LJ Vallez1, B Sun1, BD Plourde1, JP Abraham1* and CS Staniloae2
1University of St. Thomas School of Engineering, 2115 Summit Ave, St. Paul, MN 55105-1079, USA
2NYU Langone Medical Center, 333 East 38th St. New York, NY 10016, USA
Received: 03 December, 2015; Accepted: 08 December, 2015; Published: 12 December, 2015
*Corresponding author:
J.P. Abraham, University of St. Thomas School of Engineering, 2115 Summit Ave, St. Paul, MN 55105-1079; E-mail: @
Vallez LJ, Sun B, Plourde BD, Abraham JP, Staniloae CS (2015) Numerical Analysis of Arterial Plaque Thickness and its Impact on Artery Wall Compliance. J Cardiovasc Med Cardiol 2(2): 026-34. DOI: 10.17352/2455-2976.000019
© 2015 Vallez LJ, 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.

A numerical analysis was completed on the influence of plaque on the compliance of an artery wall. The analysis allowed a systematic variation of the transluminal pressure difference and plaque thickness. Included in the analysis was a sheath of surrounding tissue that provides support during the cyclical deformation. The analysis incorporated progressively thicker layers of plaque to quantify their impact on the artery compliance. It was found that the presence of plaque plays a significant role in reducing the compliance. It is also shown that modest reductions in plaque thickness can result in large changes in the compliance and a positive health outcome. The calculations also included only a health artery (absence of plaque). The healthy artery possessed the largest compliance of the cases. The healthy-artery results were compared with literature and found to be in good agreement. The calculations were made with four different transmural pressure variations to capture the entire range of possible values. The results for all pressure waveforms agreed qualitatively although there was some difference in magnitude.


Cardiovascular disease is a major cause of mortality throughout the world. The history of cardiovascular research is rich and although this study is not intended to be a review of the subject, a short summary related to the present work is necessary. Interested readers are invited to review articles such as [1-4]. These studies are representative of the literature which deal with the complexities of hemodynamics. Among the important subtopics are the relationship between the wall and the fluid. The fluid exerts a shear stress on the wall which is believed to be a causing of thickening of the wall and the initiation of cardiovascaular disease. The flow also has an impact on transport through the arterial wall [5-20].

In addition to these hemodynamic-focused studies, it has been found that arterial compliance (the distension of an artery wall during the cardiac cycle) is an important indicator of disease progression. In particular, for a diseased artery, the stiffened arterial wall or the rigid plaque layer reduces the otherwise healthy-artery response to pressure fluctuations [21-22]. In addition, the presence of a stenosis can affect the blood velocity profile [21-26] which can be measured Doppler or ultrasound techniques [27-37].

The above-mentioned fluid-wall interaction is two-way in that the fluid affects the wall and the wall in turn affects the flow. This interaction is often termed "two way fluid structural interaction (two way FSI)". Incorporating the two-way interaction is currently an active area of research [38-59].

Among these references, the majority treat the artery as an isolated structure with stress-free conditions on the exterior surface and upstream and downstream surfaces that are able to move radially but not axially. Insofar as the surrounding tissue does exert an influence on the artery as it is subjected to cardiac-cycle pressure, it may be important to include the tissue in the analysis. To the best knowledge of the authors, three studies have investigated this issue. In [57], the pressure of the surrounding tissue was incorporated into the analysis by means of an exterior non-zero pressure. In [58], the surrounding tissue was modeled with a viscoelastic support along the artery. Most recently [59], a systematic study was completed to vary the thickness of the surrounding artery wall until further changes to its thickness no longer impacted the results. It was found that the presence of plaque significantly reduces the compliance of the artery and that for a healthy artery, a 6.5 mm sheath is required. Omission of the surrounding tissue resulted in a significant over prediction of the compliance.

The present study is aimed at determining how removal of plaque changes the artery compliance. There has been a recent advance in atherectomy procedures and techniques [15,60-66]. The procedures have been shown to remove plaque, open clogged arteries, and increase flow rate. What is less known is the impact of plaque removal on the artery-wall compliance. If the effect of these treatments on compliance can be quantified, it can greatly add in the determination of treatment for patients.

Mathematical model

In the present analysis, the finite element technique will be combined with experimental data obtained from patients (pre- and post-treatment arterial and plaque dimensions). The targeted artery is the popliteal. The experiments provided pre- and post-operative flow and pressure information upstream and downstream of a superficial femoral artery lesion. In addition, the diameter and cross sectional areas upstream, downstream, and within the lesion were found.

In the present analysis, the finite element technique will be combined with experimental data obtained from patients (pre- and post-treatment arterial and plaque dimensions). The targeted artery is the popliteal. The experiments provided pre- and post-operative flow and pressure information upstream and downstream of a superficial femoral artery lesion. In addition, the diameter and cross sectional areas upstream, downstream, and within the lesion were found.

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.

The simulated geometry is shown in Figure 1. The figure shows a three-part solution domain that includes calcified plaque, artery wall tissue, and surrounding tissue. The thicknesses of the artery wall and the surrounding tissue and the material properties are from [59]. They are listed here in Table 1. Linear elastic properties have been shown to suitably calculate artery deformation [50]. In [59], it was shown that the results are not sensitive to the value of the Poisson ratio. There, it was also shown that the required thickness of the surrounding sheet was 6.5 mm and that value is used through this study. The mechanical properties are listed in Table 1.

The plaque was assumed to be uniformly presented around the lumen so that axisymmetry could be used. To reduce solution time and increase spatial resolution, an angle of 10 degrees was used. This assumption of axisymmetry is not a limitation of the modeling; if the tangential deployment of the plaque deposit is known, it can be incorporated into the analysis.

  1. Figure 1:
    Schematic of the solution domain.

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    Table 1:

    Material properties.

Boundary conditions

At the upstream and downstream ends of the artery and tissue, radial motion was allowed but axial motion was prohibited. The external conditions are less obvious; it is not clear what the value of the external pressure should be since in some cases, the artery is internal to the organisms and therefore the far-field pressure is non-zero. In other cases, the vessels are near the skin surface so not only would there be a non-axisymmetric pressure but over part of the exterior the pressure would be zero. It is recognized that the superficial femoral artery is deep within tissue and an external pressure approximately equal to the diastolic pressure would be most suitable. In an effort to provide results which are of more general value, we will define the external pressure systematically with values that vary from zero to the diastolic pressure. The pressure within the lumen is taken from measurements. To simplify the mathematics, the software will require the translumen pressure and not the separate pressures at the two surfaces. Here we define four cases which are listed in Table 2. In the Table, Pineternal refers to the pressure inside the artery.

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    Table 2:

    Definition of the translumen pressure differences used in the simulations.

The numerical model

Time discretization:he solutions were initialized to a zero stress state and five cardiac cycles (Figures 2,3) were input. The time stepping was performed with different values to ensure time-step independence. Time step values of .01, .05, and .1 seconds were used and deformation of the artery wall was found to differ by less than 1 % throughout the cycle. For the remaining solutions to be reported, time steps of .1 seconds were employed.

  1. Figure 2:
    Translumen pressure difference for the cases of Table 2 prior to plaque removal treatment.

  1. Figure 3:
    Translumen pressure difference for the cases of Table 2 after plaque removal treatment.

Spatial discretization:The finite element solution requires a spatial discretization of the solution domain. A sequence of meshes were used that were increasingly refined. The refinements incorporated a 20-fold increase in the numbers of elements. The solutions from these various levels of refinement differed by less than 2 % so that a mesh independence was obtained. The results to be presented correspond to approximately 1700 number of elements for a 10 degree wedge (approximately 62,000 for a fully circular domain). The exact number of elements varied slightly among the simulations because the size of the lesion varied. Figure 3 has been prepared to show a view of the mesh with annotations calling out the regions.

Solution domain size:One issue which needs to be addressed is the size of the solution domain which is required to ensure that boundary conditions there do not influence the results. With respect to the radial extent, it was previously shown that a 6.5 mm thick surrounding tissue zone was sufficient to ensure accuracy [59]. The axial extent (annotated in Figure 1) was determined by systematically enlarging the domain and repeating compliance calculations until further enlargements did not influence results. All the results to be provided in the following section correspond to a sufficiently large axial extent. It should be noted however that in reality, arteries are not straight. There is a tortuosity that varies from patient to patient. In addition, the axisymmetric lesion is also idealized; in reality, the lesion may vary in thickness along its length and is often deployed non-symmetric within the artery. The results obtained here are consequently idealized.

Results and Discussion

The primary result to be presented is the compliance of the artery wall for the various pressure and plaque thickness situations. The compliance is calculated by the radial deformation of the inner plaque surface. The definition of area compliance, from [66], is

Area Compliance=  ΔArea ΔPressure MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGbbGaamOCaiaadwgacaWGHbGaaiiOaiaadoeacaWGVbGaamyBaiaadchacaWGSbGaamyAaiaadggacaWGUbGaam4yaiaadwgacqGH9aqpcaGGGcWaaSaaa8aabaWdbiabfs5aejaadgeacaWGYbGaamyzaiaadggaa8aabaWdbiabfs5aejaadcfacaWGYbGaamyzaiaadohacaWGZbGaamyDaiaadkhacaWGLbaaaaaa@5442@

Where ∆Area is the increase of area during the cardiac cycle and ∆Pressure is the transmural pressure difference. Another commonly used measure of compliance is the diametrical compliance which is defined as

Diametrical Compliance=10,000  ΔD/D ΔPressure    (2) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGebGaamyAaiaadggacaWGTbGaamyzaiaadshacaWGYbGaamyAaiaadogacaWGHbGaamiBaiaacckacaWGdbGaam4Baiaad2gacaWGWbGaamiBaiaadMgacaWGHbGaamOBaiaadogacaWGLbGaeyypa0JaaGymaiaaicdacaGGSaGaaGimaiaaicdacaaIWaGaeyOiGCRaaiiOamaalaaapaqaa8qacqqHuoarcaWGebGaai4laiaadseaa8aabaWdbiabfs5aejaadcfacaWGYbGaamyzaiaadohacaWGZbGaamyDaiaadkhacaWGLbaaaiaabccacaqGGaGaaeiiaiaabIcacaqGYaGaaeykaaaa@6350@

The Diamond Back 360° Peripheral Orbital Atherectomy System (OAS; Cardiovascular Systems, Inc) was used to modify the surface of calcified plaque while preserving the more elastic arterial wall. The differential sanding property results in plaque fracture, luminal enlargement, and very low balloon inflation pressure, minimizing injury to the vessel wall. Lesion pretreatment with the OAS has translated into a decreased need for bail-out stenting, which preserves the opportunity for future revascularization treatments. For more detailed information on the peripheral orbital atherectomy device and clinical data [66].

In this case, 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 (180 seconds for the entire treatment), 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.

To aid in the discussion, tables have been created to categorize the many individual calculations. The first table (Table 3) presents results for translumen pressure variation Case 1. There it is seen that the presence of plaque has a significant impact on the deformation (plaque reduces compliance). It is also seen that the length of the lesion is not important except for very short lengths (3 mm). The results for 7, 12, and 18 mm lesions are indistinguishable.

  1. avatar

    Table 3:

    Results for transluminal pressure Case 1.

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    Table 4:

    Results for transluminal pressure Case 2.

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    Table 5:

    Results for transluminal pressure Case 3.

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    Table 6:

    Results for transluminal pressure Case 4.

With respect to improvements achieved by plaque removal, it is seen that for the pre-operative plaque thickness, the area compliance is approximately 0.1% however the post-operative case (plaque thickness of 0.825 mm, the compliance has tripled (increased by 200%). Only a modest reductions in plaque thickness with orbital atherectomy resulted in large changes in the compliance.

If attention is turned to the subsequent tables, the same conclusions can be drawn. Reductions in plaque thickness markedly alter the compliance. Furthermore, with the exception of very short lesions, the length of the lesion does not influence the results. It is also seen that the magnitude of the transmural pressure difference is positively related to the deformation with larger values of transmural pressure giving rise to larger deformation. This result is expected.

The compliance is weakly dependent on the transmural pressure at these pressure levels because of the appearance of the pressure difference in the denominator. There is a slight increase in compliance for the smallest transmural pressure differences, this finding is confirmed by literature [67].

Insofar as the popliteal artery is deep within perfused tissue, the most appropriate external pressure value is diastolic (corresponds to Case 4). The other cases correspond to arteries which would be close to the skin surface.

In these tables, the symbols ∆A, ∆P, and ∆D refer to changes of artery cross section area, change in pressure, and change in diameter during a cardiac cycle.

A graphical display of a subset of the results is provided in Figure 5. There, the results correspond to pressure Case 4 which typifies a deep-tissue artery such as the superficial femoral. There, two curves are shown that reveal the decrease of compliance with increase in calcified plaque thickness. The curves are annotated for plaque lesions 7 mm and longer and for a plaque lesion of 3 mm. It is seen that there is a very slight dependence of the results on plaque lesion length but a much more substantive dependence on plaque thickness.

  1. Figure 4:
    Solution domain and discretization mesh.

  1. Figure 5:
    Dependence of area compliance on plaque thickness for pressure variation Case 4.

Comparison of present calculations with measurements

It is important to compare these results with expectations from the literature. While there it is difficult to find a perfect comparison because of the differences from patient to patient and the paucity of high-quality quantifiable results, it was possible to find a reasonable comparison candidate [67]. A comparison of these results with [67], shows very good agreement, lending support to the present calculations.

A further comparison is made with [68]. They find a health population proximal superficial femoral artery compliance of 3.1 – 9.1% which compares well to the results of Table 6 without plaque. While it is difficult to compare the diseased population of [69], with the diseased results of this study because little information was given about the thickness, type, and length of plaque regions in [67], nevertheless, the lower range of their diametrical compliance values (3.1%) agree well with moderate diseased case (Cases 9-16 of Table 6) that were provided here.

Study limitations

While the results presented in this paper give clear guidance connecting the thickness of calcified plaque and arterial compliance, the plaque model was simplified as both axisymmetric and with a flat profile. While the internal dimensions matched those from patient measurements, in truth the non-uniform distribution is expected to lead to results that may differ from this generalized simplified case. Future stages of the work will involve the reproduction of these results with patient-specific plaque measurements and extension from the general to the specific case will be made.

Regardless of the limitations, the results from the simplified plaque model are useful to quantify the degree of compliance change which is achieved for plaque removal. In fact, if compliance targets are available, they can be directly related to plaque removal so that procedures can be tailored to specific patient compliance objectives.

Concluding Remarks

A numerical study was completed to quantify the effect of calcified plaque removal on artery compliance. The modeled artery segment was the proximal superficial femoral artery. Flow, pressure and geometry information was obtained from a patient before and after an orbital atherectomy procedure. The calculations were extended to a variety of plaque lengths and for a wide range of eternal pressure conditions which represent the surrounding tissue. The simulations included a sheath of supporting tissue as well.

It was found that progressive removal of plaque led to a large change in the compliance. It was also found that the changes were largely independence of the lesion length, however very short lesions (3 mm) exhibited a slight difference compared to longer lesions (7 mm and longer). These calculations allow clinicians to target atherectomy procedures to meet a compliance goal. For instance, for individual patients with non-compliant arteries, it is possible to create a compliance goal and to then tailor the amount of calcified plaque which must be removed to achieve that goal.

The calculations were compared with literature measurements of compliance and were found to be in agreement. To the authors' best knowledge, this is the first numerical compliance study which includes surrounding tissue and progressive changes in plaque.

The results presented here are also useful in interpreting other prior results such as [69], which use novel sensing techniques to measure intima-media thickness, diameter, strain and distensibility in arteries with a focus on potential plaque rupture.


The authors gratefully acknowledge the support of Cardiovascular Systems Inc.

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