-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 west-

ern 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 compu-

tational 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 deposi-

tion 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.

1 Introduction

For a realistic simulation of the liver behavior it is important to take into account

a detailed description of the biological processes. We present a bi-scale, tri-phasic,

continuum multi-component model simulating blood perfusion and metabolism in

T. Ricken (✉) ⋅ N. Waschinsky ⋅ D. Werner

TU Dortmund, August-Schmidt-Straße 6, Dortmund, Germany


N. Waschinsky


D. Werner


© Springer International Publishing AG 2018

P. Wriggers and T. Lenarz (eds.), Biomedical Technology, Lecture Notes in Applied

and Computational Mechanics 84, DOI 10.1007/978-3-319-59548-1_2

15

16 T. Ricken et al.

macroscopic structure mesoscopic structure microscopic structure

Fig. 1 Bi-scale approach; macroscopic structure: millimeter spatial resolution to observe the vas-

cular system from the in- and outflow of the liver to smaller and smaller branches of the vessels;

mesoscopic structure: micrometer spatial resolution to observe the perfusion through the sinusoids;

microscopic structure: biochemical description of the metabolism in the liver cells

the human liver based on the two-phasic approach, see Ricken et al. [13]. To get a

better understanding of the liver we start with a short physiological outline. For more

informations, see Hubscher et al. [8].

The liver is the main metabolic organ in the human body. Roughly, the structure

of the organ is segmented into four hepatic lobes of unequal size within the range

of centimeters; the main parts are the right lobe (lobus hepatic dexter) and left lobe

(lobus hepatis sinister). Each lobe consists of liver lobules which have the size of

about a few millimeters. They are hexagonally structured parts of the liver tissue, see

Fig. 1. Microscopic sized hepatocytes, the liver cells, are arranged in radial columns

embedded in the liver lobules. The hepatic vascular system is important for the blood

supply of the hepatocytes. For this, the vascular system consists of branches which

supply the organ from the macroscopic to microscopic anatomy of the liver. On the

macroscopic level branches of hepatic artery (oxygen rich) and portal vein (nutri-

ent rich) run together and supply the segments of the organ. Intrahepatic bile ducts

proceed together with the blood vessels and support the liver during detoxification

processes as the bile fluid is excreted with decomposed products. The microcircula-

tion takes place in the liver lobule. In each corner of the liver lobule a portal triad is

located which is composed of the three branches of hepatic artery, portal vein and

a bile duct in their smallest subdivision. The microcirculation is described by blood

vessels connecting the portal triad with the central vein which are called sinusoids.

The central vein drains the blood leading to a directed blood flow from the portal

triad to the central vein, see Fig. 1.

Under consideration of the physiological background it is unavoidable to classify

the liver in different scales to focus on scale depending processes.

We focus on a bi-scale approach; on the mesoscopic level we describe the blood

perfusion through the liver lobule and on the microscopic level the metabolism in

the cells.

The mesoscopic model is based on an ansatz for transverse isotropic permeabil-

ity relation describing the perfusion given in Ricken et al. [12]. Due to the complex

Simulation of Steatosis Zonation in Liver Lobule . . . 17

insu

lin

increase

decreaseffaext

lacext

glcext

VHGU VGS

VLP

VGLY

FFA TG

additional hypothesis:

fat metabolism

reduced metabolic model:

glucose metabolism

Fig. 2 Metabolism: the biochemical reactions are described with the metabolites of external glu-

cose (glcext ), external lactate (lacext ), external FFA (f faext ), internal triglyceride (TG) and internal

glycogen (glyc). External components are included in the fluid phase whereas internal components

are included in the solid tissue of the liver. Triglyceride describes the fat phase in the liver. Reduced

metabolic model: the glucose pathways (VHGU), (VGS) and (VGLY) are modeled by sets of ordi-

nary differential equations. WithVHGU < 0: hepatic glucose production,VHGU > 0: hepatic glucose

utilization, VGLY < 0: gluconeogenesis, VGLY > 0: glycolysis, VGS < 0: glycogenolysis, VGS > 0:

glycogenesis. Additional hypothesis: the fatty acid pathway (VLP) is modeled by assumptions con-

cerning the uptake rate of FFA. With VLP > 0: lipogenesis and VLP < 0: lipolysis

vascular system of sinusoids it is impossible to model the numerous capillaries

with an accurate discrete geometrical description. Therefore we apply a homoge-

nized multiphase approach, based on the theory of porous media (TPM) which is

a convenient tool in order to describe complex, multi-phasic and porous materials;

see de Boer [2, 3] and Ehlers [5]. This approach includes the description of three

main phases, namely the liver tissue, fat tissue and blood vessels. Additionally to the

main phases, the homogenized liver lobule includes miscible components which are

important for the biochemical description of the metabolism. A description of the

mesoscopic model is given in the appendix.

We describe the biochemical processes on the microscopic scale of our model

as the metabolism takes place in the smallest parts of the liver tissue, the cells. The

liver is crucial for metabolites regulation of glucose and free fatty acid (FFA) due

to its ability to store glucose in form of internal glycogen and FFA as internal fat

deposition. The cells can switch between glucose and FFA deposition and utilization

depending on the requirements of the rest of the body. After ingestions, glucose and

FFA concentrations coming from the digestive system increase (hyperglycemia). The

liver cells start to deposit them as internal metabolites glycogen and triglyceride.

Whereas, during fasting times (hypoglycemia) the internal metabolites are served

as a backup system and can be used to increase the low concentration of external

metabolites. We summarize the metabolism in a set of ordinary differential equations

which consider the conditions of hyperglycemia and hypoglycemia and the relations

of the internal and external metabolites. The microscopic model approach is derived

from a detailed pharmacokinetic model of hepatic glucose metabolism, described in

König et al. [10]. We presented a reduced model for glucose metabolism in Ricken

et al. [13] and add a phenomenological approach for the fat metabolism (Fig. 2).

18 T. Ricken et al.

2 Glucose and Fat Metabolism

The liver ensures biosynthetic metabolic pathways of immense complexity. In this

study we focus on the main metabolic pathway of glucose described by a simpli-

fied ordinary differential equation system. Furthermore, we add a phenomenological

approach for the fat metabolism.

Figure 2 summarizes the applied pathways for glucose and fat metabolism. The

main task of the liver is the maintenance of homeostasis for metabolites which

might be necessary for the organism. For this the liver regulates the glucose and fat

level in order to offer enough energy for the brain and muscles. The metabolites are

stored or utilized depending on hormone concentration levels. We distinguish two

conditions: the fed (hyperglycemia) and fasted state (hypoglycemia). During feed-

ing the organism is saturated of glucose and FFA leading to utilization of external

metabolites. Glycolysis and glycogenesis are two possible metabolic pathways of

glucose. The first metabolic pathway ensures the energy supply for cells and repro-

duction of lactate. The second describes energy storage in form of internal glycogen.

Lipogenesis is the fatty acid pathway describing the ester of FFA and glycerol to

triglyceride. The lipid accumulation is reinforced by high carbohydrate diets.

During the fasting state the organism utilizes hepatic glycogen, non-carbohydrate

carbon substrates and lipids. Gluconeogenesis is one pathway for the generation

of glucose. The utilization of certain non-carbohydrate carbon substrates provides

a balanced glucose level. Breaking down hepatic glycogen describes the second

mechanism to maintain glucose, namely glycogenolysis. The utilization of lipids is

part of the lipolysis pathway and the reverse of lipogenesis. Molecules of lipids are

hydrolyzed to glycerol and FFA.

For a realistic depiction of the liver functionalities, we take the above described

main metabolic pathways into account. We use a reduced metabolic model, which

has been derived from a detailed kinetic model of hepatic glucose metabolism, see

König et al. [10]. It fully represents the metabolic behavior of depletion and utiliza-

tion of glycogen. For further information see Ricken et al. [12, 13].

FFA play an important role in essential functions for the organism. They deliver

cells energy and are important constituents for esterifying lipids. For a homeostatic

circulation the organism can store, produce and consume FFA. This process is mainly

regulated in the liver by fat metabolism. Stored triglyceride in adipocytes and dietary

fat are sources, which ensure the availability of FFA in the liver. Beside the external

concentration we need to take important enzymes and insulin into account. Insulin

is a hormone which is distributed depending on the glucose concentration level in

blood. The task of insulin is to regulate glucose and fat metabolism, so the pathways

are coupled. Enzymes are responsible for hydrolysis of proteins, which bind FFA for

circulation in the blood plasma.

The effects of fatty acid metabolism are not part of the kinetic model. In the

following we present the assumptions concerning the FFA pathway. Equally to

the glucose metabolism we calculate the rates of changing concentrations for the

metabolites �̂�𝛼𝛽

contributing in the FFA pathway. We apply the influencing factors


ρ̂FFA

glc

[mm

ol/l/

s]

glucose [mmol/l]

(1) insulin dependency

Fig. 3 Insulin dependency �̂�FFAglc : uptake rate of FFA depending on the concentration of glucose

ρ̂

ρ̂FFA

ffa

[mm

ol/l/

s]

ffa [mmol/l]

FFA

nTG

[1/s

]

volume fraction fat [-]

(2a) FFA dependency: hypoglycemia (3a) fat dependency

Fig. 4 Threshold of external FFA concentration 0, 6 mmol/l which leads to a hypoglycemic con-

dition; 2a FFA dependency �̂�FFAffa : uptake rate of FFA depending on the external FFA concentration.

3a fat dependency �̂�FFAnTG : uptake rate of FFA depending on the volume fraction of fat for a stable

computation

{�̂�FFAglc ; �̂�FFAffa ; �̂�FFAnL ; �̂�FFAnTG

}. The terms are postulated via the differential equation of

growth function with

�̂�FFAglc = exp−log(2)

glc3

glcTP3

�̂�FFAffa = +∕−1 + exp−40∗ ((f fa − 0,6)∗ 2)4

�̂�FFAnL = − 1 ∗ exp−log(2)

(nL)3

0,153 + 1

�̂�FFAnTG = − 1 ∗ exp−log(2)

(nTG)3

0,23 + 1

(1)

We calculate the maximum uptake rate of FFA considering insulin influence

(�̂�FFAglc ) which couples the glucose metabolism to the fat metabolism, see Fig. 3.

Figure 4 summarizes the effects of hypoglycemic conditions which lead to

20 T. Ricken et al.

ρ̂

ρ̂

FFA

ffa

[mm

ol/l/

s]

ffa [mmol/l]

FFA

nL[1

/s]

volume fraction fluid [-]

(2b) FFA dependency: hyperglycemia (3b) fluid dependency

Fig. 5 Threshold of external FFA concentration 0, 6 mmol/l which leads to b hyperglycemic con-

dition; 2b FFA dependency �̂�FFAffa : uptake rate of FFA depending on the external FFA concentration.

3b fluid dependency �̂�FFAnL : uptake rate of FFA depending on the volume fraction of fluid for a stable

computation

production of FFA. The uptake rate (�̂�FFAffa ) is positive as the organism needs more

FFA for energy supply. The hypoglycemic condition results in phase transition from

the fat phase to the fluid phase. To get a stable computation considering the satu-

ration condition we take the influencing factor (�̂�FFAnTG ) into account which steers an

excessive increase of fluid. However, Fig. 5 shows the impact during hyperglycemic

conditions. The uptake rate (�̂�FFAffa ) is negative as the liver stores redundant FFA and

synthesizes lipids. In this case the fat phase gets additional mass and we steer an

excessive increase of fat with (�̂�FFAnL ).

The mass exchange must fulfill the restrictions of the entropy inequality (24); the

exchange can be summarized with

�̄�TG𝛽 = �̄�

TG ∶=[�̂�FFAglc ; �̂�FFAffa ; �̂�FFAnL ; �̂�FFAnTG

]. (2)

3 Numerical Example: Comparison of DifferentAssumptions for the Perfusion Coupledto the Metabolism

Many aspects of the liver are still subject of discussion. One example is the blood

supplying system of the hepatic microcirculation. In this case it is still uncertain how

the blood is emanating from the blood vessels into the liver lobule. One assump-

tion, defined by Rappaport in 1954 [11], assumes an idealized hexagonal lobule

which regards the portal triad as the ‘terminal portal venule’. This case regards

the blood supplying system of the microcirculation initiated at the corners of the

liver lobule, emanating from the portal triads. In a realistic framework, this point of

view is not rational and a different assumption has to be applied. For this case


Fig. 6 Schematic sketch of

the feeding cycle showing

duration and typical patterns

of glycogen in a single

lobule during deposition and

depletion. Figure reproduced

from Babcock and

Cardell [1]

Rappaport described additional portal venules which are derived from the portal vein

and deliver the base of the liver lobule with inflowing blood. So the blood supplying

system of the microcirculation is located at the surface of the liver lobule.

To get a better understanding of the different approaches and the influence on the

metabolism, we simulate both hypotheses for the initial point of the microcirculation:

∙ Boundary-Model A: Idealized Case—Blood is emanating from the portal triad

∙ Boundary-Model B: Realistic Case—Blood is emanating from derived portal

venules

In the numerical example we compute the microperfusion in one liver lobule cou-

pled to the metabolism similar to Ricken et al. [13]. We evaluate the storage of inter-

nal glycogen deposition resulting from glucose metabolism analog to the experi-

ments performed by Babcock and Cardell. In [1], Babcock and Cardell evaluated the

glycogen storage during 24 h in one liver lobule after a feeding period (see Fig. 6).

Furthermore we analyze the results of the fat metabolism which accrues in the hepa-

tocytes (see Figs. 11–13).

We apply external boundary conditions for the glucose and FFA concentration

which are applied in the feeding artery after one meal in 24 h (see Fig. 7). Fur-

thermore we add external conditions for FFA. (cf. Stanhope et al. and Yuen et al.

[14, 16])

The boundary conditions focus on the two different assumptions (see Fig. 8) for

the initial point of the microcirculation (see Fig. 9). Additionally to the time depend-

ing DIRICHLET boundary conditions for the external concentration glucose and

FFA, which depend on the food intake (see Fig. 7), we apply constant DIRICHLET

boundary conditions for lactate at the inflow. As the blood flow is orientated in the

direction of the pressure gradient (transverse isotropic permeability relation, Ricken

et al. [13]) we apply a pressure difference between the inflow and the outflow with

constant values (see Fig. 10).

22 T. Ricken et al.

Fig. 7 External boundary

conditions: glucose profile

corresponding to 2 h food

intake and 22 h fasting based

on 24 h profiles of plasma

glucose (red). Analog profile

for plasma FFA (blue).

Experimental data from

Stanhope et al. [14] and

Yuen et al. [16]. Evaluation

for numerical example after

6, 9 and 18 h

gluc

ose

[mm

ol/l]

time [h]

6 h 9 h 18 h

0.03

0.45

0.02

outflow

(∂pLR; ∂ μαβ ; uuu)outflow

(∂pLR; ∂ μαβ ; uuu)

inflow(∂pLR; ∂ μαβ ; uuu)

inflow(∂pLR; ∂ μαβ ; uuu)

0.5

[mm][mm]

(a) (b)

Fig. 8 Depiction of the applied geometry and boundary conditions. a Boundary-Model A

b Boundary-Model B

3.1 Discussion

The simulation provides the evaluation of metabolites (see Figs. 11, 12, 13) resulting

from glucose and fat metabolism. On the one hand we focused on the

glucose pathway which depends on the external condition of blood glucose concen-

tration. During hyperglycemic conditions the external concentration is high resulting

in storage of glycogen. Whereas, a low glucose concentration encourages glycogen

depletion. On the other hand we evaluated the phenomenological approach of the

fat metabolism which leads to accumulation of fat depending on external blood FFA

concentration. Based on the assumption that fat metabolism is inhibited by the pres-

ence of insulin we can observe the coupling between glucose and fat metabolism.

peter wriggers thomas lenarz editors biomedical technology

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