Analysis and Simulation of Blood Flow in the Portal Vein with
Uncertainty Quantification
João Pedro Carvalho Rêgo de Serra e Moura
Instituto Superior Técnico
Abstract
Blood flow simulations in CFD are seen as a very attractive solution for diagnosing diseases. The main
objective of this work is to simulate blood flow in the portal vein for patients with liver cirrhosis and to
quantify the uncertainty that surrounds blood flow. Initially all the tools required were explored: the verification and validation of the models were performed as well as convergence studies. Moreover an uncertainty
quantification process was used based on a Non-Intrusive Spectral Method. The sources of uncertainty were
researched 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
1.1
Mathematical and Physical Modelling
Governing Equations
The flow is considered to be three-dimensional, incompressible and laminar the conservation equations may
be read as 1
ρ ∂u
in Ω
∂t + u. 5 u − div σ(u, P ) = 0
(1)
div u = 0
in Ω.
In these equations, ρ is the blood density, which is considered to be constant and equal to 1060 kg/s, u is
the 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. This
behaviour is dependent on the strain rate of the fluid and it is not important in vessels where the strain rates
are over 1000 s−1 . However in this case, and since the study leans on a diseased portal vein, which not only
being small, but with decreased blood flow will show a lower strain rate in the range of 1-200 s−1 , which is
in the range where the shear-thinning will be important. Some literature also considers the viscoelasticty of
blood, however studies have shown that the predominant behaviour is the shear-thinning [1] and therefore
this will be the only one considered.
1.2
Non-Newtonian Models
There are many models to describe the shear-thinning (pseudo-plastic) Non-Newtonian behavior. At low
strain rates the blood viscosity is much higher than for high strains. These models also show a range of strain
rates where the blood viscosity enters a transition phase from high viscosity to low viscosity, ∂µ/∂ γ̇ < 0.
When considering Non-Newtonian fluids σ takes the form of equation 2
σ = −P I + 2µγ̇D
√
(2)
with γ̇ := 2D : D being the strain rate tensor modulus and D the strain rate tensor. There are models
that represent this Non-Newtonian viscosity, whose parameters allow fitting to experimental data of blood
flow.
1
In this work we will be focused mainly in he Carreau-Yasuda that is given by equation 3
µ = µ∞ + (µ0 − µ∞ )(1 + (λγ̇)a )
n−1
a
(3)
where µ0 and µ∞ are the zero and infinite strain rate limit viscosities respectively, λ is the relaxation time
constant and n is the power law index. For a=2, this model becomes the Carreau model. The Carreau and
Carreau-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 apparent
viscosity 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 in
the 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 viscosities
will be considered. However different these models may be, there seems to be no scientific consensus on
which models better represent the shear-thinning behaviour of blood ([2]). The correct specification of the
viscosity model is crucial to capture the correct rheological behavior of blood. Therefore the blood model
used in this work was a Carreau model with parameters µ0 = 0.0456 Pa.s, µ∞ = 0.0032 Pa.s, λ = 10.03 s
and n = 0.344 ([3]).
2
Verification and Validation
The Star-CCM+, numerical solver was used throughout this work including for mesh generation and CAD
model handling. This numerical code uses a SIMPLE algorithm and it was selected a 2nd order upwind convection scheme. Verification and validation , see ([4]), has been conducted for several benchmark engineering
problems with identical flow complexity to the portal vein. The verification of the numerical model was previously performed against a semi-analytical benchmark case and the validation is performed by comparing
the 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 applicability
possesses a satisfactory range of accuracy consistent with the intended application of the model ([4]). In
order 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 under
different flow conditions and graft locations. The geometry used in the modeling of the simulation can be
seen in Fig 2, where a simplified anastomosis model is represented as the intersection of two cylinders both
with a diameter of D = 3 mm at a junction angle of 45◦ . A 75% lumen axisymmetric stenosis is considered
in the host coronary and is described by a Gaussian profile.
2
Figure 2: Graft Geometry
The blood flow was modeled as being incompressible, Non-Newtonian, homogeneous, steady, threedimensional and laminar. The shear thinning behaviour, the most dominant Non-Newtonian property of
blood, 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 rate
inlets of 3 43 Q and 34 Q respectively.
Figs. 3 (x-y plane) and 4 (x-z plane) show that the results obtained in this work approximate very well
the 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 does
not show the mesh convergence, which can mean that a coarse mesh was used. Nonetheless the results are
well approximated and therefore the model can be said to be validated.
Figure 3: Velocity profiles along X in the XY plane
3
Figure 4: Velocity profiles along X in the XZ plane
Uncertainty Quantification Process
The Polynomial Chaos (PC) expansion is a non-sampling based method that uses a spectral projection of the
random variables to determine the evolution of uncertainty in a dynamical system. The PC employs orthogonal polynomials in the random space as the trial basis to expand the stochastic process. The generalized
polynomial chaos expansion can handle several random processes. From the Askey scheme, generalizing, it
is 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 using 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
3
a post-processing method in order to quantify the uncertainty propagation through the model. Consequently
no reformulation of the model’s governing equations is performed.
This method can be generalized for N independent random variables (X1 , ...XN ). For each variable there
will be an associated stochastic dimension ξi =1, ..., N , which forms a multi-dimension stochastic space.
~ can be represented using the PC expansion
Having the orthogonal polynomials, the model solution f (ξ)
~ =
f (ξ)
P
X
~
cfj Φj (ξ)
(4)
j=0
~ and P + 1 = (N + p)!/(N !p!) the total
where cfj are the unknown PC expansion mode coefficients of f (ξ)
number of terms in the PC expansion, with p equal to the maximum polynomial order of the expansion.
Thus given the orthogonality of Φj , cfj yields in:
D
E
~ j
f (ξ)Φ
(5)
cfj = 2 , j = 0, ..., P
Φj
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 type
with the PC basis Φj .
2. Determine the corresponding spectral PC expansion for each of the parameters.
s
3. Run the deterministic model for all the samples of the input parameters vector, {(X1 , ..., XN )n }n=1 ,
s
to obtain the solution for {(fd )n }n=1
4. Evaluate the expectations from equation 5 over a sufficiently large number of samples to obtain the
solution for the spectral coefficients cfj . The numerator in equation 5 is solved numerically using a
Gauss quadrature.
4
Results
This section describes the propagation of parametric uncertainty through a physical model, which is used
to investigate the problem concerning blood flow in the portal vein for people with liver cirrhosis. The
uncertainty parameters studied were based on the uncertainty on blood viscosity models and on the model’s
geometry. This section shows firstly the deterministic models with a convergence study, as well as the
geometry definition for a Newtonian and Non-Newtonian model, following the results obtained using the
NISP 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 two
branches into four different ones. The purpose of this work was to simulate a disease portal vein, a clot was
included in the geometry to simulate thrombosis in the portal vein. This clot was modeled as a cylindrical
cut through the model’s left branch (see Fig. 5).
Following some justifications from literature, the flow can be assumed steady and laminar with rigid
walls.
At the inlet a velocity profile was prescribed using an extrusion mesh to have fully-developed flow at the
entrance of the portal vein. At the outlets of the portal vein the pressure was specified taking in account
that the pressure loss throughout the liver is about 600Pa and the left part of the liver has approximately
twice the right part. These flow exits were modeled with pressure loss that is dependent on the velocity of
the blood.
The deterministic solution dependence on the mesh was performed using a velocity profile after the clot
from 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
4
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 Model
In Fig. 7 shows the strain rate range is in the range 0-60 s−1 , which leads to the assumption that the shearthinning behaviour is predominant in the flow. To verify that two models were simulated with Newtonian
and Non-Newtonian behaviour. The Newtonian model used a constant viscosity of µ = 0.0035 Pa.s, whereas
the 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
5
the absolute difference in the velocity magnitude throughout the model, being possible to see the velocity
field in Fig. 9. From this it can be seen that mostly in the right branch the differences are larger. However
there are also significant differences in the recirculation zone, which is a very important factor in blood
flow, 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 fieldbetween a Newtonian and a Non-Newtonian blood model
4.2
Figure 9: Velocity field of the Non-Newtonian blood
model
Stochastic Influence of the Blood Viscosity
Fig. 1 shows a comparison of 15 different blood viscosity models denoting large differences and it is important
to take into account this unknown into a stochastic process.
The uncertainty regarding the blood viscosity was evaluated considering three different methods: i) Multiblood 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 quantify
the uncertainty of these blood models, a function of the blood viscosity is used where φj (ξ, η) is a shape
P4
function that has values between [0, 1] and j=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 the
models 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 still
uncertainties regarding the model parameters. A study was conducted with deterministic flow solutions that
displayed the frequency of ocurence of strain rate values shown in the bar figure 7 . One may conclude that
the uncertainty may occour in the of strain rate interval [1;60]. In addition it was investigated the influence
of all the parameters that rule the shear-thinning behaviour of the Carreau model. From this study, it was
concluded that the range of viscosity could be achieved with uncertainty in µ0 , mu∞ and n. These were
taken as random variables with uniform PDFs, with µ = 0.0456, 0.004 and 0.344 and σ 2 = 0.0092, 0.00028
and 0.099 respectively.
6
Figure 10: PDF for the Shear
with a blood model mixture
as a random variable
4.2.3
Figure 11: PDF for the Shear
with the blood model parameters as a random variable
Figure 12: PDF for the Shear
with the blood model as a
random variable
Distinctive Models
Another approach to the uncertainty in the blood behaviour was performed to include the blood model as
a stochastic variable. Four different blood models were used. The stochastic variable has an uniform PDF
ranging from -1 to 1, with each model having equally spaced ranges.
4.2.4
Results
Figure 13: Average of the
velocity field with the blood
model mixture as a random
variable
Figure 14: Average of the
velocity field with the blood
model parameters as random
variables
Figure 15: Average of the
velocity field with the blood
model 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 the
right hand side.Large influence in the pressure inlet and shear happens when uncertainty is applied to the
blood viscosity and almost no influence in mass flow split. The blood model uncertainty did not change the
mass 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 Geometry
Having decided on which viscosity model was to be implemented, the introduction of uncertainty parameters
in the model’s geometry follows. In this section the radius of the thrombosis in the model’s left branch was
7
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, specially
in the wall shear stress. Adding to this, the fact that current MRI tools only have a limited accuracy, which
typically varies from 0.3mm to 0.47mm, shows that care must be taken when analyzing MRI exams from
small arteries or veins.
The stochastic analysis of the thrombosis size influence on the velocity profile behind the Clot is shown
in Fig. 16, where apart from the mean velocity profile, it is also plotted the 95% confidence interval. This
plot 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 is
caused 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 radius
uncertainty close to the clot. The velocity values around the clot were interpolated in a clot free model to
accomodate the PDF’s entire range.
Figure 16: Stochastic parameters of the velocity profile
for different small thrombosis radius
Figure 17: Stochastic parameters of the velocity profile with the thrombosis radius and the blood model
parameters as random variables
Figure 18: Average of the velocity field with the radius
as a random variable
Figure 19: Standard deviation of the velocity field with
the radius as a random variable
When comparing the confidence interval of the velocity profile of this analysis (Fig. 16) with the one
taking the radius of the clot and also the blood as random variables (Fig. 17) it is clear that they are very
much alike, showing the largest differences in the recirculation zone.
8
The combined influence of both the radius and the blood model parameters is seen in Fig. 20, clearly
showing a large influence of the µinf ty and n parameter and radius, leading to the conclusion of the importance
of 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 bigger
clot has bigger influence in the flow inside the model. This is shown mostly by the confidence interval of the
velocity 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 increasing
the chances of a larger and possibly deathly clot. The large increase in not only the shear range, but also
in its magnitude suggests an increased probability of vein rupture leading to death. Here the he mass flow
split is affected by the uncertainty in the large obstruction, whereas in the small clot it almost did not have
any influence.
Figure 21: Stochastic parameters of the velocity profile
for different small thrombosis radius
Figure 22: Shear expansion coefficients with the radius
and 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 in
the size of a large obstruction in order to see if the effect on the hemodynamic factors are also significantly
affected by the blood uncertainty.
9
The effect of uncertainty is clearly mostly due to the size of the obstruction as a random variable. Fig. 22
shows the coefficients of the shear expansion, clearly showing that the radius influence is the most important
compared to the blood model parameters.
5
Conclusions
As proposed for the objectives, a verification of the numerical code was performed using simple 2D and 3D
geometries. Thus the validation of blood flow was made using models and results accepted as accurate in
the 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 showing
different behaviours for different strain rate ranges. This investigation took into account a combination of
different blood models. The results obtained showed that even though the range of influence of the three
approaches 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 uncertainty
in the shear force in the walls of the blood vessels.
The uncertainty regarding geometry was also deeply investigated. This uncertainty was quantified with
the NISP method with two different geometries. The obtained results from the uncertainty quantification
clearly show a great influence in shear and pressure in the vein. For critical clots it is even more important
to 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 varies
greatly with the size of the clot. This way with a small clot, the influence of the blood parameters was in the
same order of magnitude as the radius influence. However when it comes to critical geometries, the radius
size has definetely larger influence in the flow development.
References
[1] D. Wang and J. Bernsdorf, “Lattice Boltzmann simulation of steady Non-Newtonian blood flow in a 3D
generic stenosis case,” Computers and Mathematics with Applications, vol. 58, pp. 1030–1034, 2009.
[2] F. Yilmaz and M. Gundogdu, “A critical review on blood flow in large arteries; relevance to blood
rheology, viscosity models, and physiologic conditions,” Korea-Australia Rheology Journal, vol. 20, no. 4,
pp. 197–211, 2008.
[3] A. Gambaruto, J. Janela, A. Moura, and A. Sequeira, “Sensitivity of hemodynamics in a patient specific
cerebral aneurysm to vascular geometry and blood rheology,” Mathematical Biosciences and Engineering,
vol. 8, no. 2, pp. 409–423, 2011.
[4] W. Oberkampf and T. Trucano, “Verification and validation in computational fluid dynamics,” Progress
in Aerospace Sciences, vol. 38, pp. 209–272, 2002.
[5] J. Chen, X. Lu, and W. Wang, “Non-newtonian effects of blood flow on hemodynamics in distal vascular
graft anastomoses,” Journal of Biomechanics, vol. 39, pp. 1983–1995, 2006.
[6] D. Xiu and G. Karniadakis, “The Wiener-Askey polynomial chaos for stochastic differential equations,”
SIAM J. SCI. COMPUT., vol. 24, no. 2, pp. 619–644, 2002.
[7] S. Acharjee and N. Zabarras, “A non-intrusive stochastic galerkin approach for modeling uncertainty
propagation in deformation processes,” Computers and Structures, vol. 85, pp. 244–254, 2007.
[8] M. Reagan, H. Najm, G. Ghanem, and O. Knio, “Uncertainty quantification in reacting-flow simulations
through non-intrusive spectral projection,” Combustion and Flame, vol. 132, pp. 545–555, 2003.
10