MICROVASCULAR
RESEARCH,
(1975)
l&246-264
1975 Eugene
M. Landis
Microcirculation
Award
Lecture as Seen by a Red Cell
Y. C. FUNG University of California-San Diego, La Jolla, California 92037
It is a great honor and pleasure to be the fortunate awardee of the Landis Award. I appreciate the encouragement you give me. I question whether I deserve it. I accept it with humbleness and consciousness about the feebleness of my contributions; I accept it with extreme gratitude. Hence let me say a most genuine “thanks” to you all from the bottom of my heart. By tradition I am supposed to describe the work I have done. My work, or rather, the work of my colleagues and mine, revolves around the tools of mathematics and mechanics. On the other hand, I think you might be curious about how an engineer became a microcirculationist. I entered aeronautical engineering by accident. When I went to apply for college entrance examination, I saw a notice that the National Central University was adding a new department of aeronautical engineering. That sounded interesting, and that was how I got into it. I got my B.S. and M.S. degrees in China, designed airplanes for two years, came to the California Institute of Technology in 1946, and got my Ph.D. two years later. Then I taught and worked on aeroelasticity and structural dynamics in Cal Tech for the next 18 years. It was during my sabbatical leave in 1958 as a Guggenheim Fellow in Goettingen, Germany, that I turned to physiology. I had always been fascinated by how our body works and had always wanted to study the subject. But the current of life had swept me along a different path. The leisure of a sabbatical provided an opportunity for me to fulfill a life-long ambition. An immediate impetus was my mother’s illness. Sometime earlier she was struck by glaucoma. Since she could not read English, I translated a book and several articles on this subject into Chinese and sent them to her weekly by mail. I was delighted in the study of physiology, but I found most works on the subject quite hard to understand. They do not have the mathematical clarity of the classical physics. Hence, I thought maybe I could use my experience in applied mechanics to help develop physiology in the direction of physics. On returning to the California Institute of Technology, I found new friendship and cooperation in Harold Wayland, Ben Zweifach, Wally Frasher, Paul Johnson, and Sid Sobin. With these friends, my predilection toward microcirculation was clear. Soon I discovered that one cannot have two loves. To conserve time and energy, to be able to concentrate, and to do some useful research, I left my old field completely, resigned my post at Cal Tech, and picked up a new trail at UCSD in 1966. This is how an engineer was turned into a microcirculationist. My mentor in aeronautical engineering, Dr. Ernest Sechler, has always maintained that “when you design an airplane, you must think like an airplane.” I have always followed his advice. Coming to microcirculation, I cannot help but think as a red blood cell. The major distinction of microcirculation is the individuality of the red blood cells. As the blood vessels become smaller and smaller, the individual red cells appear bigger Copyright 0 1975 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
246
LANDIS AWARD LECTURE
247
and bigger. The population of red blood cells in each of us is larger than the total population of people in the world. In the world each of us is insignificant. But in our own houses,and in our own gardensand local communities, we decidehow things should go. The same is with microcirculation. The individual red cells decide how blood should flow. Think what a red cell would want to know if he wanted to study himself and his environment. He is forever pushed and crowed by his neighbors. He needsto know the shape and size of himself and others around and the shape and size of the tunnels in which he is moving. He wants to know how flexible he himself is and his fellow travelers are, how hard the wall is, how sticky the ground is, how leaky the foundation is, how to get nourishment, and how to passthe food along. He wants to know how to function in the organ he is in and what happens when something goes wrong. In other words, he wants to know microcirculation. These, then, are my objectives in research. In our society, individual decisions usually show remarkable uniformity. Small changesin individual taste may changethe complexion of an entire society. Micro-circulation goesthe sameway. Decisions of individual red cells control the whole phenomenon. To illustrate the point, I made a motion picture. It is concerned with blood flow in capillary blood vessels.It deals with the questions : When a red cell reachesa crossroad, which way would it go? How would it decide?And what would its decision imply? The first part of the movie is related to flow in very narrow capillaries whosediameters are about the sameas that of the blood cells. Consider a bifurcation point. A blood cell flowing down a capillary would have to decide to which of the two daughter branches it should go in. The motion picture shows that becauseof the pressuregradient and the shear stress,the blood cell will flow into the faster branch. Thus, in very narrow capillaries, the branch in which the flow is faster gets all the blood cells. The larger the cell relative to the capillary is, the better this control is. The second part of the motion picture shows that in pulmonary alveolar sheets,the blood cell distribution reflects the velocity field. Faster channels have more red cells. However, the velocity field and the cell distribution are modulated by the apparent viscosity of blood that increaseswith hematocrit. This motion picture demonstrates that the red cell distribution in a capillary bed is not the sameas the plasma distribution. The model shows the idea in its simplest form, but it can be turned into a quantitative tool to determine the hematocrit distribution in blood vessels.This is a tool I learned to useasan engineer.The procedure is asfollows : First, we enumerate all the physical quantities that are involved in a problem; then with the so-called dimensional analysis, we determine the similarity parameters that control the phenomenon. Then we use kinematically and dynamically similar models to determine the functional relations among dimensionless parameters. Finally, we calculate what the real red cells will do in specific organs in accordance with the model experimental results. The basic idea of this approach is that if two problems are described by the same set of mathematical equations, then their solutions must be the same, no matter how different they are in size and in physical appearance. Lee and I (1967) used this method to determine the pressuredistribution on the wall of a capillary blood vesselas a red blood cell passesby. Yen and I (1972) used the samemethod to demonstrate that the pressure-flow relationship in the pulmonary alveolar sheet is linear, and we obtained the relationship between apparent viscosity and hematocrit
248
Y. C. FUNG
LANDIS AWARD LECTURE
249
250
Y. C. FUNG
by model testing. Geert Schmidt-Schoenbein, Ben Zweifach, and I (1965) used model testing to determine the sticking shear force acting on the endothelium when a white blood cell rolls on the wall of a venule. Now I wish to tell you briefly what Sid Sobin and I have beendoing recently. First, we decided to choose a subject and concentrate all our efforts on it. It is abundantly clear to all of us who work in microcirculation that our field is a fertile one. Picking problems to work on is like picking wild flowers in the spring. There are beautiful specimens in every direction. I believe, however, that the richest area lies in the organ level. I chose the lung for two reasons. One is that the lung handles gas and water, with which I am familiar. The other is that when I first got to know Sid Sobin, he had already worked out the method of catalized silicone rubber injection and obtained hundreds of beautiful histological photomicrographs of the lung that showed exquisite structures. It was then easy for us to quantitize the morphology and proceed to study the blood flow. All my work on the lung was done in cooperation with Sid. Many of the slidesI am going to show appearedin our joint papers. To Sid I am most grateful as a friend and as collaborator. First, let us look at the lung. In a photomicrograph of the vascular spacein a lung, in which red gelatin was injected into the artery and blue gelatin was injected into the vein, we can seethe arterial and venous trees. The capillary blood vesselsappear as the fluffy lacework connecting the arterial and venous trees. If we add connective tissues to this vascular space,the result will resemblevery much the foam rubber that we use in the kitchen. The holes in the foam are analogous to the alveolar air spaces;the rubber walls represent the sheetsof the capillary blood vessels. Figure 1 showsa network of capillary blood vesselsin the lung asseenin a microscope. The left-hand panel (A) shows a plan view. The right-hand panel (B) shows a cross section. Blood travels in the plane of the sheet. An anatomist may describe each segment of a capillary blood vesselas a tube. But most likely that is not how a red cell seesit. Let us imagine how the vascular bed would appear to a red cell. I am almost sure that the capillary blood vesselsin the lung would not look like long tunnels to a red cell. They would more likely appear to a red cell as an underground parking garage is to a car. To drive through the garage one has to swing right and left to avoid the posts. Imagine bumper-to-bumper traffic in such a garage, with every car moving and no one allowed to use brakes. That might be a fair picture of the blood flow in the pulmonary alveolar septa. It is a two dimensional world; hence, we named it a “sheet flow.” Figure 2 shows my interpretation of a scenein a pulmonary alveolar sheetas seenby a red cell. You can seethe garagelike columns and spaces.Red cells are floating into the garage. One on the right is caught by a post. This painting is not very faithful. To clarify the view I have drawn the red cells too small and too few. In reality the cells should be SO big as almost to fill the entire height. An analysis of the sheet-flow idea required a number of preliminary investigations. These may be summarized briefly below : 1. The geometry of the red blood cells.
-1nterferometric measurements. -Wave diffraction and holographic analysis. Microvasc. Res., 1972, 1973.
LANDIS AWARD LECTURE
251
2. How flexible the red blood cells are.
- Theoretical analyses. - Model studies. Fed. Proc., 1966; Biophys. J., 1968. 3. Elasticity of the connective tissues that make up the interalveolar septa’s walls and posts.
- Study of mesentery, skin, ureter, etc. - Concept of the pseudostrain-energy function. p. W = CExp[al eZ,+ a, e: + a4 e, e2 + a5 et,], where p,, is density in reference state, W is strain energy per unit mass, e,, e2 are normal strains, e,, is shear strain, C, aI, a4 are constants. Am. J. Physiol., 1967; ASME, 1966; J. Appl. Physiol., 1968; Am. J. Physiol., 1971; Biomechanics book, 1971; J. Biomech., 1973, 1974; Biorheology. J., 1973. 4. Morphometry of the sheet geometry.
- VSTR (Vascular SpaceTissue Ratio, % vascular spacein sheet) - Average thickness J. Appl. Physiol., 1969; Circ. Res., 1970; Circ. Res., 1972; Microvasc. Res., 1973. 5. Elasticity of the sheet (Fig. 3).
- Thickness varies linearly with respect to blood pressure: h=hO+uAp, where h is the mean sheetthickness, Ap is blood pressureminus alveolar air pressure, ho is the post length at Ap = 0, CIis the sheet-thickness compliance constant. This applies when Ap is positive and not too large. - If Ap is negative, i.e., if the blood pressureis smaller than the alveolar air pressure, then the thickness of the vascular spaceh is effectively zero. -When Ap becomes large, say, above 20 cm H20, the thickness will increase slower with increasing Ap and eventually will tend asymptotically to a constant.
-Dimension in the plane of the sheet is independent of bIood pressure. It is “rigid.” - If a pulmonary capillary is thought of as a tube, then it is linearly elastic in the direction normal to the septum, rigid in the direction parallel to the septum. Circ. Res., 1972. 6. Apparent viscosity of blood in sheetjlow.
- Testing at small Reynolds number. - Pressure-flow relationship is linear. -
f? =
Ylp1asma
(a + bH + cH2),
where q is the apparent viscosity, a, b, c are constants, His the hematocrit. ASME, 1968; Microvasc. Res., 1969;J. FluidMech., 1969; J. AppLPhysiol., 1973. 7. Starling mechanism. Permutt’s waterfall.
- Model testing at high and low Reynolds numbers. - For alveolar flow there was no ffutter. - Can be analyzed statically. Circ. Res., 1972.
252
Y. C. FUNG
Some comments on these items are given below: 1. With the interferometric method the red cell image can be analyzed as the holograph of a phase object according to the principles ofphysical optics (asopposedto geometric optics, which do not apply to such a small object asthe red cell). The sensitivity is then improved to 0.02 pm. Statistics of the geometric characteristics of the red cells are collected. The extreme statistics of Gumbel are applied to answer the question: If you were given a sample of blood containing N red cells, what is the expectedvalue of the diameter of the cell with the largest diameter, what is the expectedvolume of the cell with the largest volume, and what is the expectedvalue of the largest cell area, the largest cell thickness, the corresponding sphericity index, etc. ? These data are useful in making models for model testing and in formulating the stochastic flow behavior in microcirculation.
AF,CAPlLLARY-ALVEOLAR
PRESSURE, cm H,O
FIG. 3.The complianceof the interalveolarseptawith respectto the blood pressure.The local average thicknessof the sheet(of vascularspace)is plotted againstthe differenceof the blood pressureand the alveolar air pressure.
2. It is shown theoretically that becausethe red cell has a biconcave shape, it can have many large deformations without inducing any membrane stressin the cell membrane. These deformations, isochoric (constant volume) and “applicable” (without stretching and tearing in any direction), are unique to the biconcave geometry. “Offdesign” deformation in microcirculation may not be “applicable,” but the stretching of membrane area can be seento be everywhere small. Experiments were made on the flow of liquid-filled thin-walled rubber models of red cells in a cylindrical tube to observe the cell deformation. 3. We need information on the elasticity of the capillary blood vessel wall. Since direct experimentation with the interalveolar septa is difficult, I tested other connective tissues. My 10 years’ experience may be summarized by saying that the connective tissuesare inelastic, but in a very wide range of strain rates (which includes the physiological range), the loading process or the unloading process in any periodic motion may be approximated by an elastic one. After “preconditioning” (establishment of homeostatis), the stress-strain relationship in the loading process can be derived from a pseudo strain-energy function. (The Kirchhoff stress is equal to the partial derivative
ofthe pseudo strain+nergy function with respectto the Lagrangian or Green’s strain.)
LANDIS AWARD LECTURE
253
The best form of the strain-energy function per unit volume of the tissue in the original (unstressed)state is, for a two-dimensional specimen,the exponential form given above. This form admits orthotrophy. An unloading processcalls for a different setofconstants. Sometimes it is advantageous to supplement such an exponential function with a polynomial of strain components. 4. The geometry of the sheetmust be known before any analysis of the flow can be made. We measured the VSTR and the sheet thickness in specimensof cats and other animals. 5. To measurethe compliance of the interalveolar septa with respect to changing blood pressure, we perfused lungs with catalized silicone rubber under controlled inflation and perfusion pressures. The lungs were fixed after the polymer hardened Measurements yielded the results outlined above. (SeeFig. 3.) The resistance of sheet thickness to blood pressure changesis derived mainly from the tension in the septa due to inflation of the lung. It is analogous to a musical drum whose pitch depends on the tension in the drumhead. The drumhead deflects linearly with respectto the impact of the hammer, but the constant of proportionality depends on where the hammer strikes. The posts in the alveolar sheet,however, render this linear relationship uniformly valid over the entire sheet! 6. We now must turn our attention to the blood. We must know how the red cell suspensionbehavesin a garagelike sheetflow. For this purpose we pursued both theoretical analysis and model experiments. The theory gives us the parametric relations. The experiments yield the apparent viscosity as a function of hematocrit. 7. Finally, we experimented on the Starling mechanism becauseit was considered by Permutt and his associatesto be a principal feature of the pulmonary alveolar blood flow, particularly in West’szone II, in which the blood pressurein the pulmonary artery is higher than the alveolar air pressure,whereasthat in the pulmonary vein is lower than the air pressure. We show theoretically that the character of the flow depends on the Reynolds number. When the Reynolds number is large, the flow in an elastic vesselis governed by a wave equation. When the Reynolds number is small, it is governed by a diffussion equation. In the pulmonary alveolar septathe Reynolds number is very smalf, in the range of 10d2to 10m4.Theoretically, the flow at such a small Reynolds number is stable. This is verified by a model experiment. We show that whereasthe model flutters at large Reynolds number, it does not flutter when the Reynolds number is small. Starling mechanism operates, but it operates statically. This provides a great simplification to further analysis. Having completed these preliminary studies, we can assemblethe pieces together to look at the sheet flow. Our results may be listed as follows: 8. Governing equation of sheetflow at steady-state. V2h4=0
-
h is thickness. V2 = a2/ax2+ a2/ay2.
Basedon conservation of massand momentum, Linear thickness-pressure relationship, and pressure-flow-thickness feedback. J. Appl. Physiol., 1969; Circ. Res., 1972.
254
Y. C. FUNG
9. Distributions of velocity, pressure, and thickness within a sheet.
- Solution for any specific boundary conditions can be obtained. J. Appl. Physiol., 1969. 10. Pressure--ow relationship.
- If the flow per unit width of sheetis e, then C!= C-‘Vtr, - h&n), where C is a constant, and h,,, and h,,, are sheetthicknessatthe arteriole and venule, respectively, which are, with pa denoting alveolar air pressure, hart = ho + Npil,, -PAI
ken = ho + a(~ven -PA) Circ. Res., 1972.
Il. Input impedance. - Solution of basic equation V2h4 = 4pfu ah/at, p is blood viscosity,f, is a geometric constant, a is the compliance constant, h is sheet thickness, t is time. Ann. Biomed. Eng., 1972. 12. Fluid transfer to interstitium. - Basic equation is V2h4 = 8pkfKbt(h - h*), where Kbt is permeability constant of blood-tissue interface, h* is a constant related to tissue pressure. Microvasc. Res., 1974.
13. Transit time distribution offlow in sheet. - Convolution with flow in arteries and veins yields result for whole lung. Circ. Res., 1972.
14. Effect of gravity. - Regional blood flow. - Smooth merging of zone 2 and zone 3. Circ. Res., 1972. 15. Pulmonary blood volume.
- Correlates with arteriolar pressure. - Does not correlate with venular pressure. Circ. Res., 1972.
16. Solutesflow in interstitium. - Diffusion of permeable and nonpermeable solutes. - Edema. J. Appl. Mech., 1975.
LANDIS AWARD LECTURE
255
17. Indicator dilution in lung. - Flow in sheet with porous walls. - Longitudinal diffusion coefficient is determined. J. Appl. Mech., 1975. 18. Surfactants and stability of alveoli.
- Every alveolar septum is shared by two alveoli. - Pressureload acting on any alveolar septum
RESEARCH,
(1975)
l&246-264
1975 Eugene
M. Landis
Microcirculation
Award
Lecture as Seen by a Red Cell
Y. C. FUNG University of California-San Diego, La Jolla, California 92037
It is a great honor and pleasure to be the fortunate awardee of the Landis Award. I appreciate the encouragement you give me. I question whether I deserve it. I accept it with humbleness and consciousness about the feebleness of my contributions; I accept it with extreme gratitude. Hence let me say a most genuine “thanks” to you all from the bottom of my heart. By tradition I am supposed to describe the work I have done. My work, or rather, the work of my colleagues and mine, revolves around the tools of mathematics and mechanics. On the other hand, I think you might be curious about how an engineer became a microcirculationist. I entered aeronautical engineering by accident. When I went to apply for college entrance examination, I saw a notice that the National Central University was adding a new department of aeronautical engineering. That sounded interesting, and that was how I got into it. I got my B.S. and M.S. degrees in China, designed airplanes for two years, came to the California Institute of Technology in 1946, and got my Ph.D. two years later. Then I taught and worked on aeroelasticity and structural dynamics in Cal Tech for the next 18 years. It was during my sabbatical leave in 1958 as a Guggenheim Fellow in Goettingen, Germany, that I turned to physiology. I had always been fascinated by how our body works and had always wanted to study the subject. But the current of life had swept me along a different path. The leisure of a sabbatical provided an opportunity for me to fulfill a life-long ambition. An immediate impetus was my mother’s illness. Sometime earlier she was struck by glaucoma. Since she could not read English, I translated a book and several articles on this subject into Chinese and sent them to her weekly by mail. I was delighted in the study of physiology, but I found most works on the subject quite hard to understand. They do not have the mathematical clarity of the classical physics. Hence, I thought maybe I could use my experience in applied mechanics to help develop physiology in the direction of physics. On returning to the California Institute of Technology, I found new friendship and cooperation in Harold Wayland, Ben Zweifach, Wally Frasher, Paul Johnson, and Sid Sobin. With these friends, my predilection toward microcirculation was clear. Soon I discovered that one cannot have two loves. To conserve time and energy, to be able to concentrate, and to do some useful research, I left my old field completely, resigned my post at Cal Tech, and picked up a new trail at UCSD in 1966. This is how an engineer was turned into a microcirculationist. My mentor in aeronautical engineering, Dr. Ernest Sechler, has always maintained that “when you design an airplane, you must think like an airplane.” I have always followed his advice. Coming to microcirculation, I cannot help but think as a red blood cell. The major distinction of microcirculation is the individuality of the red blood cells. As the blood vessels become smaller and smaller, the individual red cells appear bigger Copyright 0 1975 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
246
LANDIS AWARD LECTURE
247
and bigger. The population of red blood cells in each of us is larger than the total population of people in the world. In the world each of us is insignificant. But in our own houses,and in our own gardensand local communities, we decidehow things should go. The same is with microcirculation. The individual red cells decide how blood should flow. Think what a red cell would want to know if he wanted to study himself and his environment. He is forever pushed and crowed by his neighbors. He needsto know the shape and size of himself and others around and the shape and size of the tunnels in which he is moving. He wants to know how flexible he himself is and his fellow travelers are, how hard the wall is, how sticky the ground is, how leaky the foundation is, how to get nourishment, and how to passthe food along. He wants to know how to function in the organ he is in and what happens when something goes wrong. In other words, he wants to know microcirculation. These, then, are my objectives in research. In our society, individual decisions usually show remarkable uniformity. Small changesin individual taste may changethe complexion of an entire society. Micro-circulation goesthe sameway. Decisions of individual red cells control the whole phenomenon. To illustrate the point, I made a motion picture. It is concerned with blood flow in capillary blood vessels.It deals with the questions : When a red cell reachesa crossroad, which way would it go? How would it decide?And what would its decision imply? The first part of the movie is related to flow in very narrow capillaries whosediameters are about the sameas that of the blood cells. Consider a bifurcation point. A blood cell flowing down a capillary would have to decide to which of the two daughter branches it should go in. The motion picture shows that becauseof the pressuregradient and the shear stress,the blood cell will flow into the faster branch. Thus, in very narrow capillaries, the branch in which the flow is faster gets all the blood cells. The larger the cell relative to the capillary is, the better this control is. The second part of the motion picture shows that in pulmonary alveolar sheets,the blood cell distribution reflects the velocity field. Faster channels have more red cells. However, the velocity field and the cell distribution are modulated by the apparent viscosity of blood that increaseswith hematocrit. This motion picture demonstrates that the red cell distribution in a capillary bed is not the sameas the plasma distribution. The model shows the idea in its simplest form, but it can be turned into a quantitative tool to determine the hematocrit distribution in blood vessels.This is a tool I learned to useasan engineer.The procedure is asfollows : First, we enumerate all the physical quantities that are involved in a problem; then with the so-called dimensional analysis, we determine the similarity parameters that control the phenomenon. Then we use kinematically and dynamically similar models to determine the functional relations among dimensionless parameters. Finally, we calculate what the real red cells will do in specific organs in accordance with the model experimental results. The basic idea of this approach is that if two problems are described by the same set of mathematical equations, then their solutions must be the same, no matter how different they are in size and in physical appearance. Lee and I (1967) used this method to determine the pressuredistribution on the wall of a capillary blood vesselas a red blood cell passesby. Yen and I (1972) used the samemethod to demonstrate that the pressure-flow relationship in the pulmonary alveolar sheet is linear, and we obtained the relationship between apparent viscosity and hematocrit
248
Y. C. FUNG
LANDIS AWARD LECTURE
249
250
Y. C. FUNG
by model testing. Geert Schmidt-Schoenbein, Ben Zweifach, and I (1965) used model testing to determine the sticking shear force acting on the endothelium when a white blood cell rolls on the wall of a venule. Now I wish to tell you briefly what Sid Sobin and I have beendoing recently. First, we decided to choose a subject and concentrate all our efforts on it. It is abundantly clear to all of us who work in microcirculation that our field is a fertile one. Picking problems to work on is like picking wild flowers in the spring. There are beautiful specimens in every direction. I believe, however, that the richest area lies in the organ level. I chose the lung for two reasons. One is that the lung handles gas and water, with which I am familiar. The other is that when I first got to know Sid Sobin, he had already worked out the method of catalized silicone rubber injection and obtained hundreds of beautiful histological photomicrographs of the lung that showed exquisite structures. It was then easy for us to quantitize the morphology and proceed to study the blood flow. All my work on the lung was done in cooperation with Sid. Many of the slidesI am going to show appearedin our joint papers. To Sid I am most grateful as a friend and as collaborator. First, let us look at the lung. In a photomicrograph of the vascular spacein a lung, in which red gelatin was injected into the artery and blue gelatin was injected into the vein, we can seethe arterial and venous trees. The capillary blood vesselsappear as the fluffy lacework connecting the arterial and venous trees. If we add connective tissues to this vascular space,the result will resemblevery much the foam rubber that we use in the kitchen. The holes in the foam are analogous to the alveolar air spaces;the rubber walls represent the sheetsof the capillary blood vessels. Figure 1 showsa network of capillary blood vesselsin the lung asseenin a microscope. The left-hand panel (A) shows a plan view. The right-hand panel (B) shows a cross section. Blood travels in the plane of the sheet. An anatomist may describe each segment of a capillary blood vesselas a tube. But most likely that is not how a red cell seesit. Let us imagine how the vascular bed would appear to a red cell. I am almost sure that the capillary blood vesselsin the lung would not look like long tunnels to a red cell. They would more likely appear to a red cell as an underground parking garage is to a car. To drive through the garage one has to swing right and left to avoid the posts. Imagine bumper-to-bumper traffic in such a garage, with every car moving and no one allowed to use brakes. That might be a fair picture of the blood flow in the pulmonary alveolar septa. It is a two dimensional world; hence, we named it a “sheet flow.” Figure 2 shows my interpretation of a scenein a pulmonary alveolar sheetas seenby a red cell. You can seethe garagelike columns and spaces.Red cells are floating into the garage. One on the right is caught by a post. This painting is not very faithful. To clarify the view I have drawn the red cells too small and too few. In reality the cells should be SO big as almost to fill the entire height. An analysis of the sheet-flow idea required a number of preliminary investigations. These may be summarized briefly below : 1. The geometry of the red blood cells.
-1nterferometric measurements. -Wave diffraction and holographic analysis. Microvasc. Res., 1972, 1973.
LANDIS AWARD LECTURE
251
2. How flexible the red blood cells are.
- Theoretical analyses. - Model studies. Fed. Proc., 1966; Biophys. J., 1968. 3. Elasticity of the connective tissues that make up the interalveolar septa’s walls and posts.
- Study of mesentery, skin, ureter, etc. - Concept of the pseudostrain-energy function. p. W = CExp[al eZ,+ a, e: + a4 e, e2 + a5 et,], where p,, is density in reference state, W is strain energy per unit mass, e,, e2 are normal strains, e,, is shear strain, C, aI, a4 are constants. Am. J. Physiol., 1967; ASME, 1966; J. Appl. Physiol., 1968; Am. J. Physiol., 1971; Biomechanics book, 1971; J. Biomech., 1973, 1974; Biorheology. J., 1973. 4. Morphometry of the sheet geometry.
- VSTR (Vascular SpaceTissue Ratio, % vascular spacein sheet) - Average thickness J. Appl. Physiol., 1969; Circ. Res., 1970; Circ. Res., 1972; Microvasc. Res., 1973. 5. Elasticity of the sheet (Fig. 3).
- Thickness varies linearly with respect to blood pressure: h=hO+uAp, where h is the mean sheetthickness, Ap is blood pressureminus alveolar air pressure, ho is the post length at Ap = 0, CIis the sheet-thickness compliance constant. This applies when Ap is positive and not too large. - If Ap is negative, i.e., if the blood pressureis smaller than the alveolar air pressure, then the thickness of the vascular spaceh is effectively zero. -When Ap becomes large, say, above 20 cm H20, the thickness will increase slower with increasing Ap and eventually will tend asymptotically to a constant.
-Dimension in the plane of the sheet is independent of bIood pressure. It is “rigid.” - If a pulmonary capillary is thought of as a tube, then it is linearly elastic in the direction normal to the septum, rigid in the direction parallel to the septum. Circ. Res., 1972. 6. Apparent viscosity of blood in sheetjlow.
- Testing at small Reynolds number. - Pressure-flow relationship is linear. -
f? =
Ylp1asma
(a + bH + cH2),
where q is the apparent viscosity, a, b, c are constants, His the hematocrit. ASME, 1968; Microvasc. Res., 1969;J. FluidMech., 1969; J. AppLPhysiol., 1973. 7. Starling mechanism. Permutt’s waterfall.
- Model testing at high and low Reynolds numbers. - For alveolar flow there was no ffutter. - Can be analyzed statically. Circ. Res., 1972.
252
Y. C. FUNG
Some comments on these items are given below: 1. With the interferometric method the red cell image can be analyzed as the holograph of a phase object according to the principles ofphysical optics (asopposedto geometric optics, which do not apply to such a small object asthe red cell). The sensitivity is then improved to 0.02 pm. Statistics of the geometric characteristics of the red cells are collected. The extreme statistics of Gumbel are applied to answer the question: If you were given a sample of blood containing N red cells, what is the expectedvalue of the diameter of the cell with the largest diameter, what is the expectedvolume of the cell with the largest volume, and what is the expectedvalue of the largest cell area, the largest cell thickness, the corresponding sphericity index, etc. ? These data are useful in making models for model testing and in formulating the stochastic flow behavior in microcirculation.
AF,CAPlLLARY-ALVEOLAR
PRESSURE, cm H,O
FIG. 3.The complianceof the interalveolarseptawith respectto the blood pressure.The local average thicknessof the sheet(of vascularspace)is plotted againstthe differenceof the blood pressureand the alveolar air pressure.
2. It is shown theoretically that becausethe red cell has a biconcave shape, it can have many large deformations without inducing any membrane stressin the cell membrane. These deformations, isochoric (constant volume) and “applicable” (without stretching and tearing in any direction), are unique to the biconcave geometry. “Offdesign” deformation in microcirculation may not be “applicable,” but the stretching of membrane area can be seento be everywhere small. Experiments were made on the flow of liquid-filled thin-walled rubber models of red cells in a cylindrical tube to observe the cell deformation. 3. We need information on the elasticity of the capillary blood vessel wall. Since direct experimentation with the interalveolar septa is difficult, I tested other connective tissues. My 10 years’ experience may be summarized by saying that the connective tissuesare inelastic, but in a very wide range of strain rates (which includes the physiological range), the loading process or the unloading process in any periodic motion may be approximated by an elastic one. After “preconditioning” (establishment of homeostatis), the stress-strain relationship in the loading process can be derived from a pseudo strain-energy function. (The Kirchhoff stress is equal to the partial derivative
ofthe pseudo strain+nergy function with respectto the Lagrangian or Green’s strain.)
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The best form of the strain-energy function per unit volume of the tissue in the original (unstressed)state is, for a two-dimensional specimen,the exponential form given above. This form admits orthotrophy. An unloading processcalls for a different setofconstants. Sometimes it is advantageous to supplement such an exponential function with a polynomial of strain components. 4. The geometry of the sheetmust be known before any analysis of the flow can be made. We measured the VSTR and the sheet thickness in specimensof cats and other animals. 5. To measurethe compliance of the interalveolar septa with respect to changing blood pressure, we perfused lungs with catalized silicone rubber under controlled inflation and perfusion pressures. The lungs were fixed after the polymer hardened Measurements yielded the results outlined above. (SeeFig. 3.) The resistance of sheet thickness to blood pressure changesis derived mainly from the tension in the septa due to inflation of the lung. It is analogous to a musical drum whose pitch depends on the tension in the drumhead. The drumhead deflects linearly with respectto the impact of the hammer, but the constant of proportionality depends on where the hammer strikes. The posts in the alveolar sheet,however, render this linear relationship uniformly valid over the entire sheet! 6. We now must turn our attention to the blood. We must know how the red cell suspensionbehavesin a garagelike sheetflow. For this purpose we pursued both theoretical analysis and model experiments. The theory gives us the parametric relations. The experiments yield the apparent viscosity as a function of hematocrit. 7. Finally, we experimented on the Starling mechanism becauseit was considered by Permutt and his associatesto be a principal feature of the pulmonary alveolar blood flow, particularly in West’szone II, in which the blood pressurein the pulmonary artery is higher than the alveolar air pressure,whereasthat in the pulmonary vein is lower than the air pressure. We show theoretically that the character of the flow depends on the Reynolds number. When the Reynolds number is large, the flow in an elastic vesselis governed by a wave equation. When the Reynolds number is small, it is governed by a diffussion equation. In the pulmonary alveolar septathe Reynolds number is very smalf, in the range of 10d2to 10m4.Theoretically, the flow at such a small Reynolds number is stable. This is verified by a model experiment. We show that whereasthe model flutters at large Reynolds number, it does not flutter when the Reynolds number is small. Starling mechanism operates, but it operates statically. This provides a great simplification to further analysis. Having completed these preliminary studies, we can assemblethe pieces together to look at the sheet flow. Our results may be listed as follows: 8. Governing equation of sheetflow at steady-state. V2h4=0
-
h is thickness. V2 = a2/ax2+ a2/ay2.
Basedon conservation of massand momentum, Linear thickness-pressure relationship, and pressure-flow-thickness feedback. J. Appl. Physiol., 1969; Circ. Res., 1972.
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9. Distributions of velocity, pressure, and thickness within a sheet.
- Solution for any specific boundary conditions can be obtained. J. Appl. Physiol., 1969. 10. Pressure--ow relationship.
- If the flow per unit width of sheetis e, then C!= C-‘Vtr, - h&n), where C is a constant, and h,,, and h,,, are sheetthicknessatthe arteriole and venule, respectively, which are, with pa denoting alveolar air pressure, hart = ho + Npil,, -PAI
ken = ho + a(~ven -PA) Circ. Res., 1972.
Il. Input impedance. - Solution of basic equation V2h4 = 4pfu ah/at, p is blood viscosity,f, is a geometric constant, a is the compliance constant, h is sheet thickness, t is time. Ann. Biomed. Eng., 1972. 12. Fluid transfer to interstitium. - Basic equation is V2h4 = 8pkfKbt(h - h*), where Kbt is permeability constant of blood-tissue interface, h* is a constant related to tissue pressure. Microvasc. Res., 1974.
13. Transit time distribution offlow in sheet. - Convolution with flow in arteries and veins yields result for whole lung. Circ. Res., 1972.
14. Effect of gravity. - Regional blood flow. - Smooth merging of zone 2 and zone 3. Circ. Res., 1972. 15. Pulmonary blood volume.
- Correlates with arteriolar pressure. - Does not correlate with venular pressure. Circ. Res., 1972.
16. Solutesflow in interstitium. - Diffusion of permeable and nonpermeable solutes. - Edema. J. Appl. Mech., 1975.
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17. Indicator dilution in lung. - Flow in sheet with porous walls. - Longitudinal diffusion coefficient is determined. J. Appl. Mech., 1975. 18. Surfactants and stability of alveoli.
- Every alveolar septum is shared by two alveoli. - Pressureload acting on any alveolar septum
