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J. Biomechonics Vol. 12, pp. 373 -395. Pergamon Press. 1979. Printed in Great Britain.
BLOOD FLOW IN THE LUNG*?
R. COLLINS and J. A. MACCARIO University of Compitgne. BP 233. 60206 Compiigne, France
Abstract - Pulmonary hemodynamics is studied in terms of the quasi one-dimensional unsteady nonlinear fluid flow equations which are applied to the 40-odd generations of branched arterial, capillary and venous distensible vessel segments making up the four lobes of the complete lung. An idealized pressure-area “tube law” is introduced which provides for varying degrees of vessel collapse. The model predictions agree well with experimental measurements of flow transmission as a function of pulsatile frequency. Pulmonary response is represented schematically in terms of an influence diagram. Pressure pulses are shown to increase in amplitude in the early arterial segments, with the greatest drop occurring across the capillary bed.
1. INTRODUCTION
The lung constitutes a highly complex and self- regulating system for oxygenating man’s blood and removing its waste materials. It is at the alveolar level that the respiratory and circulatory functions interact and the important exchange processes occur. Much research has been done on both aspects of pulmonary function by physiologists, medical clinicians and ap- plied mathematicians. It is evident from this work that mechanical principles play a very important role. Neural control is nonetheless present, although less prominent than in the systemic circulation. Its direct effect may enter, however, through a readjustment of the mechanical variables of the system. It is particularly the circulatory aspects of pul- monary function which are of concern in the present investigation. We address ourselves here to the general question of how the lung might adapt to external influences, caused for example, by cardiac dysfunction (mitral stenosis, left-to-right shunts), vascular obstruc- tions (emboli transported to the lungs from the systemic veins), changes in altitude (alterations in alveolar pressure in mountain-climbers and deep-sea divers) and physical work and exercise. A good quantitative understanding of pulmonary response is still lacking to this date, although a number of very imaginative experiments have been undertaken to document this behaviour. Less work has been done on the unification of this data in the form of a global quantitative model of the complete circulatory system of the lung on the basis of classical mechanical principles. Some may be of the opinion that such an undertak- ing is premature. tndeed, very little detailed and utilizable information is available for the material properties on which the results of such an analysis must depend. Direct measurements of in viva dimen- sions and mechanical properties of the intricate branching network have been accomplished only for
- Received 12 September 1978. t Reprints from Prof. R. Collins, 130 Blvd des Eta&-Unis, 60200 CompiBgne, France.
the main pulmonary artery and its early branches. Beyond, this point, the calibre of the vessels rapidly diminishes. Nor can the vascular beds be laid out in thin sheets so that their in-uiuo behaviour may be examined dynamically under a microscope. The ana- tomy of the rabbit’s ear may readily lend itself to such experimentation, but the lung is highly three- dimensional in structure. Partial circumvention of this difficulty is possible by injection into the blood-stream of a substance which is at first convected and then solidifies within the blood vessels, hopefully without changing their dimensions. Measurements are then made directly on the cast of the pulmonary tree of the sacrificed animal. Similar difficulties apply to measurements of trans- mural pressure beyond the first and last few gene- rations. Reliance on venous “wedge” pressures, ob- tained by occluding the vessel to the point of arresting the flow, is less than satisfying, for some doubt always remains about their correct interpretation. Flow measurements in the interior of the lung have been carried out recording the radiation from gaseous radioactive tracers, such as xenon-133, which may be injected into the blood-stream and detected by an external counter. One thus obtains an integrated measure across a slice of the lung. In spite of this somewhat pessimistic picture, useful estimates have been made in the intermediate regions of the lung, which serve as a starting point for a mathematical analysis. One may cite the now classical studies of lung morphology by Weibel(1963) and the more recent findings of Cummings rr al. (1969). However the results of Wiener et ul. (1966) offer the considerable advantage of accompanying their esti- mations of vessel dimensions by values of the cor- responding vessel compliances. Useful measurements of pressure and blood flow in different regions of the lung under varying conditions of flow pulsatility have been published by Attinger (1963), West et al. (1964) and Maloney er al. (1968). Mathematical modelling of the pulmonary circu- lation has been attempted at various levels of detail. Global response has been simulated by a great many investigators (for example, Rideout and Katra, 1969)
373
(^314) R. COLLINSand J. A. MACCARIO
on the basis of the analogy of the linearized equations of motion with an electrical transmission line. The notions of resistance or impedance connected with this simplified formulation have found great popularity amongst physiologists who find such physically in- tuitiveconcepts useful in their quest for an understand- ing of the fundamental underlying mechanisms of pulmonary behaviour. One must nonetheless proceed with caution in interpreting the results of such for- mulations which often do not deal adequately with important intrinsically nonlinear characteristics such as wave steepening and hence the growth of pressure amplitudes observed along the arterial segments of the circulatory system. A more realistic approach requires retaining the important nonlinear inertial terms in the fluid equa- tions of motion. The resulting system is of second order, and hyperbolic in character, provided that the transmural pressure is considered to be a function of local cross-sectional area only. It may be solved numerically by finite differences or by the method of characteristics. The latter technique has been em- ployed by Anliker et al. (1971) for the systemic circulation. Since it is difficult to include the complete systemic circulatory system in such calculations, pro- vision must be made for outflows from the model system, by means of continuous or distributed sinks. A similar procedure has been adopted here, with the added advantage that the complete pulmonary ana- tomy may be inciuded in the model, thus obviating the need for such pre-specified sink terms. It has long been recognized that distensible blood vessels may collapse, at least partially, under physi- ological flow conditions. Such vessels become appro- ximately elliptical in cross-section, the ratio of major- to-minor axes ranging in dogs from 1.25 in the main pulmonary artery to 1.91 within the next five arterial generations, These measurements of Attinger (1963) correspond to equivalent cross-sectional area ratios of 0.72-0.82 between elliptic and circular configurations. Attinger concludes that these ratios at the major branch points are considerably less than those postu- lated for optimal energy transfer. One is thus led to surmise that collapsible vessels, which disadvantage the efficient transmission of energy, may be playing another role, for which that is the price. The suggestion that the purpose of collapsible pulmonary capillaries may be to control the flow, as a ‘Starling” resistor, was first introduced by Permutt et al. (1962). They regarded such collapsible vessels, subjected to upstream arterial, downstream venous and surrounding alveolar pressures, as sensitive sluice- gates which open or close in response to these three forces. West et al. (1964) have extended this idea to define three control-zones in the lung:
upper lung: zone 1: arterial < alveolar > venous pressures - totally collapsed intermediate lung: zone 2: arterial > alveolar > venous pressures - collapsed at distal end
lower lung: zone 3: arterial > venous > alveolar pressures - fully open.
At the bottom of the lung (zone 3) where the pulmonary vessels are completely open, the flow rate depends classically upon the arterial and venous pressure difference. However the situation is rather novel in the intermediate zone 2, where the flow rate is controlled by the arterial-alveolar pressure difference, independently of the value of the venous pressure, which is less than the alveolar pressure. Permutt refers to this as the “waterfall effect”, an obvious and graphic analogy which has been generally confirmed by experiment. However, the model must be tempered somewhat by realizing that zones 1,2 and 3 merge gradually into one another, during the interplay of reflecting pressure waves within a branching system ofdistensible tubes of varying compliance. Furthermore, a “surfactant” al- veolar lining which lowers surface tension, may keep capillaries open even for slightly negative values of the transmural pressure (Bruderman et al., 1964). A rigorous analysis of the dynamics of collapsible tubes has not yet appeared in the literature, although a first attempt in this direction has been made by Collins (1978). A recent steady-state analysis by Shapiro (1977) is ofconsiderable interest, although the “sonic” type singularities inherent in that solution may possibly not be appropriate to pulsatile flows which appear to be free of the restrictive compatibility conditions invoked for steady flow. We will conclude this brief and not exhaustive summary of previously proposed mathematical mod- els of the pulmonary circulation by mentioning the very interesting “sheet flow” concept introduced by ‘Fung and Sobin (1969) to describe blood flow in the pulmonary capillary bed. The steady-state Hele-Shaw flow about the “posts” enclosed in the “sandwich” model of the idealized pulmonary vascular bed de- pends upon maintenance of very low flow Reynolds numbers for the absence of non-stationary wall flutter effects. The important question here is whether capillary vessels open and close in response to the instantaneous values of transmural pressure alone, or whether there exists a spectrum of finite opening times for collapsible vessels, as proposed by Maloney et al. (1968a). From the more general coupled solution of the fluid and wall equations (Tedgui and Collins, 1978), in lieu of the usual practice of introducing a pre-specified pressure-area law, it would appear that the Reynolds number may not constitute the only criterion by which dynamic effects are to be estimated. In addition, the residual axial wall tension and the distribution of longitudinal wall curvature may cause the vessel to “vibrate” in a manner reminiscent of a stretched violin string. But this complex behaviour will not be de- veloped further here. In the following sections, the morphology of the lung will be modelled in a discretized form, and flow
316 R. COLLINSand J. A. MACCARIO
Table 1. Idealized segments of equivalent lower right lobe on basis of Wiener et al. (1966) data Wiener data for lower right lobe Idealized segments of equivalent conduit
Generation Number of number Length Cross-section Compliance Length Cross-section Compliance vessels G L(cm x lo-‘) S(cm’) a(cm&/dyn x 106) L(cm x 10-l) S(cm*) a(cm4/dyn x 106) N
16.85 0. 13.65 0. 11.06 0. 8.956 0. 1.255 0. 5.877 1. 4.761 1. 3.856 2. 3.124 2. 2.530 3. 2.050 4. 1.660 5. 1.345 6. 1.090 8. 0.883 10. 0.715 13. 0.579 17. 0.469 22. 0.380 28.
16.85 0. 13.65 0. 11.06 0. 8.956 0. f3.1320 1.
18.0975 23
11.7410 1.9996 24.4898 130
7.5850 4.5623 37.3947 14.
4.1160 14.5623 67.4230 46.10’
EPary 22 ::
0.740 71.40 87. 0.810 290.60 373. 0.660 98.80 214.
8.503 1
32.70 218. 25.54 180. 19.95 159.
2.2100 159.92 230.23 29.10”
25 26 27 3.5439 19.1150 152.81^ 66. 28 B 29 3 30 E B
31 32 2 33 8 34 !Z 35 Lx 36 Z $ 3738 Z 39
7.1986 6.3882 76.033 29.10’
11.1440 2.7240 44.196 260
12.4664 1.4818^ 29.984^45 8.503 1 1.0284 23.761^16 10.4976 0.8032 20.276 8 12.9600 0.6274^ 17.305^4 16.0000 0.4900 14.767^2
40 41 42 Main arterial segments: pulmonary artery: L = 2.3, S = 1.33, a = 40.923 x 106; left lobar artery : L =^ 1.4,^ S = 0.60, a = 18.462^ x 106; right lobar artery: L = 2.08, S = 0.60, a = 18.462 x 106.
(1970) suggest that the most likely configuration is arterial and venous vessels respectively were main- characterized by a branching ratio (number of daugh- tained^ in^ their^ original^ configurations,^ while^ the ter vessels arising on average from each parent) of complete^ vascular^ bed was absorbed^ into^ an individual 3.26 for the right lung and 3.50 for the left. In order to segment^ of equivalent^ total^ volume.^ Table^ 1 shows^ a formulate an efficient computational procedure which typical^ restructuring^ of the^ Wiener^ data^ for the^ lower still retains the salient mechanical features of the right^ lobe^ in^ terms^ of^ the^ lengths^ L, cross-sectional pulmonary circulation, successive groups of compara- areas^ S,^ compliances^ CLand^ number^ of^ vessels^ N ble generations (i.e. of similar lengths and cross- comprising^ each^ idealized^ segment^ of the^ equivalent sections) were gathered into equivalent segments of a tube.^ Segments^ in the^ left lung^ were^ taken^ to be 30% single conduit, one for each ofthe 4 lobes. Only the first narrower than in their right-hand counterpart, rather and last few generations of the larger pulmonary than the 50% differences occasionally evident in the
Blood flow in the lung 377
data of Wiener et al. (1966). It will be shown later that the distal pressure and flow profiles in the pulmonary vein are not overly sensitive to such anomalies in an individual lobe, as they are largely compensated by confluent flows in the remainder of the pulmonary circulation. In view of the limitations of this observed pulmonary response and the rather imprecise nature of the morphological data currently available, the pre- sently proposed discretized lobular configuration would appear justifiable. The distribution of vessel lengths and cross-sections from one generation to the next may be characterized reasonably well in the form of a geometric progression. Attinger (1963) proposes common ratios for the ar- terial and venous segments of the order of 0.8 and l/0.8, respectively, while the variation of the cumulative arterial vessel cross-sections by generation has been characterized by common ratios ranging from 1.1 to 1.28 by Caro et al. (1965), Cumming et al. (1969) and Wiener et (I/. (1966). It is most important that the final equivalent “tube” configuration so developed does not contain abrupt changes in cross-sectional area or large divergence angles which could precipitate a spurious separation of the flow. Ofthe remaining two aspects of the characterization of the model lung, the pressure-area law is described in the next section, followed by the pressure and flow profiles required as boundary conditions to the numerical computation.
- MATHEMATICAL FORMULATION
Each of the four pulmonary lobes is discretized in the form of a single equivalent elastic conduit of non- uniform cross-section, by grouping successive gene- rations of branching vessels into uniform segments as described in the previous section. Care must be taken to avoid spurious flow separation between adjoining segments by limiting the axial variation of cross- sectional areas in the idealized model. With this provision, the discretized morphology can lead to a considerable economy in computational effort.
3.1 Gorerning equarions of motion
The quasi one-dimensional unsteady equations of motion for flow of a viscous incompressible fluid in a deformable conduit of varying cross-section S may be expressed as
s.5 + a(vs) o at ir,= ’
au ’
-+i(;+; at (^) 1
-F=O,
where u is the blood velocity averaged over the local cross-section, p, the locally-averaged transmural pre- ssure, p, the blood density, and F, the friction factor accounting for viscous drag between the blood and the vessel wall. Clearly this friction factor must be based upon the true vessel dimensions, and not upon the size
of the equivalent lumped conduit. The one-dimensional friction factor F is defined in terms of the shear stress T at the wail as
F=L,. pr
with T = Cr. Jpc’Re-“, (3.3)
where Re is the Reynolds number based on the true vessel radius r. For the laminar flow considered here, the parameters CJ (skin friction) and m take on the values (Kivity and Collins, 1974) m = 1. C, = 8. In terms of the true vessel cross-section .4 (= trr’). the friction factor is expressed as
and acts in a direction opposing fluid motion. If at a particular station, N such vessels of cross-section A have been combined to form the corresponding equivalent conduit of cross-section S, then F may be re-expressed in terms of the equivalent tube as
As particular vessels approach a progressively col- lapsed state during the cardiac cycle, relation (3.4) becomes modified toward an inverse quadratic law F - l/A’. However, since only a fraction of the true vessels making up the “equivalent” tube configuration will be grossly non-circular at any instant, relation (3.5) still remains a reasonable representation of the friction factor for the quasi one-dimensional flows studied here. For a system of elastic vessels, the transmural pressure can be expressed as a function of the single argument S. (Within this same condition, an appro- ximate provision may be made for vessel collapse in the event of negative transmural pressures). The resulting system of second-order linear differential equations is hyperbolic and may be re-cast into characteristic form
dpkpc’dv= &r; (3.6)
along the respective characteristic directions
dx dt = v * (‘,
where the signal propagation velocity
with
c2 =//s^ ‘as P’ **?P**
I- = pc8nN cdt. P
It is noted here that a more generalized (viscoelastic) pressure-area law for the equivalent conduit would entrain higher derivatives, invalidating a solution by the method of characteristics. These points have
Blood flow in the lung (^319)
x a-a-II- (^) b-@
Fig. 4. Characteristics net for typical interior points. Fig.^ 5.^ Characteristics^ netsegments.^ at^ junction^ between^ two^ vessel
behaviour, corresponding to pulsatile flow at different cardiac frequencies. Slightly different numerical procedures are adopted for the bifurcation points within the branching net- work of vessels. 3.2.1 Irlterior points. The solution at a new time t, (= t + At) and position x3 is computed from the known solution at time t(= tl = tz) by determining two points .Y~and .Y~(by iteration) whose characterrs- tics intersect 3 at the pre-specified time increment Ar (Fig. 4). For these three points, equation (3.6) become
cp3 - pl) + pc,(a, - u,) = -rF
3 (3.11)
In terms of T, = p1 + pc,ul and T, = pz - pc2uz, and after elimination of v3 from (3.11) the pressure at point 3 becomes
which may be solved iteratively, starting from an initial estimate of p3 and use of (3.9) or (3.10). Knowing the converged value of S3, the solution for vL)is obtained directly by eliminating p3 from equations (3.11) in the form
T, - 7-
2’3= ,c, + pc, + 2I-IS,’
3.2.2 Junction points. At the junction between seg- ments, one may encounter a jump in the cross- sectional area, to which the solution is very sensitive. It is partly for this reason that S, was introduced into the right-hand members of the difference equations (3.11). One must distinguish here between stations distal (‘) and proximal (“) to the junction (Fig. 5). In terms of the C + characteristic emanating from the proximal segment a, and the C- characteristic from the distal segment b, the difference relations take on the form :
(P; - PI) + PC,@4 - u,) = -r,:
1:; (Pi - PI) + PC,(C’; - az) = l-2 F (3.14) 3
Continuity of pressure and flow rate are expressed respectively as Pi = P’; (3.15) and 1.& = L’3s,. (3.16)
It is evident from (3.16) that a discontinuity in cross- sectional area at the junction between two discretized segments will lead to a similar jump in the flow velocity. Although such discontinuities indeed occur at physiological bifurcations of individual blood vessels, the effect may however be amplified in the idealized conduit segments, in which several generations have been combined. The solution of the system of difference equations at a junction point follows in much the same manner as described above for interior points, without any particular difficulty. 3.2.3 Ordering of computations. The slope of the .individual characteristic curves varies by a factor of approximately five within the forty-odd generations of arterial and venous branchings, attaining its highest values within the capillary bed. This feature con- siderably complicates the internal “accounting” sys- tem, by which stations within and at the extremities of vessel segments must be identified in space and time. In particular, pressure and flow rates at the junctions of adjoining segments must be matched at identical instants of time. A system of fixed-time increments is most suitable for satisfying these requirements, and is achieved by transposing flow values to the desired time levels through interpolation along therespectivecharacteris- tic directions. Since the characteristic network in certain regions advances five times more iapidly than in others. the “valleys” thus created in the com- putational net must be progressively filled in the slower wave-speed regions, before advancing the rapid zones. Without entering into the tedious details of the programming necessary to advance the flow variables in time throughout the complete pulmonary network, one may summarize the ordered procedure as in Fig. 3. As a result of the repeated interpolations required to transpose flow variables calculated at the intersections
380 R.^ COLLINSand^ J. A.^ MACCARIO
1 Settime,stepAt)
I Set LEVEL= I
-segment junction -network end
Compute u,p by
Reset u,ppt interior pants of segment and rezone If necessary
-branch pant -segment junction -network end
Compute next two levels of charocterashcs and advance time step
I
U,P calculated over complete network? I
7
4% Compute^ u, p
: Print results I
Fig. 6. Computational flow chart.
of characteristic curves to the desired fixed-time intervals, the computational net may progressively skew to the left or right. A rezoning in the spatial co- ordinate becomes useful when the ratio between the largest and small Ax increments exceeds a value of two. A detailed flow diagram representing the overall programmed computational procedures appears in Fig. 6.
4. VALIDATIONOF THE MODEL
Wiener et al. (1966) furnish pressure and flow profiles which have been measured at the proximal and distal extremities of the lung, and calculated (by linear theory) at intermediate generations. A judicious choice must be made in validating the numerical predictions of the present model with those data, which are not fully satisfactory for this purpose. In effect, the distal flow profiles (Fig. 7) reported by Wiener et (11.(1966)
I I 0.2 0.4^ 0. f, set Fig. 7. Measured flow profiles (Wiener et ol., 1966). A: pulmonary artery. B: left atrium.
382 R.^ COLLINS^ and J. A.^ MACCARIO
Fig. 10. Computed pulmonary pressure and flow fields for validation with measured data of Wiener er nl. (1966). Explanation of corresponding stations is provided in Section 5.
represent the combined output from the four lobes. We surprisingly free of the local oscillations (Fig. 9) have assigned to each lobe a fraction of the total associated with the corresponding data of Morkin et outflow in proportion to the distal venous cross- al. (1965), and one may suspect that the pulmonary section corresponding to each respective lobe. Fur- (^) venous pressures reported by Wiener et al. (1966) thermore, Wiener et al. (1966)report absolute values of (^) would better represent catheter measurements at a the intra-luminal pressures (Fig. S), but neglect to position clos.er to the left atrium, where a certain specify the extravascular levels which would permit degree of damping would have taken place. It must one to ascertain the required transmural pressures. further be assumed that the morphological data of The pressure profiles for the pulmonary vein are Wiener et u[. (1966) (Table I) refer to the right lower
Blood flow in the lung 383
t, set
Fig. 11. Computed proximal pressure profile (solid curve) using measured flow boundary conditions of Wiener et al. (1966) and comparison with their measured proximal pressure profile (dashed curve).
lobe, on the basis of the associated number of gene- rations reported. For these reasons, it is not completely clear, for the purposes of validation, whether more confidence should be placed in the pressure boundary conditions, from which the flow profiles may be calculated and compared, or vice versa. A number of cases have been computed correspond- ing to both these alternatives, for each of which the relative roles played by fluid viscosity and vessel wall collapse have been assessed. One such comparison is given in Fig. 10, for which the reduced vessel calibre of Wiener et al. (1966) was preserved. The boundary conditions imposed at the proximal and distal ends were those of flow rate, proportioned to the lower right lobe, and modified by the measurements of Morkin et al. (1965). The computed pressure profiles which result are seen to be in good qualitative agreement with the measurements of Wiener et al. (1966), once converted to their transmural counterpart (Fig. 11). The r6le of fluid viscosity (taken as 4 centipoise) appears to be essential in these calculations. In the absence of viscosity, incoherent non-cyclic oscillations develop in the calculation, partly as a result of the cumulative undamped effects of wave reflections at the junctions between adjoining vessel segments. The influence of vessel collapse cannot be ade- quately evaluated, however, for the Wiener et al. (1966) data from which one deduces transmural pressure levels which remain essentially positive. Testing of the provisions for vessel collapse in the present model must therefore be reserved for the flow distribution experiments of Maloney et al. (196Q which will be discussed in the following section.
- COMPUTATIONAL RESULTS
Following the validation in the previous section of the mathematical model by comparison with the published results of Wiener et al. (1966), we are now in a position to examine the behaviour of the solution in response to selected variations in the following four factors: (a) pulsatile frequency (at proximal end). (b) vessel wall compliance, (c) extent of vessel collapsi- bility and (d) absolute pressure level and effect of reversal of pressure gradient. Fourteen complete cases have been computed for the pressure and flow variations in the lower right lobe (typical of the four pulmonary lobes) and may be summarized as in Table 2. Of these, six have been selected to illustrate some physical characteristics of the pulmonary circulation. In these numerical results, the boundary conditions selected conform very closely with those applied in the experiments of Maloney et al. (1968). The distal end is maintained at a constant pressure level, while a sinusoidal pressurevariation is applied at the proximal extremity at frequencies varying from 0 to 5 Hz. Complete pressure and velocity fields are then com- puted. Slight incompatibilities in initial conditions are rapidly “corrected” as the solutions evolve toward a cyclic behaviour. As noted in Table 2, the 14 cases studied have been equally divided between purely distended and collapsible tubes; the former with differing values of the wall compliance. The effect of reversed flow perfusion is examined for collapsible tubes, and discussed in the framework of published experimental results. The graphs displayed in Figs. 12-17 have been plotted automatically in the computer. The left half portrays the pressure profiles in space and time as indicated by the three-dimensional surface plot, fol- lowed by three rows of p-t curves; the first row corresponding to three equi-distant stations in the arterial portion of the lobe, the second to the entrance to the precapillaries. capillaries and exit from the post- capillary vessels respectively, while in the third row are represented profiles at three equi-distant stations in the venous portion. The bottom two curves represent. respectively, the proximal and distal boundary con- ditions on the pressure. In Figs. 12-14, this second curves reduces to the abscissa p = 0; whereas thedistal pressure is maintained at a constant non-zero value (after an initial rapid ramp rise) for Figs. 15- 17. The right half of each figure depicts similar curves for the flow rate. Since each individual figure has been magnified to different degrees in order to till its respective square, the scales may vary amongst them. Nonetheless, the following constant intervals (in- dicated on each coordinate axis) are common to all figures and are sufficient to quantify the graphical results: time interval 0.2 set, pressure interval 2500 dyn/cm’, and^ volume^ flow rate^ interval^ 25cm3/sec. The boundary conditions will be chosen in order to permit direct comparison with the experimental
Blood flow in the lung 385
n
L I 1
Fig. 13. Computed pulmonary pressure and flow fields, without collapse, at 3 Hz (case 4, Table 2).
intermediate flow field, one must rely upon realistic computations which account for a variation in vessel compliance as one progresses from the pulmonary artery toward the capillary bed. As was pointed out earlier, this variation is quite significant, resulting, in the present calculations, in differences in wave speed of a factor of five within the pulmonary circuit. These results furthermore confirm the widely accep- ted hypothesis that the most significant portion of the pressure drop occurs in traversing the capillary bed.
The greatly attenuated pressure nonetheless still con- serves its pulsatilecharacter as it enters the pulmonary venous network: the abrupt drop in pressure at the capillary level is most striking in the three-dimensional surface plot of p(x,t) in the top left corner of this sequence of figures. The computed flow wave forms at the proximal and distal extremities (right-hand half of Fig. 12) indicate a time delay cf = 3 Hz) of approximately 0.1 set for a flow disturbance to traverse the complete pulmonary
R. COLLINSand J. A. MACCARIO
Fig. 14. Computed pulmonary pressure and flow fields, with doubled compliance, at 3 Hz (case 7, Table 2).
lobe, in agreement with Morkin et al. (1965). This also forms become distorted in relation to the proximal corresponds, for a transit distance of about 35 cm, to a wave shapes, due to the intervening regions of partial velocity of 350 cm/set, in excellent agreement with the collapse and wave reflections. Flow oscillations are measurements of Attinger (1963). In all figures for practically unattenuated in the pulmonary arterial forward flow in the lung, the similarity between system, but become partially damped as the blood proximal pressure and flow-rate wave forms persists. enters the capillary bed. This behaviour is in complete This is not surprising, since in the absence of collapse qualitative agreement with the observations of At- (the onset of which is delayed at the proximal end tinger (1963) for the inflated and deflated lung (cf. his where the transmural pressure is at its highest level), an (^) Figs. 8 and 9 for pulmonary venous wedge pressure). elastic relation holds. However, the distal flow wave The oscillations are again reinforced as the flow
(^388) R. COLLINS and J. A. MACCARIO
L
Fig. 16. Computed pulmonary pressure and flow fields, with collapse, at 3 Hz (case 11, Table 2).
very wide variation in flow transmission which results (Fig. 18b). Halving the compliance appears to render the vessels quasi-rigid, resulting in an almost 100% transmission of the blood flow from the pulmonary artery to the distal left atrium with virtually no losses (although viscous dissipation and a slight drop in stagnation pressure due to intermittent vortex for- mation at branch points may cause some attenuation). On the other hand, a general doubling of the vessel wall compliance leads to a marked drop in flow
transmission to a level of 20% as energy is dissipated in increased wall motion. (Attenuation in the localized capillary bed itself is, however, leas affected by changes in compliance, since the natural pressure drop there is already much greater.) But the effect of wall collapsibility will be seen to be even more significant! During collapse, the vessel momentarily assumes a high “effective compliance”. Cases 8-11 (Table 2) have been computed with a provision for vesselcohapse in a region extending from
Blood flow in the lung 389
Fig. 17. Computed pulmonary pressure and flow fields, with retrograde flow at 3 Hz (case 13, Table 2).
the pre-capillaries to the left atrium for a range of pulsatile frequencies. Results of cases 9 (1 Hz) and 11 (3 Hz) are shown in Figs. 15 and 16, respectively. The proximal (pulmonary artery) transmural pressure was maintained at zero, upon which was superimposed a fluctuating sinusoi’dal component of amplitude 5000 dyn/cm’, while the distal end was maintained at a constant level of - 10,000 dyn/cm’. Comparisons of Figs. 12 and 15 for a pulsatile flow at 1 Hz confirm that time-dependent pressure and flow oscillations are
smoothed out under conditions of partial collapse. The marked decrease in flow transmission for the col- lapsible network at 1 Hz is further corroborated in Figs. 14 and 15 for pulsatileflow at a frequency of 3 Hz. The results are general, and the variation of flow transmission with frequency for a collapsible pul- monary network is traced in Fig. 18(c), alongside the corresponding experimental results (Fig. 186) of Maloney et al. (1968), which were carried out with boundary conditions corresponding to those utilized
Blood flow in the lung^391
experiments that pressure transmission is attenuated with increases in pulsatile frequency, in a manner similar to that of flow transmission. Dawson et al. (1973) have presented a grossly- lumped model of the pulmonary vasculature consist- ing of collapsible parallel units, each made up of Starling resistors in series. They state that their numerical values were not chosen to correspond to physiological values, and that the simplified Starling resistor model “does not adequately handle the differences between forward and retrograde perfusion in the isolated lung” which their results would imply.
are fully open, and slightly distended. No backflow is evident here throughout the pulsatile cycle. (The situation may be likened to that of section F of Maloney et ul., Fig. 8.) The peak value of flow is the same in both our cases, under the influence of identical pressure gradients during the peak pressure phase.
In view of these shortcomings, and the indications of the present analysis. there would appear to be no sound basis for concluding that forward and reverse pulmonary flows should be significantly different. An interesting set of experiments aimed at determin- mg the distribution of blood flow in a vertical lung and its variation with pulsatile frequency has been de- scribed by Maloney et al. (1968a). An isolated dog’s lung was subjected successively to a pulsatile flow and pressure, at different frequencies between 0.03 and 2.3 Hz. superimposed on a steady-state level. The pulmonary blood flow was measured by a technique using injected radioactive xenon-133 which, due to its low solubility in blood, is convected into the alveoli as a gas. presumably in proportion to the local value of the blood flow rate. The investigators detected an augmentation of the flow about the height (h,, say) in the lung at which they estimate that the pulmonary vessels are just on the verge of collapsing (i.e. local blood pressure equals alveolar pressure). This flow “excess” (relative to the flow level existing in the absence of the pulsatile component of the input flow and pressure) decays as one moves above and below the it,,, level. Furthermore, the amplitude of this excess was found to decrease as the frequency increased up to 3 Hz. At these higher frequencies, the flow distribution was found to approach that of steady perfusion (with the exception of a localized region at the base of the lung in which this tendency was reversed).
It is clear that the net flow per cycle decreases in absolute value as one moves up a height of 1Ocm between these two levels in the lung and approaches the collapsed state. This is indeed what one might intuitively expect. The point being made by Maloney et al. (1968a) is that in spite of this decrease in absolute flow level, an “excess” exists relative to the correspond- ing steady state. Their interesting thesis to explain this depends essentially upon the assumption that “in such a system no backflow will occur when the arterial pressure is less than the alveolar pressure. because the vessel will collapse”. In fact, the conditions necessary for vessel collapse are not so clear cut. A detailed analysis (Tedgui and Collins, 1978) of the dynamics of collapsible vessels shows that the conditions for collapse can be com- puted on the basis of the instantaneous cross-section, the local variation of longitudinal curvature of the vessel. and its residual tension. in addition to the temporal variation of transmural pressure. One may conclude from that investigation that pulmonary blood vessels may indeed fluctuate between a state of full distension and various degrees of partial collapse. without ever closing completely. The “valvular mechanism” suggested by Maloney et al. (1968a) to explain flow excess, has intuitive appeal, but the highly complex and interactive nature of pulsatile flows through collapsible biological vessels subjected to longitudinal tension may call for some caution. The preliminary analysis developed here would appear to indicate that quantitative solutions are well within the realm of possible efficient com- putational procedures.
Although we have not yet made a complete study of this phenomenon. the results of our cases 9 and 12 (Table 2) depicted in Figs. 15 and 13 respectively, presently permit a restricted basis ofcomparison. Only absolute values of the flow rate can be evaluated here. as no computations were made for the steady flow case which has no physiological significance. We recall that the only difference between the boundary conditions of cases 9 and I2 is that for the latter, the background pressure has been raised by the equivalent of about 10 cm HzO. In other words. the results of case 12 (Fig.
- can be taken to correspond to a horizontal section of lung lying 1Ocm below that of case 9 (Fig. 15).
Finally, we note the influence of vessel collapse on the frequency at which maximum fluctuations in the flow rate occur. The maximum and minimum values of the flow rate at the inlet and outlet, respectively, of the pulmonary network appear in Fig. 19 as a function of pulsatile frequency. In the presence of vessel collapse. there is a marked resonance-like behaviour at a frequency of 2 Hz, very close to the natural cardiac frequency of the dog. If the provision for vessel collapse is suppressed in the model, these fluctuations appear to shift their maxima to a frequency of 3 Hz.
6. DISCUSSION
Under these conditions, it is noted that the upper The foregoing results have confirmed that dynamic section (Fig. 15) lies in a partially-collapsed state collapse of the pulmonary vessels is the key factor (corresponding to section D in Fig. 8 of Maloney et al., controlling the flow transmission characteristics in 1968a). Back-flow occurs since the vessels in the the lung. Its influence far outshadows variations pro- capillary region have not fully collapsed. However, for duced by physiological changes in the blood viscosity, the lower pulmonary section of our Fig. 12, the vessels pressure gradient and overall wall elasticity.
392 R. COLLINS and J. A. MACCARIO
1. Hz f, Hz Fig. 19. Maximum and minimum values offlow rate Q at the inlet (0) and outlet (0) to the lung as a function offrequency for (a) collapsible pulmonary vessels (b) no provision for collapse.
In the interest of simplicity, only an approximate treatment of collapse has been suggested here. The principal feature of the present formulation hinges on the pre-assumed pressure-area law of Fig. 2. This relationship incorporates the similarity ana- lysis of Flaherty et al. (1972) which is valid in the region near complete collapse and has been verified by experiments (Shapiro, 1977) on thin-walled latex tub- ing. Its genera1 configuration would be expected to conform reasonably well to the tube law for physio- logical vessels. The slope of the linearized segments of the pressure-area curve (Fig. 2) represents the wall com- pliance, and is inversely proportional to the cor- responding modulus of elasticity. The value of this compliance increases dramatically and abruptly as the initially distended vessel collapses, but the vessel subsequently re-assumes its lower compliance upon re-inflation. It is precisely this effective and sudden “switching” between high and low levels of “in- stantaneous elasticity” which has been shown here to account for the rapid decay in flow transmission observed by Maloney et al. (1968b). Permanently elevated values of compliance are neither effective nor realistic in explaining this attenuation with increasing frequency. Nor can the significant flow attenuation observed in the venous segments be well-reproduced in the computations without invoking vessel collapse. These results would tend to confirm the conclusions of Permutt and Riley (1963) on the relative importance of recruitment of collapsed vessels, as opposed to the further distension of open vessels.
6.1 Influence diagram Drawing upon numerous sets of laboratory and clinical observations and measurements, it is useful to summarize pictorially, by means of an “influence diagram”, the system of causal relations linking the various physical parameters of the pulmonary circu- lation. Such an attempt is represented in Fig. 20, in which the arrows indicate the direction from cause to effect. Solid lines denote positive influences, i.e. changes in the cause and effect parameters have the same sign; whereas dashed lines denote negative influences, implying changes with opposite sign. Most external influences are seen to act upon the pulmonary arterial pressure, which in turn controls the wall compliance via the position of the blood vessels in the pressure-area plane of Fig. 2. For example, on exercise, the pulmonary artery pressure increases (West et al., 1964), thus opening (or recruiting) par- tially closed vessels. The flow field is displaced toward the right of Fig. 2 implying decreased values of the compliance (a + al). Pressure and blood flow trans- mission are augmented, thus effectively enhancing the oxygenation capacity of the lung in the face of the external demand, by momentarily opening vessels normally in a partially collapsed state, and thereby extending the regions of effective blood perfusion and gas exchange within the lung. A similar response is elicited in the presence of a pulmonary occlusion or obstruction of the lumen, and the often associated condition of pulmonary hypertension. West et al. (1964) have pointed out that flow distribution in the lung can be completely accounted for by the mechani-
