## Abstract

Atherosclerosis in the superficial femoral artery (SFA) resulting in peripheral arterial disease is more common in men than women and shows a predilection for the region of the adductor canal. Blood flow patterns are related to development of atherosclerosis, and we investigated if curvature and tortuosity of the femoral artery differed between young men and women and if differences resulted in adverse flow patterns. Magnetic resonance imaging (MRI) and computational fluid dynamics (CFD) were combined in 18 young adult volunteers (9 men) to assess the relationship of flow features to likely sites of future atherosclerosis formation. Subjects underwent MRI of the right SFA, three-dimensional vascular geometry was reconstructed, and measures of tortuosity and curvature were calculated. Tortuosity and curvature were significantly greater for men than women, and this was related to increased body surface area, body mass index, or weight in men. In both sexes, “tortuosity” increased from the midthigh to the popliteal fossa. The greatest curvature was found within the distal quarter of the SFA. CFD modeling was undertaken on MRI-based reconstructions of the SFA. Wall shear stresses (WSS) were extracted from the computations. WSS showed greater spatial variation in the men than in the women, and the men exhibited lower mean WSS. These data indicate that sex differences related to body size and anatomical course of the femoral artery may contribute to the enhanced risk of focal atherosclerosis in the adductor canal.

- magnetic resonance imaging
- tortuosity
- wall shear stress
- peripheral vascular disease

peripheral arterial disease (PAD) affecting the artery of the lower limbs is common, particularly in men older than 65 yr (7). A clear sex divide is evident in the incidence of PAD, with men presenting two to five times more commonly than women. The superficial femoral artery (SFA) often develops atherosclerosis, in particular the segment of artery lying within the adductor canal at the level of the adductor hiatus. At this site, the SFA passes between the quadriceps, adductor, and sartorius muscles, with the posterior wall of the adductor canal formed predominantly by adductor longus and the anterior wall by vastus medialis. The canal commences anteriorly at the apex of the femoral triangle and terminates posteromedially above the medial condyle of the femur at the adductor hiatus where the artery enters the popliteal fossa; thus the SFA spirals across the limb as it passes through the canal (32).

The two primary risk factors for the development of PAD are cigarette smoking and diabetes mellitus (48). Other recognized risk factors include dyslipidemia and hypertension, and the majority of patients also have evidence of coexisting ischemic heart disease and atherosclerosis at other sites. However, the difference in PAD prevalence between men and women and the predisposition of the adductor canal segment of the SFA to atherosclerosis cannot be explained by these factors alone, and it is possible that anatomical differences may have an important etiological role.

It has long been known that curves and branching points constitute regions of predilection for atherosclerotic disease. This has been attributed to fluid mechanical phenomena, such as low shear stress or flow disturbance occurring at these sites (9, 11, 14). Several mechanisms have been suggested to explain these findings, including the effects of shear stress on endothelial function (4), proliferation of smooth muscle cells (25), and mass transport across the arterial wall (29, 38). Friedman et al. (10) have suggested the existence of “geometric risk factors” for atherosclerosis in the aortic bifurcation, whereas, in arterial regions with few branches, like the femoral artery, the important relevant feature may be the curvature or tortuosity of the vessel. It has been suggested that low shear stress, prominent secondary flows, or increased variation in shear stress predispose to atherosclerosis. Flow curvature is known to promote secondary motions with helical streamline patterns, resulting in variations in wall shear, with incipient or actual regions of flow separation (2, 30), and the observations of helical patterns of atheroma in the femoral artery suggest a relationship of flow patterns with subsequent atherosclerotic disease (43).

Magnetic resonance (MR) imaging is a useful tool for studying arterial geometry noninvasively, and, following three-dimensional (3D) reconstruction, MR data can be used in computational fluid dynamics (CFD) modeling, as applied in this study. The principle of CFD involves using a numerical approach to solutions of the Navier-Stokes equations that govern flow patterns. Several hemodynamic parameters, including wall shear stress (WSS), can be derived from the CFD solutions.

The purpose of this study was to measure the 3D tortuosity and curvature of the SFA above, within, and below the adductor canal in both men and women using MR imaging to describe sex and anatomical location differences for both tortuosity and curvature, and to test the hypothesis, via CFD, that these result in hemodynamic differences in the areas of the SFA prone to atherosclerosis. Because in the later stages of the disease the flow patterns would likely be modified by the presence of atheromatous plaques, studies were restricted to healthy volunteers without significant macroscopic arterial disease.

## METHODS

### Subjects

The study complied with Declaration of Helsinki and was approved by the St. Mary's Hospital Research Ethics Committee. All participating subjects gave written, informed consent. Participants were nine men and nine women (Table 1), who were young nonsmokers with no history of cardiovascular or cerebrovascular disease, hypertension, diabetes, or previous significant trauma to the right leg. Women were significantly younger, shorter, lighter, and had reduced body surface area (BSA) and body mass index (BMI) compared with men. Subjects were not taking any medication at the time of the study and were asked to refrain from caffeine-containing drinks for 24 h before investigations. BSA was calculated using the Mosteller correlation (28). and BMI was calculated as:

### Data Acquisition and Processing

Imaging of the right thigh and knee was performed in all cases using a 1.5-T whole body MR scanner (GE Medical Systems, Milwaukee, WI). For consistency, the right leg was examined in all studies, but there is no evidence that the SFA from either leg is more prone to atherosclerosis, and indeed there is evidence that both SFAs share similar morphological features (43). MR angiograms of the SFA were obtained using a two-dimensional time-of-flight gradient echo sequence with venous presaturation. A series of contiguous 2-mm-thick transverse slices was acquired from the level of the tibial plateau to the femoral triangle, just below the common femoral artery bifurcation. The SFA length was standardized for quantitative analysis by normalizing the vessel length to the subject height. Pulsed Doppler ultrasound waveforms were obtained from the origin of the SFA using an L12–5 linear array transducer and ATL HDI 5000 ultrasound machine (ATL-Phillips, Bothell, WA), and the diameters of the SFA, assumed circular, were measured by B-mode ultrasound at the same location, since this is more accurate than diameters measured from MR images. Ultrasound data were saved as cine loops and analyzed using HDI Lab version 1.9 (ATL-Phillips) and custom-written software in Matlab 7.0.4 (The Mathworks, Natick, MA).

MR data processing involved segmentation of the transverse two-dimensional time-of-flight geometric images, reconstruction of a 3D vessel surface, and mathematical treatment of the surface to correct errors due to misregistration and subject movement (26). Figure 1 shows such a reconstructed human femoral artery. Smoothing of the centerline was accomplished by applying a least squares cubic splines method, which could guarantee the existence of the second derivatives required for the calculation of curvature and tortuosity. On the basis of the MR images, there was an average taper of 5% in diameter for the whole group over the length of the artery. (True cross-sectional shapes were obtained from MR images; hence, diameters were determined as hydraulic diameter = 4*A*/*P*, where *A* is area and *P* is wetted perimeter.) This has normally been neglected in previous simplified model studies. Taper will cause flow acceleration, thus making the flow more stable in the retardation phase. It will also increase flow velocity and hence WSS in the distal region, compared with an untapered vessel. This region includes the adductor canal and emphasizes the importance of use of subject-specific anatomy in this investigation.

### Evaluation of Vessel Curvature and Tortuosity

Details of the calculation of measures of vessel curvature and tortuosity are given in the appendix. Tortuosity is a term that lacks a commonly accepted definition but is used to convey information about curvature at individual points on the centerline of a vessel segment. In addition to the measures of curvature, tortuosity was therefore also expressed as a distance factor (DF), based on the ratio of the incremental true length of the curve to the linear distance between its endpoints (appendix). Average curvature (AC), total curvature (TC), and tortuosity (DF) parameters were determined for the centerline of the SFA over its whole length. Also, to focus more closely on the adductor hiatus, the geometries of three segments of equal length (I, II, and III) were evaluated.

### Derivation of Flow and WSS Distributions

To assess the impact of the differences in curvature and tortuosity on flow patterns, the MR geometric data and Doppler proximal flow velocity data were incorporated in a numerical flow simulation, using a commercial CFD solver, CFX 4.4 (AEA Technology, Didcot, Oxfordshire, UK).

Whole field pulsatile flow solutions were obtained for two representative subjects of each sex. These subjects were chosen on the basis of their possessing median tortuosity of the group stratified by sex. For each subject, the vessel geometry was reconstructed from the processed in vivo MRI data within the CFX environment, and a 3D mesh was generated to conform to the reconstructed vessel. In each case, ∼158,400 brick elements (cells) were used for mesh generation following a grid sensitivity study. The grid was refined until the difference in maximum velocity between successive grids was <4%. They produced nearly identical WSS predictions at selected regions in the domain, and the second finest was chosen for the investigations. The cross section of the grid was nonuniform, with closer spacing near the wall to give better definition in the large wall-related gradients. The fluid was taken as incompressible (density 1,050 kg/m^{3}), assumed to be Newtonian (viscosity 0.004 Pas), and the walls were assumed rigid, generally good approximations for blood flow in the larger arteries (12).

An ensemble average was calculated from five measured Doppler flow velocity waveforms and used to derive the inlet boundary conditions for the pulsatile simulations. A single representative ensemble averaged waveform was chosen from each sex group, to remove the influence of individual variations in velocity waveform. The flow waveform was used to generate fully developed pulsatile velocity inlet profiles derived from a CFD model of a straight, rigid tube with constant area equal to the inlet of the modeled SFA domain, with the measured volumetric waveform mentioned above. The nonslip condition was applied at the walls, and uniform pressure was prescribed at the outlet. No further velocity or pressure boundary conditions were required, since there were no bifurcations and the walls were assumed rigid, an adequate approximation for the present study.

The assumption of fully developed flow as inlet boundary condition to the computational domain is the best approximation in the absence of measured velocity profiles. In practice, the profiles would be influenced by upstream vessel geometries, but the length over which pulsatile velocity profiles develop in a straight tube is quite short (12, 45). Moreover, they would respond to the local tortuosities of the presently simulated vessel over a similarly short distance, so it would be expected that the derived flow patterns over most of the vessel length will be realistic.

The Navier-Stokes equations, which govern the motion of simple fluids, were discretized using the third-order spatial differencing method QUICK (24) and the first-order fully implicit backward Euler time differencing scheme (6). Solution was via the SIMPLEC velocity-pressure coupling algorithm (5) in the CFX4 flow solver, with 100 uniform time steps per cycle. The convergence of the solution for each time step was monitored and controlled to a mass tolerance of 1 × 10^{−5}, and the flow model was solved for three cycles to obtain periodic convergence. Flow patterns and instantaneous and time-averaged WSS (TAWSS) distributions, together with the oscillatory shear index (OSI), were evaluated from the solutions.

The definition of the OSI used here is as defined by He and Ku (16), except that the multiplying coefficient, 0.5, is omitted, i.e. (1) where τ_{w} is the instantaneous WSS vector, *t* is time, and *T* is the pulse cycle period.

### Statistical Analysis

Data are presented as means ± SE. Statistical comparisons between sexes were made using a two-way Student's *t*-test, and multivariate techniques were used to adjust for possible confounders. All analyses were performed using Intercooled STATA 8.2 for Windows (StataCorp LP).

## RESULTS

### Geometry of the SFA in Men and Women

The SFA showed reverse curvature (i.e., “S” shape) along the segment within the coronal plane, accompanied by relatively smaller out-of-plane curvature in the sagittal plane (e.g., Fig. 1). Most of the increased tortuosity was attributable to curvature in the sagittal plane. In 16 of the 18 subjects, tortuosity in the sagittal plane was at least twice that in the coronal view, whereas in the coronal plane, the SFA had a less tortuous course below the adductor canal.

Measures of curvature and tortuosity in the SFA were significantly related to weight, BSA, and BMI (Table 2). SFA in men were significantly more curved and tortuous than in women (Table 3). However, after adjustment for BSA, these differences were reduced and no longer achieved statistical significance (Table 3). Differences were also not significant after adjustment for body weight or BMI (data not shown). Measures of curvature and tortuosity were unrelated to age (Table 2), and further adjustment for age in addition to BSA (or weight or BMI) had no additional effect. The mean diameter of the SFA in the group as a whole was 6.94 ± 1.31 mm, and diameters were significantly greater in men than women (Table 3).

### Three Segment Geometrical Analysis

The average geometric measures across all subjects of tortuosity and curvature for each of the three proximal to distal segments (*I*–*III*) are shown in Table 4 and for men and women separately in Fig. 2. Significantly increased curvature was found in *segment III* in the region of the adductor canal. The degree of tortuosity increased as the SFA traveled distally from the upper thigh to the knee, and the highest tortuosity was found in the most distal segment. Values of the computed parameters in all three segments were significantly higher in men than in women, although adjustment for BSA attenuated these differences such that they were no longer statistically significant.

### Flow Data

Time-mean Reynolds numbers (Re) were 116 and 117 for the women and men, respectively, while the Womersley parameter was 5.6 for the group (6.5 for the men, 4.7 for the women). The Dean number (De) (appendix), which relates the effect of curvature on secondary flow development, varied from 16.7 to 22.0 in the three segments (Table 4), based on the ACs and radius-based Re, Re_{r}, allowing for the average 5% taper down the length of the vessels. For comparison, a typical value of De for the aortic arch is 250 (44), but its effect is felt well into the descending aorta (46).

### Pulsatile Flow Simulations

#### Flow patterns.

The male and female flow velocity waveforms used for pulsatile simulations are shown in Fig. 3. Secondary flow patterns were markedly different between the male and female subjects, and between the level of the adductor canal and other sites. In a male subject, Fig. 4 shows how the pulsatile secondary flow changed during progressive phases of systolic retardation, with a double vortex during peak retardation, while at peak systole and at end systole only a single vortex was clearly seen. Much weaker secondary flow was seen in female subjects.

#### WSS.

Figure 5 compares TAWSS patterns on the inner surfaces of the vessels in the male and female subjects. The scale shows that values range from ∼0.1 to >0.65 Pa (1–6.5 dyn/cm^{2}). The scale was chosen so that spatial variations can be seen more clearly, indicating greater variation in TAWSS in the male, particularly in the distal two-thirds of the artery. Particularly low values (0.1 and 0.2 Pa) are evident in the male adductor region (arrowed). Any values >0.65 are shown in red. To determine the respective influence of geometry and flow velocity waveforms on TAWSS, this parameter was reevaluated after imposing the male flow velocity waveform onto the female geometry. In this case, patterns were qualitatively similar to those seen with the female geometry and female flow velocity, confirming it was geometric differences in the SFA that largely determined the pattern of TAWSS distribution.

Figure 6 shows the OSI in the male and female subjects. The scales show that values range from near zero to ∼0.9. While OSI is not a highly consistent predictor of atherogenesis location, the male example displays an intense narrow band of higher OSI in the adductor region and greater variation overall. The female, by contrast, shows a more diffuse region of higher OSI. These results are consistent with the trends observed in the instantaneous WSS (not shown) and TAWSS, but are complicated by the presence of significant retrograde flow in the femoral artery flow waveform (Fig. 3).

## DISCUSSION

In this study, we have determined the 3D curvature characteristics of the human SFA using data from multiple cross-sectional MR images of young volunteers and determined hemodynamic patterns in the SFA in men and women. Factors influencing flow patterns include the Re, the Womersley parameter, the curvature of the mean flow [sometimes characterized by the De (44)], and the proximity of the distal reverse curvature to the proximal segment. The intensity of flow skewing and helicity is highly dependent on the individual local vessel curvature, and the pattern will also be influenced by the planarity (1, 3, 17, 20, 21, 37, 46). All subjects showed common features of flow in the SFA, such as the occurrence of strongly skewed velocity profiles toward the outer wall of the curved regions, but there were marked sex differences in WSS patterns and the degree of development of spiral flow along the vessel. Patches of very low TAWSS were present in the adductor region of the men (∼0.1 and 0.2 Pa): these values are considerably lower than typical values (i.e., <1.0 Pa), below which it was suggested by Wootton and Ku (47) that intimal thickening was found, and below the level found at atherosclerosis-prone sites (<0.4 Pa), based on a wide-ranging survey by Malek et al. (27). Our data may, therefore, provide a link between tortuosity, flow patterns, and later development of PAD. Low WSS is implicated in atherogenesis as a consequence of its influence on endothelial function and permeability (19, 31, 38). We suggest that these regions of low TAWSS may represent regions of high risk of atherogenesis and progression of atheroma and may account for the increased prevalence of stenoses in the adductor region of the SFA (33, 40) and sex differences in PAD. Note that the pattern of atherosclerotic plaques appears to change over time (41), perhaps owing to local flow changes induced by the plaques themselves (36).

Low shear stress and the presence of flow separation may not be the only hemodynamic factors implicated in the etiology of focal atherosclerosis. Some authors have suggested that increased cyclical variation in shear stress (15, 30) may provide a stimulus for local plaque generation (23). In the present study, OSI, measure of cyclic variation in shear stress (16, 23), showed differences only in the intensity of local variation between men and women. However, OSI was introduced to characterize the separated and disturbed flow in the region of the carotid sinus (23), and the interpretation of OSI in our study is difficult, since the flow waveform contains significant retrograde flow at end systole. When the pressure gradient reverses, slow moving fluid near the walls reacts first and reverses before the faster flow near the center of the vessel and before the net flow displays reversal. During this period, therefore, the near-wall flow has some characteristics of separated flow, although with laminar, rather than disturbed, conditions (45). Moreover, this is coupled with strongly spiral flow patterns, again not necessarily associated with disturbed flow, but causing large variations in the instantaneous WSS vectors and increasing the values of the OSI. Tortuosity in the adductor region induces a helical flow pattern with a highly nonuniform, helical distribution of flow and consequently WSS. A previous study by Wensing et al. (43) noted spiral patterns of early atheroma in the femoral artery and suggested that these might be related to spiral flow. Our observations are consistent with this.

The present study has a number of limitations. Male subjects were slightly older than females, and, although differences in flow patterns were unrelated to age, it is possible that this may have influenced our findings. Predictably, men were taller and heavier than women, and statistical adjustment of the data using various measures of body size (BSA, BMI) minimized differences between the sexes. Of course, differences in size and muscularity are features of sex differences, but it seems likely that the resultant difference in SFA anatomy is the major explanation of the differences in tortuosity and flow patterns seen in this study. It should be noted that BMI is related to femoral intima-media thickness (13, 18) and PAD (8). Given the dominance of this influence, our study is too small to establish if there are other sex-related differences in tortuosity or flow patterns independent of differences in habitus. Finally, all subjects were studied supine at rest with the leg extended, and the influence of knee flexion (43) on tortuosity and flow patterns is unknown.

In conclusion, our data show a strong relationship between increased tortuosity and disturbed hemodynamic patterns in regions of the SFA. Regions of low WSS and disturbed flow were evident in the adductor canal and were more prominent in men. Sex differences in tortuosity and flow patterns in SFA are likely to be largely explained by differences related to body habitus.

## APPENDIX

### Determination of Curvature and Tortuosity Parameters

The three measures of curvature and tortuosity employed in this study were selected for their ease of interpretation and generalization to three dimensions (34, 42). Consider the vessel centerline extracted from the 3D reconstructed geometry as a curve, in each point, by the position vector **R**, which is in the form (A1) where *x*, *y*, and *z* are in the lateral, medial, and longitudinal directions; * i*,

*, and*

**j***are unit vectors; and*

**k***s*is the arc length. Curvature

*k*(below) at a point on a smooth curve can be defined as the derivative with respect to the arc length

*s*of the tangent direction and can be obtained in terms of the second derivatives with respect to

*s*(39): (A2) This means that the curvature expresses the rate of change of the tangent direction for an observer traveling on the curve at a constant velocity. The effect of curvature on the flow field in a curved tube is often characterized by the De (used for correlating secondary losses): (A3) where Re

_{r}=

*u*

_{m}

*r*/ν denotes the Re based on mean velocity

*u*

_{m}and radius, with

*r*and

*R*being the tube radius and the radius of curvature, respectively, and

*ν*= μ/ρ is the kinematic viscosity.

*R*is the inverse of the curvature.

Tortuosity is a term that lacks a commonly accepted definition but is generally used to summarize information about curvature at individual points on the centerline of a vessel segment. A natural approach is then to aggregate the curvature values by forming an average or integral along the curve. These are AC and TC, computed as a line integral of local curvature values defined as: (A4) (A5) Another intuitive approach to tortuosity is to relate the length of the vessel segment to the straight-line distance between its end points. The ratio of the incremental curve length to the straight-line distance is the DF, expressed by the formula: (A6) where *L* is an approximation of the curve length, and *d* is the distance between its two end points. With this definition, the tortuosity of a straight line is 0.

## GRANTS

This study was supported by the British Heart Foundation (PG/99057).

## Acknowledgments

The authors thank Frank Connor and Dr. Quan Long for technical MRI support, and Kim Parker for advice and assistance in determining arterial planarity.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

- Copyright © 2006 the American Physiological Society