To learn more, view our Privacy Policy. To browse Academia. Log in with Facebook Log in with Google. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up. Download Free PDF. Mechanics of a thin walled collapsible microtube Annals of Biomedical Engineering, Nico Westerhof.
A short summary of this paper. Mechanics of a thin walled collapsible microtube. Annals of Biomedical Engineering, Vol. All rights reserved. The microtubes are made by dip- ping a glass mould in a latex solution and glueing their outside ends to the inside o f glass pipettes.
The pressure-cross sectional area relation is determined both with a microplethysmograph pressure-volume relation and the microscope pressure- diameter relations. Heparinized blood is used to include the rheological properties o f blood as a perfusion medium. Static pressure-flow relations are obtained with a constant velocity piston pump f o r two values o f external pressure 0 and 10 kPa and with two downstream resistor settings 0 and kPa cm -s sec.
The calculated pressure-flow relations using length and the experimental pressure-cross sectional area relation, Poiseuille's law, and accounting f o r the diameter- and shear-dependent vis- cosity compared well with the relations obtained f r o m the experiments.
It is also found that the pressure-flow relation shows an apparent zero f l o w pressure axis inter- cept the extrapolation o f the pressure-flow relation to the pressure axis , which can therefore be explained on the basis o f the shape o f the pressure-area relations.
Keywords-Microtubes, Mechanics, Pressure-flow relation, Simulation. This approach provides information on the integrated behavior of the microvascular elements comprising the circulation. Intravital microscopy has provided information about structure and function of the individual elements of the microcirculation 9,13 : microvascular pressure-flow rela- tion low Reynold's number, quasisteady flow , microvascular pressure-cross sectional area relation 2,9,18 and the effect o f the smooth muscle contraction on these rela- tions 13, The mechanics of larger veins and thin-walled latex tubes diameter about 10 mm Acknowledgment-The authors thank Hans de Zeeuw and Connie Pieksma for typing the manuscript and Joop Meijer and Han Verbeek for preparing the figures.
Part of this work was presented at the 14th International Conference of the European Society for Mi- crocirculation, Sweden, Address correspondence to Dr. Sipkema and N. Westerhof has also been studied by many authors 16,17, One should be very cautious in applying the results on these tubes to blood vessels, because the mechanical proper- ties of veins and tubes are different as was shown by Moreno et al.
These authors showed that the relation between cross sectional area and perimeter exhibits an almost constant perimeter for latex tubes but not for veins. The reason for this difference in properties lies in the fact that the Young's modulus for latex rubber is much higher than for veins.
Therefore, the wall of latex rubber tubes resists stretching hence, the constant perimeter more firmly than of veins. The magnitude of the dif- ference in perimeter depends on the difference in Young's modulus. The theory for larger diameter vessels is described by Shapiro The standard experimental setup is a segment o f tube supported horizontally between two lengths of rigid tube and is contained in a chamber whose pressure can be given any value.
The typical result is that the pressure drop over the length of the tube initially increases with flow, then decreases negative "resistance" and then, when the tube is fully open, is proportional with flow and extrapolates back through zero 8. The Young's modulus for latex rubber and arterioles is about equal. It is therefore expected that the latex microtube results will correlate well with analogous physiological arterioles.
Recently, Westerhof et al. The smaller the diameter the more the viscous effect of the fluid plays a role and results may thus be different from results obtained for larger tubes, mainly resulting in a transmural pressure that is a function of position along the tube.
Transmural pressure is the in- travascular pressure minus the extravascular pressure. Transmural pressure deter- mines the cross sectional area which, in turn, determines resistance to flow. The purpose of the present investigation is to study the pressure-cross sectional area relation and the pressure-flow relation of a thin-walled latex tube of small diam- eter m.
Due to this small diameter there is, contrary to the situation in large diameter latex tubes, at physiological flow values, a significant axial pressure gra- dient. The static mechanical properties are determined from the effective distending pressure transmural pressure of the vessel.
A uniform external pressure applied to the entire tube, results, during flow, in a decreasing intravascular pressure and thus in a transmural pressure that is decreasing along the tube. This transmural pressure determines the cross sectional area which, in turn, determines resistance to flow. The experimental results are compared with a computer calculation based on the equa- tion of motion and the tube law.
This computer program predicts perfusion pressure- flow relations on the basis o f the measured transmural pressure-cross sectional area relation, the length of the tube and viscosity o f the perfusion fluid. This reduces the work to force blood through narrow vessels. The computer program is finally used to study the suggestion of Alexander 1 about critical closing.
Critical closure is the most commonly cited explanation for the observation that pressure-flow curves usually intercept the pressure axis at a positive value 1. The concern of Alex- ander was that the pressure-volume relationship of a constricted arteriole does not exhibit closure as pressure is reduced, and he recalculated the constricted curve with additional assumptions such that closure is obtained. The number of dippings determines the wall thickness. After setting o f the material, the microtube is carefully removed from the mould in an ultrasonic cleaning bath.
This is a level 5 arteriole according to the microvascular network description of Zamir Wall thickness and inner diameter are determined by weighing the tube with a Cahn electrobalance, accuracy: 0.
Using the specific weight of latex, the wall thickness is calculated and, by subtraction of twice the wall thickness from the outer diameter, the inner diameter is obtained. A system to handle and study isolated microvessels has been described by Duling et aL 9. The proximal end of a microvessel is mounted on a so-called double pipette Fig.
The outer pipette has a constriction and the microvessel is held in position! The latex tube is moved over the inner pipette and positioned in the constriction of the outer pipette. Panel 3 shows the outside of the latex microtube attached to the inside of the pipette. Sipkerna and N. Westerhof using slight suction between this constriction and the tapering inner pipette.
The rest of the set-up differs from Duling's technique in two ways: distal and proximal ends are handled similarly and the inner pipette is removed in this study, since the hydrau- lic resistance of this inner pipette is an order of magnitude higher than the resistance of the microvessel studied. The microvessel is positioned in the constriction of the outer pipette Fig.
Resistance of the assembly was determined by connecting the two pipettes with a tube mounted on their outside and their ends against each other. The slope of the pressure-flow relation determined in this setup was taken as resistance. The two pipettes with the latex microtube in between are mounted in a small chamber as shown in Fig. The chamber can be closed so that external pressure can be applied. The bottom and the lid of the chamber are o f high quality glass, which allows observation of the tube under the microscope.
The pressure-area relation of the microtube is determined in the no-flow situation with a microplethysmograph. This method was chosen because at negative transmu- ral pressures the determination of diameter leads to erroneous results. In the no-flow situation the microtube shows the same changes in shape over its entire length.
If one measures the volume of fluid inside the microtube as a function of transmural pres- sure, then the changes in shape from dumbbell via ellipsoidal to circular are auro- matically taken care of. The cross sectional area is found by dividing the measured volume by the length of the microtube. This procedure is described below.
Two pipettes as shown in Fig. The chamber is closed so that external pressure PQxt can be applied. The microtube can be observed under the microscope via two windows of high quality glass one in the lid and one in the bottom. Perfusion pressure is measured at the upstream and, while flow is gen- erated with a piston pump not shown. Calibrated resistors Ro,0 can be connected to the down- stream end.
The space in the chamber outside the microtube is completely filled with water and no air bubbles are allowed. One outlet o f the chamber is connected to a vertical glass capillary diameter 0. Increasing the pres- sure in the latex microtube results in a larger v o l u m e o f the tube and this causes the level o f the fluid in the glass capillary to rise. The total rise is less than 2 m m and its effect on transmural pressure is therefore neglected. The pressure-area relation is found by dividing the measured volume by the length o f the tube.
As a check, the transmural pressure is also related to diameter determined by observing the microtube through the microscope and a video camera and monitor.
This check is done for pos- itive transmural pressures only, because the microtube is circular in this transmural pressure range.
From the observed diameter one can then calculate the cross sectional area. In the results see Fig. The perfusion pressure is measured with a P23Db pressure transducer. The per- fusion flow is generated with a piston pump driven with a DC motor. The speed o f the motor can be continuously adjusted with a 10 turn dial and stepwise reductions in speed can be applied with a gear box.
Calibrated resistors stainless steel tubes o f various lengths can be connected to the downstream pipette in order to set the out- flow pressure of the latex microtube at the desired level. The fact that the tube is mounted in a container allows for application o f external pressure to the microtube.
Transmural pressure-cross sectional area relation of the microtube. The squares are the data points obtained with the microplethysmograph. The circles are the data points obtained with the video system only at positive transmural pressures. The drawn line is the tangent function fitted with a least squares method through the data points.
The parameters are given in Table 1 first column. Westerhof The perfusion fluid used in this study is heparinized blood taken from a dog Ht: The Fgthraeus-Lindqvist effect 14 and shear-dependent viscosity 6 and the pressure dependent shape of the cross section are taken into consideration. They used high shear rates, where viscosity is not depen- dent on shear rate.
These experiments were repeated by Haynes 14 and Haynes's data are used in the computer calculations see Appendix. The other important fac- tor is the shear dependence of viscosity at low shear rates. For values of the shear rate below s -1 the viscosity of blood increases as the shear rate decreases. Brooks et al.
Data of experiments of Brooks et al. The squares are the data points obtained with the microplethysmograph; the data points obtained by diameter measurement are given by the circles. At negative transmural pressures it was noted that the tube collapses and the cross section is not circular anymore, the wall of the tube touches the opposite side. The pressure area data were fitted with a shifted tangent-function fully drawn line ; the parameters of the fit are given in Table 1.
The parameters of fitted tangent function. The values in columns under Fig. The parameter C determines the slope, If C is small then the middle sec- tion is about horizontal. The bigger its value the steeper the relation- ship. Note that curve C is 4 kPa shifted upward with respect to curve b while the asymptotic values of cross sectional areas are all the same i.
Mechanics o f a Microtube the transmural pressure-cross sectional area relation. The pressure area relationship may be different from that of arteries and arterioles particularly in the plateau region.
This difference can be taken into consideration by adapting the parameter C, for example, that determines the slope in this region. This aspect is shown in the Discus- sion see Figs.
The perfusion pressure-flow relations for two external pressures 0, 10 kPa , and two downstream resistor settings 0, kPa cm -3 sec are shown in Fig.
The pressure-flow relation of the microtube with zero external pressure and without the outflow resistor squares is almost a straight line without a pressure axis intercept.
When external pressure is applied triangles the slope of the relation is high for low pressures and flows and then starts to run parallel to the one without external pres- sure. Extrapolation of this part of the relation to zero flow results in a pressure axis intercept related to but lower than the applied external pressure.
The pressure-flow relation of the microtube with external pressure equal to zero but with outflow resis- tor Fig.
Perfusion pressure-flow relations obtained in the experimental setup using the latex microtube perfused with heparinized blood at room temperature. The experimental conditions are shown in the inset. Note the apparent pressure axis intercept as external pressure is applied.
Per- fusion pressure-flow relations obtained with the computer stimulation using the experimental pres- sure-cross sectional area relation Fig. The equations of the simulation are given in the appendix.
The conditions of external pressure and outflow resistor are the same as in the experiment. Westerhof slope is increased due to the outflow resistor. When external pressure is applied plus signs the slope of the relation again is high for low flows.
For low and intermedi- ate flows this relation superimposes with the relation without outflow resistor but with external pressure. For high flows the slope changes again. This final section starts when the pressure drop across the outflow resistor is high enough to make the cross section of the entire tube circular.
The higher the outflow resistor, the lower the flow value where this occurs. No oscillations in the perfusion pressure were found in our setup No negative slope of the pressure-flow relation existed.
The pressure-flow relations produced by the computer program based on the transmural pressure-area relation and the same settings of external pressure and out- flow resistor are also given in Fig. These computed results agree with the measurements including the observation that a range of flow values exists where the presence of an outflow resistor has no effect on the perfusion pressure-flow relation.
The contribution of the FAhraeus-Lindqvist not shown effect is found to be 1. In order to compare our data with the literature where it is usual to plot the pressure drop over the tube, the difference between perfusion pressure and pressure at the end of the microtube is plotted versus flow in Fig.
Perfusion pressure minus downstream pressure versus perfusion flow obtained with the computer simulation, The conditions of external pressure and outflow resistor are the same as in Fig.
The latex microtube in this study has a small cross sectional area, so that during flow there is a considerable pressure drop along the length of the tube.
This axial pressure drop makes the present results different from findings on larger diameter latex tubes. The external pressure is con- stant along the tube so that transmural pressure is a function of the location.
With- out external pressure the pressure-flow relations are almost straight without an intercept on the pressure axis. The application o f external pressure results in a pressure-flow relationship where a region exists with an extrapolated pressure axis intercept. If the vessel is also externally pressurized, for example subject to athmospheric pressure, p is defined by subtracting the external pressure from the internal one, a difference called gage pressure.
If the external pressure is higher, as in the case of a submarine hull, the stress formulas should be applied with extreme caution because another failure mode: instability due to wall buckling, may come into play.
See Section 3. Ignoring End Effects. Features that may affect the symmetry assumptions are ignored. This includes supports and cylinder end caps. The assumption is that disturbances of the basic stress state are confined to local regions and may be ignored in basic design decision such as picking up the thickness away from such regions.
We study the two simplest geometries next. Cylindrical Vessels We consider a cylindrical vessel of radius R, thickness t loaded by internal pressure p. The resulting material element, shown in exploded view in Figure 3.
Wall material element of a pressurized cylindrical vessel referred to cylindrical coordinates. Note that thickness is grossly exaggerated for visibility. Because of symmetry assumptions on the geometry and loading no torque , this stress is zero. These stress assumptions are graphically displayed, with annotations, in Figure 3.
Both components are uniform across the wall thickness and throughout the vessel excluding possible end effects, which are discussed in Section 3. Details will be worked out in class.
Because the hoop stress is twice the axial stress, it will be the controlling one in a strength design. Spherical Pressure Vessel A similar approach can be used to derived an expression for an internally pressurized thin-wall spherical vessel. Stress Assumptions Reasoning as in the preceding case, we find that 1. Again we will neglect this value when compared to the other normal stresses and justify this assumption a posteriori.
Stress analysis of a spherical pressure vessel in spherical coordinates. Once again thickness is grossly exaggerated for visibility.
This is explained in the next section. Remarks on Pressure Vessel Design For comparable radius, wall thickness and internal pressure the maximum normal stress in a spherical pres- sure vessel is one half as large as that in a cylindrical one.
The reason can be understood by comparing Figures 3. In the spherical vessel the double curvature means that all stress directions around the pressure point contribute to resisting the pressure. The cylindrical geometry, however, can result in more efficient assignment of container space as well as stacking and better aerodynamics: a spherical rocket does not look quite right. One important point for designers is: what happens at the ends of a cylindrical vessel? Suppose for instance that a cylinder is closed by hemispherical end caps, as pictured in Figure 3.
If the cylinder and the end caps were allowed to deform independently of each other under pressurization they would tend to expand as indicated by the dashed lined in that figure. The cylinder and the ends would in general expand by different amounts. But since physical continuity of the wall must be maintained, the necessary adjustment in the displacement would produce local bending as well as shear stresses in the vicinity of the juncture, as pictured in Figure 3.
If thick plates are used instead of relatively flexible hemispherical ends, those juncture stresses would increase considerably as shown in Figure Figure 3. For this reason, the ends of cylindrical pressure vessels must be designed carefully, and flat ends are should be avoided if possible.
Most pressure vessels are fabricated from curved metal sheets that are joined by welds. Two weld types: double dillet lap joint and double welded butt joint with V grooves are shown in Figures 3.
Of these preference should be given to the latter as it avoids across-the-weld load transmission eccentricity. It should be emphasized that the formulas derived for TWPV in this Lecture should be used only for cases of internal pressure. Or, more precisely, the internal pressure exceeds the external one. If a vessel is to be designed for external pressure, as in the case of a submarine or vacuum tank, wall buckling, whether elastic or inelastic, may well become the critical failure mode.
Should that be the case, the previous wall stress formulas are only part of the design. It is reproduced here as it combines the results of this Lecture with the bolt-design-by-average-stress technique described in Lecture 2.
Each lid is bolted to the tank of Figure 3. The tank is made from sheet metal that is 12 in thick and can sustain a maximum hoop stress of 24 ksi in tension. The normal stress in the bolts is to be limited to 60 ksi in tension. A manufacturer can make tanks of diameters varying from 2 ft through 8 ft in steps of 1 ft. Develop a table that the manufacturer can use to advise custometrs of the size of the tank and the number of bolts per lid needed to hold a desired gas pressure.
Why spherical vessels are more structurally efficient.
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