Predictions of the extent of Renal Artery Stenosis in the context of rise in blood pressure.
Shukla, V. V. ; Padole, P. M. ; Sonolikar, R. L. 等
Introduction
Combination of a dormant lifestyle, a rich diet, lack of exercise
and smoking leads to an increase in blocked blood vessels, hypertension
and strokes [4-6]. A material called plaque builds up on the inner wall
of arteries cause stenosis. The plaque makes the artery wall narrow and
hard [3]. The renal arteries branch off directly from the abdominal
aorta and supply blood to the kidneys for filtration, wherein
homeostasis is maintained [1-2]. Narrowing, hardening and blocking of
renal artery due to plaque deposition is Renal Artery Stenosis (RAS).
One of the primary stages of RAS is Fibro muscular dysplasia (FMD). FMD
causes growth of fibrous tissues on the arterial wall. The most common
clinical manifestation of fibro muscular dysplasia (FMD) is
hypertension. [9-10]. RAS diminishes the blood supply to the kidneys,
which can cause them to atrophy and may ultimately lead to kidney
failure. The kidney with RAS suffers from the decreased blood flow and
often shrinks in size (ischemic nephropathy). The other kidney is at
risk for developing damage from the hypertension. This often develops
hypertensive nephrosclerosis [7-8]. The persistent elevated blood
pressures in the normal kidney can cause progressive scarring
(sclerosis) leading to progressive loss of filtering function in this
kidney as well. Both unilateral RAS and bilateral RAS can ultimately
lead to chronic renal failure. With total kidney failure, one needs
dialysis or a kidney transplant to stay alive [9].
Atherosclerotic RAS may present with hypertension, renal failure
(ischemic nephropathy), recurrent episodes of congestive heart failure
and flash pulmonary edema or may be discovered incidentally during an
imaging procedure for some other reason [11-13]. The high blood pressure
that is sometimes associated with RAS may be the first sign that it is
present, particularly if the hypertension is not responding to standard
treatment. Presence of a a swooshing sound from the artery indicates an
obstruction, may be heard through a stethoscope. It is vital to develop
effective treatments that prolong the life of a patient. Prompt
diagnosis and timely intervention done by a skilled vascular surgeon can
significantly decrease target organ damage and potentially cure high
blood pressure due to renal artery disease. Management of RAS consists
of three possible strategies: medical management, surgical management or
percutaneous therapy with balloon angioplasty and stent implantation. If
RAS is detected, the vascular surgeon will determine which of the method
of repair, Angioplasty, stent placement or arterial Bypass would be the
most appropriate and beneficial for each patient's unique situation
[14]. Renal artery stenting has replaced surgical revascularization for
most patients with atherosclerotic disease who meet the criteria for
intervention [20]. Patients with generalized atherosclerosis and renal
artery stenosis (RAS) more often die from cardiovascular causes than
renal failure. While physicians speak to patients about expanding
narrowed renal arteries and controlling blood pressure (BP), patients
desire to live longer, have fewer strokes [9].
In general, blood pressure in the circulatory system specifically
in aorta is affected by RAS; hence the relationship between rise in
blood pressure and the extent of RAS should be investigated thoroughly.
Although it is difficult to verify conclusively such relationships
without suitable in vivo studies, CFD can provide an excellent research
tool to help understand these underlying issues [15]. CFD is being
employed by several researchers to explore further the nature of flow
stagnation patterns [16-19]. Extensive work has been devoted to the
cardiovascular fluid dynamics during the last 25 years. For a general
mathematical modeling of arterial flow we refer [22-23]. Several
numerical studies dedicated to smaller (dia. Less than 2 mm) coronary,
carotid arteries and larger like AAA (Abdominal aortic Aneurysms, dia.
More than 22 mm) with their bifurcations, compliance mismatch, with and
without stent-grafts have been found in literature [21, 26-28], however
to the best of our knowledge, studies related to the influence of Renal
Artery Stenosis (RAS) on the rise of blood pressure have not yet been
reported. Arterial stenoses in the range 0-78% reduction of the cross
sectional areas have been previously investigated to estimate plaque
progression and wall stresses, both numerically and experimentally [24].
In this paper, authors have investigated, established and quantified the
relation between RAS and hypertension using Finite element method (FEM).
Materials and methods
Hemodynamic factors such as velocity gradients, shear stresses etc.
are believed to affect a number of cardiovascular diseases including
stenosis and aneurysms. Since resolving phenomenon in living body is
beyond the capabilities of in vivo measurement techniques, computer
modeling is expected to play an important role in gaining better
understanding of the relationship between cardiovascular disease and
hemodynamic factors [25]. When physical test methods are difficult (or
even impossible), computational models may sometimes be the only
alternative [24].
In Biomechanics, Finite Element Analysis (FEA) and computational
Fluid Dynamics (CFD) are increasingly used to perform strength analysis,
system response studies and design optimization of implants without the
need for time- and cost- intensive prototyping. The rapidly improving
computer performance enables a low-cost simulation of complex components
or composite structures. Different medical and technical demands on the
modeling or material laws can be examined by FEA.
Finite element method (FEM) based and not finite volume method
(FVM) based CFD is used to investigate the effect of renal artery
stenosis on the rise of blood pressure. This is because of inherited
advantages of FEM like it caters to the needs of geometric flexibility;
allow applying physical boundary conditions easily and accurately. It
satisfies global physical (linear) conservation laws automatically
especially quadratic quantities and even for which divergence theorems
are not applicable. Laplacian, divergence and gradient operators are
adjoint to each other in continuum in FEM and not in FVM. Phase speed of
FEM is always more accurate than that of FVM. Elliptic problem solutions
are more accurate in FEM than FVM.
Computational Model
a) Model geometry--Renal artery connecting aorta is almost straight
and symmetric. Therefore 2-D geometries of healthy and stenosed renal
arteries were created, as shown in Fig.1, with the FEM general purpose
computational fluid dynamics code ANSYS v 11.0 (ANSYS [R] Inc., USA).
The actual length of renal artery is less than 1 m; however blood
pressure variations are quantified on the basis of pressure drop per
meter length of the artery. Hence renal artery model constructed was 1 m
long and 6 mm in diameter. The various symmetrical stenoses model were
subsequently created at the center i.e. at 0.5 m from the inlet, by
creating arcs with three key points as shown in Fig.2. The length of all
the arcs along the inside arterial wall was 10 mm as shown in Fig.2
(inset). The minimum diameter at the site of stenoses and percentage of
stenoses based on diameters and areas calculated as shown in Table 1. In
general, Stenoses are specified relative to blocked areas, therefore
henceforth in this study, stenosis based on areas are followed.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
b) Material properties--Although blood is known to be non-Newtonian
in general we assume it to be Newtonian in this study. This is because
the renal artery considered in this study has maximum diameter of 6.0
mm. The velocity and shear rates in the larger arteries are high. The
apparent viscosity of blood in renal artery with relatively large
diameter is nearly constant and therefore non-Newtonian effects are
neglected [25]. Blood is modeled as an incompressible, homogeneous,
Newtonian viscous fluid, with a specific mass of 1050 kg/[m.sup.3] and a
constant dynamic viscosity of 3.5 X [10.sup.-3] Pa s [25]. The flow is
assumed to be steady state, Laminar and adiabatic [30].
c) Meshing-The mesh was built with 2--D Fluid 141 elements, each
having four nodes and 4 degrees of freedom (two translational velocities
and two pressures). The number of iterations are determined by using
different meshes, from coarse to progressively fine, until the inlet
pressure distribution is mesh convergent to within a prescribed
tolerance (~0.5%) [30]. The total numbers of nodes are about 23725 and
elements about 21888 for the normal configuration i.e. with 0% stenosis,
which slightly differs for the stenosed configurations due to local mesh
adaptations. While meshing the artery walls are set for desired number
of mesh divisions. The specific ratio of -2 for entry length and inlet
and outlets produces denser mesh at distal ends and near the walls
respectively. Uniform finer mesh using spacing ratio of 1 is applied for
stenosis section. The exit length mesh is obtained using spacing ratio
of 3 indicates coarser mesh towards outlet.
d) Boundary conditions and loads--A normal renal artery has a blood
flow of about 1 L/min (about 20%) of the total cardiac output from heart
[9]. The pressure profile along the intra-renal vasculature starts with
a mean arterial pressure of 100 mm Hg, and significantly drops between
the renal artery and the capillaries [1, 29].
Therefore to impose boundary conditions, the axial inlet velocity
of 0.57 m/s was assigned in X-direction and zero transverse velocity
components at the entrance of the vessel. No slip boundary conditions
were imposed on the impermeable, rigid vessel walls. At the outlet zero
atmospheric pressure was imposed [30]. The numerical simulations were
carried out for different values of Reynolds number (Re) ranging from
1125 to 1421.The external forces, such as those due to gravity or human
motion are assumed to be not significant and are neglected [25].
e) Methodology--To enable efficient and accurate numerical solution
to the modeled boundary value problem two key factors must be addressed:
the modeling phase and the solution phase. Each have these phases have
uncertainties linked with them and thus sensible selection of suitable
techniques, mathematical and numerical, must be taken to guarantee the
final approximate solution accurately represents the initial physical
system. The rise in the blood pressure depends not only on the viscosity
and density of the blood but also on the extent of the stenosis.
Therefore simulation of flow of water ([rho] = 1000 Kg/[m.sup.3] and
[mu] = 0.798 X [10.sup.-3] Pa-s) through various stenoses have also been
carried out.
f) Validation experiments--
Simple experimental setup has been designed to investigate the
effect of stenoses on the rise of pressure. The experimental setup as
shown in Fig. 3 consists of a positive displacement gear pump with
torque of 0.04 Nm. The outlet of the pump was connected to a
polyethylene tube resembling artery of diameter 6 mm and length 1 m
connected by means of a Tee. One end of the Tee is attached to the
polyethylene tube while the other is connected to a vertical tube of
diameter 12 mm and length 2 meters. This vertical tube acts as both a
piezometer and a surge tank in the experiment. Nine numbers of
converging-diverging Nylon66 specimens of length 10 mm were
manufactured; these represent variable stenosis (30.55% to 99.82%). The
specimens were fitted in the polyethylene tube one at a time. During the
experiment care is taken to maintain the flow rate through the
polyethylene tube to be fixed and is 1.64 x [10.sup.-3] [m.sup.3]/s.
[FIGURE 3 OMITTED]
The excessive power consumed by the pump and excessive head acting
on the pump is estimated by Eq. (1) and Eq. (2)
Power = 2[pi]NT/60 - - - - - - - - - - - (1)
Hf = Power/[rho]gQ + Piezometer correction - (2)
Where P--Power (Watt), N--speed of the pump (rpm), [rho]--Density
of liquid (Kg/[m.sup.3]), Q--Flow rate ([m.sup.3]/s), Hf--Excessive Head
(m), T--Torque (Nm)
Governing equations of motion
The blood flow in the renal arteries is assumed to be laminar
[25,38]. Flow motion was described by Navier-Stokes equations (principle
of momentum conservation and continuity equations) of incompressible
flow. For a steady-state analysis, the flow motion governing equations,
in the Cartesian coordinates.
[rho]([partial derivative]u/[partial derivative]t + u x [gradient]
u) - [gradient] x [sigma] = 0 - - - - - - - - - - (3a)
[gradient] x u = 0 - - - - - - - - - - - - - - - - - - - (3b)
Equation for fully developed laminar flow through a straight tube
(Hagen- Poiseullie flow) is given in Eq. (3c)
Q = [pi][D.sup.4]/128 [micro] ([partial derivative]p/[partial
derivative]z - - - - - - - - - - - - - - - - (3c)
Results and discussion
The effect of stenosis on the pressure distribution, velocity
profile, wall shear stress, stream function, Reynolds number and
pressure coefficients are studied. However the main focus was to study
the rise of pressure at inlet of the flow for water as well as the
blood. The flow of water was simulated using ANSYS [R] Inc. FLOTRAN CFD
and validated against the experimental observations. The flow of blood
was simulated and could not be verified by the experimentation. But we
comprehend that the results can be justified on the basis of
experimental validation of water flow and also against the published
data. In addition, a few medical cases have been examined successfully
with appreciable conformance.
Simulation Results
a) Flow of water
The flow of water through tube without blockage is turbulent (Re =
4648). The velocity profile as shown in Fig 4(a) indicates highest
velocity of 0.6182 m/s at the center of tube and fairly follows power
law near the wall. The highest velocity of 4.665 m/s (Re = 4676) is
observed in 99% stenosis at 0.16 mm from the wall. It is shown in Fig.
4(b); it fairly remains the same within the core of the tube except at
central region of the stenosis where velocity drops little to 4.147 m/s.
The CFD models of the tube with increasing size of blockages were then
solved using similar manner. The trends for increasing velocity,
increasing pressure drops, increasing fluid shear stress near wall and
constant pressure coefficients were observed, they are listed in Table
2. Fluctuations of Reynolds number have been observed, this may be
because water flow is turbulent in nature and very close to transition
zone. The velocity profile of tube with maximum blockage (99%) is shown
in Fig. 5(a). The pressure loss distribution is shown in Fig. 5(b). It
shows there is a loss of about 508.743 N/[m.sup.2] (3.9 mm of Hg)
between the entry and exit of the tube for zero stenosis. This means
that if there is gauge pressure of 13243.5 N/[m.sup.2] (100 mm of Hg) at
inlet, the gauge pressure at the outlet would be 12734.757 N/[m.sup.2]
(96.1 mm of Hg). The pressure loss distribution is shown in Fig. 5(c).
It shows there is a loss of about 4353 N/[m.sup.2] (32.8 mm of Hg)
between the entry and exit of the 99% blocked tube. Maximum fluid shear
stress (very small value of 1.439 Pa) is found near the wall at the
inlet of the flow. Maximum fluid shear stress (103.554 Pa) is found at
the center of stenosis wall as shown in Fig. 5(d).
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
a) Flow of blood The flow of blood through renal artery without
blockage is Laminar (Re = 1421.1). The velocity profile as shown in Fig
6(a) indicates highest velocity of 0.789 m/s at the center of artery and
is parabolic. The velocity profile of tube with maximum blockage (99%)
is shown in Fig. 6(b). The highest velocity of 4.688 m/s (Re = 1125.12)
is observed at 0.16 mm from the wall. And as shown in Fig. 7(a), it
fairly remains the same within the core of the renal artery. There exist
recirculations both up and down stream of the stenosis as shown in Fig.
7(a). The stenosis forms a jet and the impasse of the jet results in
recirculation upstream and the jet flow induces recirculation
downstream. One of the important things noticed in the study is little
reduction in velocities downstream and far away from the stenosis
region. The CFD model of the renal artery with increasing size of
stenoses are then solved using similar manner. The trends for increasing
velocity, increasing pressure drops, increasing fluid shear stress near
wall and constant pressure coefficients are observed, they are listed in
Table.3. The pressure loss distribution is shown in Fig. 7(b). It shows
there is a loss of about 684.377 N/[m.sup.2] (6.97 mm of Hg) between the
entry and exit of the artery. This means that if there is gauge pressure
of 13243.5 N/[m.sup.2] (100 mm of Hg) at inlet, the gauge pressure at
the outlet would be 12599.123 N/[m.sup.2] (94.8 mm of Hg). Maximum fluid
shear stress (4.19 Pa) is found near the wall at the inlet of the flow.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
Reynolds number in blood flow simulation is observed to be
gradually reducing from 1421 to 1125 for 0% to 99% stenoses. This could
be because the average velocity does not increase by the rate at which
diameter of the flow reduces and also there may be slight increase in
effective viscosity. This suggests that reducing velocity minimizes the
blood flow to kidney because of the presence of stenosis. The pressure
loss distribution is shown in Fig. 7(c). It shows there is a loss of
about 5829 N/[m.sup.2] (44 mm of Hg) between the entry and exit of the
99% stenosed artery. Maximum fluid shear stress (183.856 Pa) is found at
the center of stenosis wall as shown in Fig. 7(d).
The percent pressure rise as shown in table 2 and 3 are computed by
Equation (4).
% pressure rise at inlet = [(13243.5 + Pr essuredifference) -
13243.5]/13243.5 X 100 (4)
And relative % pressure rise at inlet as shown in table 2 and 3 are
computed by the Equation (4)
Relative % pressure rise at inlet = {(% pressure rise at inlet} -
{(% pressure rise at inlet for 0% stenosis} (5)
And relative % pressure rise at inlet as shown in table 2 and 3 are
computed by the Equation (2) Equation (4) helps in computing relative
percent increase over the initial pressure of 100 mm of Hg at inlet.
Whereas Equation (2) helps in setting the initial relative percentage of
pressure rise to zero in order to standardize the reference relative
percentage of pressure. Thus the results of pressure difference from
each CFD analysis is substituted in Eq. (4) and then in Eq. (5) in order
to compute Relative percentage of pressure rise for rest of the cases.
Referring Table 2 and Table 3, Pressure drop increases and therefore
pressure at inlet rises in both the cases. Rise of pressure in blood
flow is found higher as compared to water flow. This is because of
viscosity of water is approximately 4.3 times to that of blood.
Also this is quite expected as water flow is turbulent where as
blood flow is laminar and similar to Poiseullie flow. Velocities in both
cases increases gradually with increasing extent of the blockage in both
the cases and are almost equal, this is because the flow rate and cross
sectional areas are identical. Variations of velocity and 2-D stream
function are shown in Fig. 8.
[FIGURE 8 OMITTED]
A very interesting flow structure is that the secondary blood flow
from aorta while entering renal artery (0% stenosis) forms vortices near
the inlet wall of renal artery. Also the wall shear stresses at Re =
1421 at the entry point are considerably high. This is supposed to be
the favorable location for plaque deposition and arterial remodeling and
would eventually lead to stenosis location. The trends of rise of
pressure are found similar but different regression coefficients for
blood flow and water flow; based on CFD results have been determined.
The comparison of the CFD results for relative percentage of
pressure rise is shown in Fig. 9. It is observe that for the same extent
of stenosis, the blood pressure rise at inlet of the renal artery is
more than that of water pressure rise in the tube. Also up to 82% there
is about 5.5% rise in blood pressure but then onwards blood pressure
shoots up rapidly.
[FIGURE 9 OMITTED]
Limitations
The present CFD study of blood flow through renal artery considers
steady-state boundary condition, while blood flow in arteries with
larger diameters is highly pulsatile. Steady flow results are not
physiologically relevant (16-19); however they demonstrate clinically
acceptable patterns. In future, the condition of pulsatile blood flow
through renal artery based on patient specific data needs to addressed
using transient analysis [30, 33, and 36]. Also the displacement
variation of the left and right kidney during normal respiration [29,
34, and 35] is to be accounted in computing arterial wall parameters.
Experimental Results
It is found that, to maintain the flow rate constant, the pump
draws increasingly higher power i.e. speed increases. Therefore it was
necessary to track pump speed for each of the specimen fitted. When the
pump was switched on, water flows through the polyethylene tube. The
readings of rise of water in piezometer tube and pump speed were noted
for full open and thereafter for each fit of the nylon specimen. The
pump speed for each of stenoses and rest of the determined parameters
are shown in Table 4. Equation (3) is used to compute the power consumed
by the pump and Equation (4) is used to compute pressure (Head) Loss for
each run. The piezometer correction is a small difference in the rise of
water level due to maintained constant flow rate and the blockage
fitting in piezometer tube, measured in unit time interval.
The experimental results are in good agreement with CFD results of
water. The comparison for experimental pressure difference of water at
the inlet and outlet of the polyethylene tube and the CFD pressure
difference is shown in Fig. 10.
[FIGURE 10 OMITTED]
Closing Remarks--Based on the CFD analyses, data obtained for % RAS
and blood pressure rise. A mathematical model is researched to fit this
data. Sigmoidal family of curves producing growth curves is common in a
wide variety of applications such as biology and engineering.. These
curves start at a fixed point and increase their growth rate
monotonically to reach an inflection point. After this, the growth rate
approaches a final value asymptotically. Selected Logistic Model has the
equation (6) Covariance matrix is computed and listed in Table 4
y = a/(1 + b * [e.sup.-cx]) - (6)
With a Standard error of 3.42 and a correlation coefficient of
0.97, the values of the constants to be used in Eq. (6) are computed as
shown in Table 6. Model verification is shown in Fig. 11.
[FIGURE 11 OMITTED]
Conclusion
The objective of this study was to establish the effect of RAS upon
the rise in blood pressure acting through aorta and thus on the aortic
valve of heart. The percent increase in stenosis increases pumping load
on heart, this ultimately reduces the volume of blood from heart to the
tissues beyond the stenosis. The reduced volume flow to kidney results
in ischemic nephropathy. For a single flow conduit this can be justified
by the basic laws of science, however human body is comprised of several
arteries and veins and thus complex blood flow networks exists. In such
multifarious case it would be precarious to state that pumping load on
heart alone will rise or in general blood pressure will increase. Rather
the test case dealt here assumes 100% stenosis elsewhere in the
circulatory system, which cannot be true. On the contrary, when there is
rise in blood pressure the Entire healthy circulatory system soothe out
this effect by distributing the blood pressure rise proportionately,
however, total nullification of the blood pressure rise effect is also
impossible. And therefore, it is possible that this analysis may not be
accurate, however it gives an overall idea and the trends towards the
future risk to the patients, diagnosis and treatments to general medical
practitioners. Although this basic work is one of its kinds, the
numerical technique is validated reasonably, against the clinical data
and published results. The present study provides easy to use
information and one of the clinical steps to general medical
practitioners to diagnose Renal Artery Stenosis (RAS). The information
could be used in the tabular or graphical form to conjecture the
likelihood of RAS based on measured blood pressure of the patient.
Following conclusions can be drawn from the computed results
(a) As stenosis increases there is rise in blood pressure. The rise
in blood pressure is steeper towards the far end of the graph,
indicating rate of increase in blood pressure/gradients are higher for
stenoses above 88% and precipitously rises thereafter.
(b) For and above 88% stenosis, velocity vectors are sharp and if
oriented towards artery can lead to erosion kind of forces, which would
facilitate in remodeling artery leading to further stenosis at that
site.
(c) Reynolds number throughout the renal artery ranges from 1125 to
1421 clearly indicates laminar flow.
(d) Blood Pressure rise due to severe renal artery stenosis
(99.82%) are observe to be maximum (about 38%). Stagnation pressure
remains little higher than the static pressure in the stenosed renal
artery.
(e) The simulation has revealed that under steady flow conditions,
the size of flow recirculation zone increases with flow Reynolds'
number.
(f) Higher shear stress at the entry of the normal renal artery
indicates favorable location to form stenosis.
Acknowledgements
The authors wish to thank Dr. S. S. Gokhale (Director, V.N.I.T),
Nagpur, Dr. Shirish Deshpande (Radiologist, CIIMS Hospital), Dr. H. M.
Mardikar (Spandan Hospital) Nagpur, for guidance, verification and
validation of the research work.
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pulmonary arterial bifurcation, Journal of Biomechanics (2207) article
in press
V. V. Shukla (a), *, P. M. Padole (a), R. L. Sonolikar (a), Shirish
Deshpande (b) and H. M. Mardikar (c)
(a) Dept. of Mech. Engg., Visvesvaraya National Institute of Tech.,
Nagpur, M.S., India.
(b) Dept. of Radiology, Central Indian Institute of Medical
Science, Nagpur, M.S., India.
(c) Spandan Heart Institute & Research Center, Nagpur, M.S.,
India.
Table 1: Percentage of stenosis for stenosed arterial diameters and
areas.
Inside Inside cross Stenosis based Stenosis based
diameter Area ([m.sup.2]) on the inside on the cross
(m) diameter sectional area
0.006 0.00002826 0% 0%
0.005 0.000019625 16.66% 30.55%
0.0043 1.45147E-05 28.33% 48.64%
0.0035 9.61625E-06 41.66% 65.97%
0.0031 7.54385E-06 48.33% 73.3%
0.0025 4.90625E-06 58.33% 82.63%
0.002 0.00000314 66.66% 88.88%
0.0015 1.76625E-06 75% 93.75%
0.001 0.000000785 83.33% 99.72%
0.0008 5.024E-07 86.66% 99.82%
Table 2: CFD results of investigated parameters for water flow.
% Pressure Max. Shear pressure
Sten difference Velocity Stress coefficient
osis (N/[m.sup.2]) (m/s) (N/[m.sup.2])
00 508.743 0.618 1.439 0.01077
30 549.150 0.777 2.391 0.01906
48 559.928 0.881 3.469 0.01906
65 583.204 1.075 5.547 0.01874
75 605.108 1.213 7.177 0.01877
82 668.572 1.525 11.21 0.01903
99 2820 3.756 67.88 0.01904
% 2-D % Relative Reynolds
Sten stream pressure % Number
osis function rise at pressure
inlet rise
00 2.836 3.841 0.000 4648
30 3.307 4.146 0.305 4873
48 3.343 4.227 0.386 4747
65 3.306 4.403 0.562 4714
75 3.337 4.569 0.727 4712
82 3.397 5.048 1.206 4777
99 3.445 21.293 17.452 4706
Table 3: CFD results of investigated parameters for blood flow.
% Pressure Max. Shear pressure
Sten difference Velocity Stress coefficient
osis (N/[m.sup.2]) (m/s) (N/[m.sup.2])
00 684.377 0.7895 4.190 0.0550
30 767.821 0.8925 8.978 0.0779
48 776.281 0.9782 8.978 0.0779
65 803.69 1.142 8.978 0.0781
75 854.692 1.266 12.201 0.0781
82 1030 1.538 19.37 0.0781
99 3910 3.759 117.10 0.0781
% 2-D % Relative Reynolds
Sten stream pressure % Number
osis function rise at pressure
inlet rise
00 3.290 5.1676 0.000 1421
30 3.472 5.797 0.630 1338
48 3.500 5.861 0.693 1261
65 3.470 6.068 0.900 1199
75 3.579 6.453 1.286 1177
82 3.816 7.777 2.609 1153
99 3.614 29.523 24.356 1127
Table 4: Experimental results for the pump speed during induced
stenoses (0 to 99%).
% Pump Piezo meter Power Head loss Pressure differ
Stenosis speed correction P(watt) Hf (m) (N/[m.sup.2])
N (rpm) (m)
0 198 0.001 0.8289 0.052 513
30 217 0.0015 0.90850 0.058 566
48 224 0.002 0.9378 0.060 589
65 231 0.0025 0.9671 0.062 611
73 241 0.003 1.0089 0.065 642
82 269 0.005 1.1262 0.075 732
88 299 0.007 1.2518 0.084 828
93 571 0.009 2.3905 0.157 1539
99 1742 0.015 7.2931 0.466 4575
Table 5: Covariance matrix for the Logistic Model.
F parameter a 21.7343 829.771 -0.140
parameter b 829.771 32035.258 -5.410
parameter c -0.14 -5.410 0.0091
Table 6: Best fit Constants of Logistic model Eq. (6).
a -4.06294
b -56.1312
c 0.0392398
