^{1}University of St. Thomas, School of Engineering, 2115 Summit Ave, St. Paul and Minneapolis, Minnesota 55105-1079, United States
^{2}NYU Langone Medical Center, 333 East 38th St, New York 10016, United States
Received date: December 07, 2015; Accepted date: December 15, 2015; Published date: December 21, 2015
Citation: Plourde BD, Vallez LJ, Sun B, Abraham JP, Staniloe CS (2015) The Effect of Plaque Removal on Pressure Drop and Flow Rate through an Idealized Stenotic Lesion. Biol Med (Aligarh) 8:261. doi:10.4172/0974-8369.1000261
Copyright: © Plourde, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.
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Numerical calculations have been performed to quantify the importance of plaque removal on blood flow. The artery under consideration is the popliteal artery which is susceptible to plaque lesions. An orbital artherectomy device was used to partially remove a calcified plaque layer. Measurements taken before and after the treatment were used in idealized calculations and pressure losses through the lesion were determined. It was found that the removal of plaque by orbital atherectomy increases the blood flowrate through the artery. At the same time, there is a major reduction of pressure loss through the lesion. After treatment, the systolic pressure drop was 2.5 times less than prior to treatment. The cycle-averaged pressure drop was improved by a factor of 3.5. The results are similar for a wide range of plaque lesion lengths (from 3 mm to 18 mm). A deeper investigation into the source of pressure loss reveals that the majority of the loss is confined to the entrance of the lesion and is caused by flow acceleration (and later deceleration) rather than by friction. The calculations were repeated with three increasingly complex numerical methods (steady laminar, unsteady laminar, and unsteady transitional). It was found that all methods were in good agreement so that more computationally expensive techniques are not required in order to obtain accurate results. The results of the simulation were compared with clinical pressure measurements before and after treatment. The two results were found to be in good agreement.
Cardiovascular disease; Plaque; Blood flow; Stenosis; Computational fluid dynamics
Cardiovascular disease is a major health risk throughout the world and the leading cause of death in some developed countries. Aside from risks to the coronary system, peripheral arteries can be compromised by plaque buildup. The topic of blood flow through arteries and the impact of the flow on the health of living organisms has been dealt with extensively in the literature and for brevity, review articles are referenced here for interested readers [1-4]. Prior research has included both numerical and clinical/experimental information. Research has progressed from steady flow in circular geometries [5-7]; flow through patient-specific arteries created from medical images [8-9]; unsteady flow driven by fluctuating pressure waveforms [10] with both Newtonian and non-Newtonian constitutive models for blood viscosity [11-13].
Aside from the relation of pressure to flow through arteries, the constitution of the wall has a significant impact on mass transport through the tissue [14-21]. Furthermore, the presence or absence of plaque has an impact on the compliant response of the arterial wall which, in turn, affects the blood flow. In the recent literature, studies have been completed to quantify this two-way interaction [22-42].
In the present study, a new question will be addressed related to the impact of plaque removal on the flow/pressure relationship. In particular, calcified plaque will be removed using a technique called orbital atherectomy [43-45]. The device is designed to be used either as a stand-alone procedure or more commonly in conjunction with balloon angioplasty. The preferential target is calcified plaque in the peripheral arteries for the treatment of peripheral arterial disease.
The device used for the procedure was the Diamondback 360 Peripheral Orbital Atherectomy System (OAS; Cardiovascular Systems, Inc.). The device is designed to remove calcified plaque while preserving the compliant tissue. For more information on the procedure, readers are invited to [41]. For the present study, the pre-treated lesion had a minimum diameter of 3.39 mm and the post treated lesion opening was 5.05 mm measured by intravascular ultrasound. These measurements were incorporated into an idealized stenotic geometry to be discussed shortly.
In this particular case, standard lower extremity angiography of the entire vascular tree was performed. The Combo Wire^{®} (Volcano Corporation) was normalized at the level of the common femoral artery, and then placed in the distal popliteal artery, just proximal to the take-off of the anterior tibial artery. The politeal artery was selected not only because it is a location that is often occluded by plaque, but also because its relatively constant diameter and straight length make it a candidate for an idealized simulation such as that presented here.
The resting gradient was recorded. A blood pressure cuff was positioned at the calf level and inflated for 1 minute at 10 mmHg above the systolic blood pressure. After cuff deflation, the hyperemic gradient was recorded. At that time, an IVUS catheter (Volcano Corporation) was passed across the lesion. The reference vessel diameter was measured in the normal arterial segment just proximal to the lesion. The minimal luminal area was measured at the narrowest point inside the lesion. Once the procedure was completed, the pressure wire was repositioned in the same segment of the popliteal artery, and measurements were repeated. Case files were collected and archived on the ComboMap® console from Volcano Corporation.
An image of the situation is shown in Figure 1. The figure shows a close-up view of a plaque lesion which protrudes into a circular artery. The top image shows the idealized lesion prior to treatment and the bottom image is after treatment. It is seen that the lumen of the artery is enlargened by the treatment as confirmed by ultrasound. The lengths of the lesion will be systematically changed from 3 mm to 18 mm in the simulations to quantify the effect of length on the pressure drop through the lesion.
Numerical simulations of blood flow through the artery will be completed and the pressure drop through the lesion is calculated. Comparisons of steady flow with unsteady pulsatile flow will be made as will a comparison between the calculated and experimental results.
It is recognized that this plaque scenario is idealized in its axisymmetry and its shape. However idealization of the shape will allow generalized conclusions to be made which characterize the flow phenomena. Those generalizations can be extended to patient-specific situations in future work.
Numerical model
The numerical model is based on the control-volume calculation technique wherein the fluid domain is subdivided into a large multitude of control volumes. Equations of mass and momentum conservation are applied to each control volume (Equations 1 & 2). To make the present results most useful, both steady and unsteady calculations will be made. The steady calculations are made solely for the systolic peak velocity whereas the unsteady calculations are made for four cycles of the pulsating flow so that periodic steady flow is obtained.
Flow measurements made before and after treatment allowed the construction of the velocity waveforms which were applied at the inlet to the fluid domain. The flow profile at the inlet was fully developed and the upstream extension was sufficient that the results were not dependent on its length.
At the downstream location, zero second derivatives were employed on all transported variables. As with the inlet, the downstream extension was sufficiently long so that the results did not depend on its location.
The velocities recorded upstream of the lesion are displayed for one cardiac cycle in Figures 2 and 3.
The governing differential equations are:
(1)
for mass conservation and
(2)
for conservation of momentum. To simplify the expressions, tensor notation has been used. The symbols u, ρ, x, p and μ represent velocity, density, location, pressure, and viscosity, respectively. The subscript t refers to turbulent quantities.
To allow for the potential transition from laminar-to-turbulent flow, the shear stress transport model of Menter [46] is utilized which involves two turbulent transport equations for turbulent kinetic energy k and specific rate of turbulent dissipation ω (Equations 3 & 4).
(3)
and
(4)
The symbol γ is an intermittency term which takes on values of 0 when the flow is laminar and 1 when the flow is fully turbulent. Intermittency is solved from two connected transport equations (Equations 5 & 6).
(5)
and
(6)
which are taken from Menter [47-49], but modified to account for internal flows. This method has been shown to correctly capture very low Reynolds numbers transitional flows in the literature [50-59].
The blood viscosity was modeled with a non-Newtonian model described in [9] as shown in Equation (7).
(7)
where K = 0.0147 (kg/m-s^{1.22}), S is the strain rate, and n = 0.78 [60].
The spatial discretization was performed in a sequence of increasingly refined stages wherein the elements were reduced in size until the results were independent of mesh. The mesh was deployed finely near the artery wall where large gradients of velocity exist. A side view and end view of the mesh are shown in Figure 4.
The figure also shows an internal view of the mesh with a callout that magnifies the mesh across the stenosis lumen. The magnification shows the refinement of elements in the near wall region. The final number of elements depended on the length of the lesion but were approximately 50,000.
Similarly, the time step size was modified until results were independent of time step. The final time step used for the calculations was 0.01 seconds. In total, three different calculations were made for each lesion. One calculation was for laminar flow at peak systolic velocity. The second calculation was pulsatile (four cardiac cycles) using a laminar solver, and the third calculation was pulsatile (four cardiac cycles) where laminar-to-turbulent transition was included. A comparison between these three calculations will be provided later and it will be seen that they are in good agreement with each other. Since it was found that the steady laminar simulation provided adequate flow/pressure results, it is possible to save computational resources for future calculations.
The first set of results to be discussed is the pressure losses through the stenosis for the three calculation methods (steady laminar, unsteady laminar, and unsteady transitional). The comparison will be made at peak systole for the unsteady cases. Comparisons will be made both before and after the atherectomy treatment.
The plaque was removed through the use of the Diamondback 360 orbital atherectomy device (Cardiovascular Systems, Inc). We used a 2.0mm crown which was initially activated at 60,000 rotation/min. After 2 passes, the orbital speed was increased to 90,000 rotations/min. The plaque removal procedure was terminated after the final 2 passes at 120,000 rotations/min. The activation time was 30 seconds per pass, with a pause of 30 seconds in between passes. The atherectomy result was checked via angiogram, and the residual stenosis was measured with the IVUS. Table 1 shows a summary of the results for the pre-operative case. In the table, pressure drop through the lesion for the steady laminar, unsteady laminar, and unsteady transitional. The unsteady values were obtained at the systolic flow.
In the clinical experiment, the lesion length was estimated to be 1 cm and the measured systolic pressure drop was 12 torr. This value agrees to within the accuracy of the measurements to the calculated results. The results of Table 1 are graphically displayed in Figure 5.
Case | Pre/Post | Lesion Length (mm) | Steady Laminar ΔP (torr) | Unsteady Laminar ΔP (torr) | Unsteady Transitional ΔP (torr) |
---|---|---|---|---|---|
1 | Pre | 3 | 10.9 | 11.6 | 10.9 |
2 | Pre | 6 | 11.3 | 12.0 | 11.3 |
4 | Pre | 12 | 12.0 | 12.8 | 12.4 |
5 | Pre | 18 | 12.8 | 13.9 | 13.1 |
Table 1: Pre-treatment systolic pressure drop through lesion as a function of lesion length.
Similarly, Table 2 presents the cardiac-cycle average pressure drop through the lesion. The measured value for the 1 cm lesion was 2 torr, again, in good agreement with the calculations. The date from Table 2 is displayed graphically in Figure 6.
Case | Pre/Post | Lesion Length (mm) | Unsteady Laminar ΔP (torr) | Unsteady Transitional ΔP (torr) |
---|---|---|---|---|
1 | Pre | 3 | 2.26 | 2.11 |
2 | Pre | 6 | 2.37 | 2.28 |
4 | Pre | 12 | 2.93 | 2.81 |
5 | Pre | 18 | 3.07 | 2.95 |
Table 2: Pre-treatment mean pressure drop through lesion as a function of lesion length.
From Tables 1 and 2 and from Figures 5 and 6, it is seen that there is a slight difference between the three calculations methods (steady laminar, unsteady laminar, unsteady transitional) with the systolic pressure drop for the transitional calculation very closely matching that of the steady laminar case. The unsteady laminar situation slightly overpredicts the systolic pressure by less than 10%. It is therefore open to question whether the added complexity of the transitional simulation is worthwhile.
For the cardiac-cycle averages listed in Table 2, again the unsteady laminar calculation slightly overpredicts the unsteady transitional by approximately 5%. Again, this motivates the question of whether the complexity of transitional calculations is necessary for meaningful results.
Another conclusion from the results is that the length of the lesion does not dominate the pressure drop; despite a six-fold increase in the length of the lesion, there is approximately a 20% increase in systolic pressure drop and a 40% increase in cycle-average pressure drop. This issue will be explored in more detail later in this report but it indicates that the pressure drop caused by constriction (and flow acceleration) exceeds that caused by friction within the lesion. It is important to note that while flow acceleration results in a pressure decrease (Bernoulli Effect), it is possible for a pressure recovery to occur if the flow is decelerated carefully.
After the orbital atherectomy treatment, the pressure drop through the lesion is significantly reduced (even through the flowrate is increased by 61% at peak systolic flow and by 20% for the full cardiac cycle). Tabulated results are provided in Table 3. The pressure drop through the stenosis at systolic conditions is reduced by a factor of approximately 2.5 and the mean cardiac-cycle pressure is reduced by a factor of approximately 3.5.
Case | Pre/Post | Lesion Length (mm) | Steady Laminar Δ P (torr) | Unsteady LaminarΔ P (torr) | Unsteady Transitional Δ P (torr) |
---|---|---|---|---|---|
6 | Post | 3 | 3.92 | 4.12 | 3.81 |
7 | Post | 6 | 4.16 | 4.43 | 4.19 |
8 | Post | 12 | 5.30 | 5.87 | 5.59 |
9 | Post | 18 | 6.00 | 6.15 | 5.84 |
Table 3: Post-treatment systolic pressure drop through lesion as a function of lesion length.
In the clinical experiment, the post-operative systolic pressure drop was measured to be 4 torr, again in agreement with the calculations.
The results of Tables 3 and 4 are displayed graphically in Figures 7 and 8. It is again seen that there is good agreement between the three calculation methods and transitional calculations are not necessary for such calculations.
Case | Pre/Post | Lesion Length (mm) | Unsteady Laminar Δ P (torr) | Unsteady Transitional Δ P (torr) |
---|---|---|---|---|
6 | Post | 3 | 0.63 | 0.58 |
7 | Post | 6 | 0.67 | 0.61 |
8 | Post | 12 | 0.88 | 0.83 |
9 | Post | 18 | 0.90 | 0.86 |
Table 4: Post-treatment mean pressure drop through lesion as a function of lesion length.
It is also seen that the impact of lesion length on pressure drop is not very large, a six-fold change in the lesion length changes the systolic and mean pressure drop by approximately 50%. This finding again indicates that the pressure drop caused by constriction (and flow acceleration without downstream recovery) exceeds that caused by friction within the lesion.
To explore more fully the pressure variations within a lesion, two representative images have been prepared. The first, Figure 9, shows velocity contours along a central plane which bisects the lumen. In the figure, significant flow acceleration occurs at the entrance to the lesion. A strong jet emerges from the downstream end before spreading out to fill the cross section.
The acceleration causes a concomitant pressure loss (as expected from fluid mechanics) and the pressure drop is seen confined to the lesion entrance. The lack of a controlled downstream deceleration renders this pressure decrease permanent. There is a further decrease in pressure through the lesion but it is small compared to that at the lesion entrance. The lesion shown in Figures 9 and 10 was pre-operative but it is representative of the fluid flow behavior for the other cases. This finding leads to the conclusion that the dominating factor to lesion pressure drop is the degree of acceleration (and consequently the decrease in pressure) rather than friction.
Concluding remarks
A numerical simulation has been performed to assess the impact of plaque removal on blood flow and pressure loss. Plaque removal was performed using an orbital atherectomy procedure on calcified plaque. Changes to the open cross section were measured by intravenous ultrasound and were used to construct an idealized computational model. It was found that the removal of plaque increased the flowrate through the stenosis while simultaneously decreasing the pressure drop. The peak systolic pressure drop decreased by a factor of 2.5 and the cycle-mean pressure decreased by a factor of 3.5. The results are consistent for lengths of plaque lesions that varied from 3 mm to 18 mm; the impact of lesion length on pressure drop was much smaller than the impact of plaque thickness.
An exploration of velocity and pressure variation throughout the interior space revealed that the major component of pressure drop was from the blood flow acceleration at the stenosis inlet and the lack of a pressure recovery with deceleration. Only secondarily did friction within the lesion contribute to pressure variations.
The calculations were performed using three different numerical schemes of increasing complexity. The first was a steady laminar flow calculation at systolic conditions. The second was an unsteady laminar flow of four cardiac cycles to ensure that periodic steady conditions were reached. The final calculation approach utilized a new and complex transitional modeling code that accounts for laminar-to-turbulent fluctuations. All methods were in good agreement so that it is not necessary to use more computational expensive methods.
Comparisons between the simulations and experiments were made and found to be in very good agreement both before and after the operation. This agreement lends to the veracity of the calculations.
The next stages of the work involve obtaining patient-specific plaque profiles and repeating the calculations to determine whether the generalized findings presented here can also be applied to individual patients. It is also of interest how these results complement other works which show, for instance, effects of heart rate and the evolution of arterial disease and cardiac health [61-62]. This connection is only recently explored.
The authors gratefully acknowledge the support of Cardiovascular Systems Inc.