Analysis and Simulation of Blood Flow in the Portal Vein withUncertainty Quantification

João Pedro Carvalho Rêgo de Serra e MouraInstituto Superior Técnico

AbstractBlood flow simulations in CFD are seen as a very attractive solution for diagnosing diseases. The mainobjective of this work is to simulate blood flow in the portal vein for patients with liver cirrhosis and toquantify the uncertainty that surrounds blood flow. Initially all the tools required were explored: the verifi-cation and validation of the models were performed as well as convergence studies. Moreover an uncertaintyquantification process was used based on a Non-Intrusive Spectral Method. The sources of uncertainty wereresearched and quantified as the geometry and blood model were assumed as the main random variables.Key Words: Blood flow, CFD, uncertainty quantification, Non-Intrusive Spectral Projection.

1 Mathematical and Physical Modelling

1.1 Governing EquationsThe flow is considered to be three-dimensional, incompressible and laminar the conservation equations maybe read as 1

ρ(

∂u∂t + u.5 u

)− div σ(u, P ) = 0 in Ω

div u = 0 in Ω. (1)

In these equations, ρ is the blood density, which is considered to be constant and equal to 1060 kg/s, u isthe velocity and P the pressure, which are both unknowns and σ(u,P) is the Cauchy stress tensor.

The blood shows a shear-thinning behaviour and is often modeled as a Non-Newtonian fluid. Thisbehaviour is dependent on the strain rate of the fluid and it is not important in vessels where the strain ratesare over 1000 s−1. However in this case, and since the study leans on a diseased portal vein, which not onlybeing small, but with decreased blood flow will show a lower strain rate in the range of 1-200 s−1, which isin the range where the shear-thinning will be important. Some literature also considers the viscoelasticty ofblood, however studies have shown that the predominant behaviour is the shear-thinning [1] and thereforethis will be the only one considered.

1.2 Non-Newtonian ModelsThere are many models to describe the shear-thinning (pseudo-plastic) Non-Newtonian behavior. At lowstrain rates the blood viscosity is much higher than for high strains. These models also show a range of strainrates where the blood viscosity enters a transition phase from high viscosity to low viscosity, ∂µ/∂γ < 0.

When considering Non-Newtonian fluids σ takes the form of equation 2

σ = −PI + 2µγD (2)

with γ :=√

2D : D being the strain rate tensor modulus and D the strain rate tensor. There are modelsthat represent this Non-Newtonian viscosity, whose parameters allow fitting to experimental data of bloodflow.

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In this work we will be focused mainly in he Carreau-Yasuda that is given by equation 3

µ = µ∞ + (µ0 − µ∞)(1 + (λγ)a)n−1

a (3)

where µ0 and µ∞ are the zero and infinite strain rate limit viscosities respectively, λ is the relaxation timeconstant and n is the power law index. For a=2, this model becomes the Carreau model. The Carreau andCarreau-Yasuda are the models that best fit reported experimental results.

Many different blood models are used in the literature. Fig. 1 presents a comparison of the apparentviscosity see ([2]) for some detailed parameters.

At low strain rate strain rate ranges, say in between 0.1 and 100 there is a considerable variance inthe viscosity models values. As it can be seen for different strain rate ranges, there are many models that

Figure 1: Strain Rate vs Apparent Viscosity

show considerable variance with each other, which tells that for different models very different viscositieswill be considered. However different these models may be, there seems to be no scientific consensus onwhich models better represent the shear-thinning behaviour of blood ([2]). The correct specification of theviscosity model is crucial to capture the correct rheological behavior of blood. Therefore the blood modelused in this work was a Carreau model with parameters µ0 = 0.0456 Pa.s, µ∞ = 0.0032 Pa.s, λ = 10.03 sand n = 0.344 ([3]).

2 Verification and ValidationThe Star-CCM+, numerical solver was used throughout this work including for mesh generation and CADmodel handling. This numerical code uses a SIMPLE algorithm and it was selected a 2nd order upwind con-vection scheme. Verification and validation , see ([4]), has been conducted for several benchmark engineeringproblems with identical flow complexity to the portal vein. The verification of the numerical model was pre-viously performed against a semi-analytical benchmark case and the validation is performed by comparingthe Physical model results with other blood flow simulations, as presented below.

A model validation is the substantiation that a computerized model within its domain of applicabilitypossesses a satisfactory range of accuracy consistent with the intended application of the model ([4]). Inorder to fulfill this requirement the work of [5] was reproduced.

In this study, steady Non-Newtonian flow in a simplified geometry for coronary bypass is simulated underdifferent flow conditions and graft locations. The geometry used in the modeling of the simulation can beseen in Fig 2, where a simplified anastomosis model is represented as the intersection of two cylinders bothwith a diameter of D = 3 mm at a junction angle of 45. A 75% lumen axisymmetric stenosis is consideredin the host coronary and is described by a Gaussian profile.

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Figure 2: Graft Geometry

The blood flow was modeled as being incompressible, Non-Newtonian, homogeneous, steady, three-dimensional and laminar. The shear thinning behaviour, the most dominant Non-Newtonian property ofblood, was modeled with a Carreau-Yasuda model (equation 3) with µ∞ = 0.0022 Pa.s, µ0 = 0.022 Pa.s, λ= 0.110 s, a = 0.644 and n = 0.392 and the blood density is considered to be ρ = 1410 kg/m3.

The outlet of the host artery has a prescribed boundary condition as mass flow rate outlet of Q =1.708×10−3 kg/s. As for the inlet of the host artery and graft, the boundary conditions are mass flow rateinlets of 33

4Q and 34Q respectively.

Figs. 3 (x-y plane) and 4 (x-z plane) show that the results obtained in this work approximate very wellthe results obtained in [5], except for the coarse model (the velocity scale is not plotted for clarity sake).However the results are not exactly the same. This can be explained by the fact that the data collected from[5] was interpolated as the author could not supply the actual results and by the fact that that article doesnot show the mesh convergence, which can mean that a coarse mesh was used. Nonetheless the results arewell approximated and therefore the model can be said to be validated.

Figure 3: Velocity profiles along X in the XY plane Figure 4: Velocity profiles along X in the XZ plane

3 Uncertainty Quantification ProcessThe Polynomial Chaos (PC) expansion is a non-sampling based method that uses a spectral projection of therandom variables to determine the evolution of uncertainty in a dynamical system. The PC employs orthog-onal polynomials in the random space as the trial basis to expand the stochastic process. The generalizedpolynomial chaos expansion can handle several random processes. From the Askey scheme, generalizing, itis possible to obtain a set of orthogonal polynomials from a given measure/PDF, see, e.g., [6].

In the Non-Intrusive Spectral Projection (NISP) method, the output stochastic process is constructed us-ing deterministic functions evaluations at an optimal number of points defined in the input support space ([7]).This way the deterministic model is evaluated for different samples of the uncertain parameters, which follow

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a post-processing method in order to quantify the uncertainty propagation through the model. Consequentlyno reformulation of the model’s governing equations is performed.

This method can be generalized for N independent random variables (X1, ...XN ). For each variable therewill be an associated stochastic dimension ξi=1, ..., N , which forms a multi-dimension stochastic space.

Having the orthogonal polynomials, the model solution f(~ξ) can be represented using the PC expansion

f(~ξ) =P∑

j=0

cfj Φj(~ξ) (4)

where cfj are the unknown PC expansion mode coefficients of f(~ξ) and P + 1 = (N + p)!/(N !p!) the totalnumber of terms in the PC expansion, with p equal to the maximum polynomial order of the expansion.Thus given the orthogonality of Φj , c

fj yields in:

cfj =

⟨f(~ξ)Φj

⟩⟨Φ2

j

⟩ , j = 0, ..., P (5)

In general the NISP method is developped through the following process [8].

1. Define the PDFs for the uncertainty parameters Xi, i = 1, ..., N , thus associating the distribution typewith the PC basis Φj .

2. Determine the corresponding spectral PC expansion for each of the parameters.

3. Run the deterministic model for all the samples of the input parameters vector, (X1, ..., XN )nsn=1,to obtain the solution for (fd)nsn=1

4. Evaluate the expectations from equation 5 over a sufficiently large number of samples to obtain thesolution for the spectral coefficients cfj . The numerator in equation 5 is solved numerically using aGauss quadrature.

4 ResultsThis section describes the propagation of parametric uncertainty through a physical model, which is usedto investigate the problem concerning blood flow in the portal vein for people with liver cirrhosis. Theuncertainty parameters studied were based on the uncertainty on blood viscosity models and on the model’sgeometry. This section shows firstly the deterministic models with a convergence study, as well as thegeometry definition for a Newtonian and Non-Newtonian model, following the results obtained using theNISP method for both the blood and geometry uncertainties.

The idealised portal vein model is described by a main vein that branches into two, and those twobranches into four different ones. The purpose of this work was to simulate a disease portal vein, a clot wasincluded in the geometry to simulate thrombosis in the portal vein. This clot was modeled as a cylindricalcut through the model’s left branch (see Fig. 5).

Following some justifications from literature, the flow can be assumed steady and laminar with rigidwalls.

At the inlet a velocity profile was prescribed using an extrusion mesh to have fully-developed flow at theentrance of the portal vein. At the outlets of the portal vein the pressure was specified taking in accountthat the pressure loss throughout the liver is about 600Pa and the left part of the liver has approximatelytwice the right part. These flow exits were modeled with pressure loss that is dependent on the velocity ofthe blood.

The deterministic solution dependence on the mesh was performed using a velocity profile after the clotfrom the left branch. Fig. 6 shows five different profiles with different number of elements in the mesh.Assuming that the more refined model (5.4 million elements) is the closest to the right solution, the mesh

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Figure 5: Geometry of the Portal Vein Model

Figure 6: Convergence Graphic of a Velocity profile in the left branch after the clot for different sized meshes

with 3.1 million elements shows a very good fitting in the velocity profile showing it is well converged,therefore this mesh was the one chosen throughout the rest of the work.

Figure 7: Bar Chart of the Strain Rate values in the model

4.1 Newtonian and Non-Newtonian Deterministic ModelIn Fig. 7 shows the strain rate range is in the range 0-60 s−1, which leads to the assumption that the shear-thinning behaviour is predominant in the flow. To verify that two models were simulated with Newtonianand Non-Newtonian behaviour. The Newtonian model used a constant viscosity of µ = 0.0035 Pa.s, whereasthe Non-Newtonian used a Carreau model for the fluid viscosity with parameters µ0 = 0:0456 Pa.s, µ∞ =0.0032 Pa.s and n = 0.344. The radius of the clot was constant and equal to 2.833 mm. In Fig. 8 is plotted

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the absolute difference in the velocity magnitude throughout the model, being possible to see the velocityfield in Fig. 9. From this it can be seen that mostly in the right branch the differences are larger. Howeverthere are also significant differences in the recirculation zone, which is a very important factor in bloodflow, due to the fact that if a recirculation bubble persist for a long time, the slowed RBCs will aggregate,increasing the chances of increasing the blood clot.

Figure 8: Absolute difference of the velocity fieldbe-tween a Newtonian and a Non-Newtonian blood model

Figure 9: Velocity field of the Non-Newtonian bloodmodel

4.2 Stochastic Influence of the Blood ViscosityFig. 1 shows a comparison of 15 different blood viscosity models denoting large differences and it is importantto take into account this unknown into a stochastic process.

The uncertainty regarding the blood viscosity was evaluated considering three different methods: i) Multi-blood models Uncertainty in the blood behavior from a sample of a mixture of blood models considered;ii) Uncertainty regarding the non-linear square fit method used in a Carreau blood viscosity model; iii)uncertainty of a single-blood model describing blood viscosity.

4.2.1 Model Uncertainty from a Mixture of Models

The Carreau and Carreau-Yasuda models are vastly used throughout the literature. In order to quantifythe uncertainty of these blood models, a function of the blood viscosity is used where φj(ξ, η) is a shapefunction that has values between [0, 1] and

∑4j=1 φj(ξ, η) = 1. ξ and η are two random variables with an

uniform PDF varying from [0; 1]. Three different Carreau models and one Carreau-Yasuda model are themodels chosen for this study.

4.2.2 Model Parameters Uncertainty

On the other hand, assuming that the blood viscosity is given only by a specific model, there are stilluncertainties regarding the model parameters. A study was conducted with deterministic flow solutions thatdisplayed the frequency of ocurence of strain rate values shown in the bar figure 7 . One may conclude thatthe uncertainty may occour in the of strain rate interval [1;60]. In addition it was investigated the influenceof all the parameters that rule the shear-thinning behaviour of the Carreau model. From this study, it wasconcluded that the range of viscosity could be achieved with uncertainty in µ0, mu∞ and n. These weretaken as random variables with uniform PDFs, with µ = 0.0456, 0.004 and 0.344 and σ2 = 0.0092, 0.00028and 0.099 respectively.

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Figure 10: PDF for the Shearwith a blood model mixtureas a random variable

Figure 11: PDF for the Shearwith the blood model param-eters as a random variable

Figure 12: PDF for the Shearwith the blood model as arandom variable

4.2.3 Distinctive Models

Another approach to the uncertainty in the blood behaviour was performed to include the blood model asa stochastic variable. Four different blood models were used. The stochastic variable has an uniform PDFranging from -1 to 1, with each model having equally spaced ranges.

4.2.4 Results

Figure 13: Average of thevelocity field with the bloodmodel mixture as a randomvariable

Figure 14: Average of thevelocity field with the bloodmodel parameters as randomvariables

Figure 15: Average of thevelocity field with the bloodmodel as a random variable

Figs. 10, 12 and 11 show the PDF for shear at the branch with the clot presenting great uncertainty.Despite the differences in the curve’s shape, the same range is covered with lower probability density in theright hand side.Large influence in the pressure inlet and shear happens when uncertainty is applied to theblood viscosity and almost no influence in mass flow split. The blood model uncertainty did not change themass split at the first bifurcation due to the strong outlet pressure drop.

Having consequences in the velocity field variance, the influence in the mean velocity field is not significant,as can be seen in Figs. 13, 14 and 15.

4.3 Stochastic Influence of the Idealized Thrombosis Radius GeometryHaving decided on which viscosity model was to be implemented, the introduction of uncertainty parametersin the model’s geometry follows. In this section the radius of the thrombosis in the model’s left branch was

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taken as a random variable.

4.3.1 Small Obstruction Clot

The size of the thrombosis has a great influence in the haemodynamic characteristics of the flow, speciallyin the wall shear stress. Adding to this, the fact that current MRI tools only have a limited accuracy, whichtypically varies from 0.3mm to 0.47mm, shows that care must be taken when analyzing MRI exams fromsmall arteries or veins.

The stochastic analysis of the thrombosis size influence on the velocity profile behind the Clot is shownin Fig. 16, where apart from the mean velocity profile, it is also plotted the 95% confidence interval. Thisplot shows that the uncertainty in the clot radius greatly influences this velocity profile.

One of the most important influences, might be the uncertainty in the size of the recirculation bubble.Also, the maximum velocity magnitude as well as its location changes in a significant manner, which iscaused by the change in the size of the recirculation buble.

When looking at Figs. 18 and 19 it can be seen that the velocity is mostly affected by the radiusuncertainty close to the clot. The velocity values around the clot were interpolated in a clot free model toaccomodate the PDF’s entire range.

Figure 16: Stochastic parameters of the velocity profilefor different small thrombosis radius

Figure 17: Stochastic parameters of the velocity pro-file with the thrombosis radius and the blood modelparameters as random variables

Figure 18: Average of the velocity field with the radiusas a random variable

Figure 19: Standard deviation of the velocity field withthe radius as a random variable

When comparing the confidence interval of the velocity profile of this analysis (Fig. 16) with the onetaking the radius of the clot and also the blood as random variables (Fig. 17) it is clear that they are verymuch alike, showing the largest differences in the recirculation zone.

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The combined influence of both the radius and the blood model parameters is seen in Fig. 20, clearlyshowing a large influence of the µinfty and n parameter and radius, leading to the conclusion of the importanceof considering uncertainty in the blood model.

Even with the combination of random variables, the mass flow split becomes almost unchanged.

Figure 20: Shear expansion coefficients with the radius and the blood model parameters as random variables

4.3.2 Large Obstruction Clot

For large obstructions human life becomes at a great risk. As expected the geometry uncertainty in the biggerclot has bigger influence in the flow inside the model. This is shown mostly by the confidence interval of thevelocity profile (Fig. 21), clearly showing a large uncertainty regarding the maximum velocity in that zone.Also the recirculation zone is longer, which will increase the “roulleaux” formation and therefore increasingthe chances of a larger and possibly deathly clot. The large increase in not only the shear range, but alsoin its magnitude suggests an increased probability of vein rupture leading to death. Here the he mass flowsplit is affected by the uncertainty in the large obstruction, whereas in the small clot it almost did not haveany influence.

Figure 21: Stochastic parameters of the velocity profilefor different small thrombosis radius

Figure 22: Shear expansion coefficients with the radiusand the blood model parameters as random variables

It is also important to take a look at the combined effect of the blood uncertainty with uncertainty inthe size of a large obstruction in order to see if the effect on the hemodynamic factors are also significantlyaffected by the blood uncertainty.

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The effect of uncertainty is clearly mostly due to the size of the obstruction as a random variable. Fig. 22shows the coefficients of the shear expansion, clearly showing that the radius influence is the most importantcompared to the blood model parameters.

5 ConclusionsAs proposed for the objectives, a verification of the numerical code was performed using simple 2D and 3Dgeometries. Thus the validation of blood flow was made using models and results accepted as accurate inthe scientific community.

The model of the portal vein was studied with a clot in one branch.A NISP method was implemented in the geometries examined. A thorough study was developed on the

influence of blood viscosity in blood flow. On this note, several blood viscosity models were studied showingdifferent behaviours for different strain rate ranges. This investigation took into account a combination ofdifferent blood models. The results obtained showed that even though the range of influence of the threeapproaches was similar, the shape of the PDF was very different leading to different uncertainty behaviours.It can be concluded from this study that the uncertainty in the the blood model can lead to great uncertaintyin the shear force in the walls of the blood vessels.

The uncertainty regarding geometry was also deeply investigated. This uncertainty was quantified withthe NISP method with two different geometries. The obtained results from the uncertainty quantificationclearly show a great influence in shear and pressure in the vein. For critical clots it is even more importantto have accurate images of the geometry, thus uncertainty should be definetely taken into account.

When combining uncertainty from geometry and the blood, the influence of each random variable variesgreatly with the size of the clot. This way with a small clot, the influence of the blood parameters was in thesame order of magnitude as the radius influence. However when it comes to critical geometries, the radiussize has definetely larger influence in the flow development.

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