Ultrasound in Med. & Biol., Vol. 41, No. 3, pp. 760–774, 2015 Copyright Ó 2015 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/$ - see front matter
http://dx.doi.org/10.1016/j.ultrasmedbio.2014.10.012
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Original Contribution COMPARISON OF VORTICAL STRUCTURES INDUCED BY ARTERIOVENOUS GRAFTS USING VECTOR DOPPLER ULTRASOUND EFSTRATIOS KOKKALIS,*y ANDREW N. COOKSON,z PETER A. STONEBRIDGE,y GEORGE A. CORNER,x J. GRAEME HOUSTON,y and PETER R. HOSKINS{ * Institute for Medical Science and Technology, University of Dundee, Dundee, UK; y Cardiovascular and Diabetes Medicine, Ninewells Hospital and Medical School, University of Dundee, Dundee, UK; z Department of Biomedical Engineering, King’s College London, London, UK; x Medical Physics, Ninewells Hospital and Medical School, Dundee, UK; and { Centre for Cardiovascular Science, University of Edinburgh, Edinburgh, UK (Received 17 March 2014; revised 18 August 2014; in final form 18 October 2014)
Abstract—Arteriovenous prosthetic grafts are used in hemodialysis. Stenosis in the venous anastomosis is the main cause of occlusion and the role of local hemodynamics in this is considered significant. A new spiral graft design has been proposed to stabilize the flow phenomena in the host vein. Cross-flow vortical structures in the outflow of this graft were compared with those from a control device. Both grafts were integrated in identical in-house ultrasound-compatible flow phantoms with realistic surgical configurations. Constant flow rates were applied. Inplane 2-D velocity and vorticity mapping was developed using a vector Doppler technique. One or two vortices were detected for the spiral graft and two to four for the control, along with reduced stagnation points for the former. The in-plane peak velocity and circulation were calculated and found to be greater for the spiral device, implying increased in-plane mixing, which is believed to inhibit thrombosis and neo-intimal hyperplasia. (E-mail: [email protected] or [email protected]) Ó 2015 World Federation for Ultrasound in Medicine & Biology. Key Words: Color Doppler, Vector Doppler ultrasound, Velocity, Vorticity, Circulation, Flow phantom, Arteriovenous prosthetic grafts, Vortical structures, Spiral flow, Flow mixing and stagnation.
2010) and graft anastomoses (Doorly et al. 2002; Lee et al. 2005). The display of vortical phenomena requires at least two components of velocity. The evolving understanding of the role of local hemodynamics in endothelial function and the proliferation of vascular smooth muscle (Caro 2009; Malek et al. 2013; Slager et al. 2005a, 2005b) serves to increase the interest in more detailed velocity profiles and fluid dynamic assessments. During the last few decades, a number of vector Doppler approaches have been proposed (Fei et al. 1994; Garcia et al. 2010; Hoskins et al. 1994; Jensen and Munk 1998; Maniatis et al. 1994), and recently, a real-time technique was integrated into a conventional system (Pedersen et al. 2012). The rotational nature of blood flow in vessels is widely accepted. In 1990, Frazin et al. detected spiral flow in the thoracic aorta using color Doppler. Stonebridge and Brophy (1991) were the first to suggest the blood flow in arteries is spiral in nature. Subsequent ultrasound studies confirmed the presence of spiral flow in arteries (Frazin et al. 1996; Hoskins et al. 1994; Tanaka et al. 2010; Udesen et al. 2007), veins (Marie
INTRODUCTION Color Doppler ultrasound has been found to be of great value in clinical diagnosis where blood flow can be visualized and quantified to assess cardiovascular complications. However, conventional color Doppler imaging is limited to displaying two spatial components and one velocity component. The velocity measurements are vulnerable to errors mainly because of the angle dependency between the ultrasound beam and the direction of the target (Evans et al. 2011; Hoskins 2010). In addition, the complex vortical structures that occur in the cardiovascular system increase the angle dependency and cannot be quantified. These structures can be intense in bifurcations (Udesen et al. 2007), poststenotic regions (Hoskins 1997), the heart (Garcia et al.
Address correspondence to: Efstratios Kokkalis, Ninewells Hospital and Medical School, Mail Box 1, Level 7, University of Dundee, Dundee DD1 9 SY, UK. E-mail: [email protected] or [email protected] Conflicts of interest: This work was supported by Vascular Flow Technologies Ltd. of which J.G.H and P.A.S are founder members. 760
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et al. 2014; Rosenthal et al. 1995) and the heart (Garcia et al. 2010; Mehregan et al. 2014). It has also been verified with magnetic resonance imaging (Kilner et al. 1993; Markl et al. 2005) and computational modeling (Gallo et al. 2012; Hardman et al. 2013). The formation of vortical structures in blood originates from the rotational compression of the heart (Jung et al. 2006) and is propagated because of the curvature, branching, non-planarity and spiral folds in the arteries (Caro et al. 1996; Stonebridge 2011; Stonebridge and Brophy 1991). Caro et al. (1996, 2005) hypothesized that helical flow in arteries supports in-plane mixing and uniform distribution of wall shear and inhibits blood stagnation, separation and instability. Similarly Stonebridge et al. (2004) maintained that single-spiral induces flow stability and coherence, and Cookson et al. (2009) verified and explained an increase of in-plane mixing as a result of helical flow. Recently, Marie et al. (2014) reported single-spiral flow as a predictor of successful arteriovenous (AV) fistula maturation. As an alternative to AV fistulas, prosthetic grafts can be used for vascular access. The most frequent complication of AV prostheses is stenosis, which can lead to thrombosis and occlusion. Stenoses can develop in the venous or arterial anastomosis from neo-intimal hyperplasia or the mid-graft section as a result of ingrowth of fibrous tissue through puncture holes. The former is the most common incident, with disturbed hemodynamics being one of the causes (Gessaroli and Massini 2012; Manos et al. 2010; Mickley 2004; Tricht et al. 2005). An AV graft can be implanted in either the arm or the thigh, with patency depending on the location. In an AV graft review Akoh (2009) reported 1- and 2-y cumulative patency rates of 59%–90% and 47%–85% in the arm and 41%–68% and 26%–43% in the thigh, respectively, indicating the need for steady and enhanced patency. A novel AV prosthetic graft specifically engineered to introduce a stabilized spiral flow has recently been developed and introduced for vascular access applications. Clinical data are not yet available but a similar peripheral vascular spiral graft has been compared with a control device in vitro exhibiting improved secondary flow phenomena (Kokkalis et al. 2013) and has been applied in vivo with enhanced patency rates in a
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30-mo follow-up (Stonebridge et al. 2012). These promising results motivated us to investigate the hemodynamics of the AV spiral graft and the underlying mechanism. The aim of this article is to describe and compare the rotational structures in the outflow of the spiral AV graft and a non-spiral control AV graft, using vector Doppler ultrasound. Two-dimensional velocity and vorticity maps were developed; peak velocity and circulation were identified for quantification. METHODS Arteriovenous access grafts The spiral and control grafts were made of expanded polytetrafluoroethylene (ePTFE) and had a 6-mm inner diameter (ID) (Vascular Flow Technologies, Dundee, UK). The former device is supported by an injection molded polyurethane spiral flow inducer, which forms an internal non-planar helical geometry along the distal end of the graft (Fig. 1a). The control device was identical in all parameters, but with a smooth planar internal geometry along its length. Flow phantom setup The proximal end of each graft was connected to C-Flex (Cole–Parmer, London, UK) tubing, which simulated the arterial flow and the distal end with an in-house venous mimic with end-to-side anastomoses. The venous mimic was made of polyvinyl alcohol cryogel (PVA-c). This vascular-graft model was embedded in a liquid medium tissue mimic and housed in an acrylic tank (485 3 180 3 60 mm). The tissue mimic composition by volume was 9% glycerol and 91% distilled water. PVC tubing was used to connect the outflow of a flow pump with the C-Flex tubing (artery) and the PVA-c venous mimic with the inflow of the pump. The connections between the PVC tubing and vascular mimics on the wall of the tank were supported by straight, rigid 60-mm-long PVC screw connectors. An ultrasound Doppler test fluid (Model 707, ATS Laboratories, Bridgeport, CT, USA) was used as blood mimic. The flow pump was a UHDC computer controlled piston system (Shelley Medical Imaging Technologies, London,
Fig. 1. (a) Distal end of the 6-mm-inner-diameter arteriovenous spiral graft. (b) Side view diagram of the distal cuff where angle 4 5 15 .
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ON, Canada) with steady cycle-to-cycle reproducibility and a standard deviation less than 1% for constant flow rate up to 35 mL s21. All used tubes and connectors had a 6-mm-ID matching that of the grafts. Geometry of vascular-graft models Two pairs of vascular-graft models were created. Each pair consisted of a spiral graft model and a nonspiral graft model with identical geometries. The looped surgical configuration was simulated in one of the pairs and the straight in the other (Fig. 2) (Akoh 2009; Brown and Mitchell 2014). The proximal end of the looped grafts was cut at 15 , and for the straight grafts, at 90 . The distal cuff of all grafts was trimmed at 15 (Fig. 1b). The lengths of both spiral and control grafts in each pair of models were identical, as were the lengths of vascular mimics. The location and type of the proximal (arterial) and distal (venous) anastomoses in each pair of models were also identical. The overall geometry of the spiral and non-spiral models in each surgical configuration was the same. The anastomoses were made with superglue and reinforced with adhesive epoxy. Their quality and similarity to clinical practice were verified by a consultant vascular surgeon. The distal end of the C-Flex tubing and proximal end of the PVA-c venous mimic (in relation to the flow direction) were blocked, forcing the blood mimic to follow the grafted region and venous flow direction. Figure 2 illustrates the overall
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geometry of the models and terminology in the distal end-to-side anastomoses. The 3-D flow profile in a fluid circuit is strongly influenced by the tubing geometry. The pump, tubing and vascular-graft models were positioned at the same height to create a planar flow design excluding the non-planar spiral inducer region in the spiral graft. The inlet length distance was taken into consideration proximal to the inlet of each graft to ensure a fully developed parabolic profile and minimize flow variations (Kokkalis et al. 2013; Nichols and O’Rourke 2005). Acoustic properties of mimicking materials An important parameter of the mimicking components in an ultrasound flow phantom is their acoustic properties. These components need to be similar and also to agree with those observed in the human body. Differences in the ultrasound beam propagation speed need to be minimized because they cause refraction of the beam and distortion of the detected Doppler frequency shift (Hoskins 1994, 2007, 2008). Production of the in-house PVA-c venous mimic and its acoustic properties, as well as the acoustic properties of blood and tissue mimic, were described in detail in our previous work (Kokkalis et al. 2013). These properties were based on the specifications of the International Electrotechnical Commission (2001). During the study, temperature was monitored and maintained at 21 6 1 C to avoid changes in acoustic properties. Vorticity and circulation Two-dimensional velocity information was obtained and visualized in each scanning location using a previously described dual-beam vector Doppler technique, which is approximately 98% accurate providing that the Doppler frequency shifts are above the clutter filter and not aliased (Hoskins 1997; Hoskins et al. 1994; Kokkalis et al. 2013). Similar velocity maps and a comparison of peak cross-flow velocity were previously used to characterize spiral flow patterns induced by peripheral vascular grafts (Kokkalis et al. 2013). This comparison gives an indication of the velocity range, but it is based on a single point. Quantities from fluid mechanics were investigated to quantify the total 2-D cross-flow spiral field. Circulation, which is such a parameter, was calculated as an index of comparison between the spiral and non-spiral grafts. The calculation of circulation was based on vorticity, as described below.
Fig. 2. (a) Looped configuration spiral graft model. (b) Straight configuration spiral graft model. The respective control models were identical. Scan plane 1 was positioned 0.1 cm distal from the graft outflow. Terminology for the end-to-side venous anastomosis is provided. AV 5 arteriovenous.
Vorticity. It is a vector field, or more precisely a pseudo-vector, that shows the spinning motions of particles in a fluid as a result of the shearing forces applied on them. Vorticity, ! u , is defined as the curl of the fluid velocity field:
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Fig. 3. Color Doppler images of spiral flow in the cross-flow view. (a) Single spiral. (b) Double spiral. (c) Four-spiral pattern.
! u 5 V3! y
(1)
where ! y is the velocity vector. Providing that the 2-D flow maps of this study were parallel to the x–y plane of a Cartesian coordinate system, it is the axial component of the vorticity vector in which we are interested. This is defined as vyy vyx 2 uz k 5 k (2) vx vy where yx and yy are the components of the velocity vector ! y and were calculated with the vector Doppler technique, and k is the unit vector in the axial direction. In this case, the vorticity vector is always perpendicular to the examined plane of the flow or parallel to axis z (Morrison 2013; Tritton 2007). The relation between the direction of rotation of an element and the direction of the vorticity vector is described by the right-hand rule. The unit of vorticity is radians per second (Morrison 2013). A validation of the vorticity algorithm and its relation to shearing processes is provided in Appendix. Circulation. Circulation, G, is the line integral around any arbitrary closed curve C of the velocity field ! y in a fluid. If dL is an element of curve C, then this line integral is defined by the first part of eqn (3). If S is the surface that has the closed curve C as boundary and dS is an element of this surface, then from Stoke’s theorem it can be seen that circulation is also equal to the integral of vorticity ! u in this area, as in the last part of the equation I ð ! y ,dl 5 ! u ,dS (3) Gc 5 c
s
From the above is derived that the units of circulation are length squared over time (m2 s21). This parameter allows quantification of the strength of a vortex (Morrison 2013; Tritton 2007).
Data acquisition Color Doppler cine loops were recorded for both pairs of phantoms at 0.1, 0.5, 2, 5 (Fig. 2a, b, scan planes 1–4) and 10 cm distal from the graft outflow, to study the development of vortical structures. The abnormal hemodynamics that an AV graft introduces in the vein are linked with stenosis primarily in the junction and the first 2–5 cm distal of it (Haruguchi and Teraoka 2003), but also downstream of the venous flow (Doelman et al. 2005). These scan planes were transverse to the flow in both anteroposterior and right-to-left directions. In this way, helical patterns in a vessel or vessel mimic can be visualized as ‘‘red–blue splits,’’ representing flow toward and away from the transducer (Frazin et al. 1990; Hoskins et al. 1994). The number of splits in a scan plane is dependent on the number of vortices in it (Fig. 3). Constant flow rates of 240, 360, 480, 600 and 720 mL min21, or corresponding Reynolds numbers 570, 850, 1140, 1420 and 1700, respectively, were applied. The Reynolds number was based on the mean flow velocity, ID of the phantom and kinematic viscosity of the blood mimic. Flow rates less than 600 mL min21 are linked with thrombosis; 300 mL min21 is the threshold for adequate dialysis (Akoh 2009). In each scan plane, two cine loops were recorded, one with the beam-target angle at 120 and one at 220 in relation to the linear transmission. Each recording was 115 frames, filling the limit of the memory buffer. Steady transducer positioning and relatively accurate displacements were enabled with a retort clamp adjusted in a micromanipulator. Each scanning was repeated three times after transducer repositioning to confirm repeatability. The data were transferred for post-processing with an in-house suite of programs in MATLAB R2012a (The MathWorks, Natick, MA, USA). Each cine loop was averaged to reduce noise from small flow pattern variations and color speckle. The averaged velocity information from the left and right steered projections were
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centralized to avoid lateral displacement and subjected to triangulation with our vector Doppler algorithm (Kokkalis et al. 2013). Thus the 2-D velocity vector was identified in each scan plane, and velocity direction, velocity magnitude and vorticity maps were developed. The ultrasound system employed was an HDI 5000 (ATL Ultrasound, Bothell, WA, USA) with a linear transducer (L12-5). The color Doppler settings were persistence low, smoothing low, sensitivity high, wall filter low and dynamic motion differentiation off. B-Mode imaging was used to measure the ID of the PVA-c venous mimic. Data selection criteria A significant flow pattern variation from frame to frame was observed for the non-spiral models at the 10-cm scan plane for all applied flow rates. For low flow rates, the in-plane tangential velocity signal was not only unstable but also weak, and the Doppler frequency shift was low. Reduced pulse repetition frequency and increased color gain were tried for the detection of these weak secondary flow motions, but clutter signal was induced from the venous and tissue mimic regions. These observations may suggest a decline of rotational structures at cross-flow plane 4 under low flow rates for the control models. On the contrary, a clear and steady flow pattern was detected at the respective scanning location and flow parameters for the spiral graft models. Because of the lack of satisfactory Doppler signal and flow stability from the control models at the 10-cm scan plane, the location was excluded from the study. Data analysis Peak velocity comparison. The angle independent peak cross-flow velocity was detected in each scanning location under all applied flow parameters. These measurements were normalized by the mean axial velocity, y, as indicated in the equation by 5
y y
(4)
where by is the normalized velocity, and y 5 Q=A, with Q being the average of the five applied flow rates and A the area of the venous mimic, and a comparison between the spiral and control graft models was applied. Circulation detection and comparison. The vorticity map was generated for each 2-D velocity data set. Vorticity is the kinematic representation of the shearing processes in a flow field and, thus, may appear to change direction not only because of counter-rotated vortices but also within the area of one vortex. To calculate the strength of the vortical structures in the whole area of a scanning location, circulation was computed for both
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positive and negative areas of vorticities. The contiguous regions of positive and negative vorticities were integrated over their areas, and their absolute values were summed. This was applied for all scan planes and flow parameters, for both pairs of phantoms. Circulation, G, was normalized by the mean circulation, G, for the looped and straight pair of phantoms b 5G G G
(5)
b is the normalized circulation. where G Statistical analysis The significance of the differences in peak velocity and circulation in the comparison between spiral and non-spiral models was determined with unpaired twotailed t-tests. A p-value , 0.05 was considered to indicate statistically significant differences; a p-value # 0.01, highly significant differences; and a p-value # 0.001, very highly significant differences. The t-tests were executed with SPSS Version 21 (IBM, Armonk, NY, USA).
RESULTS Color Doppler ultrasound was used for initial observation of vortical structures in the cross-flow view and for their understanding using conventional color Doppler. Examples of single-, double- and four-spiral patterns are illustrated in Figure 3a–c, respectively. Two-dimensional velocity and vorticity maps were developed based on vector Doppler imaging and are illustrated in Figures 4–7. These figures represent the four scan planes from the proximal (scan plane 1 at 0.1 cm) to the distal (scan plane 4 at 5 cm) outflow of the looped configuration models (Fig. 2a). Spiral and control graft outflow mapping is included in each one for Reynolds numbers 570, 1140 and 1700. The orientation of the maps is such that the left and right sides are the floor and toe wall sides of the vessel mimic respectively, and the top and bottom sides are the anterior and posterior wall sides respectively (see Fig. 2a). Velocity direction maps are shown in the first row of each device, velocity magnitude in the second and vorticity in the third. The color distribution in velocity direction maps represents the direction of the velocity vector, with dark blue and dark red the limits from 0 to 360 . The size of the vectors is proportional to velocity magnitude. In velocity magnitude maps, dark red represents maximum value and dark blue zero. The vorticity maps correspond to axial vorticity, uZ . Thick black contour lines at uZ 5 0 separate the regions of positive and negative vorticity. Positive
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Fig. 4. In-plane velocity direction (first and fourth rows), velocity magnitude (second and fifth rows) and axial vorticity uz maps (third and sixth rows) from the spiral and control looped vascular-graft models at scan plane 1. Reynolds numbers (Re) 570, 1140 and 1700 are shown for each model. The boundaries of zero vorticity are represented by a thick black contour line. The orientation of each map in relation to the anastomotic locations is: left 5 floor, right 5 toe, top 5 anterior and bottom 5 posterior.
regions are associated with counterclockwise vorticity, and negative regions with clockwise vorticity. The number of spirals in Figures 4–7 can be seen from the velocity directional map, and their centers of
rotation, from the velocity magnitude and vorticity maps. The direction maps in scan plane 1 reveals a double spiral for both grafts (Fig. 4). For the spiral device, it is composed of a prevalent spiral on the
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Fig. 5. In-plane velocity direction (first and fourth rows), velocity magnitude (second and fifth rows) and axial vorticity uz maps (third and sixth rows) from the spiral and control looped vascular-graft models at scan plane 2. Reynolds numbers (Re) 570, 1140 and 1700 are shown for each model. The boundaries of zero vorticity are represented by a thick black contour line. The orientation of each map in relation to the anastomotic locations is: left 5 floor, right 5 toe, top 5 anterior and bottom 5 posterior.
anterior side (top) and a smaller spiral in the posterior (bottom), whereas for the control, it has a symmetric double-spiral pattern or Dean-type flow. In scan plane 2 (Fig. 5), the patterns for the spiral graft remain
similar to those observed in scan plane 1 (Fig. 4), whereas for the control, there is a three-spiral pattern for Reynolds numbers 1140 and 1700, respectively. In scan plane 3 of the spiral model (Fig. 6a–i), the second
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Fig. 6. In-plane velocity direction (first and fourth rows), velocity magnitude (second and fifth rows) and axial vorticity uz maps (third and sixth rows) from the spiral and control looped vascular-graft models at scan plane 3. Reynolds numbers (Re) 570, 1140 and 1700 are shown for each model. The boundaries of zero vorticity are represented by a thick black contour line. The orientation of each map in relation to the anastomotic locations is: left 5 floor, right 5 toe, top 5 anterior and bottom 5 posterior.
small vortex can be seen under low Reynolds numbers, but is coalesced with the main vortex under the peak Reynolds number. This process is more easily observed from the velocity directional maps (Fig. 6a–c). In scan
plane 3 of the non-spiral graft (Fig. 6j–r), Dean flow is seen for Reynolds number 570 and a four-spiral pattern for Reynolds numbers 1140 and 1700 (Fig. 6j–r). In scan plane 4, one uniform vortex is illustrated for the
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Fig. 7. In-plane velocity direction (first and fourth rows), velocity magnitude (second and fifth rows) and axial vorticity uz maps (third and sixth rows) from the spiral and control looped vascular-graft models at scan plane 4. Reynolds numbers (Re) 570, 1140 and 1700 are shown for each model. The boundaries of zero vorticity are represented by a thick black contour line. The orientation of each map in relation to the anastomotic locations is: left 5 floor, right 5 toe, top 5 anterior and bottom 5 posterior.
spiral device (Fig. 7a–i). For the control model in plane 4, Dean flow is seen for Reynolds number 570, a fourspiral pattern for 1140 and a disturbed pattern for 1700 (Fig. 7j–r).
In the vorticity maps of Figures 4–7, the core of each vortex is clearly illustrated along with regions of counterrotation near the wall and between counter-rotating vortices in multispiral patterns. A counter-rotation region
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Fig. 8. Progression of peak in-plane velocity at scan planes 1 to 4 downstream from the outflow of the spiral and control looped grafts under all applied Reynolds numbers. *p , 0.05, **p # 0.01, ***p # 0.001.
in the area of one vortex is the result of opposite shearing forces in the flow trajectories of this region. Solutions of this nature have been reported by Cookson et al. (2009). For more, see the Appendix (particularly Fig. 13c). Figures 8 and 9 illustrate the comparison of peak inplane velocity for the looped and straight configuration models, respectively. The velocity is consistently higher for the spiral prosthesis apart from Figure 9, plane 1, Reynolds number 1420. This difference is significantly higher in almost all cases for the looped models and in almost all cases from plane 2 and beyond for the straight models. Figures 10 and 11 illustrate the comparison of circulation for the looped and straight configuration models, respectively. Circulation is constantly higher for the spiral device, except for Figure 11, plane 1, Reynolds numbers 850 and 1140. This difference is significant in all scan planes for the looped models, apart from plane 1, Reynolds number 1140. In the straight surgical configuration, the difference is significant from plane 2 and beyond. The differences in Figures 8–11 increased with the increase in Reynolds number, and the level of significance increased with distance from the anastomosis. There is a match in the ranking and level of significance between the peak velocity and circulation comparisons in each pair of phantoms. DISCUSSION The comparison of peak in-plane velocity and circulation in Figures 8–11, revealed increased values for the spiral
graft. This quantitative evaluation indicated increased in-plane mixing for the spiral device. In-plane mixing has been proposed to protect anastomotic regions from neo-intimal hyperplasia and thrombosis (Caro et al. 2005). The rotational patterns within the control graft– venous mimic model revealed a constant Dean flow for Reynolds number 570, which implied a plane of symmetry about the model centerline (Figs. 4–7j, m, p). The number of spiral cells from plane 2 and beyond was increased from 2 to 4 as the Reynolds number was increased. At peak Reynolds numbers, in scan plane 4, the pattern had a disturbed or chaotic profile, as it was presented with both the velocity and vorticity maps. These results may imply flow separation and instability in the outflow of the planar control graft, which may be detrimental for the vascular function in vivo (Caro et al. 2005; Stonebridge 2011). Moreover, all the velocity magnitude maps presented low velocities or cross-flow stagnation points in the wall sides of the floor and toe, locations where adjacent vortexes were mixed. Such points were also seen in anteroposterior locations, although we focus on the floor and toe sides, which are widely associated with neo-intimal hyperplasia in patients (Doorly et al. 2002; Sherwin et al. 2000; Tricht et al. 2005; Walsh et al. 2003). In contrast, the rotational patterns in each scan plane of the spiral graft had relatively steady profiles, stabilizing the flow even at peak Reynolds numbers. The proximal outflow of the spiral device (planes 1 and 2) revealed a main and a secondary smaller spiral. This
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Fig. 9. Progression of peak in-plane velocity at scan planes 1 to 4 downstream from the outflow of the spiral and control straight grafts under all applied Reynolds numbers. *p , 0.05, **p # 0.01, ***p # 0.001.
smaller spiral was coalesced with the main one in scan plane 3 (Fig. 6a–i), and as a result, a steady uniform bulk rotation was presented in plane 4. The ability of the spiral graft to develop a single spiral adds to its stabilizing effect (Fig. 7a–i). In terms of stagnation points in the floor and toe wall sides, they were detected in the floor of planes 1 and 2, as seen in the velocity magnitude maps. The above flow pattern contrast was supported by the results from the excluded scan plane 5 at 10 cm distal the venous anastomosis, where a steady single-spiral was found for the spiral graft and a weak temporally unsteady or disturbed flow pattern was found for the non-spiral (see Data Selection Criteria). Doorly et al. (2002) and Sherwin et al. (2000) have previously compared planar and non-planar anastomotic models using computational and experimental methods with low Reynolds numbers (Re # 600). Their models can be compared with the control and spiral models of this study, respectively. In the outflow of the planar anastomotic model, they noted Dean flow related to a centerline plane of symmetry and floor–toe located stagnation points. In the outflow of the non-planar anastomotic model, they reported a single recirculation, associated with reduced peak wall shear stress magnitude at the floor of the anastomosis. They supported that a non-planar helical graft breaks the potential symmetric flow, reduces stagnation points and increases flow mixing. In addition, the single-spiral flow has been previously linked with reduced turbulent kinetic energy (Paul and Larman
2009; Stonebridge et al. 2004) and enhanced wall shear rate in comparison to Hagen–Poiseuille parabolic flow (Zhang et al. 2008), both of which are considered atheroprotective. The flow patterns and fluid dynamic parameters that were detected in the outflow of the spiral graft exhibited reduced flow separation and stagnation and increased inplane mixing, stability and coherence downstream from the venous anastomosis, in comparison to the control graft. The induction of hemodynamics with these characteristics in the outflow of AV grafts in patients may prevent the development of neo-intimal hyperplasia and thrombosis, which would lead to increased patency rates (Caro et al. 2005; Cookson et al. 2009; Stonebridge 2011). The proposed 2-D analysis could be applied in a clinical study to address the effectiveness of the spiral graft in vivo. Another parameter, which is considered detrimental for vascular implants and is related to hemodynamics, is pressure drop. A numerical study found that the pressure drop arises in helical tubes because of the energy required to drive vortical structures and the curvature of the device (Cookson et al. 2009). Therefore, it is possible that the spiral graft increases not only mixing, but also pressure drop. Moreover, the pressure drop in a tube is inversely proportional to the diameter. The spiral inducer ridge decreases the diameter, and as a consequence, it may increase the pressure drop. Determination of whether the spiral graft gives rise to the pressure drop and if this is significant or not would require direct measurements
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Fig. 10. Progression of circulation at scan planes 1 to 4 downstream from the outflow of the spiral and control looped grafts under all applied Reynolds numbers; *p , 0.05, **p # 0.01, ***p # 0.001.
and consideration in relation to the advantages from enhanced mixing and flow stability. To our knowledge this is the first time that whole-area vorticity maps based on color Doppler imaging have been generated. A similar approach was presented by Mehregan et al. (2014) to quantify the formation of intraventricular eddies during left ventricular filling. They pro-
posed a core vorticity technique where the center of each vortex was color mapped in red or blue hues in relation to vortex direction. Vorticity maps provide further understanding of spiral flow and an option to extract fluid dynamic parameters such as circulation. The evolving understanding of the clinical impact vortices have in the cardiovascular system could make 2-D vector Doppler,
Fig. 11. Progression of circulation at scan planes 1 to 4 downstream from the outflow of the spiral and control straight grafts under all applied Reynolds numbers. *p , 0.05, **p # 0.01, ***p # 0.001.
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Fig. 12. Numerical simulations of vortices moving in (a) rigid-body rotational flow pattern and (b) irrotational flow pattern. From left to right are velocity direction, velocity magnitude and vorticity maps. In direction maps, the length of the arrows follows the magnitude.
vorticity and extracted hemodynamic parameters important and low-cost tools in clinical diagnosis. The ID of the PVA-c venous mimic was measured using B-mode imaging. It was found to be 5.4 mm instead of 6 mm, which was the diameter of the mold. This suggests shrinkage of the PVA-c tubing during the molding process. The flow rate in the outflow of a physiologic AV graft varies because of the pulsatile cardiac waveform. Pulsatile flow can play an important role in blood mixing (Caro et al. 2005; Cookson et al. 2009). The peak flow rate in an AV graft depends on the patient and the patency of the prosthesis. In this study, a number of constant flow rates or Reynolds numbers were used, and their values were within the clinically observed range. The spiral graft was able to stabilize the induced flow patterns better than the control graft under the
applied flow rates. The ability of the spiral graft to stabilize the flow may add further to its protective effect under the physiologic pulsatile arterial circulation. Use of a liquid medium tissue mimic was possible because the constant flow rates prevented vibrations of the vascular-graft models. The application of a physiologic pulsatile waveform would require a solid tissue mimic to prevent expansions of the vessel mimic. Such expansions would affect the internal diameter of the vessel mimic, damp the waveform and induce clutter signal from the liquid medium tissue mimic. In this study we assumed that the arterial flow (C-Flex tubing) was blocked distal from the inlet graft anastomosis, and the venous flow, proximal from the graft outflow anastomosis (in relation to the flow direction). Clinically there is low flow in both locations for blood
Fig. 13. Schematic diagram of three characteristic single vortices. (a) Rigid-body rotation: both particles rotate about their own axis with the same rate and direction during time, and their vorticity is constant. (b) Irrotational flow: the particles do not change direction during time, and their vorticity is zero. (c) The first particle rotates around its own axis in a counterclockwise direction resulting in positive vorticity, and the second particle rotates around its own axis in a clockwise direction resulting in negative vorticity. The plus and minus signs indicate the direction of vorticity.
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circulation in the hand. We expect that this would have an impact on the secondary flow motions of access vascular graft outflows. CONCLUSIONS Vorticity mapping and circulation measurement were used, on the basis of vector Doppler ultrasound, to characterize vortical structures in AV graft outflows; these approaches could provide an alternative insight into blood rotational flow. One to two spirals were detected from the proximal to distal outflow of the spiral AV graft and two to four spirals for the control, respectively. The multispiral patterns were related to more stagnation points and flow separation. In-plane peak velocity and circulation were constantly higher for the spiral device, indicating increased in-plane mixing. These results indicate that the geometry of the graft affects the flow field and that the AV spiral graft may have a beneficial impact on the pathophysiology of the host vein and may result in increased clinical patency rates. Acknowledgments—We thank Professor Spencer J. Sherwin from Imperial College London for his suggestions in fluid mechanics. We are grateful to the Scottish Universities Physics Alliance (SUPA) INSPIRE program (supported by the Scottish Funding Council) and Vascular Flow Technologies Ltd. for funding this work. We also thank the Scottish Imaging Network: A Platform for Scientific Excellence (SINAPSE) for their support.
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Mehregan F, Tournoux F, Muth S, Pibarot P, Rieu R, Cloutier G, Garcia D. Doppler vortography: a color Doppler approach to quantification of intraventricular blood flow vortices. Ultrasound Med Biol 2014;40:210–221. Mickley V. Stenosis and thrombosis in haemodialysis fistulae and grafts: The surgeon’s point of view. Nephrol Dial Transplant 2004;19:309–311. Morrison FA. An introduction to fluid mechanics. New York: Cambridge University Press; 2013. Nichols WW, O’Rourke MF. McDonald’s blood flow in arteries. London: Hodder Arnold; 2005. Paul MC, Larman A. Investigation of spiral blood flow in a model of arterial stenosis. Med Eng Phys 2009;31:1195–1203. Pedersen MM, Pihl MJ, Haugaard P, Hansen JM, Hansen KL, Nielsen MB, Jensen JA. Comparison of real-time in vivo spectral and vector velocity estimation. Ultrasound Med Biol 2012;38:145–151. Rosenthal SJ, Harrison LA, Baxter KG, Wetzel LH, Cox GG, Batnitzky S. Doppler US of helical flow in the portal-vein. Radiographics 1995;15:1103–1111. Sherwin SJ, Shah O, Doorly DJ, Peiro J, Papaharilaou Y, Watkins N, Caro CG, Dumoulin CL. The influence of out-of-plane geometry on the flow within a distal end-to-side anastomosis. J Biomech Eng 2000;122:86–95. Slager CJ, Wentzel JJ, Gijsen FJ, Schuurbiers JC, van der Wal AC, van der Steen AF, Serruys PW. The role of shear stress in the generation of rupture-prone vulnerable plaques. Nat Clin Pract Cardiovasc Med 2005a;2:401–407. Slager CJ, Wentzel JJ, Gijsen FJH, van der Wal AC, Schaar JA, Serruys PW. The role of shear stress in the destabilization of vulnarable plaques and related therapeutic implications. Nature Clin Pract Cardiovasc Med 2005b;2:456–464. Stonebridge PA. Three-dimensional blood flow dynamics: Spiral/helical laminar flow. Methodist Debakey Cardiovasc J 2011;7:21–26. Stonebridge PA, Brophy CM. Spiral laminar flow in arteries? Lancet 1991;338:1360–1361. Stonebridge PA, Buckley C, Thompson A, Dick J, Hunter G, Chudek JA, Houston JG, Belch JJ. Non spiral and spiral (helical) flow patterns in stenoses: In vitro observations using spin and gradient echo magnetic resonance imaging (MRI) and computational fluid dynamics modeling. Int Angiol 2004;23:276–283. Stonebridge PA, Vermassen F, Dick J, Belch JJ, Houston JG. Spiral laminar flow prosthetic bypass graft: Medium-term results from a first-in-man structured registry study. Ann Vasc Surg 2012;26:1093–1099. Tanaka M, Sakamoto T, Sugawara S, Nakajima H, Kameyama T, Katahira Y, Ohtsuki S, Kanai H. Spiral systolic blood flow in the ascending aorta and aortic arch analyzed by echo-dynamography. J Cardiol 2010;56:97–110. Tricht I, Wachter D, Tordoir J, Verdonck P. Hemodynamics and complications encountered with arteriovenous fistulas and grafts as vascular access for hemodialysis: A review. Ann Biomed Eng 2005;33:1142–1157. Tritton DJ. Physical fluid dynamics. New York: Oxford University Press; 2007. Udesen J, Nielsen MB, Nielsen KR, Jensen JA. Examples of in vivo blood vector velocity estimation. Ultrasound Med Biol 2007;33:541–548. Walsh MT, Kavanagh EG, O’Brien T, Grace PA, McGloughlin T. On the existence of an optimum end-to-side junctional geometry in peripheral bypass surgery-a computer generated study. Eur J Vasc Endovasc Surg 2003;26:649–656. Zhang Z, Fan Y, Deng X, Wang G, Zhang H, Guidoin R. Simulation of blood flow in a small-diameter vascular graft model with a swirl (spiral) flow guider. Sci China C Life Sci 2008;51:913–921.
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APPENDIX VALIDATION OF VORTICITY AND ITS RELATION TO SHEARING PROCESSES Vorticity provides information on the rotation of the fluid from the point of view of a particle moving in the fluid, as a result of the shearing process. An element of fluid may be considered to be translating (moving with no rotation) and rotating (rotating but with no translation). Vorticity is concerned only with the rotational component, so that a fluid element, which is translating but not rotating, has zero vorticity. Two characteristic vortical types in the cross-flow view of a tube, which were originally simulated to verify the vector Doppler algorithm, were used to validate vorticity and understand how it relates to shearing forces. The first is a rigid-body rotation, where the particles translate in counterclockwise circular trajectories with speed proportional to their distance from the center (Fig. 12a). At the same time, they rotate about their own axis in counterclockwise direction and at a constant rate because of the shear forces between adjacent circular trajectories. This flow is expected to have a steady positive vorticity (according to the right-hand rule) because of the constant rate of counterclockwise rotation of the particles about their own axis, as verified in Figure 12a. The second example is an irrotational flow pattern, where the particles move with speed inversely proportional to their distance from the center (Fig. 12b). As a result, they translate in a circular pattern, but the shear forces keep them in a steady orientation and they do not rotate about their own axis. This is why the flow is called irrotational and vorticity is zero as verified in Figure 12b (Morrison 2013). The existence of vorticity in the center of the vortex is the result of zero velocity in it. Further understanding of these shearing processes is provided in the schematic diagram of Figure 13, which illustrates three characteristic examples of vorticity in a counterclockwise single vortex. The black arrows represent the magnitude of translational velocity in each trajectory, and two particles are illustrated in three different times (t1– t3). The blue and red arrows in the center of the first and second particles, respectively, indicate their direction in space as they move in circular counterclockwise paths. Figure 13a illustrates a rigid-body vortex as described for Figure 12a. The shear forces, induced by the proportional increase in translational velocity with distance from the centre of rotation, rotate both particles about their own axis having the same direction during time (translation with steady rate of rotation) and their vorticity remains constant. Figure 13b is an irrotational vortex as described for Figure 12b. The shearing process, induced by the proportional decrease in translational velocity with distance from the centre of rotation, keeps the orientation of both particles steady during time (translation with no rotation) and their vorticity is zero. In Figure 13c, the translational velocity of the vortex is increased from the center of the spiral to the third trajectory. This rotates the first particle about its own axis in counterclockwise direction and its vorticity is positive. After the third trajectory, the translational velocity is decreased because of the no-slip boundary condition and becomes zero relative to the wall. The inverse shear forces that are applied in the second particle rotate it about its own axis in clockwise direction, and its vorticity has opposite direction in relation to the first particle. Thus, the vorticity of this single vortex appears to change direction near the wall. Figure 7g–i is a characteristic case of this example.
http://dx.doi.org/10.1016/j.ultrasmedbio.2014.10.012
d
Original Contribution COMPARISON OF VORTICAL STRUCTURES INDUCED BY ARTERIOVENOUS GRAFTS USING VECTOR DOPPLER ULTRASOUND EFSTRATIOS KOKKALIS,*y ANDREW N. COOKSON,z PETER A. STONEBRIDGE,y GEORGE A. CORNER,x J. GRAEME HOUSTON,y and PETER R. HOSKINS{ * Institute for Medical Science and Technology, University of Dundee, Dundee, UK; y Cardiovascular and Diabetes Medicine, Ninewells Hospital and Medical School, University of Dundee, Dundee, UK; z Department of Biomedical Engineering, King’s College London, London, UK; x Medical Physics, Ninewells Hospital and Medical School, Dundee, UK; and { Centre for Cardiovascular Science, University of Edinburgh, Edinburgh, UK (Received 17 March 2014; revised 18 August 2014; in final form 18 October 2014)
Abstract—Arteriovenous prosthetic grafts are used in hemodialysis. Stenosis in the venous anastomosis is the main cause of occlusion and the role of local hemodynamics in this is considered significant. A new spiral graft design has been proposed to stabilize the flow phenomena in the host vein. Cross-flow vortical structures in the outflow of this graft were compared with those from a control device. Both grafts were integrated in identical in-house ultrasound-compatible flow phantoms with realistic surgical configurations. Constant flow rates were applied. Inplane 2-D velocity and vorticity mapping was developed using a vector Doppler technique. One or two vortices were detected for the spiral graft and two to four for the control, along with reduced stagnation points for the former. The in-plane peak velocity and circulation were calculated and found to be greater for the spiral device, implying increased in-plane mixing, which is believed to inhibit thrombosis and neo-intimal hyperplasia. (E-mail: [email protected] or [email protected]) Ó 2015 World Federation for Ultrasound in Medicine & Biology. Key Words: Color Doppler, Vector Doppler ultrasound, Velocity, Vorticity, Circulation, Flow phantom, Arteriovenous prosthetic grafts, Vortical structures, Spiral flow, Flow mixing and stagnation.
2010) and graft anastomoses (Doorly et al. 2002; Lee et al. 2005). The display of vortical phenomena requires at least two components of velocity. The evolving understanding of the role of local hemodynamics in endothelial function and the proliferation of vascular smooth muscle (Caro 2009; Malek et al. 2013; Slager et al. 2005a, 2005b) serves to increase the interest in more detailed velocity profiles and fluid dynamic assessments. During the last few decades, a number of vector Doppler approaches have been proposed (Fei et al. 1994; Garcia et al. 2010; Hoskins et al. 1994; Jensen and Munk 1998; Maniatis et al. 1994), and recently, a real-time technique was integrated into a conventional system (Pedersen et al. 2012). The rotational nature of blood flow in vessels is widely accepted. In 1990, Frazin et al. detected spiral flow in the thoracic aorta using color Doppler. Stonebridge and Brophy (1991) were the first to suggest the blood flow in arteries is spiral in nature. Subsequent ultrasound studies confirmed the presence of spiral flow in arteries (Frazin et al. 1996; Hoskins et al. 1994; Tanaka et al. 2010; Udesen et al. 2007), veins (Marie
INTRODUCTION Color Doppler ultrasound has been found to be of great value in clinical diagnosis where blood flow can be visualized and quantified to assess cardiovascular complications. However, conventional color Doppler imaging is limited to displaying two spatial components and one velocity component. The velocity measurements are vulnerable to errors mainly because of the angle dependency between the ultrasound beam and the direction of the target (Evans et al. 2011; Hoskins 2010). In addition, the complex vortical structures that occur in the cardiovascular system increase the angle dependency and cannot be quantified. These structures can be intense in bifurcations (Udesen et al. 2007), poststenotic regions (Hoskins 1997), the heart (Garcia et al.
Address correspondence to: Efstratios Kokkalis, Ninewells Hospital and Medical School, Mail Box 1, Level 7, University of Dundee, Dundee DD1 9 SY, UK. E-mail: [email protected] or [email protected] Conflicts of interest: This work was supported by Vascular Flow Technologies Ltd. of which J.G.H and P.A.S are founder members. 760
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et al. 2014; Rosenthal et al. 1995) and the heart (Garcia et al. 2010; Mehregan et al. 2014). It has also been verified with magnetic resonance imaging (Kilner et al. 1993; Markl et al. 2005) and computational modeling (Gallo et al. 2012; Hardman et al. 2013). The formation of vortical structures in blood originates from the rotational compression of the heart (Jung et al. 2006) and is propagated because of the curvature, branching, non-planarity and spiral folds in the arteries (Caro et al. 1996; Stonebridge 2011; Stonebridge and Brophy 1991). Caro et al. (1996, 2005) hypothesized that helical flow in arteries supports in-plane mixing and uniform distribution of wall shear and inhibits blood stagnation, separation and instability. Similarly Stonebridge et al. (2004) maintained that single-spiral induces flow stability and coherence, and Cookson et al. (2009) verified and explained an increase of in-plane mixing as a result of helical flow. Recently, Marie et al. (2014) reported single-spiral flow as a predictor of successful arteriovenous (AV) fistula maturation. As an alternative to AV fistulas, prosthetic grafts can be used for vascular access. The most frequent complication of AV prostheses is stenosis, which can lead to thrombosis and occlusion. Stenoses can develop in the venous or arterial anastomosis from neo-intimal hyperplasia or the mid-graft section as a result of ingrowth of fibrous tissue through puncture holes. The former is the most common incident, with disturbed hemodynamics being one of the causes (Gessaroli and Massini 2012; Manos et al. 2010; Mickley 2004; Tricht et al. 2005). An AV graft can be implanted in either the arm or the thigh, with patency depending on the location. In an AV graft review Akoh (2009) reported 1- and 2-y cumulative patency rates of 59%–90% and 47%–85% in the arm and 41%–68% and 26%–43% in the thigh, respectively, indicating the need for steady and enhanced patency. A novel AV prosthetic graft specifically engineered to introduce a stabilized spiral flow has recently been developed and introduced for vascular access applications. Clinical data are not yet available but a similar peripheral vascular spiral graft has been compared with a control device in vitro exhibiting improved secondary flow phenomena (Kokkalis et al. 2013) and has been applied in vivo with enhanced patency rates in a
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30-mo follow-up (Stonebridge et al. 2012). These promising results motivated us to investigate the hemodynamics of the AV spiral graft and the underlying mechanism. The aim of this article is to describe and compare the rotational structures in the outflow of the spiral AV graft and a non-spiral control AV graft, using vector Doppler ultrasound. Two-dimensional velocity and vorticity maps were developed; peak velocity and circulation were identified for quantification. METHODS Arteriovenous access grafts The spiral and control grafts were made of expanded polytetrafluoroethylene (ePTFE) and had a 6-mm inner diameter (ID) (Vascular Flow Technologies, Dundee, UK). The former device is supported by an injection molded polyurethane spiral flow inducer, which forms an internal non-planar helical geometry along the distal end of the graft (Fig. 1a). The control device was identical in all parameters, but with a smooth planar internal geometry along its length. Flow phantom setup The proximal end of each graft was connected to C-Flex (Cole–Parmer, London, UK) tubing, which simulated the arterial flow and the distal end with an in-house venous mimic with end-to-side anastomoses. The venous mimic was made of polyvinyl alcohol cryogel (PVA-c). This vascular-graft model was embedded in a liquid medium tissue mimic and housed in an acrylic tank (485 3 180 3 60 mm). The tissue mimic composition by volume was 9% glycerol and 91% distilled water. PVC tubing was used to connect the outflow of a flow pump with the C-Flex tubing (artery) and the PVA-c venous mimic with the inflow of the pump. The connections between the PVC tubing and vascular mimics on the wall of the tank were supported by straight, rigid 60-mm-long PVC screw connectors. An ultrasound Doppler test fluid (Model 707, ATS Laboratories, Bridgeport, CT, USA) was used as blood mimic. The flow pump was a UHDC computer controlled piston system (Shelley Medical Imaging Technologies, London,
Fig. 1. (a) Distal end of the 6-mm-inner-diameter arteriovenous spiral graft. (b) Side view diagram of the distal cuff where angle 4 5 15 .
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ON, Canada) with steady cycle-to-cycle reproducibility and a standard deviation less than 1% for constant flow rate up to 35 mL s21. All used tubes and connectors had a 6-mm-ID matching that of the grafts. Geometry of vascular-graft models Two pairs of vascular-graft models were created. Each pair consisted of a spiral graft model and a nonspiral graft model with identical geometries. The looped surgical configuration was simulated in one of the pairs and the straight in the other (Fig. 2) (Akoh 2009; Brown and Mitchell 2014). The proximal end of the looped grafts was cut at 15 , and for the straight grafts, at 90 . The distal cuff of all grafts was trimmed at 15 (Fig. 1b). The lengths of both spiral and control grafts in each pair of models were identical, as were the lengths of vascular mimics. The location and type of the proximal (arterial) and distal (venous) anastomoses in each pair of models were also identical. The overall geometry of the spiral and non-spiral models in each surgical configuration was the same. The anastomoses were made with superglue and reinforced with adhesive epoxy. Their quality and similarity to clinical practice were verified by a consultant vascular surgeon. The distal end of the C-Flex tubing and proximal end of the PVA-c venous mimic (in relation to the flow direction) were blocked, forcing the blood mimic to follow the grafted region and venous flow direction. Figure 2 illustrates the overall
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geometry of the models and terminology in the distal end-to-side anastomoses. The 3-D flow profile in a fluid circuit is strongly influenced by the tubing geometry. The pump, tubing and vascular-graft models were positioned at the same height to create a planar flow design excluding the non-planar spiral inducer region in the spiral graft. The inlet length distance was taken into consideration proximal to the inlet of each graft to ensure a fully developed parabolic profile and minimize flow variations (Kokkalis et al. 2013; Nichols and O’Rourke 2005). Acoustic properties of mimicking materials An important parameter of the mimicking components in an ultrasound flow phantom is their acoustic properties. These components need to be similar and also to agree with those observed in the human body. Differences in the ultrasound beam propagation speed need to be minimized because they cause refraction of the beam and distortion of the detected Doppler frequency shift (Hoskins 1994, 2007, 2008). Production of the in-house PVA-c venous mimic and its acoustic properties, as well as the acoustic properties of blood and tissue mimic, were described in detail in our previous work (Kokkalis et al. 2013). These properties were based on the specifications of the International Electrotechnical Commission (2001). During the study, temperature was monitored and maintained at 21 6 1 C to avoid changes in acoustic properties. Vorticity and circulation Two-dimensional velocity information was obtained and visualized in each scanning location using a previously described dual-beam vector Doppler technique, which is approximately 98% accurate providing that the Doppler frequency shifts are above the clutter filter and not aliased (Hoskins 1997; Hoskins et al. 1994; Kokkalis et al. 2013). Similar velocity maps and a comparison of peak cross-flow velocity were previously used to characterize spiral flow patterns induced by peripheral vascular grafts (Kokkalis et al. 2013). This comparison gives an indication of the velocity range, but it is based on a single point. Quantities from fluid mechanics were investigated to quantify the total 2-D cross-flow spiral field. Circulation, which is such a parameter, was calculated as an index of comparison between the spiral and non-spiral grafts. The calculation of circulation was based on vorticity, as described below.
Fig. 2. (a) Looped configuration spiral graft model. (b) Straight configuration spiral graft model. The respective control models were identical. Scan plane 1 was positioned 0.1 cm distal from the graft outflow. Terminology for the end-to-side venous anastomosis is provided. AV 5 arteriovenous.
Vorticity. It is a vector field, or more precisely a pseudo-vector, that shows the spinning motions of particles in a fluid as a result of the shearing forces applied on them. Vorticity, ! u , is defined as the curl of the fluid velocity field:
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Fig. 3. Color Doppler images of spiral flow in the cross-flow view. (a) Single spiral. (b) Double spiral. (c) Four-spiral pattern.
! u 5 V3! y
(1)
where ! y is the velocity vector. Providing that the 2-D flow maps of this study were parallel to the x–y plane of a Cartesian coordinate system, it is the axial component of the vorticity vector in which we are interested. This is defined as vyy vyx 2 uz k 5 k (2) vx vy where yx and yy are the components of the velocity vector ! y and were calculated with the vector Doppler technique, and k is the unit vector in the axial direction. In this case, the vorticity vector is always perpendicular to the examined plane of the flow or parallel to axis z (Morrison 2013; Tritton 2007). The relation between the direction of rotation of an element and the direction of the vorticity vector is described by the right-hand rule. The unit of vorticity is radians per second (Morrison 2013). A validation of the vorticity algorithm and its relation to shearing processes is provided in Appendix. Circulation. Circulation, G, is the line integral around any arbitrary closed curve C of the velocity field ! y in a fluid. If dL is an element of curve C, then this line integral is defined by the first part of eqn (3). If S is the surface that has the closed curve C as boundary and dS is an element of this surface, then from Stoke’s theorem it can be seen that circulation is also equal to the integral of vorticity ! u in this area, as in the last part of the equation I ð ! y ,dl 5 ! u ,dS (3) Gc 5 c
s
From the above is derived that the units of circulation are length squared over time (m2 s21). This parameter allows quantification of the strength of a vortex (Morrison 2013; Tritton 2007).
Data acquisition Color Doppler cine loops were recorded for both pairs of phantoms at 0.1, 0.5, 2, 5 (Fig. 2a, b, scan planes 1–4) and 10 cm distal from the graft outflow, to study the development of vortical structures. The abnormal hemodynamics that an AV graft introduces in the vein are linked with stenosis primarily in the junction and the first 2–5 cm distal of it (Haruguchi and Teraoka 2003), but also downstream of the venous flow (Doelman et al. 2005). These scan planes were transverse to the flow in both anteroposterior and right-to-left directions. In this way, helical patterns in a vessel or vessel mimic can be visualized as ‘‘red–blue splits,’’ representing flow toward and away from the transducer (Frazin et al. 1990; Hoskins et al. 1994). The number of splits in a scan plane is dependent on the number of vortices in it (Fig. 3). Constant flow rates of 240, 360, 480, 600 and 720 mL min21, or corresponding Reynolds numbers 570, 850, 1140, 1420 and 1700, respectively, were applied. The Reynolds number was based on the mean flow velocity, ID of the phantom and kinematic viscosity of the blood mimic. Flow rates less than 600 mL min21 are linked with thrombosis; 300 mL min21 is the threshold for adequate dialysis (Akoh 2009). In each scan plane, two cine loops were recorded, one with the beam-target angle at 120 and one at 220 in relation to the linear transmission. Each recording was 115 frames, filling the limit of the memory buffer. Steady transducer positioning and relatively accurate displacements were enabled with a retort clamp adjusted in a micromanipulator. Each scanning was repeated three times after transducer repositioning to confirm repeatability. The data were transferred for post-processing with an in-house suite of programs in MATLAB R2012a (The MathWorks, Natick, MA, USA). Each cine loop was averaged to reduce noise from small flow pattern variations and color speckle. The averaged velocity information from the left and right steered projections were
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centralized to avoid lateral displacement and subjected to triangulation with our vector Doppler algorithm (Kokkalis et al. 2013). Thus the 2-D velocity vector was identified in each scan plane, and velocity direction, velocity magnitude and vorticity maps were developed. The ultrasound system employed was an HDI 5000 (ATL Ultrasound, Bothell, WA, USA) with a linear transducer (L12-5). The color Doppler settings were persistence low, smoothing low, sensitivity high, wall filter low and dynamic motion differentiation off. B-Mode imaging was used to measure the ID of the PVA-c venous mimic. Data selection criteria A significant flow pattern variation from frame to frame was observed for the non-spiral models at the 10-cm scan plane for all applied flow rates. For low flow rates, the in-plane tangential velocity signal was not only unstable but also weak, and the Doppler frequency shift was low. Reduced pulse repetition frequency and increased color gain were tried for the detection of these weak secondary flow motions, but clutter signal was induced from the venous and tissue mimic regions. These observations may suggest a decline of rotational structures at cross-flow plane 4 under low flow rates for the control models. On the contrary, a clear and steady flow pattern was detected at the respective scanning location and flow parameters for the spiral graft models. Because of the lack of satisfactory Doppler signal and flow stability from the control models at the 10-cm scan plane, the location was excluded from the study. Data analysis Peak velocity comparison. The angle independent peak cross-flow velocity was detected in each scanning location under all applied flow parameters. These measurements were normalized by the mean axial velocity, y, as indicated in the equation by 5
y y
(4)
where by is the normalized velocity, and y 5 Q=A, with Q being the average of the five applied flow rates and A the area of the venous mimic, and a comparison between the spiral and control graft models was applied. Circulation detection and comparison. The vorticity map was generated for each 2-D velocity data set. Vorticity is the kinematic representation of the shearing processes in a flow field and, thus, may appear to change direction not only because of counter-rotated vortices but also within the area of one vortex. To calculate the strength of the vortical structures in the whole area of a scanning location, circulation was computed for both
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positive and negative areas of vorticities. The contiguous regions of positive and negative vorticities were integrated over their areas, and their absolute values were summed. This was applied for all scan planes and flow parameters, for both pairs of phantoms. Circulation, G, was normalized by the mean circulation, G, for the looped and straight pair of phantoms b 5G G G
(5)
b is the normalized circulation. where G Statistical analysis The significance of the differences in peak velocity and circulation in the comparison between spiral and non-spiral models was determined with unpaired twotailed t-tests. A p-value , 0.05 was considered to indicate statistically significant differences; a p-value # 0.01, highly significant differences; and a p-value # 0.001, very highly significant differences. The t-tests were executed with SPSS Version 21 (IBM, Armonk, NY, USA).
RESULTS Color Doppler ultrasound was used for initial observation of vortical structures in the cross-flow view and for their understanding using conventional color Doppler. Examples of single-, double- and four-spiral patterns are illustrated in Figure 3a–c, respectively. Two-dimensional velocity and vorticity maps were developed based on vector Doppler imaging and are illustrated in Figures 4–7. These figures represent the four scan planes from the proximal (scan plane 1 at 0.1 cm) to the distal (scan plane 4 at 5 cm) outflow of the looped configuration models (Fig. 2a). Spiral and control graft outflow mapping is included in each one for Reynolds numbers 570, 1140 and 1700. The orientation of the maps is such that the left and right sides are the floor and toe wall sides of the vessel mimic respectively, and the top and bottom sides are the anterior and posterior wall sides respectively (see Fig. 2a). Velocity direction maps are shown in the first row of each device, velocity magnitude in the second and vorticity in the third. The color distribution in velocity direction maps represents the direction of the velocity vector, with dark blue and dark red the limits from 0 to 360 . The size of the vectors is proportional to velocity magnitude. In velocity magnitude maps, dark red represents maximum value and dark blue zero. The vorticity maps correspond to axial vorticity, uZ . Thick black contour lines at uZ 5 0 separate the regions of positive and negative vorticity. Positive
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Fig. 4. In-plane velocity direction (first and fourth rows), velocity magnitude (second and fifth rows) and axial vorticity uz maps (third and sixth rows) from the spiral and control looped vascular-graft models at scan plane 1. Reynolds numbers (Re) 570, 1140 and 1700 are shown for each model. The boundaries of zero vorticity are represented by a thick black contour line. The orientation of each map in relation to the anastomotic locations is: left 5 floor, right 5 toe, top 5 anterior and bottom 5 posterior.
regions are associated with counterclockwise vorticity, and negative regions with clockwise vorticity. The number of spirals in Figures 4–7 can be seen from the velocity directional map, and their centers of
rotation, from the velocity magnitude and vorticity maps. The direction maps in scan plane 1 reveals a double spiral for both grafts (Fig. 4). For the spiral device, it is composed of a prevalent spiral on the
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Fig. 5. In-plane velocity direction (first and fourth rows), velocity magnitude (second and fifth rows) and axial vorticity uz maps (third and sixth rows) from the spiral and control looped vascular-graft models at scan plane 2. Reynolds numbers (Re) 570, 1140 and 1700 are shown for each model. The boundaries of zero vorticity are represented by a thick black contour line. The orientation of each map in relation to the anastomotic locations is: left 5 floor, right 5 toe, top 5 anterior and bottom 5 posterior.
anterior side (top) and a smaller spiral in the posterior (bottom), whereas for the control, it has a symmetric double-spiral pattern or Dean-type flow. In scan plane 2 (Fig. 5), the patterns for the spiral graft remain
similar to those observed in scan plane 1 (Fig. 4), whereas for the control, there is a three-spiral pattern for Reynolds numbers 1140 and 1700, respectively. In scan plane 3 of the spiral model (Fig. 6a–i), the second
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Fig. 6. In-plane velocity direction (first and fourth rows), velocity magnitude (second and fifth rows) and axial vorticity uz maps (third and sixth rows) from the spiral and control looped vascular-graft models at scan plane 3. Reynolds numbers (Re) 570, 1140 and 1700 are shown for each model. The boundaries of zero vorticity are represented by a thick black contour line. The orientation of each map in relation to the anastomotic locations is: left 5 floor, right 5 toe, top 5 anterior and bottom 5 posterior.
small vortex can be seen under low Reynolds numbers, but is coalesced with the main vortex under the peak Reynolds number. This process is more easily observed from the velocity directional maps (Fig. 6a–c). In scan
plane 3 of the non-spiral graft (Fig. 6j–r), Dean flow is seen for Reynolds number 570 and a four-spiral pattern for Reynolds numbers 1140 and 1700 (Fig. 6j–r). In scan plane 4, one uniform vortex is illustrated for the
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Fig. 7. In-plane velocity direction (first and fourth rows), velocity magnitude (second and fifth rows) and axial vorticity uz maps (third and sixth rows) from the spiral and control looped vascular-graft models at scan plane 4. Reynolds numbers (Re) 570, 1140 and 1700 are shown for each model. The boundaries of zero vorticity are represented by a thick black contour line. The orientation of each map in relation to the anastomotic locations is: left 5 floor, right 5 toe, top 5 anterior and bottom 5 posterior.
spiral device (Fig. 7a–i). For the control model in plane 4, Dean flow is seen for Reynolds number 570, a fourspiral pattern for 1140 and a disturbed pattern for 1700 (Fig. 7j–r).
In the vorticity maps of Figures 4–7, the core of each vortex is clearly illustrated along with regions of counterrotation near the wall and between counter-rotating vortices in multispiral patterns. A counter-rotation region
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Fig. 8. Progression of peak in-plane velocity at scan planes 1 to 4 downstream from the outflow of the spiral and control looped grafts under all applied Reynolds numbers. *p , 0.05, **p # 0.01, ***p # 0.001.
in the area of one vortex is the result of opposite shearing forces in the flow trajectories of this region. Solutions of this nature have been reported by Cookson et al. (2009). For more, see the Appendix (particularly Fig. 13c). Figures 8 and 9 illustrate the comparison of peak inplane velocity for the looped and straight configuration models, respectively. The velocity is consistently higher for the spiral prosthesis apart from Figure 9, plane 1, Reynolds number 1420. This difference is significantly higher in almost all cases for the looped models and in almost all cases from plane 2 and beyond for the straight models. Figures 10 and 11 illustrate the comparison of circulation for the looped and straight configuration models, respectively. Circulation is constantly higher for the spiral device, except for Figure 11, plane 1, Reynolds numbers 850 and 1140. This difference is significant in all scan planes for the looped models, apart from plane 1, Reynolds number 1140. In the straight surgical configuration, the difference is significant from plane 2 and beyond. The differences in Figures 8–11 increased with the increase in Reynolds number, and the level of significance increased with distance from the anastomosis. There is a match in the ranking and level of significance between the peak velocity and circulation comparisons in each pair of phantoms. DISCUSSION The comparison of peak in-plane velocity and circulation in Figures 8–11, revealed increased values for the spiral
graft. This quantitative evaluation indicated increased in-plane mixing for the spiral device. In-plane mixing has been proposed to protect anastomotic regions from neo-intimal hyperplasia and thrombosis (Caro et al. 2005). The rotational patterns within the control graft– venous mimic model revealed a constant Dean flow for Reynolds number 570, which implied a plane of symmetry about the model centerline (Figs. 4–7j, m, p). The number of spiral cells from plane 2 and beyond was increased from 2 to 4 as the Reynolds number was increased. At peak Reynolds numbers, in scan plane 4, the pattern had a disturbed or chaotic profile, as it was presented with both the velocity and vorticity maps. These results may imply flow separation and instability in the outflow of the planar control graft, which may be detrimental for the vascular function in vivo (Caro et al. 2005; Stonebridge 2011). Moreover, all the velocity magnitude maps presented low velocities or cross-flow stagnation points in the wall sides of the floor and toe, locations where adjacent vortexes were mixed. Such points were also seen in anteroposterior locations, although we focus on the floor and toe sides, which are widely associated with neo-intimal hyperplasia in patients (Doorly et al. 2002; Sherwin et al. 2000; Tricht et al. 2005; Walsh et al. 2003). In contrast, the rotational patterns in each scan plane of the spiral graft had relatively steady profiles, stabilizing the flow even at peak Reynolds numbers. The proximal outflow of the spiral device (planes 1 and 2) revealed a main and a secondary smaller spiral. This
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Fig. 9. Progression of peak in-plane velocity at scan planes 1 to 4 downstream from the outflow of the spiral and control straight grafts under all applied Reynolds numbers. *p , 0.05, **p # 0.01, ***p # 0.001.
smaller spiral was coalesced with the main one in scan plane 3 (Fig. 6a–i), and as a result, a steady uniform bulk rotation was presented in plane 4. The ability of the spiral graft to develop a single spiral adds to its stabilizing effect (Fig. 7a–i). In terms of stagnation points in the floor and toe wall sides, they were detected in the floor of planes 1 and 2, as seen in the velocity magnitude maps. The above flow pattern contrast was supported by the results from the excluded scan plane 5 at 10 cm distal the venous anastomosis, where a steady single-spiral was found for the spiral graft and a weak temporally unsteady or disturbed flow pattern was found for the non-spiral (see Data Selection Criteria). Doorly et al. (2002) and Sherwin et al. (2000) have previously compared planar and non-planar anastomotic models using computational and experimental methods with low Reynolds numbers (Re # 600). Their models can be compared with the control and spiral models of this study, respectively. In the outflow of the planar anastomotic model, they noted Dean flow related to a centerline plane of symmetry and floor–toe located stagnation points. In the outflow of the non-planar anastomotic model, they reported a single recirculation, associated with reduced peak wall shear stress magnitude at the floor of the anastomosis. They supported that a non-planar helical graft breaks the potential symmetric flow, reduces stagnation points and increases flow mixing. In addition, the single-spiral flow has been previously linked with reduced turbulent kinetic energy (Paul and Larman
2009; Stonebridge et al. 2004) and enhanced wall shear rate in comparison to Hagen–Poiseuille parabolic flow (Zhang et al. 2008), both of which are considered atheroprotective. The flow patterns and fluid dynamic parameters that were detected in the outflow of the spiral graft exhibited reduced flow separation and stagnation and increased inplane mixing, stability and coherence downstream from the venous anastomosis, in comparison to the control graft. The induction of hemodynamics with these characteristics in the outflow of AV grafts in patients may prevent the development of neo-intimal hyperplasia and thrombosis, which would lead to increased patency rates (Caro et al. 2005; Cookson et al. 2009; Stonebridge 2011). The proposed 2-D analysis could be applied in a clinical study to address the effectiveness of the spiral graft in vivo. Another parameter, which is considered detrimental for vascular implants and is related to hemodynamics, is pressure drop. A numerical study found that the pressure drop arises in helical tubes because of the energy required to drive vortical structures and the curvature of the device (Cookson et al. 2009). Therefore, it is possible that the spiral graft increases not only mixing, but also pressure drop. Moreover, the pressure drop in a tube is inversely proportional to the diameter. The spiral inducer ridge decreases the diameter, and as a consequence, it may increase the pressure drop. Determination of whether the spiral graft gives rise to the pressure drop and if this is significant or not would require direct measurements
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Fig. 10. Progression of circulation at scan planes 1 to 4 downstream from the outflow of the spiral and control looped grafts under all applied Reynolds numbers; *p , 0.05, **p # 0.01, ***p # 0.001.
and consideration in relation to the advantages from enhanced mixing and flow stability. To our knowledge this is the first time that whole-area vorticity maps based on color Doppler imaging have been generated. A similar approach was presented by Mehregan et al. (2014) to quantify the formation of intraventricular eddies during left ventricular filling. They pro-
posed a core vorticity technique where the center of each vortex was color mapped in red or blue hues in relation to vortex direction. Vorticity maps provide further understanding of spiral flow and an option to extract fluid dynamic parameters such as circulation. The evolving understanding of the clinical impact vortices have in the cardiovascular system could make 2-D vector Doppler,
Fig. 11. Progression of circulation at scan planes 1 to 4 downstream from the outflow of the spiral and control straight grafts under all applied Reynolds numbers. *p , 0.05, **p # 0.01, ***p # 0.001.
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Fig. 12. Numerical simulations of vortices moving in (a) rigid-body rotational flow pattern and (b) irrotational flow pattern. From left to right are velocity direction, velocity magnitude and vorticity maps. In direction maps, the length of the arrows follows the magnitude.
vorticity and extracted hemodynamic parameters important and low-cost tools in clinical diagnosis. The ID of the PVA-c venous mimic was measured using B-mode imaging. It was found to be 5.4 mm instead of 6 mm, which was the diameter of the mold. This suggests shrinkage of the PVA-c tubing during the molding process. The flow rate in the outflow of a physiologic AV graft varies because of the pulsatile cardiac waveform. Pulsatile flow can play an important role in blood mixing (Caro et al. 2005; Cookson et al. 2009). The peak flow rate in an AV graft depends on the patient and the patency of the prosthesis. In this study, a number of constant flow rates or Reynolds numbers were used, and their values were within the clinically observed range. The spiral graft was able to stabilize the induced flow patterns better than the control graft under the
applied flow rates. The ability of the spiral graft to stabilize the flow may add further to its protective effect under the physiologic pulsatile arterial circulation. Use of a liquid medium tissue mimic was possible because the constant flow rates prevented vibrations of the vascular-graft models. The application of a physiologic pulsatile waveform would require a solid tissue mimic to prevent expansions of the vessel mimic. Such expansions would affect the internal diameter of the vessel mimic, damp the waveform and induce clutter signal from the liquid medium tissue mimic. In this study we assumed that the arterial flow (C-Flex tubing) was blocked distal from the inlet graft anastomosis, and the venous flow, proximal from the graft outflow anastomosis (in relation to the flow direction). Clinically there is low flow in both locations for blood
Fig. 13. Schematic diagram of three characteristic single vortices. (a) Rigid-body rotation: both particles rotate about their own axis with the same rate and direction during time, and their vorticity is constant. (b) Irrotational flow: the particles do not change direction during time, and their vorticity is zero. (c) The first particle rotates around its own axis in a counterclockwise direction resulting in positive vorticity, and the second particle rotates around its own axis in a clockwise direction resulting in negative vorticity. The plus and minus signs indicate the direction of vorticity.
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circulation in the hand. We expect that this would have an impact on the secondary flow motions of access vascular graft outflows. CONCLUSIONS Vorticity mapping and circulation measurement were used, on the basis of vector Doppler ultrasound, to characterize vortical structures in AV graft outflows; these approaches could provide an alternative insight into blood rotational flow. One to two spirals were detected from the proximal to distal outflow of the spiral AV graft and two to four spirals for the control, respectively. The multispiral patterns were related to more stagnation points and flow separation. In-plane peak velocity and circulation were constantly higher for the spiral device, indicating increased in-plane mixing. These results indicate that the geometry of the graft affects the flow field and that the AV spiral graft may have a beneficial impact on the pathophysiology of the host vein and may result in increased clinical patency rates. Acknowledgments—We thank Professor Spencer J. Sherwin from Imperial College London for his suggestions in fluid mechanics. We are grateful to the Scottish Universities Physics Alliance (SUPA) INSPIRE program (supported by the Scottish Funding Council) and Vascular Flow Technologies Ltd. for funding this work. We also thank the Scottish Imaging Network: A Platform for Scientific Excellence (SINAPSE) for their support.
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Mehregan F, Tournoux F, Muth S, Pibarot P, Rieu R, Cloutier G, Garcia D. Doppler vortography: a color Doppler approach to quantification of intraventricular blood flow vortices. Ultrasound Med Biol 2014;40:210–221. Mickley V. Stenosis and thrombosis in haemodialysis fistulae and grafts: The surgeon’s point of view. Nephrol Dial Transplant 2004;19:309–311. Morrison FA. An introduction to fluid mechanics. New York: Cambridge University Press; 2013. Nichols WW, O’Rourke MF. McDonald’s blood flow in arteries. London: Hodder Arnold; 2005. Paul MC, Larman A. Investigation of spiral blood flow in a model of arterial stenosis. Med Eng Phys 2009;31:1195–1203. Pedersen MM, Pihl MJ, Haugaard P, Hansen JM, Hansen KL, Nielsen MB, Jensen JA. Comparison of real-time in vivo spectral and vector velocity estimation. Ultrasound Med Biol 2012;38:145–151. Rosenthal SJ, Harrison LA, Baxter KG, Wetzel LH, Cox GG, Batnitzky S. Doppler US of helical flow in the portal-vein. Radiographics 1995;15:1103–1111. Sherwin SJ, Shah O, Doorly DJ, Peiro J, Papaharilaou Y, Watkins N, Caro CG, Dumoulin CL. The influence of out-of-plane geometry on the flow within a distal end-to-side anastomosis. J Biomech Eng 2000;122:86–95. Slager CJ, Wentzel JJ, Gijsen FJ, Schuurbiers JC, van der Wal AC, van der Steen AF, Serruys PW. The role of shear stress in the generation of rupture-prone vulnerable plaques. Nat Clin Pract Cardiovasc Med 2005a;2:401–407. Slager CJ, Wentzel JJ, Gijsen FJH, van der Wal AC, Schaar JA, Serruys PW. The role of shear stress in the destabilization of vulnarable plaques and related therapeutic implications. Nature Clin Pract Cardiovasc Med 2005b;2:456–464. Stonebridge PA. Three-dimensional blood flow dynamics: Spiral/helical laminar flow. Methodist Debakey Cardiovasc J 2011;7:21–26. Stonebridge PA, Brophy CM. Spiral laminar flow in arteries? Lancet 1991;338:1360–1361. Stonebridge PA, Buckley C, Thompson A, Dick J, Hunter G, Chudek JA, Houston JG, Belch JJ. Non spiral and spiral (helical) flow patterns in stenoses: In vitro observations using spin and gradient echo magnetic resonance imaging (MRI) and computational fluid dynamics modeling. Int Angiol 2004;23:276–283. Stonebridge PA, Vermassen F, Dick J, Belch JJ, Houston JG. Spiral laminar flow prosthetic bypass graft: Medium-term results from a first-in-man structured registry study. Ann Vasc Surg 2012;26:1093–1099. Tanaka M, Sakamoto T, Sugawara S, Nakajima H, Kameyama T, Katahira Y, Ohtsuki S, Kanai H. Spiral systolic blood flow in the ascending aorta and aortic arch analyzed by echo-dynamography. J Cardiol 2010;56:97–110. Tricht I, Wachter D, Tordoir J, Verdonck P. Hemodynamics and complications encountered with arteriovenous fistulas and grafts as vascular access for hemodialysis: A review. Ann Biomed Eng 2005;33:1142–1157. Tritton DJ. Physical fluid dynamics. New York: Oxford University Press; 2007. Udesen J, Nielsen MB, Nielsen KR, Jensen JA. Examples of in vivo blood vector velocity estimation. Ultrasound Med Biol 2007;33:541–548. Walsh MT, Kavanagh EG, O’Brien T, Grace PA, McGloughlin T. On the existence of an optimum end-to-side junctional geometry in peripheral bypass surgery-a computer generated study. Eur J Vasc Endovasc Surg 2003;26:649–656. Zhang Z, Fan Y, Deng X, Wang G, Zhang H, Guidoin R. Simulation of blood flow in a small-diameter vascular graft model with a swirl (spiral) flow guider. Sci China C Life Sci 2008;51:913–921.
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APPENDIX VALIDATION OF VORTICITY AND ITS RELATION TO SHEARING PROCESSES Vorticity provides information on the rotation of the fluid from the point of view of a particle moving in the fluid, as a result of the shearing process. An element of fluid may be considered to be translating (moving with no rotation) and rotating (rotating but with no translation). Vorticity is concerned only with the rotational component, so that a fluid element, which is translating but not rotating, has zero vorticity. Two characteristic vortical types in the cross-flow view of a tube, which were originally simulated to verify the vector Doppler algorithm, were used to validate vorticity and understand how it relates to shearing forces. The first is a rigid-body rotation, where the particles translate in counterclockwise circular trajectories with speed proportional to their distance from the center (Fig. 12a). At the same time, they rotate about their own axis in counterclockwise direction and at a constant rate because of the shear forces between adjacent circular trajectories. This flow is expected to have a steady positive vorticity (according to the right-hand rule) because of the constant rate of counterclockwise rotation of the particles about their own axis, as verified in Figure 12a. The second example is an irrotational flow pattern, where the particles move with speed inversely proportional to their distance from the center (Fig. 12b). As a result, they translate in a circular pattern, but the shear forces keep them in a steady orientation and they do not rotate about their own axis. This is why the flow is called irrotational and vorticity is zero as verified in Figure 12b (Morrison 2013). The existence of vorticity in the center of the vortex is the result of zero velocity in it. Further understanding of these shearing processes is provided in the schematic diagram of Figure 13, which illustrates three characteristic examples of vorticity in a counterclockwise single vortex. The black arrows represent the magnitude of translational velocity in each trajectory, and two particles are illustrated in three different times (t1– t3). The blue and red arrows in the center of the first and second particles, respectively, indicate their direction in space as they move in circular counterclockwise paths. Figure 13a illustrates a rigid-body vortex as described for Figure 12a. The shear forces, induced by the proportional increase in translational velocity with distance from the centre of rotation, rotate both particles about their own axis having the same direction during time (translation with steady rate of rotation) and their vorticity remains constant. Figure 13b is an irrotational vortex as described for Figure 12b. The shearing process, induced by the proportional decrease in translational velocity with distance from the centre of rotation, keeps the orientation of both particles steady during time (translation with no rotation) and their vorticity is zero. In Figure 13c, the translational velocity of the vortex is increased from the center of the spiral to the third trajectory. This rotates the first particle about its own axis in counterclockwise direction and its vorticity is positive. After the third trajectory, the translational velocity is decreased because of the no-slip boundary condition and becomes zero relative to the wall. The inverse shear forces that are applied in the second particle rotate it about its own axis in clockwise direction, and its vorticity has opposite direction in relation to the first particle. Thus, the vorticity of this single vortex appears to change direction near the wall. Figure 7g–i is a characteristic case of this example.
