Blood Flow Modeling: A review: Comparison
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Please note this is a comparison between Version 1 by Violeta Carvalho and Version 3 by Dean Liu.

Blood flow modeling consists of using computational techniques to investigate the blood flow behavior in a rapid and accurate fashion. This has become an area of extensive research due to the prevalence of cardiovascular diseases, responsible for a critical number of deaths every year worldwide, most of which are associated with atherosclerosis, a disease that causes unusual hemodynamic conditions in arteries. In the present review,  the application of computational simulations by using different physiological conditions of blood flow, several rheological models, and boundary conditions, were discussed.

  • atherosclerosis
  • coronary arteries
  • hemodynamics
  • numerical methods

1. Introduction

Despite the progress done in experimental studies and blood flow measurement techniques, there are still some challenges associated with them[1] [19]. For instance, in vitro wall shear stress (WSS) measurements are extremely difficult to perform and the velocity measurements have high associated errors. These, combined with other complications of directly measuring quantities of interest, have motivated the use of computer simulations to predict them in silico[2] [69].

The earliest numerical detailed studies solving the flow problem in constricted tubes were conducted by Lee and Fung (1970)[3] [70]. After that, other studies in this field conducted by Caro et al., (1971)[4] [71], Glagov et al., (1989)[5] [72], and Ku et al., (1985)[6] [73] are important references in this area and should be highlighted. Ever since, CFD approaches have been progressively adopted by most researchers as the preferred technique for numerical modeling of hemodynamics. Owing to the continued growth of computational power, these have become an increasingly reliable tool for measuring biomechanical factors vital for clinical decision-making and surgical planning. However, the proper selection of the flow boundary conditions has to be done, otherwise, the findings can be considered uncertain, weak, and unrealistic[7] [74]. In this regard, the different geometries, boundary conditions, and flow characteristics applied by some researchers in the last ten years are summarized in Table 1.

Table 1. Numerical studies of hemodynamics and the respective assumptions for numerical simulations.

GeometrySchematic RepresentationModeling ApproachesFluidBoundary ConditionsAuthors
WallInletOutlet
Idealized
Fluids 06 00053 i001LaminarNon-Newtonian (Carreau-Yasuda)RigidTime-dependent velocity profileZero gauge pressureKashyap et al., (2020) [8]
LaminarNon-Newtonian (Carreau-Yasuda)RigidTime-dependent velocity profileZero gauge pressureKashyap et al., (2020) [6]
Idealized
Fluids 06 00053 i002LaminarNewtonianRigidTime-dependent mass flow profileZero surface tensionBiglarian et al., (2019) [9]
LaminarNewtonianRigidTime-dependent mass flow profileZero surface tensionBiglarian et al., (2019) [75]
Idealized
Fluids 06 00053 i003LaminarNon-Newtonian (Cross model)Rigid and FlexibleConstant inlet velocity
LaminarNon-Newtonian (Cross model)
Constant pressure outlet (10 kPa)
Rigid and Flexible
Mulani et al., (2015) [10]
Constant inlet velocityConstant pressure outlet (10 kPa)Mulani et al., (2015) [57]
Idealized
Fluids 06 00053 i004LaminarNewtonianRigid and FlexibleTime-dependent flowrate profileTime-dependent pressure profileWu et al., (2015) [11]
LaminarNewtonianRigid and FlexibleTime-dependent flowrate profileTime-dependent pressure profileWu et al., (2015) [58]
Idealized
Fluids 06 00053 i005LaminarNewtonianRigidConstant inlet velocity (fully developed parabolic profile)Constant pressure outlet (13 kPa)Kenjereš et al., (2019) [12]
LaminarNewtonianRigidConstant inlet velocity (fully developed parabolic profile)Constant pressure outlet (13 kPa)Kenjereš et al., (2019) [76]
Idealized
Fluids 06 00053 i006LaminarNewtonianRigidConstant inlet velocityZero gauge pressureCarvalho et al., (2020) [13]
LaminarNewtonianRigidConstant inlet velocityZero gauge pressureCarvalho et al., (2020) [47]
Idealized
Fluids 06 00053 i007k-ω turbulent modelNon-Newtonian (Carreau model)RigidSpiral boundary conditionwith a parabolic velocity profileZero gauge pressureKabir et al., (2018) [14]
k-ω turbulent modelNon-Newtonian (Carreau model)RigidSpiral boundary conditionwith a parabolic velocity profileZero gauge pressureKabir et al., (2018) [77]
Idealized
Fluids 06 00053 i008k-ω turbulent model (SST)Non-Newtonian (Carreau model)RigidTime-dependent velocity profile
k-ω turbulent model (SST)Non-Newtonian (Carreau model)
Zero gauge pressure
Rigid
Carvalho et al., (2020) [15]
Time-dependent velocity profileZero gauge pressureCarvalho et al., (2020) [42]
Idealized
Fluids 06 00053 i009k-ω turbulent model (SST)Non-Newtonian (Carreau model)RigidTime-dependent velocity profileZero gauge pressureCarvalho et al., (2020) [16][17]
k-ω turbulent model (SST)Non-Newtonian (Carreau model)RigidTime-dependent velocity profileZero gauge pressureCarvalho et al., (2020) [59,78]
Idealized
Fluids 06 00053 i010
N.A
1NewtonianFlexibleTime-dependent velocity profileTime-dependent pressure profile
FlexibleTime-dependent velocity profile
Jahromi et al., (2019) [
Time-dependent pressure profile
18]
NewtonianJahromi et al., (2019) [79]
Idealized
Fluids 06 00053 i011LaminarNewtonianRigidTime-dependent velocity profileFlow partition implied in Murray’s lawDoutel et al., (2018) [19]
LaminarNewtonianRigidTime-dependent velocity profileFlow partition implied in Murray’s lawDoutel et al., (2018) [11]
Patient-specific
Fluids 06 00053 i012LaminarNon-Newtonian (Generalized power-law model) and NewtonianRigidTime-dependent flow rate profileTime-dependent pressure profileChaichana et al., (2012) [20]
LaminarNon-Newtonian (Generalized power-law model) and NewtonianRigidTime-dependent flow rate profileTime-dependent pressure profileChaichana et al., (2012) [60]
Patient-specific
Fluids 06 00053 i013LaminarNon-Newtonian (Carreau model)RigidTime-dependent velocity profileTime-dependent pressure profileLiu et al., (2015) [21]
LaminarNon-Newtonian (Carreau model)RigidTime-dependent velocity profileTime-dependent pressure profileLiu et al., (2015) [80]
Patient-specific
Fluids 06 00053 i014LaminarNewtonianRigid and FlexibleTime-dependent pressure profile
LaminarNewtonian
Parabolic velocity profile
Rigid and Flexible
Siogkas et al., (2014) [22]
Time-dependent pressure profileParabolic velocity profileSiogkas et al., (2014) [81]
Patient-specific
Fluids 06 00053 i015N.ANewtonianRigidTime-dependent pressure profileConstant pressure outlet (9.85 kPa)Zhao et al., (2019) [23]
N.ANewtonianRigidTime-dependent pressure profileConstant pressure outlet (9.85 kPa)Zhao et al., (2019) [82]
Patient-specific
Fluids 06 00053 i016LaminarNon-Newtonian (Carreau model)RigidTime-dependent velocity profileFlow partition implied in Murray’s lawPandey et al., (2020) [24]
LaminarNon-Newtonian (Carreau model)RigidTime-dependent velocity profileFlow partition implied in Murray’s lawPandey et al., (2020) [43]
Patient-specific
Fluids 06 00053 i017LaminarNon-Newtonian (Carreau model)RigidVarious time-dependent velocity profilesFlow partition implied in Murray’s lawRizzini et al., (2020) [7]
LaminarNon-Newtonian (Carreau model)RigidVarious time-dependent velocity profilesFlow partition implied in Murray’s lawRizzini et al., (2020) [74]
Patient-specific
Fluids 06 00053 i018N.ANon-Newtonian (Power-law model)RigidTime-dependent velocity profilePressure outlet (N.A)Zhang et al., (2020) [25]
N.ANon-Newtonian (Power-law model)RigidTime-dependent velocity profilePressure outlet (N.A)Zhang et al., (2020) [83]
Patient-specific
Fluids 06 00053 i019k-ω turbulent model (SST)Non-Newtonian (Bird-Carreau model)RigidTime-dependent velocity profile
k-ω turbulent model (SST)Non-Newtonian (Bird-Carreau model)
Constant pressure outlet (10 kPa)
Rigid
Kamangar et al., (2019) [26]
Time-dependent velocity profileConstant pressure outlet (10 kPa)Kamangar et al., (2019) [64]
Patient-specific
Fluids 06 00053 i020LaminarNewtonianRigidTime-dependent flow rate profileTwo-Element Windkessel ModelLo et al., (2019) [27]
LaminarNewtonianRigidTime-dependent flow rate profileTwo-Element Windkessel ModelLo et al., (2019) [84]
Patient-specific and Idealized
Fluids 06 00053 i021LaminarNewtonian and Non-Newtonian (Carreau model)RigidConstant inlet velocity and Time-dependent velocity profileN.ADoutel et al., (2019) [28]
LaminarNewtonian and Non-Newtonian (Carreau model)RigidConstant inlet velocity and Time-dependent velocity profileN.ADoutel et al., (2019) [85]
Patient-specific and Idealized
Fluids 06 00053 i022k-ω turbulent model (SST)Non-Newtonian (Carreau model)RigidTime-dependent velocity profileOutflow conditionMahalingam et al., (2016) [29]
k-ω turbulent model (SST)Non-Newtonian (Carreau model)RigidTime-dependent velocity profileOutflow conditionMahalingam et al., (2016) [86]
Patient-specific and Idealized
Fluids 06 00053 i023N.ANon-Newtonian (Carreau model)RigidTime-dependent velocity profileConstant pressure outlet (10 kPa)Rabbi et al., (2020)[30]
N.ANon-Newtonian (Carreau model)RigidTime-dependent velocity profileConstant pressure outlet (10 kPa)Rabbi et al., (2020) [87]
Patient-specific and Idealized
Fluids 06 00053 i024LaminarNewtonianRigidConstant inlet mass flow and Time-dependent flow rate
LaminarNewtonian
Zero gauge pressure
Rigid
Malota et al., (2018) [31]
Constant inlet mass flow and Time-dependent flow rateZero gauge pressureMalota et al., (2018) [88]

N.A—Not available. All images were adapted from [8][19][15][24][13][10][11][16][20][26][7][9][12][14][17][18][21][22][23][25][27][28][29][30][31][6,11,42,43,47,57,58,59,60,64,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

In general, from the above-mentioned investigations, it can be seen that, regardless of the type of geometry, the majority of authors consider that the blood is a non-Newtonian fluid, usually approximated by the Carreau model, with a laminar behavior. Regarding the boundary conditions, in most cases, the wall is considered rigid, and at the inlet, a pulsatile velocity is applied. At the outlet, the condition set mainly depends on the study, but either the default conditions are maintained, or pressures are applied, time-dependent or constant values. In the following section, the main observations drawn from these studies are presented.

2. Concluding Rremarks and Ffuture Pperspectives

Although huge advancements have been made in imaging techniques to obtain patient-specific images, this step is time-consuming, and it is still a challenging task for all researchers. For this reason, nowadays, idealized models continue to be widely used by researchers, since these allow to obtain important and relevant results, without requiring much computational time and without the need to collect the medical images, which is highly time-consuming. In this regard, a promising study was proposed by Doutel et al., (2018) wherein artificial, but realistic stenosis can be generated.

Although huge advancements have been made in imaging techniques to obtain patient-specific images, this step is time-consuming, and it is still a challenging task for all researchers. For this reason, nowadays, idealized models continue to be widely used by researchers, since these allow to obtain important and relevant results, without requiring much computational time and without the need to collect the medical images, which is highly time-consuming. In this regard, a promising study was proposed by Doutel et al., (2018) [1] wherein artificial, but realistic stenosis can be generated.

It was also noted that, although the modelling of blood as a Newtonian fluid is a good approximation for large vessels with high shear rates, the assumption of non-Newtonian behavior of blood flow has been increasingly used in the presence of stenosis. From the overall studies, the most used models are the Carreau and the Carreau-Yasuda, and these have also been indicated as the most appropriate to simulate the blood rheology by Razavi et al., (2011) [2]. Nevertheless, currently, one cannot say which is the right model, because there is not yet enough evidence in the literature to prove which model fully expresses the complex nature of blood rheology and its dependence on many biological factors [3]. Accordingly, it is of great importance to obtain proper models for CFD analysis that take into account the non-Newtonian behavior of blood. For this purpose, more experimental studies are needed. Regarding the boundary conditions, few studies have evaluated the impact of using different inlet and outlet boundary conditions [4][5], and therefore, it would be interesting, in future studies, to investigate what are the profiles more adequate to study the blood flow behavior in coronary arteries.

Despite the great efforts that have been made so far, the blood has been mainly modeled as a single-phase fluid. However, blood is a mixture of plasma, red blood cells, white blood cells, and platelets. Therefore, the consideration of multiphase models is of great importance when modeling atherosclerotic lesions. Although some studies have already applied these models [6][7][8][9], the research is still in the beginning. Moreover, it should be also noted that the use of these models is a promising option for studying nanoparticle-mediated targeted drug delivery treatment of atherosclerosis. In this context, a promising study was conducted by Zhang et al., (2020)[7]. The authors used an Eulerian-Lagrangian approach coupled with FSI to investigate the impact of plaque morphology on magnetic nanoparticles targeting under the action of an external field.

It was also noted that, although the modelling of blood as a Newtonian fluid is a good approximation for large vessels with high shear rates, the assumption of non-Newtonian behavior of blood flow has been increasingly used in the presence of stenosis. From the overall studies, the most used models are the Carreau and the Carreau-Yasuda, and these have also been indicated as the most appropriate to simulate the blood rheology by Razavi et al., (2011) . Nevertheless, currently, one cannot say which is the right model, because there is not yet enough evidence in the literature to prove which model fully expresses the complex nature of blood rheology and its dependence on many biological factors . Accordingly, it is of great importance to obtain proper models for CFD analysis that take into account the non-Newtonian behavior of blood. For this purpose, more experimental studies are needed. Regarding the boundary conditions, few studies have evaluated the impact of using different inlet and outlet boundary conditions , and therefore, it would be interesting, in future studies, to investigate what are the profiles more adequate to study the blood flow behavior in coronary arteries.

Due to the continuous improvements acquired in computational methods, in the following years more amazing and complex hemodynamic studies will be performed. The work of Zhao et al. (2019) [10] should be highlighted since their numerical approach has a great potential to achieve more realistic simulations. They have simulated 4D hemodynamic profiles of time-resolved blood flow. The results proved that these simulations can provide extensive information about blood flow, both qualitatively and quantitatively that may be advantageous for future investigations of clinical diagnosis and treatment of atherosclerosis.

Despite the great efforts that have been made so far, the blood has been mainly modeled as a single-phase fluid. However, blood is a mixture of plasma, red blood cells, white blood cells, and platelets. Therefore, the consideration of multiphase models is of great importance when modeling atherosclerotic lesions. Although some studies have already applied these models , the research is still in the beginning. Moreover, it should be also noted that the use of these models is a promising option for studying nanoparticle-mediated targeted drug delivery treatment of atherosclerosis. In this context, a promising study was conducted by Zhang et al., (2020). The authors used an Eulerian-Lagrangian approach coupled with FSI to investigate the impact of plaque morphology on magnetic nanoparticles targeting under the action of an external field.

To conclude, although computational methods have been extensively used for atherosclerosis investigations in recent years, they are expected to become more popular and more effective to simulate the blood flow in the cardiovascular system, and consequently, they will promote medical innovation at an affordable cost. However, to this end, active collaborations between engineers and medical staff are needed to assure the successful application of this technique in atherosclerosis treatment.

Due to the continuous improvements acquired in computational methods, in the following years more amazing and complex hemodynamic studies will be performed. The work of Zhao et al. (2019) should be highlighted since their numerical approach has a great potential to achieve more realistic simulations. They have simulated 4D hemodynamic profiles of time-resolved blood flow. The results proved that these simulations can provide extensive information about blood flow, both qualitatively and quantitatively that may be advantageous for future investigations of clinical diagnosis and treatment of atherosclerosis.

Reference (Editors will rearrange the references after the entry is submitted)

To conclude, although computational methods have been extensively used for atherosclerosis investigations in recent years, they are expected to become more popular and more effective to simulate the blood flow in the cardiovascular system, and consequently, they will promote medical innovation at an affordable cost. However, to this end, active collaborations between engineers and medical staff are needed to assure the successful application of this technique in atherosclerosis treatment.

 

References

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