Numerical Study of the Blood Flow in a Deformable Human Aorta

applied sciences

Article Numerical Study of the Blood Flow in a Deformable Human Aorta

Marwa Selmi 1,2,*, Hafedh Belmabrouk 2,3 and Abdullah Bajahzar 4

1 Department of Radiological Sciences and Medical Imaging, College of Applied Medical Sciences, Majmaah University, Majmaah 11952, Saudi Arabia 2 Laboratory of Electronics and Microelectronics, Faculty of Science of Monastir, University of Monastir, Environment Boulevard, Monastir 5019, Tunisia; [email protected] 3 Department of Physics, College of Science, Majmaah University, Zulﬁ 11932, Saudi Arabia 4 Department of Computer Science and Information, College of Science, Majmaah University, Zulﬁ 11932, Saudi Arabia; [email protected] * Correspondence: [email protected]; Tel.: +966-563447961

Received: 17 February 2019; Accepted: 19 March 2019; Published: 22 March 2019

Featured Application: This article developed a computer simulation based on the finite element method to investigate the interaction between blood flow and arterial wall deformation. The numerical results show that the blood flow has an unsteady and complex behavior. The velocity field, pressure, and displacement distributions were investigated during a cardiac cycle. The method described in this study will be useful for investigating cardiovascular diseases and performing hemodynamics analyses.

Abstract: In this work, we present a numerical investigation of blood flow in a portion of the human vascular system. More precisely, the present work analyzed the blood flow in the upper portion of the aorta. The aorta and its ramified blood vessels are surrounded by the cardiac muscle. The blood flow generates pressure on the internal surfaces of the artery and its ramifications, thereby causing deformation of the cardiac muscle. The numerical analysis used the Navier–Stokes equations as the governing equations of blood flow for the calculation of the velocity field and pressure distribution in the blood. The neo-Hookean hyperelastic model was used for the description of the behavior of the vessel walls. The velocity and pressure distributions were analyzed. The deformation of the vessel was also investigated. The numerical results could be used to better understand and predict the factors that trigger cardiovascular diseases and distortions of the aorta and as a diagnostic tool in clinical applications.

Keywords: numerical simulation; blood ﬂow; ﬂuid dynamics; computational modeling; vascular system; deformation

1. Introduction The cardiovascular system consists of the heart and vessels (arteries and veins). It has the function to distribute to the organs oxygen and nutrients essential for their life through the blood. The heart and its various organs can be affected by different pathologies, called cardiovascular diseases, namely, stenosis, aneurysm, occlusion, atherosclerosis, etc. According to recent statistics, cardiovascular diseases are the third leading cause of death in the western world [1]. The understanding of the vascular function requires knowledge of the behavior and the material properties of blood vessels [2,3]. In addition, the deep characterization of blood ﬂow is crucial for the understanding of the behavior of the whole human system and for the improvement of human health [4,5]. Even though a few

Appl. Sci. 2019, 9, 1216; doi:10.3390/app9061216 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1216 2 of 11 experimental works are available on the histological structure of arteries, they are still limited by the difficulty of making rapid analyses on the heart [6,7]. Numerical simulation is a widely used tool for understanding many phenomena and especially, it was confirmed to be useful in studying the blood flow in the cardiovascular system, to diagnose cardiovascular diseases [8–12], evaluate the hemodynamics of healthy and diseased blood vessels [13–15] and design and optimize vascular medical devices [16,17]. Over the last decades, various computational techniques have been used extensively by a large number of researchers seeking to simulate and analyze blood flow and to understand the relationship between vascular diseases and hemodynamics [18,19]. In addition, numerical simulation and computer models provide a fast, simple, safe way to study phenomena involved in vascular diseases. Recently, there have been numerous studies of blood flow related to the aorta that use computer simulation to better investigate blood flow variables. Panta et al. [20] analyzed, through their numerical simulation, blood velocity as a function of the shape of human arteries, namely straight, bent, T-shaped, and the main arterial network models. Wood et al. [21] combined magnetic resonance imaging (MRI) and computational fluid dynamic (CFD) simulation of flow to analyze the human descending aorta. Their objective was to investigate the potential of combining CFD simulations with in vivo MRI scans. Tokuda et al. [22] simulated blood flow numerically using the finite element method to cognize the mechanism of stroke during cardio-pulmonary bypass (CPB). The location of turbulence, wall pressure and flow distribution within the aortic arch were investigated. They succeeded in simulating and visualizing the blood flow during CPB using CFD techniques. Torii et al. [23] developed a computer modeling technique based on the deforming-spatial-domain/stabilized space–time method to study fluid–structure interactions between flowing blood and arterial walls. They demonstrated that the wall shear stress is a very important factor in cardiovascular diseases, and their results proved that numerical modeling is an operative tool in hemodynamic studies of cardiovascular diseases. Bazilevs et al. [24] presented a computational framework for the simulation of patient-specific cerebral aneurysm configurations and also described a new approach for the computation of blood vessel tissue prestress. They studied some variables, namely, hemodynamics, wall shear stress, and wall tension, in order to check the relevance of fluid–structure interactions when adapting the rigidity of the arterial wall and the relevance of a flexible wall. Crosetto et al. [25] performed a numerical algorithm based on finite elements to evaluate the mean velocity and temporal evolution of pressure and the distribution of wall shear stress on diverse sections of the aorta. More specifically, they reported the results of 3D simulations using a physiological geometry over several heartbeats. They changed the inlet condition of flux by pressure which caused a backward diastolic flow through the section and thereafter led to the closure of the aortic valve and the circulation in the coronary arteries. However, these observations were only predicted numerically. Garje et al. [26] studied numerically the changes in blood flow vessels and the growth of a blockage, which is an occlusion of the coronary arteries typical of coronary artery disease (CAD), due to an accumulation of cholesterol. Their work was focused on the size and position of the blockage. Priyadharshini and Ponalagusamy [27] studied numerically the effects of magneto-hydrodynamics on blood flow parameters in a stenotic artery under an applied external magnetic field and body acceleration. In this work, we intended to simulate the interaction of blood flow with the artery wall. We simulated the pressure along the surface of the artery, the displacements and stresses of the walls of part of a child’s aorta and some of its ramifications embedded in the heart muscle when subjected to the pressure caused by heart beating. In Section2, the geometry of the aorta and its ramifications are presented. Then, the Navier–Stokes equations and the equation governing the displacement of the blood vessels are introduced. The initial and boundary conditions, as well as the numerical method, are also presented. The results are discussed in Section3. Appl. Sci. 2019, 9, 1216 3 of 11

2. MethodsAppl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 12 2. Methods 2.1. Simulated Geometry Figure2.1. Simulated1 shows Geometry the simulated geometry. The domain was divided into three parts: blood, aorta and its ramificationsFigure 1 shows (branching the simulated vessels), geometry. and The cardiac domain muscle. was divided The cardiacinto three muscle parts: blood, is very aorta important becauseand itits protects ramifications the artery (branching from vessels), the deformation and cardiac duemuscle. to theThe pressure. cardiac muscle The inletis very and important outlet orifices are alsobecause shown it protects in this the figure. artery from the deformation due to the pressure. The inlet and outlet orifices are also shown in this figure.

Figure 1. Schematic showing a view of the aorta, its ramifications (branching vessels), and the cardiac muscle. 2.2. GoverningFigure Equations1. Schematic showing a view of the aorta, its ramifications (branching vessels), and the cardiac muscle. In this study, we aimed to compute the flow of blood in the aorta and to investigate the deformation2.2. Governing of the Equations cardiac muscle due to blood pressure and the behavior of the vessel walls. The numericalIn this study study, consisted we aimed of to two compute parts. the flow of blood in the aorta and to investigate the deformation of the cardiac muscle due to blood pressure and the behavior of the vessel walls. The Part 1:numericalFluid dynamics study consisted analysis of two which parts. required to solve the Navier–Stokes equations. It included the calculation of the velocity field and the distribution of blood pressure (variable over time and Part 1: Fluid dynamics analysis which required to solve the Navier–Stokes equations. It included the in space).calculation of the velocity field and the distribution of blood pressure (variable over time and in Part 2: Mechanicalspace). analysis of the deformation of the cardiac tissue and artery. Part 2: Mechanical analysis of the deformation of the cardiac tissue and artery. Navier–Stokes equations: The blood flow fluid dynamics was described using the time-dependentNavier–Stokes Navier–Stokes equations: equations The blood and flow the fluid equations dynamics for was the conservationdescribed using of the mass. time- The fluid is assumeddependent to beNavier–Stokes incompressible equations and and homogeneous, the equations withfor the a conser Newtonianvation of behavior. mass. The The fluid blood is was assumed to be incompressible and homogeneous, with a Newtonian behavior. The blood was assumed to be Newtonian in this study, as its viscosity is almost constant in arteries with relatively assumed to be Newtonian in this study, as its viscosity is almost constant in arteries with relatively largelarge diameters diameters (5 mm); (5 mm); therefore, therefore, non-Newtonian non-Newtonian effects effects were were neglected neglected [28]. [28]. The Navier–StokesThe Navier–Stokes equations equations may may be written as as follows: follows: 𝛁. 𝒖 =0, (1) ∇.u = 0, (1) 𝜕𝒖 𝜌 +𝒖.∇𝒖=𝛁𝝈 (2) 𝜕𝑡∂u ρ + u.∇u = ∇σ (2) where 𝜌 is the blood density, 𝒖 is the blood∂t velocity, and 𝝈 is the stress tensor [29]. The relationship describing the stress tensor and the strain rate tensor is defined as: where ρ is the blood density, u is the blood velocity, and σ is the stress tensor [29]. The relationship describing the stress tensor and the strain𝝈 rate = −𝑝𝑰 tensor is+ 𝜇(𝛁𝒖+ defined(𝛁𝒖) ) as: (3) where 𝑝 is the blood pressure, 𝜇 is the dynamic viscosity of the blood, 𝑰 is the identity tensor, and denotes the transpose tensor. σ = −pI + µ ∇u + (∇u)T (3) The fluid properties such as the density and the dynamic viscosity were assumed to be constant. Their values were, respectively 𝜌 = 1060 𝑘𝑔/𝑚 and 𝜇 = 0.005 𝑃𝑎. 𝑠. where p is the blood pressure, µ is the dynamic viscosity of the blood, I is the identity tensor, and T denotes the transpose tensor. The fluid properties such as the density and the dynamic viscosity were assumed to be constant. 3 Their values were, respectively ρblood = 1060 kg/m and µblood = 0.005 Pa.s. Appl. Sci. 2019, 9, 1216 4 of 11

Structural mechanics: The motion of the vessel wall can be mathematically described by the following equation [30]: 2 ∂ usolid ρw − ∇σs = ρwf (4) ∂t2 s where ρw is the wall density, fs represents the external body forces acting on the solid, and σs is the Cauchy stress tensor. The wall can be the artery or the cardiac muscle. The value of the wall density is 3 3 ρartery = 960 kg/m for the artery and ρmuscle = 1200 kg/m for the cardiac muscle. It should be noted that u is a velocity (unit m/s); however, usolid is a displacement (unit meter). The artery is assumed to be subjected to large strains and undergo large deformations. Its behavior is non-linear. More precisely, it behaves as a hyperelastic neo-Hookean material. The constitutive law of this material involves the material shear modulus µs and the bulk modulus κs. The values of these parameters are provided in Table1. In the present study, the material is assumed to be isothermal, and no effect of the temperature is considered.

Table 1. Mechanical parameters of the artery.

Parameter Value 3 Density ρartery = 960 kg/m Shear modulus µs = 62.04 × 105 N/m2 Bulk modulus κs = 124 × 105 N/m2

For a three-dimensional, compressible, neo-Hookean material such as the artery, the strain energy density is given by [24]:

1 1 1 1 W = µs tr(C) − 3 + κs J2 − 1 − ln(J) (5) 2 J2/3 2 2 where J = detF, F is the deformation gradient tensor, and C = FTF is the Cauchy–Green deformation gradient tensor. From the above constitutive law, the expressions of various tensors can be deduced by means of more or less arduous calculations.

2.3. Initial and Boundary Conditions As mentioned above, the model uses three domains, namely, the blood, the artery and the cardiac muscle. The numerical simulation considers the solution of the 3D Navier–Stokes equations and the ﬂuid–structure interactions. The initial and boundary conditions required to solve the equations of the model are given hereafter. The human heart beat is cyclically time-varying. Thus, the velocity at the inlet of the aorta is not set to be constant. It has a periodic proﬁle as a function of time. During a cycle, the velocity varies between a minimal and a maximal value. The duration of each period is 0.8 s. We introduced the following function [31]: ( 0.8 sin(2.857πt) 0 ≤ t ≤ 0.35 f (t) = (6) 0.6 sin(2.22π(t − 0.35)) 0.35 ≤ t ≤ 0.8

The inlet velocity for the artery in this simulation was fixed to 0.3 [m/s]*f(t). The temporal evolution of the inlet velocity is represented in Figure2. The outlet condition pressure was considered to be 125 [mmHg]*f(t). These conditions are similar to those adopted by Choudhari and Panse [31]. This function has two peaks which have different amplitudes and different periods. On the aortic walls, a non-slip boundary condition is applied, and the wall is assumed to be impermeable since the fluid is supposed not to penetrate through the walls as it flows along the vessel. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 12

0.25

Appl. Sci. 2019, 9, 1216 5 of 11 0.2 Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 12

0.25 0.15

0.2 0.1 Velocity (m/s) Velocity

0.15 0.05

0.1 Velocity (m/s) Velocity 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (s) 0.05 Figure 2. Flow velocity waveform in the inlet used in the simulation for the flow analysis.

On the aortic walls, a0 non-slip boundary condition is applied, and the wall is assumed to be impermeable since the fluid0 is supposed0.1 0.2 not 0.3to penetrate0.4 0.5through0.6 the 0.7walls 0.8as it flows along the Time (s) vessel. FigureFigure 2. 2.Flow Flow velocity velocity waveform waveform inin thethe inletinlet used in the simulation simulation fo forr the the flow flow analysis. analysis. 2.4. Numerical Method 2.4. NumericalOn the Methodaortic walls, a non-slip boundary condition is applied, and the wall is assumed to be In this study, the finite element method was used to solve the set equations describing the impermeableIn this study, since the the finite fluid element is supposed method not to was pe usednetrate to through solve the the setwalls equations as it flows describing along the the problem [32]. The simulated geometry was meshed in tetrahedral discretization as shown in Figure problemvessel. [ 32]. The simulated geometry was meshed in tetrahedral discretization as shown in Figure3a. 3a. Figure 3b depicts the radial discretization of the blood aorta. The central part stands for the blood. Figure3b depicts the radial discretization of the blood aorta. The central part stands for the blood. The2.4. artery Numerical and Methodits thickness appear clearly in this figure. However, the muscle was considered as Thehaving artery no andthickness, its thickness and only appear a boundary clearly condit in thision figure. was applied However, to represent the muscle this wassurface. considered Indeed, as In this study, the finite element method was used to solve the set equations describing the havingthe muscle no thickness, was considered and only a free a boundarysurface, with condition no external was force applied applied to represent on it. this surface. Indeed, problem [32]. The simulated geometry was meshed in tetrahedral discretization as shown in Figure the muscleThe model was considered used to asolve free surface,the governing with no externalequations force is two-way applied oncoupled. it. Actually, any 3a. Figure 3b depicts the radial discretization of the blood aorta. The central part stands for the blood. deformationThe model of usedblood to vessel solve walls the governing has an effe equationsct on the fluid is two-way flow and coupled. vice versa Actually, [33,34]. any deformation of bloodThe artery vessel and walls its thickness has an effect appear on theclearly fluid in flow this andfigure. vice However, versa [33 the,34 ].muscle was considered as having no thickness, and only a boundary condition was applied to represent this surface. Indeed, the muscle was considered a free surface, with no external force applied on it. The model used to solve the governing equations is two-way coupled. Actually, any deformation of blood vessel walls has an effect on the fluid flow and vice versa [33,34].

FigureFigure 3. 3.( (aa)) GeometryGeometry mesh, ( (bb)) zoom zoom of of the the mesh mesh of of th thee inlet/outlet inlet/outlet surface. surface.

MeshMesh sizes sizes are are summarizedsummarized in Table 22.. The total number number of of elements elements were were 195,050, 195,050, and and the the total total numbernumber of of nodes nodes was was 33,506.33,506.

Figure 3.Table (a) Geometry 2. Finite-element mesh, (b) mesh zoomsizes of the used mesh in of this the simulation. inlet/outlet surface.

Fluid Elements Solid Elements Total Elements Total Nodes Mesh sizes are summarized in Table 2. The total number of elements were 195,050, and the total number of nodes85,083 was 33,506. 109,967 195,050 33,506 Appl. Sci. 2019, 9, 1216 6 of 11

Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 12 3. Results Table 2. Finite-element mesh sizes used in this simulation. The results concern mainly the temporal evolution of the velocity field, the pressure, and the wall displacement. Fluid Solid Total Total Figure4 shows the bloodElements speed distribution Elements inside Elements the vessel Nodes at different times. To discuss this figure, one should keep in mind85083 the profile109967 of the inlet195050 velocity 33506 depicted in Figure2. At t = 0.1 s, the fluid was mainly directed to the first inlet. The section between the first and second outlets started to3. Results move. However, the remaining part of the aorta was almost at rest. At t = 0.2 s, the fluid also reached the second and the third exits. Progressively, the blood flowed into the vessel accelerating or The results concern mainly the temporal evolution of the velocity field, the pressure, and the decelerating. The un-steady flow remained laminar. wall displacement.

FigureFigure 4.4. BloodBlood velocityvelocity ﬁeldfield distribution inside the vessel at different times; ( (aa)) t=0.1 t = 0.1 s ( sb ()b t=0.2) t = 0.2s (c s) (c) t = 0.4 s (d) t = 0.6 s (e) t = 0.7t=0.4 s and s (d ()f )t=0.6 t = 0.8 s (e s.) t=0.7 s and (f) t=0.8 s.

Figure 4 shows the blood speed distribution inside the vessel at different times. To discuss this figure, one should keep in mind the profile of the inlet velocity depicted in Figure 2. At t = 0.1 s, the Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 12

fluid was mainly directed to the first inlet. The section between the first and second outlets started to move. However, the remaining part of the aorta was almost at rest. At t = 0.2 s, the fluid also reached Appl.the second Sci. 2019, 9and, 1216 the third exits. Progressively, the blood flowed into the vessel accelerating7 of or 11 decelerating. The un-steady flow remained laminar. FromFrom thethe resultsresults ofof Figure Figure4 4,, one one can can also also compute compute thethe flowflow raterate inin each each outputoutput asas wellwell asas the the strokestroke volume.volume. FigureFigure5 5 exhibits exhibits thethe temporal temporal evolutionevolution ofof the the maximalmaximal velocity.velocity. The The maximum maximum valuevalue reachedreached variedvaried fromfrom 0.10.1 toto 0.32 0.32 m/sm/s depending on time. The relative curve is coarsely similar to that of thethe inputinput velocity.velocity. ItIt isis constituted by two peaks that that have have different different amplitudes amplitudes and and different different frequencies. frequencies. AA least-squareleast-square fitfit allowedallowed toto obtainobtain thethe amplitudeamplitude ofof thesethese peaks.peaks. WeWe obtainedobtained 0.35 0.35 m/sm/s for the firstfirst peakpeak andand 0.26 0.26 m/sm/s for the second one.one. The discrepancydiscrepancy between the computation results and thethe theoreticaltheoretical resultsresults obtainedobtained byby thethe least-squareleast-square fitfit waswas inin aa tolerabletolerable limit.limit. TheThe meanmean uncertaintyuncertainty waswas aboutabout 10%.10%.

0.35 computational results least square fit 0.30

0.25

0.20

0.15

Maximalspeed (m/s) 0.10

0.05

0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (s)

Figure 5. Temporal evolution of the maximal velocity.

ItIt waswas alsoalso interestinginteresting toto analyzeanalyze thethe pressurepressure distributiondistribution versusversus time.time. Figure6 6 exhibits exhibits thethe pressurepressure distributiondistribution atat tt == 0.1 s and t = 0.7 s. It is clear that the pressure has a complex distribution. Appl. Furthermore,Furthermore,Sci. 2019, 9, x FOR this this PEER distributiondistribution REVIEW dependsdepends stronglystrongly onon time,time, sincesince thethe inputinput conditionsconditions areare unsteady.unsteady.8 of 12 The values of the wall shear stress applied to the blood vessel walls during a cardiac cycle vary strongly in time and position. This has an important effect on cardiovascular diseases.

FigureFigure 6. Pressure 6. Pressure distribution distribution inside inside the thevessel vessel at different at different times; times; (a) (t=0.1a) t = s 0.1and s ( andb) t=0.7 (b) t s. = 0.7 s.

Figure 7 shows the temporal evolution of the maximal pressure. This figure confirms the large variation in pressure during a cardiac cycle. Evidently, this has an important effect on the wall shear stress and consequently on cardiovascular diseases. Like the maximal velocity, the temporal evolution of the maximal pressure consisted of two pulses. The amplitudes of the pulses were 12.65 kPa and 9.62 kPa. The mean uncertainty between the computational results and the data obtained from the least-square fit was lesser than 8%.

14 computational results 12 least square fit

Maximal pressure (kPa) pressure Maximal 2

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 Time (s) Figure 7. Temporal evolution of the maximal pressure.

Figure 8 depicts the displacement of the blood vessel wall at t = 0.1 s and t = 0.7 s. It is clear that a large deformation of the vessel occurred mainly in the vicinity of the embranchments. However, all the vessels may be subjected to large deformation. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 12

Appl. Sci. 2019, 9, 1216 8 of 11

TheFigure values 6. of Pressure the wall distribution shear stress inside applied the vessel to the at blood different vessel times; walls (a) during t=0.1 s and a cardiac (b) t=0.7 cycle s. vary strongly in time and position. This has an important effect on cardiovascular diseases. Figure 7 shows the temporal evolution of the maximal pressure. This figure confirms the large Figure7 shows the temporal evolution of the maximal pressure. This figure confirms the large variationvariation in in pressure pressure during a cardiaccardiac cycle.cycle. Evidently,Evidently, this this has has an an important important effect effect on on the the wall wall shear shear stressstress and consequentlyconsequently on on cardiovascular cardiovascular diseases. diseases. Like the Like maximal the maximal velocity, the velocity, temporal the evolution temporal evolutionof the maximal of the pressuremaximal consistedpressure ofconsisted two pulses. of two The pulses. amplitudes The amplitudes of the pulses of were the pulses 12.65 kPawere and 12.65 kPa9.62 and kPa. 9.62 The kPa. mean The uncertainty mean uncertainty between thebetween computational the computational results and results the data and obtained the data from obtained the fromleast-square the least-square fit was lesser fit was than lesser 8%. than 8%.

14 computational results 12 least square fit

4 aximal pressure (kPa) pressure aximal

M 2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (s)

Figure 7. Temporal evolution evolution of of the the maximal maximal pressure. pressure.

Figure8 depicts the displacement of the blood vessel wall at t = 0.1 s and t = 0.7 s. It is clear that Figure 8 depicts the displacement of the blood vessel wall at t = 0.1 s and t = 0.7 s. It is clear that a large deformation of the vessel occurred mainly in the vicinity of the embranchments. However, a large deformation of the vessel occurred mainly in the vicinity of the embranchments. However, all Appl.all theSci. 2019 vessels, 9, x FOR may PEER be subjected REVIEW to large deformation. 9 of 12 the vessels may be subjected to large deformation.

FigureFigure 8. 8. DisplacementsDisplacements in in the the blood blood vessel vessel using using a a hyperelastic hyperelastic model; ( a)) t=0.1 t = 0.1 s and s and (b ()b t=0.7) t = 0.7s. s.

TheThe temporal temporal evolution evolution of of maximal maximal displacement displacement during during a a cardiac cardiac cycle cycle was investigated. Figure 9 9 presentspresents the the related related results. results. Similarly Similarly to theto maximalthe maximal velocity velocity and the and maximal the maximal pressure, pressure, the temporal the temporal evolution of the maximal pressure consisted of two pulses. Using the least-square fit, we found that the amplitudes were 8.33 μm and 6.26 μm. The mean uncertainty due to the fit was about 3%.

8 computational results least square fit

6 cement (µm) 4

2 Maximal displa

0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 Time (s) Figure 9. Temporal evolution of the maximal displacement.

It appears clearly that the maximal displacement depended on time and could reach 8 μm. These values are in accordance with the important shear stress on the vessel. This has an impact on cardiovascular diseases such as stenosis. Our results have the same order of magnitude as previous works [31,35–37]. For instance, Choudhari and Panse obtained a radial deformation ranging from 1.5 to 2.2 μm and a maximal velocity about 0.6 m/s. In this study, the pressure, the velocity, and the deformation were predicted with respect to a given blood flow velocity. It could be interesting, in future work, to analyze the inverse problem. A numerical method such as the iterative gradient search method is expected to be useful to perform this analysis [38,39]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 12

Figure 8. Displacements in the blood vessel using a hyperelastic model; (a) t=0.1 s and (b) t=0.7 s.

The temporal evolution of maximal displacement during a cardiac cycle was investigated. Figure Appl.9 presents Sci. 2019 ,the9, 1216 related results. Similarly to the maximal velocity and the maximal pressure,9 ofthe 11 temporal evolution of the maximal pressure consisted of two pulses. Using the least-square fit, we evolutionfound that of the the amplitudes maximal pressure were 8.33 consisted μm and of6.26 two μm. pulses. The mean Using uncertainty the least-square due to ﬁt, the we fit foundwas about that the3%. amplitudes were 8.33 µm and 6.26 µm. The mean uncertainty due to the ﬁt was about 3%.

8 computational results least square fit

6 cement (µm) 4

2 Maximal displa

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (s)

Figure 9. Temporal evolutionevolution ofof thethe maximalmaximal displacement.displacement.

ItIt appears clearly clearly that that the the maximal maximal displacement displacement depended depended on time on time and could and could reach reach8 μm. 8Theseµm. Thesevalues values are in are accordance in accordance with withthe theimportant important shea shearr stress stress on onthe the vessel. vessel. This This has has an an impact onon cardiovascularcardiovascular diseasesdiseases suchsuch asas stenosis.stenosis. OurOur resultsresults havehave thethe samesame orderorder ofof magnitudemagnitude asas previousprevious worksworks [31[31,35–37].,35–37]. For instance, ChoudhariChoudhari andand Panse Panse obtained obtained a radial a radial deformation deformation ranging ranging from from 1.5 to 1.5 2.2 µtom 2.2 and μ am maximal and a maximal velocity aboutvelocity 0.6 about m/s. 0.6 m/s. InIn thisthis study,study, thethe pressure,pressure, thethe velocity,velocity, andand thethe deformationdeformation werewere predictedpredicted withwith respectrespect toto aa givengiven blood flow ﬂow velocity. It It could could be be interesting, interesting, in in future future work, work, to to analyze analyze the the inverse inverse problem. problem. A Anumerical numerical method method such such as as the the iterative iterative gradient gradient se searcharch method method is is expected expected to to be useful to performperform thisthis analysisanalysis [[38,39].38,39].

4. Conclusions In this work, we developed a computer simulation based on the finite element method to investigate the interaction between blood flow and arterial wall deformation. This simulation tool was applied to the upper part of a child’s aorta. The numerical results showed that the blood flow had an unsteady and complex behavior. The velocity field, pressure, and displacement distributions were investigated during a cardiac cycle. Further, the present study reveals that modeling the blood flow is helpful to the investigation of cardiovascular diseases and it can be an effective tool in hemodynamics studies of cardiovascular diseases. Experimental validation on real human aorta and a silicon rubber model would be very useful. We intend to perform this validation in a future work.

Author Contributions: M.S. proposed the idea, contributed to data acquisition, performed the simulation. H.B. contributed to data analysis and result discussion. A.B. performed the algorithm construction. Acknowledgments: The authors thank the Deanship of Scientific Research and the Deanship of Community Service at Majmaah University, Saudi Arabia, for supporting this work. The authors thank Shatha Mohammed Al-Mutairy, Danah Bander Al Mutairi, Reef Saleh Al-Aybani, Sarah Ibrahim Altwaijry for participating in this work. Conflicts of Interest: Competing Interests: The authors declare no competing financial interests. Appl. Sci. 2019, 9, 1216 10 of 11

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