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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Biomech. Author manuscript; available in PMC 2010 July 22.
Published in final edited form as:
PMCID: PMC2777988

Optimization of Intravascular Shear Stress Assessment in Vivo


Hemodynamic forces, namely, fluid shear stress, regulate the biological activities of the vascular endothelial cells (EC) (14) and vessel wall remodeling in the resistant arteries and aorta (9, 37). Sedentary lifestyle promotes atherogenic hemodynamics in arterial bifurcations where oscillatory and low shear stress predisposes the development of atherosclerosis (8). The advent of Microelectromecahnical Systems (MEMS) provides an entry point to characterize the spatial and temporal variations of the WSS in various arterial configurations(12, 15, 2325). Previously, silicon nitride was applied as an insulating layer on the diaphragm to enhance unidirectional heat transfer (40). In the new generation of MEMS sensors, biocompatible parylene C coating not only insulated the microelectronics but also provided flexibility for packaging the sensors to the catheter/coaxial wire.

Numerous CFD codes have been developed to assess WSS with relevance to cardiovascular disease (2022). Steinman and Ethier investigated the effects of wall distensibility in a 2-D end-to-side anastomosis (32). Lei et al. further described the applications of CFD to design end-to-side anastomoses (16). Taylor et al. developed a stabilized FEM to simulate blood flow in the abdominal aorta under resting and exercise-induced pulsatile flow conditions (3335). Hence, CFD codes predicted arterial regions where elevated temporal oscillations in shear stress are prone to vascular oxidative stress and inflammatory responses (1).

Despite numerous CFD solutions for WSS, there has been a paucity of experimental data. Conventionally, shear stress has been assessed by the direct and indirect methods (28). The former is based on both piezoresistive (30) and capacitive readout schemes in a floating element sensing (28). The latter is based on convective heat transfer by correlating convective cooling of a heated element to shear stress. Indirect methods also include ultrasonic Doppler and Magnetic Resonance Imaging (MRI). Both methods require a known geometry to correlate flow rates with shear stress. While ultrasonic Doppler and MRI are non-invasive, the spatial resolution near the wall is difficult to establish.

To assess shear stress in the complicated arterial geometry, we developed the polymer-based sensors that are flexible, biocompatible, and deployable into the arterial system. A Titanium (Ti) and Platinum (Pt) layer embedded in the flexible polymer was used as the sensing elements. Based on heat transfer principle (42), shear stress can be inferred from the heat loss from the heated sensing element to the flow in terms of the measured changes in voltage. The sensor was fabricated via surface micromachining technique utilizing parylene C as electrical insulation layer. The resistance of the sensing element was measured at approximately 1.0 kOhm and the signal-to-noise ratio was 4.8 (42). Direct measurement of wall shear stress in the aorta thereby enables us to predict the arterial regions exposed to low or oscillatory shear stress that is linked with the development of plaque formation.

To optimize intravascular shear stress (ISS) assessment, we performed both theoretical and CFD analyses. These analyses predicted the extent to which experimental parameters governed the deployment of the catheter/coaxial wire-based MEMS sensors into the aorta of NZW rabbits; namely, the diameter ratios of the aorta to the coaxial wires, position of sensors, and entrance length (Le) in relation to the Reynolds numbers.


1. Packaging and Deployment of MEMS Thermal Sensors

We have developed biocompatible and flexible intravascular MEMS sensors to assess real-time ISS (Figs. 1a and 1b). The detailed operational principle and fabrication process of the MEMS sensor were previously described (41). The diameter of the coaxial wire was 0.4 mm and the length of the sensor was 4 cm (Fig. 1b). The biocompatible epoxy (EPO-TEK 301: Epoxy Technology, Billerica, MA, USA) was used to anchor the sensor to the coaxial wire (Fig. 1c). The sensor was packaged to the coaxial wire, which was deployed into the abdominal aorta of adult NZW rabbits under fluoroscopic guidance.

Fig. 1
Biocompatible and flexible intravascular sensors. (a) The sensor array on the wafer prior to release into individual devices. (b) The sensor could be bended or folded in a zigzag fashion without structural or functional damage. The Ti/Pt sensing element ...

The sensors were first tested in the tube model and the shear stress measurements were validated by CFD simulation (Fig. 2a). A range of steady flow rates were generated by a flow system consisting of a digital modular drive (Master Flex L/S 77300-80, Cole-Palmer) and a pump drive (Master Flex L/S 7518-12, Cole-Palmer). The pump setting was calibrated using an electromagnetic flow meter (MAGFLO®, Danfoss A/S, DK). Once validated in the in vitro model, the MEMS sensors were deployed into the aorta of the rabbits through the co-axial wire via femoral cut-down procedure. Using the 3-D fluoroscopic guidance (Phillips BV-22HQ C-arm), the guide wire and MEMS sensors were visualized in the abdominal aorta. The fluoroscope could be rotated to acquire the images in various planes. By injecting the contrast dye from the carotid artery, the operators were able to delineate the geometry of the rabbit aorta and the position of the coaxial wire in relation to the inner diameter of the aorta (Fig. 2b).

Fig. 2
The sensors were tested in a 3-D fluidic tube model. (a) The measurements were validated by CFD simulation with the geometry reconstructed from photos taken by a digital camera (b) Using fluoroscopic guidance in the animal angiographic lab, the operator ...

2. Theoretical and Computational Analyses of Intravascular Shear Stress

2.1. Theoretical Formulation to Predict WSS in the Presence of a Catheter

Analogous to intravascular ultrasound (IVUS), the deployment of coaxial wires would inevitably influence the flow field(4, 5, 13, 18, 19, 25, 38). To optimize the MEMS sensor measurement, we formulated Navier-Stokes equations analogous to flow in an annular duct to predict the flow disturbance when the catheter was positioned at the center (Fig. 3a). The viscosity, μ, was treated as a constant. The velocity profiles obtained for annulus and circular pipe flow were expressed as (27):



where U and u represent the velocity in the annulus and circular pipe, respectively. R and Ri are the radii of the pipe and coaxial wire, respectively. n is the radius or diameter ratio between the vessel and the catheter , n=R/Ri, and dP/dx and dp/dx are the pressure gradient in the annulus and the circular pipe, respectively. The flow rate, Q, and WSS, τw, in the absence of coaxial wire were expressed as:

Q=0Ru 2πr dr=π8μdpdxR4



Fig. 3
The schematic diagram illustrates the geometry and boundary conditions for computational simulation. These conditions provided a basis to validate the in vivo intravascular shear stress (ISS) in relation to the WSS. The parameters, u , v and w, are the ...

2.2. Flow Rates and Wall Shear Stress

Assuming the effect of the coaxial wire on the flow was negligible (10, 26), the flow rate in the presence of coaxial wire would be defined as:

Q=RiRU 2πr dr=π8μdpdxR4n4lnn(n41)lnn(n21)2

From Eq. (5), the following relation was obtained:


where Ep is defined as the pressure elevation factor (PEF):


From the velocity profile, U , the WSS in the presence of coaxial wire, τw_wire, was expressed as:


where Ewss is the WSS elevation factor (WEF):


Similarly, the shear stress on the MEMS sensor, τi_wire, was evaluated at r=Ri:


where τw is the WSS in the absence of coaxial wire and Eiss is defined as the ISS elevation factor (IEF):


2.3. Computational Analyses

CFD analyses were performed to investigate the effects of the flow disturbance and the entrance length on the coaxial wire required to mount the MEMS sensors. The governing equations were solved for laminar, incompressible, and non-Newtonian flow. In our CFD model, the shear rate-dependent dynamic viscosity was implemented for the shear-thinning behavior of the rabbit blood (6, 7). The simulations were performed for the geometry as shown in figure 3b.

In general, entrance length, Le, is the distance required from the terminal end of the coaxial wire to the point where fully developed flow develops. The definition of the Le (31) was also based on the length required to achieve a local incremental pressure drop of ~ 98% of the fully developed region. In our study, shear stress was used to predict Le. The flow disturbance in the Le region influenced heat transfer from the sensor to the flow stream (17). In this context, predicting the entrance length in relation to the Reynolds numbers guided the packaging of MEMS sensors to the coaxial wire.

The inlet Reynolds number, Re, was defined by the mean velocity:


where Umean is the mean flow velocity at the inlet.

3.1 Generation of 3-D Geometries and Meshes

The geometries of the tube model and the rabbit abdominal aorta model in the presence of coaxial wire or catheter were reconstructed in Pro Engineer Wildfire V.3.0 (Parametric Technology, Needham, MA) using boundary conditions from experimental measurements. The models were imported into GAMBIT for mesh generation (Fluent Inc., Gambit 2.3.16, Lebanon, NH, USA). The meshed models were imported into the main CFD solver (Fluent Inc., Fluent 6.2.16, Lebanon, NH, USA) for further flow simulation. The CFD model was constructed with 78,868 cells, which were primarily the Tet/Hybrid elements. Since our focus was on the shear stress on the sensor, fine elements immediately adjacent to the coaxial wire were constructed to obtain sufficient information to characterize the large fluid velocity gradients near the wall.

In the abdominal aorta model, the flow field was solved under three catheter positioning schemes as described previously (2). The vessel inner diameter, Daorta, was measured to be 2.4 mm from the angiogram (Fig. 3). The MEMS sensor was located 4.0 cm from the terminal end of the catheter/coaxial wire. Three different diameters (Dcatheter) namely, 0.25 mm (mouse catheter 0.010”OD), 0.4 mm (coaxial wire), and 0.97 mm (dog catheter 0.038”OD), were studied in the CFD simulations. The geometry of the computational model of rabbit abdominal aorta is shown in figure 3b, where u and v represent the axial and radial velocities, and Dcatheter represents the catheter diameter.

3.2. Boundary conditions

The steady flow rates from the in vitro experiment were applied as the inlet flow boundary condition for the corresponding 3-D computational model. The ultrasound transducer (Philips SONOS 5500 at 12 MHz) was used to obtain the pulsatile velocity waveform. The recorded profile of the center-line velocity, Uc, as shown in figure 4a, was reconstructed by applying the Fourier analysis using 12 harmonics (Fig. 4b). The peak and time-averaged velocity were 57.7 cm·s−1 and 14.6 cm·s−1, respectively. The period of one cardiac cycle, T, was 0.33 seconds corresponding to 180 beats·min−1. From Eq. (12), the mass flux at the inlet in the axial direction (nx) was obtained as:


Fig. 4
The pulsatile velocity waveform was derived from Doppler ultrasound measurement and reconstructed using Fourier analysis with 12 harmonics for computational simulation. (a) The center-line velocity profile, including the peak velocity, was recorded. ...

The calculated mass flux was applied as the inlet boundary condition and implemented by a user defined C++ code in FLUENT.

The traction-free condition was assumed at flow outlet. No-slip boundary condition was applied to the inner walls and the surface of the coaxial wire.

The spatial WSS, τw, was calculated for incompressible fluids as:


where ut is the velocity tangential to the wall and n is the unit vector perpendicular to the wall.

A range of steady flow rates were also studied in order to establish the relationship between the entrance length and flow rate. The mean flow velocity was maintained at 0.158, 0.288, 0.625, 0.92654, and 1.567m·sec−1, yielding Reynolds numbers of 116, 210, 330, 439, 680, 910, 1150, 1350 and 1550. The Reynolds numbers of 116 and 439 represented the time-averaged value and the maximum value in the rabbit thoracic aorta, respectively. Comparatively, the Reynolds number of 1150 represented the maximal value in the human descending aorta.

Reynolds numbers were determined in the presence or absence of coaxial wire. The hydraulic diameter, Dh = D − Di, was used to normalize the Reynolds number and the entrance length, Le. In the absence of a catheter, the Reynolds number was expressed in terms of the flow rate, Q:


Given that in the presence of catheter,


the Reynolds number, Rewire, was calculated from the mean velocity, Umean_wire:


We assumed the effect of the catheter on the flow to be negligible (10, 26):



The normalized entrance length, Le_wire as a function of the normalized Reynolds number, Rewire, was derived using the Curve Fitting Toolbox of MATLAB 7.1 (Math-works, Natick, USA).


3.1. Theoretical Prediction of WSS in the Presence of a Coaxial Wire

The diameter ratio influenced the extent to which the coaxial wire increased the pressure and WSS measurements in a 3-D tube. This change in velocity profile reversed the direction of shear stress, leading to an increase in shear stress on the coaxial wire by an elevation factor (Eiss). Figure 5a illustrated an increase in pressure by an elevation factor (Ep) as a function of the diameter ratio. The magnitude of Ep decreased significantly from 24.2 to 2.5 as the ratio increased from 1.5 to 4.5 with an asymptotic value of ~1.1 at a diameter ratio approaching 100. The pressure was predicted to increase by 20% at a diameter ratio of 100.

Fig. 5
Variations in elevation factors as a function of the diameter ratio, n. (a) Pressure elevation factor, Ep , as a function of n. (b) WSS elevation factor, Ewss , as a function of n. (c) ISS elevation factor, Eiss , as a function of n.

Wall shear stress and WSS elevation factor (Ewss) were also dependent on the diameter ratio (Fig. 5b). Ewss decreased from 7.6 to 1.7 as the diameter ratio increased from 1.5 to 4.5. The elevation factor for ISS on the coaxial wire (Eiss) was reduced from 8.747 to its minimum of 3.024 as the diameter ratio increased from 2 to 4.5 (Fig. 5c). In humans, the large diameter ratio would minimize the WSS elevation to ~10%. Taken together, a large catheter (n < 2) would introduce significant elevations in pressure and shear stress whereas n > 4.5 would minimize these effects (3).

3.2. Shear Stress Assessment in a 3-D Tube Model

CFD simulation was performed to analyze both velocity profiles and shear stress distribution on the catheter/coaxial wire in response to three distinct Reynolds numbers; namely, 23, 91, and 133 (Fig. 6). The effects of flow disturbance on the MEMS sensors were negligible in the presence of a coaxial wire until the Reynolds number reached to above 133 (Fig. 6a and 6b). At a Reynolds number of 23, shear stress on the coaxial wire was 0.59 dyn·cm−2; at a Reynolds number of 91, shear stress was 3.13 dyn·cm−2; and at 133, shear stress was 5.61 dyn·cm−2. Next, real-time shear stress assessment by the MEMS sensor was compared with that of CFD solutions (Fig. 7). The relationship between the voltage changes and flow rates (Fig. 7a) were converted to shear stress using the calibration curve (Fig. 7b) (41). Real-time shear stress measurement was in good agreement with that of CFD at flow rates greater than 6 ml·s−1.

Fig. 6
CFD simulation was performed at various flow rates to analyze both velocity profiles and shear stress distributions on the coaxial wire. (a) The velocity profiles in response to three distinct flow rates (Re = 23, 91, and 133, respectively). (b) The shear ...
Fig. 7
The experimental ISS as detected by the MEMS sensor was compared with the computational solutions. (a) The relationship between the voltage changes and flow rates. (b) The converted shear stress using our previously reported calibration curve was compared ...

3.3. Entrance Length and Reynolds Numbers

The normalized entrance length, Le_wire, and its corresponding normalized Reynolds number, Rewire, were determined based on the CFD results (Fig. 8). A relation between Rewire and Le_wire derived from CFD was expressed as:


Le_wire increased in a relatively linear fashion as a function of Rewire. This relation enabled us to mount the MEMS sensors downstream from the terminal end of coaxial wire to optimize real-time shear stress assessment.

Fig. 8
The normalized entrance length, e*, and its corresponding normalized Reynolds number, Re*, were determined based on the CFD results.

3. 4. Shear Stress Assessment in the Abdominal Aorta of NZW Rabbits

The entrance lengths were determined as a function of flow rates by the computational results. The MEMS sensor was positioned at 4 cm from the terminal end of the coaxial wire. The inflow boundary conditions obtained from the Doppler ultrasound were incorporated into the CFD simulations.

The position of the coaxial wire and the orientation of the sensors facing the flow field significantly influenced ISS assessment. At the diameter ratio of 6, Le of 1.18 cm, and position of the coaxial wire at the center, the time-averaged shear stress (τave) on the coaxial wire was computed to be 31.2 dyn·cm−2 with a systolic peak at 102.8 dyn·cm−2 (corresponding to a maximum Reynolds number of 212). Despite the identical afore mentioned physical parameters, a shift in the coaxial position at 0.2 mm off the center resulted in an asymmetric distribution of shear stress on the coaxial wire from τ 0 degree = 97.8 dyn·cm−2 at 0°, τ 90 degree = 107.6 dyn·cm−2 at 90°, to τ 180 degree = 112.8 dyn·cm−2 at 180° corresponding to a maximum Reynolds number of 212. Similarly, positioning the coaxial wire at 0.1 mm away from the vessel wall, the shear stress at the peak of systole was τ 0 degree = 6.9 dyn·cm−2 at 0°, τ 90 degree = 53.6 dyn·cm−2 at 90°, and τ 180 degree = 98.3 dyn·cm−2 at 180°.

CFD analysis revealed that the presence of a coaxial wire (Daorta / Dcoaxial wire = 6 and Le = 1.18 cm) increased the time-averaged WSS (τave) from 10.06 dyn·cm−2 to 15.5 dyn·cm−2 and the systolic peak from 33.2 dyn·cm−2 to 51.3 dyn·cm−2 (Fig. 9). Experimentally, under fluoroscopic visualization, real-time ISS assessment by the coaxial wire near the wall with the MEMS sensor facing the flow field revealed τave value of 11.92 dyn·cm−2 with a systolic peak at 47.04 dyn·cm−2. The difference between CFD and experimental τave was 18.5%. Taken together, our findings demonstrated that deploying the coaxial wire using a steerable catheter under 3-D fluoroscopic guidance along with the angiogram would optimize real-time shear stress assessment.

Fig. 9
Measured shear stress tracing in one cardiac cycle was compared with CFD solutions. (a) Superimposed on the real-time ISS profile (red dashed line) are the CFD solutions of WSS before (blue) and after (magenta) the coaxial wire was positioned at the center. ...


In this study, we performed theoretical and CFD analyses to optimize real-time intravascular assessment in the aorta of NZW rabbits. CFD analyses revealed that an entrance length of 2.9 mm and the diameter ratio (n = 4.5) would minimize the pressure and shear stress elevation in the rabbit aorta. When the catheter was positioned off the center of vessel, Velusamy and Garg (36) reported that an eccentricity of the velocity profiles developed, and shear stress on the catheter and the vessel wall varied circumferentially (2) (Fig 3b). In humans, the diameter ratio could approach to 100 for an aortic inner diameter of 2.5cm with a catheter outer diameter of 0.254 mm. Hence, ISS assessment could be further optimized by positioning the catheter near the vessel wall using a steerable catheter with a large diameter ratio, such as a large animal model.

Several equations for estimating entrance length had been proposed (29, 39). Among these was the one proposed by Wojtkowiak and Popiel (39) for flow in an annular duct. The slope of their linear equation, 0.0161, was about twice as our estimation at 0.008184. One reason for this deviation was the criteria used to define the fully developed flow. We adopted the notion that the wall shear rate converges to its fully developed value at about half the length at which the centerline velocity converges to its fully developed value (14).

Despite numerous CFD solutions for wall shear stress, there has been a paucity of experimental data. A 30% distortion is commonly encountered for experimental intravascular measurements (14). The boundary conditions and dimensions implemented in the CFD model might not precisely reflect the experimental boundary conditions. Therefore, the center-positioning scheme and the CFD results provided a basis to compare with direct sensor measurement.

Our computational results indicated that the entrance length in response to the maximal inlet Reynolds numbers in the rabbits (Re max_rabbit=459) and human abdominal aorta (Re max_human=1150) were Le rabbit =1.244 cm and Le human=1.985 cm, respectively. For a Womersley number greater than 1, the above Le values could be overestimated. Sufficient time was allowed for the flow to develop towards a parabolic velocity profile during individual cardiac cycles (2) and the pulsatile flow behaved in a quasi-steady manner (11). The MEMS sensing element were mounted at 4.0 cm downstream from the terminal end of the coaxial wire; thereby, optimizing intravascular measurements.

In conclusion, ISS assessment was validated in vitro by deploying the catheter-based MEMS sensors in a 3-D model. Theoretical and computational models further provided physical parameters to optimize real-time shear stress assessment. These fundamental analyses provided a basis to further investigate the application of MEMS devices (24) for in vivo studies.


This work was supported by American Heart Association GIA 0655051Y (TKH), NIH R01 HL083015 (TKH), NIH K08 HL068689 (TKH), and American Heart Association Post-Doctoral Fellowship 0725016Y (HY). The authors would like to thank Ping Sun for assistance with signal processing.


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Contributor Information

Lisong Ai, Department of Biomedical Engineering and Cardiovascular Medicine, University of Southern California, Los Angeles, California 90089-1111.

Hongyu Yu, School of Earth and Space Exploration and the Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287.

Wakako Takabe, Department of Biomedical Engineering and Cardiovascular Medicine, University of Southern California, Los Angeles, California 90089-1111.

Anna Paraboschi, Department of Biomedical Engineering and Cardiovascular Medicine, University of Southern California, Los Angeles, California 90089-1111.

Fei Yu, Department of Biomedical Engineering and Cardiovascular Medicine, University of Southern California, Los Angeles, California 90089-1111.

E. S. Kim, Department of Electrical Engineering and Electrophysics, University of Southern California, Los Angeles, California 90089-1111.

Rongsong Li, Department of Biomedical Engineering and Cardiovascular Medicine, University of Southern California, Los Angeles, California 90089-1111.

Tzung K. Hsiai, Department of Biomedical Engineering and Cardiovascular Medicine, University of Southern California, Los Angeles, California 90089-1111.


1. Ai L, Rouhanizadeh M, Wu JC, Takabe W, Yu H, Alavi M, Li R, Chu Y, Miller J, Heistad D, Hsiai TK. Shear stress influences spatial variations in vascular Mn-SOD expression: implication for LDL nitration. Am. J. Physiol. Cell Physiol. 2008:C1576–C1585. [PMC free article] [PubMed]
2. Ai L, Yu, Dai WH, Hale S, Kloner R, Hsiai TK. Real-time Intravascular Shear Stress in the Rabbit Abdominal Aorta. IEEE Transactions on Biomedical Engineering (submitted) 2008 [PMC free article] [PubMed]
3. Anderson HV, Zaatari GS, Roubin GS, Leimgruber PP, Gruentzig AR. Steerable fiberoptic catheter delivery of laser energy in atherosclerotic rabbits. Am Heart J. 1986;111:1065–1072. [PubMed]
4. Back LH. Estimated Mean Flow Resistance Increase during Coronary-Artery Catheterization. Journal of Biomechanics. 1994;27:169–175. [PubMed]
5. Banerjee RK, Back LH, Back MR, Cho YI. Catheter obstruction effect on pulsatile flow rate-pressure drop during coronary angioplasty. Journal of Biomechanical Engineering-Transactions of the ASME. 1999;121:281–289. [PubMed]
6. Bird RB, ARC, Hassager O. Fluid Mechanics. Vol.1. New York: Wiley; 1977. Dynamics of Polymeric Liquids; p. 470.
7. Boger DV. Viscoelastic Flows through Contractions. Annual Review of Fluid Mechanics. 1987;19:157–182.
8. Chatzizisis YS, Coskun AU, Jonas M, Edelman ER, Feldman CL, Stone PH. Role of endothelial shear stress in the natural history of coronary atherosclerosis and vascular remodeling - Molecular, cellular, and vascular behavior. Journal of the American College of Cardiology. 2007;49:2379–2393. [PubMed]
9. Chatzizisis YS, Jonas M, Coskun AU, Beigel R, Stone BV, et al. Prediction of the localization of high-risk coronary atherosclerotic plaques on the basis of low endothelial shear stress - An intravascular ultrasound and histopathology natural history study. Circulation. 2008;117:993–1002. [PubMed]
10. Doucette JW, Corl PD, Payne HM, Flynn AE, Goto M, et al. Validation of a Doppler Guide Wire for Intravascular Measurement of Coronary-Artery Flow Velocity. Circulation. 1992;85:1899–1911. [PubMed]
11. Fung YC. Biomechanics:Circulation. Springer; 1997. pp. 130–132.
12. Hsiai TK, Cho SK, Wang PK, Ing MH, Salazar A, et al. Micro Sensors: Linking Vascular Inflammatory Responses with Real-Time Oscillatory Shear Stress. Ann Biomed Eng. 2004;32:189–201. [PubMed]
13. Karahalios GT. Some Possible Effects of a Catheter on the Arterial-Wall. Medical Physics. 1990;17:922–925. [PubMed]
14. Ku DN. Blood Flow in Arteries. Annu. Rev. Fluid Mech. 1997;29:399–434.
15. Lasheras JC. The biomechanics of arterial aneurysms. Annual Review of Fluid Mechanics. 2007;39:293–319.
16. Lei M, Archie JP, Kleinstreuer C. Computational design of a bypass graft that minimizes wall shear stress gradients in the region of the distal anastomosis. Journal of Vascular Surgery. 1997;25:637–646. [PubMed]
17. Liu C, Huang JB, Zhu ZJ, Jiang FK, Tung S, et al. Micromachined flow shear-stress sensor based on thermal transfer principles. Journal of Microelectromechanical Systems. 1999;8:90–99.
18. Macdonald DA. Fully-Developed Incompressible-Flow between Non-Coaxial Circular-Cylinders. Zeitschrift Fur Angewandte Mathematik Und Physik. 1982;33:737–751.
19. Macdonald DA. Pulsatile Flow in a Catheterized Artery. Journal of Biomechanics. 1986;19:239–249. [PubMed]
20. Moore JA, Steinman DA, Ethier CR. Computational blood flow modelling: Errors associated with reconstructing finite element models from magnetic resonance images. Journal of Biomechanics. 1998;31:179–184. [PubMed]
21. Pedersen EM, Agerbaek M, Kristensen IB, Yoganathan AP. Wall shear stress and early atherosclerotic lesions in the abdominal aorta in young adults. European Journal of Vascular and Endovascular Surgery. 1997;13:443–451. [PubMed]
22. Perktold K, Resch M, Peter RO. 3-Dimensional Numerical-Analysis of Pulsatile Flow and Wall Shear-Stress in the Carotid-Artery Bifurcation. Journal of Biomechanics. 1991;24:409–420. [PubMed]
23. Rouhanizadeh M, Lin TC, Arcas D, Hwang J, Hsiai TH. Spatial variations in shear stress in a 3-D bifurcation model at low Reynolds numbers. Annals of Biomedical Engineering. 2005;33:1360–1374. [PubMed]
24. Rouhanizadeh M, Soundararajan G, Ascara D, Lo R, Browand F, Hsiai T. MEMS sensors to resolve spatial variations in shear stress in a 3-D bifurcation model. IEEE Sensors. 2006;6:78–88.
25. Roy AS, Back LH, Banerjee RK. Guidewire flow obstruction effect on pressure drop-flow relationship in moderate coronary artery stenosis. Journal of Biomechanics. 2006;39:853–864. [PubMed]
26. Roy AS, Banerjee RK, Back LH, Back MR, Khoury S, Millard RW. Delineating the guide-wire flow obstruction effect in assessment of fractional flow reserve and coronary flow reserve measurements. Am J Physiol Heart Circ Physiol. 2005;289:H392–H397. [PubMed]
27. Schetz JA, Fuhs AE, editors. Fundamentals of Fluid Mechanics. New York: John Wiley & Sons; 1999.
28. Schmidt MA, Howe RT, Senturia SD, Haritonidis JH. Design and Calibration of a Microfabricated Floating-Element Shear-Stress Sensor. Ieee Transactions on Electron Devices. 1988;35:750–757.
29. Shah RK. Correlation for Laminar Hydrodynamic Entry Length Solutions for Circular and Noncircular Ducts. Journal of Fluids Engineering-Transactions of the Asme. 1978;100:177–179.
30. Shajii J, Ng KY, Schmidt MA. A microfabricated floating element shear stress sensor using wafer-bonding technology. J. Microelectromech. Syst. 1992:89–94.
31. Sparrow EM, Lin SH, Lundgren TS. Flow Development in the Hydrodynamic Entrance Region of Tubes and Ducts. Physics of Fluids. 1964;7:338–347.
32. Steinman DA, Ethier CR. The Effect of Wall Distensibility on Flow in a 2-Dimensional End-to-Side Anastomosis. Journal of Biomechanical Engineering-Transactions of the Asme. 1994;116:294–301. [PubMed]
33. Taylor CA, Hughes TJR, Zarins CK. Finite element modeling of blood flow in arteries. Computer Methods in Applied Mechanics and Engineering. 1998;158:155–196.
34. Taylor CA, Hughes TJR, Zarins CK. Finite element modeling of three-dimensional pulsatile flow in the abdominal aorta: Relevance to atherosclerosis. Annals of Biomedical Engineering. 1998;26:975–987. [PubMed]
35. Taylor CA, Hughes TJR, Zarins CK. Effect of exercise on hemodynamic conditions in the abdominal aorta. Journal of Vascular Surgery. 1999;29:1077–1089. [PubMed]
36. Velusamy K, Garg VK. Entrance Flow in Eccentric Annular Ducts. International Journal for Numerical Methods in Fluids. 1994;19:493–512.
37. Volokh KY, Vorp DA. A model of growth and rupture of abdominal aortic aneurysm. J Biomech. 2008;41:1015–1021. [PubMed]
38. Wentzel JJ, Krams R, van der Steen AFW, Li W, Cespedes EI, Bom N, Slager CJ. Disturbance of 3-D velocity profiles induced by IVUS catheter, evaluation with computational fluid dynamics. Comput Cardiol IEEE. 1997:597–599.
39. Wojtkowiak J, Popiel CO. Inherently linear annular-duct-type laminar flowmeter. Journal of Fluids Engineering-Transactions of the Asme. 2006;128:196–198.
40. Xu Y, Lin Q, Lin GY, Katragadda RB, Jiang FK, et al. Micromachined thermal shear-stress sensor for underwater applications. Journal of Microelectromechanical Systems. 2005;14:1023–1030.
41. Yu H, Ai L, Rouhanizadeh M, Patel D, Kim ES, Hsiai TK. Flexible Polymer Sensors for in Vivo Intravascular Shear Stress Analysis. IEEE/ASME J MEMS. 2008 in press.
42. Yu H, Ai L, Rouhanizadeh M, Patel D, Kim ES, Hsiai TK. Flexible Polymer Sensors for In Vivo Intravascular Shear Stress Analysis. IEEE/ASME J. MEMS. 2008;17:1178–1186.