The simulation of complex biological systems such as the description of blood flow in organs requires a lot of computational power as well as a detailed description of the organ physiology. We present a novel Least-Squares discretization method for the simulation of sinusoidal blood flow in liver lobules using a porous medium approach for the liver tissue. The scaling of the different Least-Squares terms leads to a robust algorithm and the inherent error estimator provides an efficient refinement strategy.Peer Reviewe

Bertrand, Fleurianne

Lambers, Lena

Ricken, Tim

arXiv

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Received: 15 July 2020 Accepted: 9 November 2020DOI: 10.1002/pamm.202000306Least Squares Finite Element Method for Hepatic Sinusoidal Blood FlowFleurianne Bertrand1,∗, Lena Lambers2, and Tim Ricken21 Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany2 Institute of Mechanics, Structural Analysis and Dynamics, Faculty of Aerospace Engineering and Geodesy, University ofStuttgart, 70569 Stuttgart, GermanyThe simulation of complex biological systems such as the description of blood flow in organs requires a lot of computationalpower as well as a detailed description of the organ physiology. We present a novel Least-Squares discretization method forthe simulation of sinusoidal blood flow in liver lobules using a porous medium approach for the liver tissue. The scaling ofthe different Least-Squares terms leads to a robust algorithm and the inherent error estimator provides an efficient refinementstrategy.© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH1 IntroductionModeling the hepatic blood perfusion is challenging [4] but a crucial part for the understanding of the complex microcir-culation processes and the development of treatment of possible liver diseases. Performing large scale simulations of themicrocirculation arises in applications of paramount importance involving the supply of hepatocytes (liver cells) with oxygen,nutrients or pharmaceuticals. However, a fully spatially resolved model of the complex microcirculation of the hepatic lobule,the functional unit of the liver, on a very fine mesh would exceed the limits of current computational power and time. An inher-ent error estimator as in the Least-Squares method then constitutes a major advantage. The continuum biomechanical modelis based on [14] and [15], where a multicomponent, multiscale and multiphase homogenization model in the framework of theTheory of Porous Media (TPM) for blood micro-circulation in hepatic lobules is presented. Taking into account the porousstructure and the hexagonal shape with inflow at the outer edges (portal triad) and an outflow at the center (central vein), cf.Figure 1c, a Least-Squares approach with suitable boundary conditions is validated using a single liver lobule. Least-Squaresfinite element methods are an attractive class of methods for the numerical solution of partial differential equations, as theyproduce symmetric and positive definite discrete systems, possess an inherent error estimator and are relatively easy to applyon nonlinear systems. We refer to [3] for a comprehensive overview.ΩpΩfa)ΩpΩfb) c)Fig. 1: a and b : different views of free-flow domain Ωf and porous domain Ωp and c Different size scales of the human liver namely theorgan scale including the macroscopic vascular, the porous lobule scale with a hexagonal shape and blood flow through the sinusoids andthe cell scale.∗Corresponding author: email fb@math.hu-berlin.deThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distributionin any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.PAMM · Proc. Appl. Math. Mech. 2020;20:1 e202000306. www.gamm-proceedings.com 1 of 4https://doi.org/10.1002/pamm.202000306 © 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH2 of 4 Young Researchers’ Minisymposium 6: Novel discretization methods for phase-field modeling in fracture mechanics2 Modeling Hepatic Lobular Sinusoidal Blood FlowThe multi-scale structure of the liver consists of cells, lobules, segments, and lobes. The liver cells (hepatocytes) are arrangedalong so-called sinusoids. Sinusoids are capillary-like small blood vessels in the liver and form the liver lobules, the functionalunit of the liver. The liver structure is a highly complex structure that can be described as a porous medium similar to a capillarybed as in [11]. There a porous medium is typically a multiphase system where a mixture ϕ = ∪α∈{S,F} with a solid phase ϕSand the pore fluid ϕF is assumed. Each constituent α ∈ {S, F} of the mixture is described by an individual motion functionχα and velocity x′α and has a partial density ρα. The local composition of the mixture is described by partial volumes V α andvolume fractions nα. In our transvascular model we have constant solidity nS and the porosity nF , as well as∑α nα = 1and ρ =∑α ρα. The mass balance of a constituent α reads ∂ρα∂t +∇ · (ραx′α) = ρ̂α = 0 where ρ̂α is a production term thataccounts for interaction with the other constituents. Since here the two constituents are immiscible, ρ̂α vanishes. We furtherassume a rigid body motion as well as a quasi static description with x′′α = 0. The balance of momentum for the constituent αreadsρα(∂x′α∂t+ x′α · ∇x′α)= ∇ ·Tα + ραfα + p̂α + ρ̂αx′α (1)where p̂α accounts for the momentum production by interaction with other constituents. The stress tensor for a general fluidphase is given by TF = TFµ − nF pF I = 2µFDF + λ (DF · I) I − nF pF I with the second Lamé constant λ. The fluidin the porous medium follows the constitutive law and thus is given by p̂F = pF∇nF − nF 2µFK−1 (x′F − x′S) , with thefluid pressure pF , nF denotes the porosity, µF the dynamic viscosity of the interstitial fluid and K the permeability of theporous medium. The mass balance of the interstitial fluid reduces to ∇ ˙(nFx′F ) = 0. Thus, the momentum balance for theinterstitial fluid leads to (nFx′F ) = − KµF ∇pF . The domain representing the porous capillary bed is denoted by Ωp, while thedomain Ωf represents the blood vessel. Moreover, we denote Ω = Ωf ∪ Ωp and Γ = Ωf ∩ Ωp. The blood in the sinusoid isa mixture of several components and its behavior is generally non-Newtonian but can be assumed to behave as a Newtonianfluid. Moreover, as the linearisation of Least-Squares methods is mostly straight-forward, we assume that the shear stresstensor τ is given by τ = 2µD(vf ) such that we obtain−2µ∇ ·D(vf ) +∇pf = 0 ∇ · vf = 0 D(vf ) =12(∇vf +∇Tvf)in Ωf (2)with the hydrostatic pressure pf . It remains to find interface conditions to couple the domain Ωf and Ωp representing aselective permeable membrane and leading to the well-posedness of the problem. Following [11] we describe the vessel wallwithout spatially resolving it : we neglect the slip velocity vf · τ = 0, consider the continuity of the normal velocity such thatvf · n = (nFx′F ) · n and the Beavers-Joseph-Saffman condition−2µD (vf )n · n + pf =µF εKMvf · n + pFwith the thickness of the vessel wall / sinusoid ε and the intrinsic permeability of the capillaries KM . We obtain the followingcoupled system.−2µ∇ ·D (vf ) +∇pf = 0 in Ωf−∇ · vf = 0 in Ωf vf · n = (nFx′F ) · n on ΓµFK(nFx′F ) +∇pF = 0 in Ωp −2µD (vf )n · n + pf =µF εKMvf · n + pF on Γ−∇ · (nFx′F ) = 0 in Ωp vf · τ = 0 on Γ(3)This is in fact a Darcy- Stokes system as reviewed in [5]. Although for the Least-Squares method this is similar to [12], theadditional coupling term µiεKMvf · n has to be taken into account. Moreover, the different scaling require special care.3 Least Squares Finite Element MethodIn order to apply a Least-Squares Method to a first-order system corresponding to the system (3), we introduce the stresstensor σf = 2µD (vf )− pfI. Therefore, the first equation in (3) becomes ∇ · σf = 0. Moreover, with the deviator operatordevσ = σ − (1/2)(trσ)I the definition of the stress tensor becomes devσf = 2µD (vf ) due to the incompressibility ofvf . The incompressibility of vf also implies that the hydrostatic pressure is directly related to the trace of the stress tensor :pF = −(1/2)(trσf ). In fact, both equations are sufficient to ensure the well-posedness of the method, see e.g. [2]. Finally,© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH www.gamm-proceedings.comPAMM · Proc. Appl. Math. Mech. 20:1 (2020) 3 of 4we replace the filter velocity (nFx′F ) by vp. The first-order system therefore reads−∇ · σf = 0 in Ωf devσf = −2µ∇ ·D (vf ) in Ωf−∇ · vf = 0 in Ωf vf · n = vp · n on ΓµFKvp +∇pF = 0 in Ωp −2µD (vf )n · n−12(trσf ) =µF dMKMvf · n + pF on Γ−∇ · vp = 0 in Ωp vf · τ = 0 on Γ(4)We use the standard notation and definition for the Sobolev spaces L2(Ω), Hs(Ω) for s ≥ 0 andH(div; Ω) ={τ ∈ L2(Ω)d : ∇ · τ ∈ L2(Ω)}.We assume that the boundary of the domain Ω is split into a Dirichlet part ΓD and a Neumann part ΓN and write ΩF,Dfor the intersection of ΓD with Ωp where an effective pressure p0f is imposed and Ωf,D for the intersection of ΓD with ΩSwhere an effective pressure p0F is imposed. In the numerical example, p0f and p0F are piecewise constants such that theseconditions can be built directly in the finite element space. Therefore, the Least-Squares methods seeks (σf ,vf ,vF , pF ) ∈H := HΓN (div; Ωf )2 ×H1ΓD (Ωf )2 ×HΓN (div; Ωp)×H1ΓD (Ωp) such that the Least-Square FunctionalF(σf ,vf ,vF , pF ) =‖∇ · σf‖2Ωf + ‖devσf + 2µ∇ ·D (vf ) ‖2Ωf + ‖∇ · vf‖2Ωf + ‖∇ · vp‖2Ωp‖µFKvp +∇pF ‖2Ωp+ ‖2µD (vf )n · n +12(trσf ) +µF dMKMvf · n + pF ‖2Γ + ‖vf · n− vp · n‖2− 12 ,Γ(5)is minimized in H. Note that the condition vf · τ = 0 is imposed directly in the space H1ΓD (Ωf )2. The underlying result forthe success of the numerical method is the following continuity and ellipticity of the Least-Squares functional.Theorem 3.1 The Least-Squares functional defined in (5) is continuous and elliptic in H, equipped with the norm|||(σf ,vf ,vF , pF )|||2 := ‖σf‖2div,Ωf + ‖vf‖21,Ωf + ‖vp‖2div,Ωp + ‖pF ‖21,Ωp . (6)P r o o f. The proof is similar to the one in [2], [12] and [13]. Special care is needed for the term ‖2µD (vf )n · n +12 (trσf ) +µF dMKMvf · n + pF ‖2Γ in the ellipticity proof. In fact, it follows from [13] Lemma 3.2. that1η(‖2µD (vf )n · n +12(trσf ) +µF dMKMvf · n + pF ‖2Γ + ‖vf · n− vp · n‖2− 12 ,Γ)+ η‖vf · n‖0,Γ + η‖pF ‖0,Γ≥− 2〈2µD (vf )n · n +12(trσf ),µF dMKMvf · n〉0,Γ − 2〈pF ,vp · n0,Γ〉+1η|Γ|((1− ρ)(∫Γpds)2− 1− ρρ(∫Γn · (σf · n)ds)2)(7)for any sufficiently small η > 0 and ρ ∈ (0, 1). This can be inserted in the lower bounds for the two functional of the domainsΩp and Ωf and completes the proof.A further major advantage of this result is the inherent error estimator for any conforming discretization, in particular fordiscretization in the spaceHh := (RT k−1(Ωf ))2 × P k(Ωf )2 ×RT k−1(Ωp)× P k(Ωp) (8)for any k ≥ 1, as considered in the next section.4 Numerical ResultsThis section is concerned with the validation of the Least-Squares approach by numerical results. The first numerical exampledescribes a capillary of length 1 mm before the bifurcation and surrounded by the tissue of the capillary bed, in analogyto [11]. Since the different terms of the Least-Squares Functional have different scaling, the domain is rescaled such thatL = 2 and an auxiliary solution is computed. A simple mapping transforms the solution to the original domain back. Theboundary conditions are chosen from [11]: the blood is flowing in the capillary through ΓD,f,1 where the effective pressure isset to 400 Pa to ΓD,f,2x and ΓD,f,3 where it is set to −1600 Pa. On ΓD,F , the effective pressure pF is set to −933 Pa. Thethickness of the vessel wall ε is set to 6.0 · 10−7m. The second numerical example is constructed in a similar way, but takesthe structure of the liver lobule into account. In both cases, the results are shown in figure 5 and confirm the convergence ofthe method. The Least-Squares functional as an error estimator leads to a mesh refinement around the interface as expected.www.gamm-proceedings.com © 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH4 of 4 Young Researchers’ Minisymposium 6: Novel discretization methods for phase-field modeling in fracture mechanics3⇡4L N,2 N,1  D,f,1 D,F,1  D,F,2 D,F,3 N,3 N,4 D,f,2  D,f,3 N,5 N,6`Fig. 2: Geometry for the bifurcation case. Fig. 3: Initial meshes for Ωf and Ωp Fig. 4: Effective pressureFig. 5: Initial meshes for Ωf and Ωp representing thehexagonal-shaped liver lobule and the sinusoids therein.Fig. 6: Effective pressure in the liver lobule with peripor-tal inflow and outflow at the central vein.Acknowledgements Funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s ExcellenceStrategy – EXC 2075 – 390740016.Open access funding enabled and organized by Projekt DEAL.References[1] K. Baber, Coupling free flow and flow in porous media in biological and technical applications: From a simple to a complex interfacedescription. Institut für Wasser und Umweltsystemmodellierung Mitteilungen, (2014).[2] F. Bertrand, First-Order System Least-Squares for Interface Problems. SIAM Journal on Numerical Analysis 56.3 (2018).[3] P. Bochev and M. Gunzburger, Least-squares finite element methods. Vol. 166. Springer Science & Business Media, 2009.[4] B. Christ, U. Dahmen, K.-H. Herrmann, M. König, J. Reichenbach, T. Ricken, el al., Computational Modeling in Liver Surgery,Frontiers in Physiology 8 (2017)[5] M. Discacciati, and A. Quarteroni. Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Com-plut 22.2 (2009)[6] L. Formaggia, A. M. Quarteroni and A. Veneziani, Cardiovascular mathematics. Number CMCS-BOOK-2009-001. Springer, (2009a).[7] W. Ehlers and A. Wagner, Constitutive and computational aspects in tumor therapies of multiphasic brain tissue. In Computer Modelsin Biomechanics. Springer, Dordrecht. (2013)[8] W. Ehlers and J. Bluhm Porous media: theory, experiments and numerical applications. Springer. (2002).[9] K. Erbertseder, J. Reichold, B. Flemisch, P. Jenny and R. Helmig, A coupled discrete/continuum model for describing cancer-therapeutic transport in the lung. (2012)[10] L. Formaggia, A. Quarteroni, and A. Veneziani, Cardiovascular Mathematics: Modeling and simulation of the circulatory system (Vol.1). Springer Science & Business Media.[11] T. Koch, Coupling a vascular graph model and the surrounding tissue to simulate flow processes in vascular networks, (2014)[12] S. Münzenmaier, First-Order system least squares for generalized-Newtonian coupled Stokes-Darcy flow. Numerical Methods forPartial Differential Equations, (2015)[13] S. Münzenmaier, and G. Starke, First-order system least squares for coupled Stokes-Darcy flow, SIAM J Numer Anal 49 (2011).[14] T. Ricken, U. Dahmen, and O. Dirsch, A biphasic model for sinusoidal liver perfusion remodeling after outflow obstruction, BiomechModel Mechanobiol. 9 (2010)[15] T. Ricken, D. Werner, H. G. Holzhütter, M.König, U. Dahmen, and O. Dirsch, Modeling function-perfusion behavior in liver lobules in-cluding tissue, blood, glucose, lactate and glycogen by use of a coupled two-scale PDE-ODE approach, Biomech Model Mechanobiol14 (2011).© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH www.gamm-proceedings.com

Least Squares Finite Element Method for Hepatic Sinusoidal Blood Flow

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