Pressure Drops Through Arterial Stenosis Models in Steady Flow Condition
Table 1 L, [mm] 15 30
Stenosis A Stenosis B
> [mm] 10 •1.5
S. Cavalcanti1, P. Bolelli1, and E. Belardinelli1 Upstream
Measures of pressure drops were made in two different plexiglass models of axial-symmetric arterial stenoses. The stenosis models had the same are reduction {86 percent) but were of different length so as to have a different tapering degree. Pressures were measured in steady flow condition at three equidistant points of the stenosis: upstream, in the middle, and downstream. Results indicate that: the upstream-middle pressure drop is independent of tapering degree but is highly influenced by area reduction; moreover it is much greater than the middle-downstream drop. The upstream-middle pressure drop can be accurately predicted by means of a relationship deduced by the momentum equation.
Introduction Atherosclerotic plaque formation in an artery gives rise to a local reduction of vessel lumen. The severity of stenosis depends on the flow reduction involved. A basic role in stenosis hemodynamics and, consequently, in flow reduction, is played by the vessel outline [1, 2]. The local area reduction makes convergent the vessel upstream segment of the stenosis, divergent the downstream one. Convective accelerations induced by tapering increase the fluid element velocities between upstream- and mid-stenosis with a consequent increase in the pressure drop [3, 4], When the area reduction is severe the pressure drop increases and the pressure in the mid-stenosis is much lower than in the upstream one [10]. Although the percent area reduction is the most widely used clinical parameter in evaluating the importance of vessel disease, from a functional view-point evaluation of the pressure drop through the stenosis is more relevant [6]. In order to evaluate how a relationship, based on the one-dimensional flow assumptions, can predict the pressure drop through a stenosis, we have performed several flow-pressure measures in two plexiglass models of arterial stenosis.
Methods Stenosis Models. Two rigid-walled plexiglass stenosis models, with an axial-symmetric geometry and an area reduction of 86 percent, were made. To connect the pressure transducers three holes were bored: one in the middle and the others upstream and downstream of the stenosis (see Fig. 1). The L\ and L 2 lengths are reported in Table 1. Stenosis A has a 9.5 deg taper angle (a) whereas stenosis B has a 4.8 deg taper angle. Hydraulic Equipment. The main conduit of the circuit (see Fig. 2) was a polyethylene tube of constant diameter connecting the test section to the constant level tank and the downstream reservoir. The one-way valves allowed the setting of the steady
'Department of Electronics, Computer Science and Systems, University of Bologna, Bologna, Italy. Contributed by the Bioengineering Division of THE AMERICAN SOCIETY OF
MECHANICAL ENGINEERS. Manuscript received by the Bioengineering Division, June 1, 1991; revised manuscript received September 28, 1991. Associate Technical Editor: L. Talbot.
Middle
Downstream
F^^^'^^2^ R_= 1.5mm
Tr77777/7777777rA £1
Fig. 1 Geometry of stenosis model. Lengths Li and L2 are different in the two models (see Table 1)
Costant head t a n k Pump
^ XValve ] = [ ZTestZsection Z r = MValve =
Fig. 2 Scheme of hydraulic experimental apparatus
flow rate in a range similar to the physiological one for a human medium artery (1 -f- 28 ml/s) [5], which correspond Reynold numbers between 61 -*• 1500. The distances between the valves and the test section were sufficient to ensure that the entrance effect would be negligible. The fluid used for all tests was a water-glycol ethylenic mixture with an absolute viscosity p = 3 • 10 ~3 Kg/ms and a density p = 1.068-103 kg/m3 at 28 °C, which are close to the real values of blood. The main instruments employed were: an integrated amplified pressure transducer (Honeywell 142PC15G) equipped with a further amplifier circuit made in our laboratory; a digital multimeter with thermometric probe (Philips PM2519 and PM9249). The flow rate was measured by the weighting method. Theory. The motion equation of fluid is deduced by assuming that it is incompressible and newtonian, and supposing a one-dimensional steady flow condition. Fluid motion is, then, governed by the momentum Eq. (7), which, in a cylindrical reference frame (r, z) can be written as
2
•KR1
dz
'I(M
- 1-KVR
dw ~dr
(1)
with the following meaning of the symbols: p density, p = ^/ p cinematic viscosity; w = w (r, z) fluid element velocity, while p = p(z) is the pressure andR = R(z) the radius. To integrate Eq. (1) with respect to r the following velocity profile is assumed. w(r, z)
k+\ k
Q TTR2
1--
R2k
(2)
where Q is the flow rate and k a integer parameter that allows the profile shape to be arranged: when k = 1 the profile is
4 1 6 / V o l . 114, AUGUST 1992
Transactions of the ASME
Copyright © 1992 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 06/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use
parabolic while by increasing k the profile assumes a flat shape. The validity of (2) is confirmed by the experimental results (see Discussion). The momentum equation can then be rearranged in the form dp_ dz"
2k+ 2 ~2k+l
p
d_ / l _
(TTR)2 dz
\R
G2-
(* + D^e.
(3)
To obtain the pressure drop between the upstream and middle sections Eq. (3) must be integrated with respect to z. Let Ru and Rm denote, respectively, the inner radius in the upstream (z - 0) and in the middle (z = Lt + L2) sections. Thus we have AP„.
1+:
1 2k+l
W +
k+\
(4)
RPQ,
with 2-K1RI
Table 2
etcc/s]
P„ [mmHg]
P,„ [mmHg]
Pd [mmHg]
1.16 2.40 3.16 3.84 5.44 6.42 8.25 10.83 12.50 13.33 14.32 16.70 19.40 23.30 27.00 28.00
133.14 132.88 132.78 132.70 132.40 132.14 131.70 131.30 130.19 129.83 129.55 129.05 128.09 126.85 125.26 124.96
132.94 131.94 131.34 130.63 128.55 126.77 123.46 118.27 112.96 110.63 107.40 100.11 89.64 74.67 52.62 46.04
132.94 131.84 131.18 130.36 128.01 126.19 122.56 116.98 111.29 108.77 105.26 97.22 85.95 69.68 47.53 40.55
eicc/s] 0.96
P„ [mmHg] 133.47 133.29 133.31 , 133.11 132.63 132.12 131.67 131.48 130.48 130.25 129.41 128.65 128.43 127.85 127.20 126.87
P,„ [mmHg] 133.37 132.33 131.87 130.25 126.95 123.28 118.56 114.31 105.74 102.86 94.82 87.68 81.83 74.65 63.20 52.06
P„ [mmHg] 133.39 132.37 131.92 130.46 127.30 123.97 119.40 115.40 107.09 105.24 98.68 92.93 87.80 81.85 73.24 62.57
si" 1 ) '"d 8JA
Rn = —
Table 3 R u + RUR,„ + R',
7T
L,+-
R'.
The first term on the right side of (4) takes the inertial forces into account and is the same as the Bernoulli pressure drop when k is very great (flat velocity profile); the second term takes the viscous forces into account and is the same as the Poiseuille drop when k = 1. For low values of k (k < 10) the inertial prevail over viscous forces and, above all, the sensitivity of (4) with respect to k is very low; i.e., in this condition the k value has little influence on the total pressure drop.
Results The first columns of Table 2 and Table 3 are the sixteen steady flow values, respectively, in stenosis A and B. The corresponding measured pressure in the upstream, middle, and downstream sections are indicated in the other columns. Figure 3 shows the pressure drops between the upstream and middle sections (star for stenosis A and circle for B) and those between the middle and downstream sections (X for A and plus for E). Comparison of the pressure drop calculated by relationship (4) with the measured data (circle) is shown in Fig. 4. Computed and measured pressure drops were normalized by the dynamic pressure 7&Q2. The continuous curves are obtained for k = 1 while the dashed curves are relative to k = 4. The two lines at the bottom of the right picture are viscous drops, calculated by means of the second terms on the right side of (4). Comparison between calculated and measured pressure drops for stenosis B is shown in Fig. 5 in the same way as in Fig. 4.
2.70 3.40 5.00 7.34 9.20 11.28 13.20 15.95 16.40 18.95 20.70 22.70 24.20 25.35 28.11
80 60
* x Stenosis A o + Stenosis B
40 d*
20-
-20
20 Flow [cc/s]
10
30
Fig. 3 Pressure drops between upstream-middle sections (star and circle) and middle-downstream (Xand plus)
middle-downstream
Discussion Figure 3 clearly shows that the degree of tapering of the upstream region has no influence on the upstream-middle pressure drop (star for stenosis A and circle for B). For the downstream region the same is not true: in this region, by decreasing the diverging degree, the sign of pressure drop changes so that downstream pressure is lower than middle pressure for stenosis A (plus in Fig. 3), whereas it is higher than middle pressure for stenosis B(X'm Fig. 3). It is well known that in a tapering vessel the inertial forces associated with convective accelerations compel the velocity profile to assume a flat instead of a parabolic shape [8]. Nevertheless, relationship (4) with k = 1 predicts the upstream-middle Journal of Biomechanical Engineering
Fig. 4 Comparison of experimental data (circle) measured in stenosis A and pressure drop predicted by relationship (4) (continuous and dashed curves). The stenosis has a 9.5 deg taper angle.
pressure drop satisfactorily in both stenoses (Figs. 4 and 5). Actually, the estimate of the pressure drop, assuming a parabolic profile (k = 1), introduces an error in both terms of relationship (4) with tend to balance each other, so that it has AUGUST 1992, Vol. 114/417
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 06/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use
upstream-middle
Stenosis B Q*
middle-downstream
Pretension Critically Affects the Incremental Strain Field on Pressure-Loaded Cell Substrate Membranes G. W. Brodland,1 A. T. Dolovich,2 and J. E. Davies3
Flow [cc/s]
Flow [cc/s]
Fig. 5 Comparison of experimental data (circle) measured in stenosis Sand pressure drop predicted by relationship (4)(continuous and dashed curves). The stenosis has a 4.8 deg taper angle.
little influence on the total pressure drop evaluation. All this is confirmed by the low sensitivity of (4) with respect to k for small values of k itself. So, a parabolic profile hypothesis is acceptable if we want to calculate the upstream-middle pressure drop. It is emphasized that the error of prediction increases when the flow increases and the taper and angle decreases. The most accurate pressure drop evaluation is obtained when k = 3 for stenosis A and k = 2 for stenosis B. A few studies involving the development of turbulence in stenosis have been reported [1, 2, 9, 11]. The results clearly show that: the disturbed region is downstream of the stenosis and the critical Reynolds number decreases as the diverging degree increases. In particular, in a diverging vessel a flow separation with a vortex formation occurs and near the wall there is a thin anular layer of reverse flow [8, 10]. Then, for the downstream region, it is impossible to make a correct pressure drop prediction by means of a one-dimensional flow equation. In this case it is necessary to assume at least a twodimensional flow, i.e., fluid element with two velocity components which are respectively directed as axial and radial coordinates. However, for both the stenoses considered here the pressure drop is almost completely concentrated between the upstream and middle sections. Therefore, relationship (4) can be expected to serve as a guide to evaluate pressure drop through a stenosis when both the mean flow in the cardiac cycle as well as the stenosis radius and length are measured.
Introduction Strainable cell substrates have found increasing application in cell culture experiments, where they are used to mechanically stimulate cells. For a review, see Williams et al. (1992). A number of solutions to the associated mechanics problem for a circular membrane exist (Hencky, 1915; Green and Adkins, 1960). None of these addresses the important issue of membrane pretension or prestrain. Here, for circular membranes, we derive a solution which produces formulas which are easy to use and which show clearly the interrelationships between inflation pressure, center deflection, and incremental strain fields. Formulas giving the variation of the radial and hoop strains over the membrane, an important issue recently addressed by Williams et al. (1992), are also presented. In addition, the critical importance of membrane pretension is demonstrated, and factors which can be used to manipulate the strain field are identified. Analysis A thin circular disk, such as a pressure-loaded cell substrate, which is supported at its edge and which is transversely deflected by more than approximately three times its thickness behaves mechanically like a membrane (Brodland, 1986). Consider then a thin circular membrane of radius, a, and thickness, h, which is made of a linear material having Young's modulus, E, and Poisson's ratio, v. Let the membrane carry a transverse pressure, p. See Fig. 1. The deflection of a pressure-loaded membrane which has uniform in-plane tension, T, is governed by the linear differential equation (Sokolnikoff, 1956)
For a circular membrane subject to the boundary condition References 1 Yongchareon, W., and Young, D. F., "Initiation of Turbulence in Models of Arterial Stenoses," J. Biomech., Vol. 12, 1979, pp. 185-196. 2 Hutchison, K., and Karpinski, E., "In Vivo Demonstration of Flow Recirculation and Turbulence Downstream of Graded Stenoses in Canine Arteries,'' J. Biomech., Vol. 18, 1985, pp. 285-296. 3 Young, D. F., Cholvin, N. R., and Roth, A., "Pressure Drop Across Artificially Induced Stenoses in the Femoral Artery of Dogs," Circ. Res., Vol. 36, 1975, p. 735. 4 Deshpande, M. D., Giddens, D. P., and Mabon, R. F., "Steady Laminar Flow Through Modelled Vascular Stenoses," J. Biomech., Vol. 9, 1976, pp. 165-174. 5 Milnor, W. R., Hemodynamics, Williams & Wilkins, 1982, Baltimore. 6 Cho, Y. I., Back, L. H., Crawford, D. W., and Cuffel, R. F., "Experimental Study of Pulsatile and Steady Flow Through a Smooth Tube and an Atherosclerotic Coronary Artery Casting of Man." J. Biomech., Vol. 16, 1983, pp. 933-946. 7 Pedley, T. Y., The Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge, 1980. 8 Belardinelli, E., and Cavalcanti, S., "A New Nonlinear Two-Dimensional Model of Blood Motion in Tapered and Elastic Vessels," Comput. Biol. Med., Vol. 21, 1991, pp. 1-13. 9 Ahmed, S. A., and Giddens, D. P., "Pulsatile Poststenotic Flow Studies with Laser Doppler Anemometry," J. Biomech., Vol. 17, 1984, pp. 695-705. 10 Deshpande, M. D., Giddens, D. P., and Mabon, R. F., "Steady Laminar Flow Through Modelled Vascular Stenoses," J. Biomech., Vol. 9, 1976, pp. 165-174. 11 Back, L. H., Radbill, J. R., Cho, Y. I., and Crawford, D. W., "Measurement and Prediction of Flow Through a Replica Segment of a Mildly Atherosclerotic Coronary Artery of Man.," J. Biomech., Vol. 19, 1986, pp. 1-17.
418 / Vol. 114, AUGUST 1992
w(«) = 0,
(2)
the corresponding solution for transverse deflection, w, is (Volterra and Gaines, 1971) w(r) = ^ ; ( f l 2 - r 2 ) ,
(3)
or
Mr) = wmJl-(-)j.
(4)
The closely related problem of circular plates has been studied extensively. See Brodland (1986, 1988). Some of the most useable of the circular plate solutions are formulated using the nonlinear Kirchhoff theory on which the von Karman equations are based, but begin with the assumption that the trans-
"Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, N2L3G1, Canada. department of Mechanical Engineering, University of Saskatchewan, Saskatoon, SK, S7N 0W0, Canada. 3 Center for Biomaterials, University of Toronto, Toronto, Ontario, M5S 1 A l , Canada. Contributed by the Bioengineering Division of THE AMERICAN SOCIETY OF
MECHANICAL ENGINEERS. Manuscript received by the Bioengineering Division July 29, 1991; revised manuscript received September 25, 1991.
Transactions of the ASME
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 06/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Table 1 L, [mm] 15 30
Stenosis A Stenosis B
> [mm] 10 •1.5
S. Cavalcanti1, P. Bolelli1, and E. Belardinelli1 Upstream
Measures of pressure drops were made in two different plexiglass models of axial-symmetric arterial stenoses. The stenosis models had the same are reduction {86 percent) but were of different length so as to have a different tapering degree. Pressures were measured in steady flow condition at three equidistant points of the stenosis: upstream, in the middle, and downstream. Results indicate that: the upstream-middle pressure drop is independent of tapering degree but is highly influenced by area reduction; moreover it is much greater than the middle-downstream drop. The upstream-middle pressure drop can be accurately predicted by means of a relationship deduced by the momentum equation.
Introduction Atherosclerotic plaque formation in an artery gives rise to a local reduction of vessel lumen. The severity of stenosis depends on the flow reduction involved. A basic role in stenosis hemodynamics and, consequently, in flow reduction, is played by the vessel outline [1, 2]. The local area reduction makes convergent the vessel upstream segment of the stenosis, divergent the downstream one. Convective accelerations induced by tapering increase the fluid element velocities between upstream- and mid-stenosis with a consequent increase in the pressure drop [3, 4], When the area reduction is severe the pressure drop increases and the pressure in the mid-stenosis is much lower than in the upstream one [10]. Although the percent area reduction is the most widely used clinical parameter in evaluating the importance of vessel disease, from a functional view-point evaluation of the pressure drop through the stenosis is more relevant [6]. In order to evaluate how a relationship, based on the one-dimensional flow assumptions, can predict the pressure drop through a stenosis, we have performed several flow-pressure measures in two plexiglass models of arterial stenosis.
Methods Stenosis Models. Two rigid-walled plexiglass stenosis models, with an axial-symmetric geometry and an area reduction of 86 percent, were made. To connect the pressure transducers three holes were bored: one in the middle and the others upstream and downstream of the stenosis (see Fig. 1). The L\ and L 2 lengths are reported in Table 1. Stenosis A has a 9.5 deg taper angle (a) whereas stenosis B has a 4.8 deg taper angle. Hydraulic Equipment. The main conduit of the circuit (see Fig. 2) was a polyethylene tube of constant diameter connecting the test section to the constant level tank and the downstream reservoir. The one-way valves allowed the setting of the steady
'Department of Electronics, Computer Science and Systems, University of Bologna, Bologna, Italy. Contributed by the Bioengineering Division of THE AMERICAN SOCIETY OF
MECHANICAL ENGINEERS. Manuscript received by the Bioengineering Division, June 1, 1991; revised manuscript received September 28, 1991. Associate Technical Editor: L. Talbot.
Middle
Downstream
F^^^'^^2^ R_= 1.5mm
Tr77777/7777777rA £1
Fig. 1 Geometry of stenosis model. Lengths Li and L2 are different in the two models (see Table 1)
Costant head t a n k Pump
^ XValve ] = [ ZTestZsection Z r = MValve =
Fig. 2 Scheme of hydraulic experimental apparatus
flow rate in a range similar to the physiological one for a human medium artery (1 -f- 28 ml/s) [5], which correspond Reynold numbers between 61 -*• 1500. The distances between the valves and the test section were sufficient to ensure that the entrance effect would be negligible. The fluid used for all tests was a water-glycol ethylenic mixture with an absolute viscosity p = 3 • 10 ~3 Kg/ms and a density p = 1.068-103 kg/m3 at 28 °C, which are close to the real values of blood. The main instruments employed were: an integrated amplified pressure transducer (Honeywell 142PC15G) equipped with a further amplifier circuit made in our laboratory; a digital multimeter with thermometric probe (Philips PM2519 and PM9249). The flow rate was measured by the weighting method. Theory. The motion equation of fluid is deduced by assuming that it is incompressible and newtonian, and supposing a one-dimensional steady flow condition. Fluid motion is, then, governed by the momentum Eq. (7), which, in a cylindrical reference frame (r, z) can be written as
2
•KR1
dz
'I(M
- 1-KVR
dw ~dr
(1)
with the following meaning of the symbols: p density, p = ^/ p cinematic viscosity; w = w (r, z) fluid element velocity, while p = p(z) is the pressure andR = R(z) the radius. To integrate Eq. (1) with respect to r the following velocity profile is assumed. w(r, z)
k+\ k
Q TTR2
1--
R2k
(2)
where Q is the flow rate and k a integer parameter that allows the profile shape to be arranged: when k = 1 the profile is
4 1 6 / V o l . 114, AUGUST 1992
Transactions of the ASME
Copyright © 1992 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 06/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use
parabolic while by increasing k the profile assumes a flat shape. The validity of (2) is confirmed by the experimental results (see Discussion). The momentum equation can then be rearranged in the form dp_ dz"
2k+ 2 ~2k+l
p
d_ / l _
(TTR)2 dz
\R
G2-
(* + D^e.
(3)
To obtain the pressure drop between the upstream and middle sections Eq. (3) must be integrated with respect to z. Let Ru and Rm denote, respectively, the inner radius in the upstream (z - 0) and in the middle (z = Lt + L2) sections. Thus we have AP„.
1+:
1 2k+l
W +
k+\
(4)
RPQ,
with 2-K1RI
Table 2
etcc/s]
P„ [mmHg]
P,„ [mmHg]
Pd [mmHg]
1.16 2.40 3.16 3.84 5.44 6.42 8.25 10.83 12.50 13.33 14.32 16.70 19.40 23.30 27.00 28.00
133.14 132.88 132.78 132.70 132.40 132.14 131.70 131.30 130.19 129.83 129.55 129.05 128.09 126.85 125.26 124.96
132.94 131.94 131.34 130.63 128.55 126.77 123.46 118.27 112.96 110.63 107.40 100.11 89.64 74.67 52.62 46.04
132.94 131.84 131.18 130.36 128.01 126.19 122.56 116.98 111.29 108.77 105.26 97.22 85.95 69.68 47.53 40.55
eicc/s] 0.96
P„ [mmHg] 133.47 133.29 133.31 , 133.11 132.63 132.12 131.67 131.48 130.48 130.25 129.41 128.65 128.43 127.85 127.20 126.87
P,„ [mmHg] 133.37 132.33 131.87 130.25 126.95 123.28 118.56 114.31 105.74 102.86 94.82 87.68 81.83 74.65 63.20 52.06
P„ [mmHg] 133.39 132.37 131.92 130.46 127.30 123.97 119.40 115.40 107.09 105.24 98.68 92.93 87.80 81.85 73.24 62.57
si" 1 ) '"d 8JA
Rn = —
Table 3 R u + RUR,„ + R',
7T
L,+-
R'.
The first term on the right side of (4) takes the inertial forces into account and is the same as the Bernoulli pressure drop when k is very great (flat velocity profile); the second term takes the viscous forces into account and is the same as the Poiseuille drop when k = 1. For low values of k (k < 10) the inertial prevail over viscous forces and, above all, the sensitivity of (4) with respect to k is very low; i.e., in this condition the k value has little influence on the total pressure drop.
Results The first columns of Table 2 and Table 3 are the sixteen steady flow values, respectively, in stenosis A and B. The corresponding measured pressure in the upstream, middle, and downstream sections are indicated in the other columns. Figure 3 shows the pressure drops between the upstream and middle sections (star for stenosis A and circle for B) and those between the middle and downstream sections (X for A and plus for E). Comparison of the pressure drop calculated by relationship (4) with the measured data (circle) is shown in Fig. 4. Computed and measured pressure drops were normalized by the dynamic pressure 7&Q2. The continuous curves are obtained for k = 1 while the dashed curves are relative to k = 4. The two lines at the bottom of the right picture are viscous drops, calculated by means of the second terms on the right side of (4). Comparison between calculated and measured pressure drops for stenosis B is shown in Fig. 5 in the same way as in Fig. 4.
2.70 3.40 5.00 7.34 9.20 11.28 13.20 15.95 16.40 18.95 20.70 22.70 24.20 25.35 28.11
80 60
* x Stenosis A o + Stenosis B
40 d*
20-
-20
20 Flow [cc/s]
10
30
Fig. 3 Pressure drops between upstream-middle sections (star and circle) and middle-downstream (Xand plus)
middle-downstream
Discussion Figure 3 clearly shows that the degree of tapering of the upstream region has no influence on the upstream-middle pressure drop (star for stenosis A and circle for B). For the downstream region the same is not true: in this region, by decreasing the diverging degree, the sign of pressure drop changes so that downstream pressure is lower than middle pressure for stenosis A (plus in Fig. 3), whereas it is higher than middle pressure for stenosis B(X'm Fig. 3). It is well known that in a tapering vessel the inertial forces associated with convective accelerations compel the velocity profile to assume a flat instead of a parabolic shape [8]. Nevertheless, relationship (4) with k = 1 predicts the upstream-middle Journal of Biomechanical Engineering
Fig. 4 Comparison of experimental data (circle) measured in stenosis A and pressure drop predicted by relationship (4) (continuous and dashed curves). The stenosis has a 9.5 deg taper angle.
pressure drop satisfactorily in both stenoses (Figs. 4 and 5). Actually, the estimate of the pressure drop, assuming a parabolic profile (k = 1), introduces an error in both terms of relationship (4) with tend to balance each other, so that it has AUGUST 1992, Vol. 114/417
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 06/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use
upstream-middle
Stenosis B Q*
middle-downstream
Pretension Critically Affects the Incremental Strain Field on Pressure-Loaded Cell Substrate Membranes G. W. Brodland,1 A. T. Dolovich,2 and J. E. Davies3
Flow [cc/s]
Flow [cc/s]
Fig. 5 Comparison of experimental data (circle) measured in stenosis Sand pressure drop predicted by relationship (4)(continuous and dashed curves). The stenosis has a 4.8 deg taper angle.
little influence on the total pressure drop evaluation. All this is confirmed by the low sensitivity of (4) with respect to k for small values of k itself. So, a parabolic profile hypothesis is acceptable if we want to calculate the upstream-middle pressure drop. It is emphasized that the error of prediction increases when the flow increases and the taper and angle decreases. The most accurate pressure drop evaluation is obtained when k = 3 for stenosis A and k = 2 for stenosis B. A few studies involving the development of turbulence in stenosis have been reported [1, 2, 9, 11]. The results clearly show that: the disturbed region is downstream of the stenosis and the critical Reynolds number decreases as the diverging degree increases. In particular, in a diverging vessel a flow separation with a vortex formation occurs and near the wall there is a thin anular layer of reverse flow [8, 10]. Then, for the downstream region, it is impossible to make a correct pressure drop prediction by means of a one-dimensional flow equation. In this case it is necessary to assume at least a twodimensional flow, i.e., fluid element with two velocity components which are respectively directed as axial and radial coordinates. However, for both the stenoses considered here the pressure drop is almost completely concentrated between the upstream and middle sections. Therefore, relationship (4) can be expected to serve as a guide to evaluate pressure drop through a stenosis when both the mean flow in the cardiac cycle as well as the stenosis radius and length are measured.
Introduction Strainable cell substrates have found increasing application in cell culture experiments, where they are used to mechanically stimulate cells. For a review, see Williams et al. (1992). A number of solutions to the associated mechanics problem for a circular membrane exist (Hencky, 1915; Green and Adkins, 1960). None of these addresses the important issue of membrane pretension or prestrain. Here, for circular membranes, we derive a solution which produces formulas which are easy to use and which show clearly the interrelationships between inflation pressure, center deflection, and incremental strain fields. Formulas giving the variation of the radial and hoop strains over the membrane, an important issue recently addressed by Williams et al. (1992), are also presented. In addition, the critical importance of membrane pretension is demonstrated, and factors which can be used to manipulate the strain field are identified. Analysis A thin circular disk, such as a pressure-loaded cell substrate, which is supported at its edge and which is transversely deflected by more than approximately three times its thickness behaves mechanically like a membrane (Brodland, 1986). Consider then a thin circular membrane of radius, a, and thickness, h, which is made of a linear material having Young's modulus, E, and Poisson's ratio, v. Let the membrane carry a transverse pressure, p. See Fig. 1. The deflection of a pressure-loaded membrane which has uniform in-plane tension, T, is governed by the linear differential equation (Sokolnikoff, 1956)
For a circular membrane subject to the boundary condition References 1 Yongchareon, W., and Young, D. F., "Initiation of Turbulence in Models of Arterial Stenoses," J. Biomech., Vol. 12, 1979, pp. 185-196. 2 Hutchison, K., and Karpinski, E., "In Vivo Demonstration of Flow Recirculation and Turbulence Downstream of Graded Stenoses in Canine Arteries,'' J. Biomech., Vol. 18, 1985, pp. 285-296. 3 Young, D. F., Cholvin, N. R., and Roth, A., "Pressure Drop Across Artificially Induced Stenoses in the Femoral Artery of Dogs," Circ. Res., Vol. 36, 1975, p. 735. 4 Deshpande, M. D., Giddens, D. P., and Mabon, R. F., "Steady Laminar Flow Through Modelled Vascular Stenoses," J. Biomech., Vol. 9, 1976, pp. 165-174. 5 Milnor, W. R., Hemodynamics, Williams & Wilkins, 1982, Baltimore. 6 Cho, Y. I., Back, L. H., Crawford, D. W., and Cuffel, R. F., "Experimental Study of Pulsatile and Steady Flow Through a Smooth Tube and an Atherosclerotic Coronary Artery Casting of Man." J. Biomech., Vol. 16, 1983, pp. 933-946. 7 Pedley, T. Y., The Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge, 1980. 8 Belardinelli, E., and Cavalcanti, S., "A New Nonlinear Two-Dimensional Model of Blood Motion in Tapered and Elastic Vessels," Comput. Biol. Med., Vol. 21, 1991, pp. 1-13. 9 Ahmed, S. A., and Giddens, D. P., "Pulsatile Poststenotic Flow Studies with Laser Doppler Anemometry," J. Biomech., Vol. 17, 1984, pp. 695-705. 10 Deshpande, M. D., Giddens, D. P., and Mabon, R. F., "Steady Laminar Flow Through Modelled Vascular Stenoses," J. Biomech., Vol. 9, 1976, pp. 165-174. 11 Back, L. H., Radbill, J. R., Cho, Y. I., and Crawford, D. W., "Measurement and Prediction of Flow Through a Replica Segment of a Mildly Atherosclerotic Coronary Artery of Man.," J. Biomech., Vol. 19, 1986, pp. 1-17.
418 / Vol. 114, AUGUST 1992
w(«) = 0,
(2)
the corresponding solution for transverse deflection, w, is (Volterra and Gaines, 1971) w(r) = ^ ; ( f l 2 - r 2 ) ,
(3)
or
Mr) = wmJl-(-)j.
(4)
The closely related problem of circular plates has been studied extensively. See Brodland (1986, 1988). Some of the most useable of the circular plate solutions are formulated using the nonlinear Kirchhoff theory on which the von Karman equations are based, but begin with the assumption that the trans-
"Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, N2L3G1, Canada. department of Mechanical Engineering, University of Saskatchewan, Saskatoon, SK, S7N 0W0, Canada. 3 Center for Biomaterials, University of Toronto, Toronto, Ontario, M5S 1 A l , Canada. Contributed by the Bioengineering Division of THE AMERICAN SOCIETY OF
MECHANICAL ENGINEERS. Manuscript received by the Bioengineering Division July 29, 1991; revised manuscript received September 25, 1991.
Transactions of the ASME
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 06/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use
