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Three-Dimensional Numerical Simulation of Plaque Formation And View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PUblication MAnagement 272 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE, VOL. 16, NO. 2, MARCH 2012 ARTreat Project: Three-Dimensional Numerical Simulation of Plaque Formation and Development in the Arteries Nenad Filipovic, Member, IEEE, Mirko Rosic, Irena Tanaskovic, Zarko Milosevic, Dalibor Nikolic, Nebojsa Zdravkovic, Aleksandar Peulic, Milos R. Kojic, Dimitris I. Fotiadis, Senior Member, IEEE, and Oberdan Parodi Abstract—Atherosclerosis is a progressive disease characterized elements in artery walls. Over the past decade, scientists come to by the accumulation of lipids and fibrous elements in arteries. It appreciate a prominent role for inflammation in atherosclerosis. is characterized by dysfunction of endothelium and vasculitis, and Atherosclerosis develops from oxidized low-density lipopro- accumulation of lipid, cholesterol, and cell elements inside blood vessel wall. In this study, a continuum-based approach for plaque tein (LDL) molecules. When oxidized LDL evolves in plaque formation and development in 3-D is presented. The blood flow formations within an artery wall, a series of reactions occur is simulated by the 3-D Navier–Stokes equations, together with to repair the damage to the artery wall caused by oxidized the continuity equation while low-density lipoprotein (LDL) trans- LDL. The body’s immune system responds to the damage to port in lumen of the vessel is coupled with Kedem-Katchalsky the artery wall caused by oxidized LDL by sending specialized equations. The inflammatory process was solved using three addi- tional reaction–diffusion partial differential equations. Transport white blood cells-macrophages (Mphs) to absorb the oxidized- of labeled LDL was fitted with our experiment on the rabbit ani- LDL forming specialized foam cells. Macrophages accumu- mal model. Matching with histological data for LDL localization late inside arterial intima. Also, smooth muscle cells (SMC) was achieved. Also, 3-D model of the straight artery with initial accumulated in the atherosclerotic arterial intima, where they mild constriction of 30% plaque for formation and development is proliferate and secrete extracellular matrix to form a fibrous presented. cap [1]. Unfortunately, macrophages are not able to process Index Terms—Atherosclerosis, low-density lipoprotein (LDL), the oxidized LDL, and ultimately grow and rupture, deposit- plaque, wall shear stress. ing a larger amount of oxidized cholesterol into the artery wall. There has been considerable effort investigated LDL transport and atheroslerosis process through the artery [2]–[5]. Some au- I. INTRODUCTION thors [2] found that macrophage proliferation and constant sig- naling to the endothelial cells drive lesion instability, rather than THEROSCLEROSIS is a progressive disease character- an increasing influx of modified LDL. Pazos et al. [3] used to- ized in particular by the accumulation of lipids and fibrous A gether finite-element simulations of inflation experiments with nonlinear least-squares algorithm to estimate the material model parameters of the wall and of the inclusion. They fitted the sim- ulated experiment and the real experiment with the parameter Manuscript received April 15, 2011; revised July 26, 2011; accepted estimation algorithm. Hoi et al. [4] found that oscillatory and August 30, 2011. Date of publication September 19, 2011; date of current version March 7, 2012. This work was supported by the European Commission retrograde flow induced in the mice may exacerbate or accel- under Project ARTreat, ICT 224297 and the Serbian Ministry of Science under erate lesion formation, but the distinct anatomical curvature of Project III-41007 and Project ON-174028. the mouse aorta is responsible for the spatial distribution of N. Filipovic is with the Faculty of Mechanical Engineering, University of Kragujevac, 34000 Kragujevac, Serbia (e-mail: fi[email protected]). lesions. Olgac et al. [5] found that hypertension leads to an in- A. Peulic is with Technical Faculty Cacak, University of Kragujevac, 32000 creased number of regions with high LDL concentration where Cacak, Serbia (e-mail: [email protected]). the endothelium is represented by a three-pore model. M. Rosic, I. Tanaskovic, and N. Zdravkovic are with the Faculty of Medicine, University of Kragujevac, 34000 Kragujevac, Serbia (e-mail: We are using similar approach for LDL transport through the [email protected]; [email protected]; [email protected]). wall domain as homogenous wall model [6]. For the moment, Z. Milosevic D. Nikolic are with Bioengineering Research and De- we did not consider multilayer wall model [7]. The Kedem– velopment Center, 34000 Kragujevac, Serbia (e-mail: [email protected]; [email protected]). Katchalsky equations for mass transport between lumen and M. Kojic is with the Methodist Hospital System, Houston, TX 77030 USA, wall domains are used. The main difference is that we added with the Harvard School of Public Health, Boston, MA 02115 USA, and also additional reaction–diffusion equations for process of inflam- with the Bioengineering Research and Development Center, 34000 Kragujevac, Serbia (e-mail: [email protected]). mantion and plaque development. D. I. Fotiadis is with University of Ioannina, 45110 Ioannina, Greece (e-mail: This study describes a new computer model for plaque for- [email protected]). mation and development. The potential benefit for the present O. Parodi is with CNR Clinical Physiology Institute, Via Moruzzi 1, Pisa 56124, Italy (e-mail: [email protected]). computational model is to simulate plaque progression based on Color versions of one or more of the figures in this paper are available online real patient data from clinical measurements. The first section at http://ieeexplore.ieee.org. is devoted to the LDL model of transport from the lumen to Digital Object Identifier 10.1109/TITB.2011.2168418 1089-7771/$26.00 © 2011 IEEE FILIPOVIC et al.: ARTREAT PROJECT: THREE-DIMENSIONAL NUMERICAL SIMULATION 273 intima and a detailed 3-D model for inflammatory and process. is the pressure drop across the endothelium, Δπ is the oncotic The next section describes labeled LDL transport and matching pressure difference across the endothelium, σd is the osmotic with histological finding as well as benchmark example 3-D reflection coefficient, σf is the solvent reflection coefficient, P is model of plaque formation and development. Finally, the main the solute endothelial permeability, and c¯is the mean endothelial conclusions of the work are addressed. concentration. The incremental iterative form of finite-element equations of II. METHODS balance is obtained by including the diffusion equations and transforming them into incremental form. The final equations In this section, a continuum-based approach for plaque for- are, as shown in (7) at the bottom of this page, where the matrices mation and development in 3-D is presented. All algorithms are are incorporated in program PAK-Athero from the University of Kragujevac, Kragujevac [8]. (M ) = ρN N dV The governing equations and numerical procedures are given. v jjKJ K J V The blood flow is simulated by the 3-D Navier–Stokes equations, together with the continuity equation (Mc )jjKJ = NK NJ dV V 2 − μ∇ ul + ρ (ul ·∇) ul + ∇pl =0 (1) n+1 (i−1) Kcc = DNK,jNJ,j dV ∇ul =0 (2) jjKJ V − where ul is the blood velocity in the lumen, pl is the pressure, n+1 (i 1) Kμv = μNK,jNJ,j dV μ is the dynamic viscosity of the blood, and ρ is the density of jjKJ V the blood. n+1 (i−1) n+1 (i−1) Mass transfer in the blood lumen is coupled with the Kcv = NK c,j NJ dV jjKJ V blood flow and modeled by the convection–diffusion equation as follows: n+1 (i−1) n+1 (i−1) Kvv = ρNK vj NJ,j dV jjKJ V ∇·(−Dl ∇cl + cl ul )=0 (3) n+1 (i−1) n+1 (i−1) Jcc = ρNK vj NJ,j dV in the fluid domain, where cl is the solute concentration in the jjKJ V blood lumen and Dl is the solute diffusivity in the lumen. n+1 (i−1) ˆ Mass transfer in the arterial wall is coupled with the trans- Kvp = ρNK,j NJ dV jjKJ V mural flow and modeled by the convection–diffusion-reaction equation as follows: n+1 (i−1) n+1 Jvv = ρNK vj,kNJ dV jkKJ V ∇·(−Dw ∇cw + kcw uw )=rw cw (4) and the vectors are in the wall domain, where cw is the solute concentration in the − − 1 arterial wall, Dw is the solute diffusivity in the arterial wall, n+1F(i 1) = n+1F + n+1F(i 1) − M c q sc Δt c K is the solute lag coefficient, and rw is the consumption rate constant. × n+1 (i−1) − n − n+1 (i−1) n+1 (i−1) C C Kcv V LDL transport in lumen of the vessel is coupled with Kedem–Katchalsky equations − n+1 (i−1) n+1 (i−1) Kcc C − Jv = Lp (Δp σd Δπ) (5) n+1 B n+1 (i−1) Fq = NK q dV F Js = P Δc +(1− σf ) Jv c¯ (6) K sc V where Lp is the hydraulic conductivity of the endothelium, Δc is n+1 (i−1) = DNK ∇ c · ndS. the solute concentration difference across the endothelium, Δp S ⎡ ⎤ 1 n+1 (i−1) n+1 (i−1) n+1 (i−1) n+1 (i−1) ⎢ Mv + Kvv + Kμv + Jvv Kvp 0 ⎥ ⎢ Δt ⎥ ⎢ ⎥ ⎢ KT 00⎥ ⎣ vp ⎦ − 1 n+1K(i 1) 0 M + n+1K(i−1) + n+1J(i−1) cv Δt c cc cc ⎧ ⎫ ⎧ ⎫ (i) ⎪ n+1 (i−1) ⎪ ⎨⎪ ΔV ⎬⎪ ⎨⎪ Fv ⎬⎪ × (i) n+1 (i−1) ⎪ ΔP ⎪ = ⎪ Fp ⎪ (7) ⎩ ⎭ ⎩⎪ ⎭⎪ (i) n+1 (i−1) ΔC Fc 274 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE, VOL. 16, NO. 2, MARCH 2012 Fig. 1. Schematic presentation of the isolated blood vessel segment in the water bath.
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