Influence of bubble size and blood perfusion on absorption of gas bubbles in tissues

Respiration Physiology (1969) 7, 111-121; North-Holland Publishing Company, Amsterdam

INFLUENCE OF BUBBLE SIZE AND BLOOD PERFUSION ABSORPTION OF GAS BUBBLES IN TISSUES

ON

HUGH D. VAN LIEW AND MICHAEL P. HLASTALA Department of Physiology, School of Medicine, State University of New York at Buffalo, Bufalo, New York 14214, U.S.A.

Abstract. We have developed a mathematical theory to predict behavior of tissue gas bubbles such as those occurring in decompression sickness. According to the theory, rate of change of diameter of a spherical bubble is proportional to the sum of a) a factor that accounts for the divergence of the paths of molecules as they diffuse outward from the bubble, and b) a factor that accounts for blood perfusion of the tissues around the bubble. For bubbles of radius below 10 microns, the perfusion factor is negligible in comparison to the divergence factor. This finding and assumptions that unsteady-state effects and surface tension are negligible allow the rate of growth or decay of small bubbles to be approximated by a relatively simple differential equation. According to the epuation, rate of radius change is directly proportional to the inert gas partial pressure difference between the inside and outside of the bubble, inversely proportional to the pressure of the gas inside the bubble, and inversely proportional to the bubble radius. Decompression Gas bubbles

sickness

Gas diffusion Perfusion

One of the unique aspects of animal tissue- the perfusion of blood through capillaries - has been ignored in previous mathematical formulations concerning growth or absorption of the bubbles of decompression sickness (BATEMAN, 1951; HARVEY, 1951; HILLS, 1966; NIMMS, 1951; RUFF and M~~LLER, 1966; VAN LIEW, 1967). Diffusion of gas out of decompression bubbles can be expected to differ from diffusion in many commonly-known systems, for there is no discrete barrier across which molecules move from one well-stirred compartment, a source, to another well-stirred compartment, a sink. Instead, gas diffusing through living tissue meets capillaries at various depths. Multiple small sinks for the gas are within the diffusion barrier (VAN LIEW, 1968a). Accepted for publication 7 January 1969. 1 These studies were aided by Contract NOOO14-68-A-0216, (NR 102-722), between the Office of Naval Research, Department of the Navy, and the State University of New York at Buffalo. 111

112

HUGHD. VANLIEWANDMICHAELP. HLASTALA

Experimental data are available for bubble absorption in in vitro systems @ERTZ and LOESCHCKE, 1954; LIEBERMANN,1957; PIIPER, HUMPHREYand RAHN, 1962; WYMANet al., 1952) and for large quantities of gas in contact with perfused tissues (PIIPER,CANFIELDand RAHN, 1962; VANLIEW and PASSKE,1967; VANLIEW, SCHOENFISCHand OLSZOWKA,1968/1969). Unfortunately, it is technically difficult to study the small bubbles in perfused tissue that may be most interesting in decompression sickness (BUCKLES,1968). FUNDAMENTAL CONCEPTS When a spherical bubble serves as a source of gas, two factors can be expected to cause the gas concentration to decrease with distance from the gas-tissue interface : a) Blood leaving capillaries removes gas from the system. According to a recentlydeveloped model (VANLIEW, 1968a, 1968b), the partial pressure of dissolved gas in the tissue is relatively high near the gas source; blood in capillaries tends to equilibrate with the high partial pressure, and each unit of blood carries away a relatively large quantity of gas. At a greater distance from the interface, partial pressure is low because of the gas that has been removed by capillaries which are between the interface and the point in question. b) Partial pressure decreases with distance because of the divergence of radii of the sphere of tissue around the bubble. Fixed increments of distance from the interface represent increasing volumes of tissue so that even if there were no uptake by blood, the concentration of the gas molecules must decrease with distance. In large bubbles, the second factor is unimportant because there is little divergence over the small distance in which diffusion can occur. Conversely, it is shown below that in bubbles the size of small blood vessels, the factor of divergence is so important that the perfusion factor is negligible by comparison. Derivation

The gradient in the tissue around the bubble determines the rate at which gas diffuses out. In the paragraphs below, an equation for the gradient is derived, which is then used to obtain an equation for bubble radius. The radius equation is constituted so that it applies for any degree of compression; its significance is illustrated by showing its relation to equations based on other premises and by graphs showing its evaluation under different conditions of initial size of the bubble and perfusion rate. GRADIENT EQUATION Naturally-occurring bubbles in soft tissue are probably approximately spherical and consist mainly of inert gas, such as Ns or He. Diffusion flux of an inert gas through tissue across the surface of any sphere which is concentric with the center of the bubble is given by Fick’s law in spherical coordinates (assume spherical symmetry): (1)

dV,/dt = - a, DAdP/dr

where V, is volume of the inert gas expressed at standard conditions, ml STPD; t is

EFFECTS OF SIZE AND PERFUSION

ON BUBBLE BESOLUTION

113

time, min; a, is solubility of the gas in tissue at body temperature, mm Hg-’ (ml of gas per ml of tissue per mm Hg); D is diffusivity of the gas in tissue at body temperature, cm2 min- 1; A is surface area of the sphere, cm 2 ; P is partial pressure of the inert gas dissolved in tissue, mm Hg; and r is radius, cm. Consider an infinitesimally small spherical shell of tissue with inner and outer radii of r and r+Ar, respectively. The difference of the gradient of partial pressure per unit distance between the two sides of the shell (the divergence of the gradient) is V’P. In spherical coordinates with spherical symmetry:

(2)

v2

p

=

1

d(r2dP/W

7

dr

*

To facilitate manipulation, restate eq. (2) in terms of the basic definition of a derivative : (r+Ar)‘dP/drl,+,,-r’dP/drl, Ar

(3)

I.

Substitute for dP/dr from eq. (l), with the formula for surface of a sphere in place ofA:

(4)

=

V2P = lim

APOT"

lim

Ar-*0

1

(r+Ar)2dV/dtl,+A, - qD4x(r f Ar)’ Ar

dV/dtl,+,,-dV/dtl, -4nr’Arcr,D

r2dVjdtl, -gD4ar2

1 .

1’

The difference in flux across the r surface and the r + Ar surface (the numerator of the last form of eq. (4)) is the amount of gas absorbed by blood in the shell, (a,kQP’ 4xr2 Ar), where at, is solubility in blood, mm Hg-l ; the product (kQ) is effective blood perfusion, min-’ (ml of blood per ml of tissue per min); Q is actual perfusion, min-’ ; k is a coefficient of end-capillary equilibration (VAN Lmw, 1968a), dimensionless; P’ is difference between P, partial pressure of the inert gas dissolved in the shell and P,, pressure in the incoming arterial blood (P’ =P-P,), mm Hg; and 4nr2 Ar is the volume of the shell, ml. Equation (4) can therefore be rewritten:

(5)

V2P’ = $$P. t

To bring eq. (5) to a convenient form, set va,,kQ/tltD equal to A. (6)

v2P’=12P’.

Equation (6) in spherical coordinates is: (7)

114

HUGH D. VAN LIEW AND MICHAEL P. HLASTALA

A change of variable, P’=u/r, gives an equation that is easily integrated:

(8)

2 - A2u=0.

The general solution for eq. (8) is: (9)

u = P’r =AeLr+Be-lr.

Constants A and B are evaluated by boundary conditions: As r + co, P’ and u approach zero, and therefore A = 0. At r = R, P = PBso P’ = P, - P,, and u = (P, - P,) R. This gives B = (P, - P,) RelR. The solution of eq. (7) is :

(lo)

P=P,+(P,-

P,)

e-‘(r-R) for r>R.

Equation (10) can be rewritten in terms of physiological parameters of the system by replacing R.

(11)

P=P,+(P,-P,)

(F)exp(-(sf(r-R)]for

r>R.

BUBBLE RADIUS EQUATION

The flux of gas out of the bubble can be predicted by evaluating eq. (1) at the gasstisue interface, that is, where r = R. A convenient form of eq. (1) comes from assumptions that the inert gas pressure inside the bubble is in a steady state and that the bubble is spherical: a) replace the volume of inert gas at standard pressure with its volume at ambient pressure (by this means the equation can be used at any degree of compression), b) replace volume of the inert gas with an expression for total volume (vol of inert gas = fraction of inert gas x total vol), and c) state total volume and area of the spherical bubble in terms of R, its radius, to get the final equation: (12)

dR/dt = -cr,D(P,/PJdP/dr,

when r=R.

In eq. (12), P, is standard pressure, 760 mm Hg, and P, has the same meaning as in earlier equations, partial pressure of inert gas in the bubble. The final goal, to predict rate of decrease of bubble radius, can be attained by taking the first derivative of eq. (11) for the condition r = R and substituting it for the dP/dr term of eq. (12) : (13)

dR/dt = (a,DP,)(l-P,/P,)(l/R+il)

where, as before, A = v(abk@l(a,D)

.

The three parentheses of eq. (13) separate different factors which are concerned with bubble resolution; they contain, from left to right, respectively: a) constants or coefficients, b) partial pressure of the inert gas in the bubble and in the blood and

EFFECTSOF SIZEANDPERFUSION ON BUBBLEBBSOLUTION

115

tissue, and c) the reciprocal of the bubble radius plus a factor which contains the square root of the effective blood flow, kQ. Diussion

SPECIALCASES The right-hand parentheses of eq. (13) show that the effects of blood perfusion and geometry (divergence) are additive. The importance of this can be illustrated by showing that three published mathematic treatments for bubble absorption are special cases of eq. (13): a) If perfusion is zero, eq. (13) is essentially the same as the equation derived by EPSTEINand PLESSET(1950) for steady state absorption of bubbles in a nonstirred fluid : (14)

dR/dt=(a,D

P,) (1 -Pa/P,)

(l/R).

b) With a large bubble that approximates a planar interface, the l/R term of eq. (13) falls out as R gets very large. With reintroduction of volume and area as indicators of bubble size, eq. (13) becomes equivalent to the equation used with data obtained from large subcutaneous pockets (VAN LIEW, 1968a): (15)

dV,/dt = Ava,a,DkQ

(PB- P,) .

c) The equation previously presented for absorption of gas from decompression sickness bubbles under conditions of varying pressures and inert gas partial pressures (VAN LIEW, 1967) has the same form as eq. (13) except that the value of (l/R+J) was given as l/L, on the assumption that a nonstirred layer of invariant thickness L exists around a bubble and that there is no divergence effect. The present work and the derivations of EPSTEINand PLESSET(1950) show that if one conceives of a virtual nonstirred layer around small bubbles, it must be considered as varying in thickness. Because of the divergence effect, the thickness of a hypothetical nonstirred layer increases as the bubble radius increases except for very large bubbles in which L is large compared to l/R. EVALUATION Simple calculations of the relative values of l/R and I show that the 1 factor is not important for small bubbles. For Nz gas in human tissues, the value of I may range from 0.3 to 20.0 cm-‘. (High estimate for well-perfused, watery tissue-use a, =a,,, D=.2x10e3 cm’ min-I, and kQ=.l mm-‘; low estimate for poorly-perfused fatty tissue-use at=5ab, D=2 x 10e3 cm2 min-‘, kQ=O.OOl min-‘.) Therefore when A is 20 cm- ‘, 1 and l/R are equal for a bubble with R of .05 cm (500 microns), but for a bubble with R of JO05 cm (5 microns), L is only 1% of the total of the third parenthesis of eq. (13). When Iz is lower than 20 cm- ‘, the bubble size for which it can be considered negligible is greater. The conclusion is that even for very high Iz

116

HUGHD. VANLIEWANDMICHAELP. HLASTALA

p/min

460 Radius

*A0 (RI , microns

12bo

Fig. 1. Rate of change of radius as a function of bubble radius. It is assumed that C%b = at, 1d-2 ml gas@ tissuelatm, k = 1, and D = 1.3 x 1O-s cm*/min. To indicate maximal bubble absorption rate (VAN LIEW, 1967), Pa is assumed to be zero.

values (high blood flows and high ab/cl, ratios), the 1 factor is negligible below radii of 5 microns. This means that blood flow has a negligible direct effect on resolution of bubbles the size of capillaries (radius 5 microns or less). The influence of bubble size on rate of bubble resolution is illustrated in fig. 1, which shows solutions of eq. (13). The three curves represent different perfusion rates, but at the left of the diagram below bubble radius of 100 p, the curves come together; the blood flow effect is negligible and the divergence effect dominates. At the right of the diagram, the curves tend to level off at a value determined by perfusion, since the divergence effect becomes unimportant when radius is large. If Q is zero, dR/dt is always inversely proportional to the radius, and the rate approaches zero for large bubbles. Note that the curve for 0 = .l levels at approximately 2 microns per min; this multiplied by 50 cm2 (surface area of the pockets) yields 1 x 10e2 ml/mm, the observed rate for N, exit from large subcutaneous gas pockets in live rats when there was a 1 atm difference of PNI between the pocket and the tissue (VAN LIEW et al., 1968/1969). Total time required for a bubble to disappear can be obtained from integration of the inverse of eq. (13). (16)

T =

AR,,- ln(1 + lRO) G~~DP,

Figure 2 shows solutions: A bubble of 800 p radius will take 250 min to be completely absorbed if Q is zero, 175 min if Q is 0.1 mm-‘, and 110 min if Q is 1.0 min-‘.

EFFECTS OF SIZE AND PERFUSIONON BUBBLE RESOLUTION

117

300-

: .3 F . F i=

200-

g=o.1

//

I

/

loo-

I

I

f3bO 460 Initial Radius (Ro) , microns

I

Fig. 2. Total time of resolution as a function of initial bubble radius. Curves for three different perfusion rates are shown. Assumptions are the same as in fig. 1.

Curves with a non-zero blood flow approach a constant slope at the right since absorption of large bubbles is governed by perfusion alone. Evaluations of eq. (11) give the profile of P us distance (r - R) out into the tissue. The profile is of interest because by eq. (12) rate of radius change is proportional to the slope of the profile at the gas-tissue interface (where r = R). For a very large bubble R/r approaches unity and eq. (11) approaches as a limit the form: (17)

P=P,+(P,-P,)

exp {-(-$.fx),

where x corresponds to (r - R), distance out from the interface. At any (r - R), the other curves for smaller bubbles are less than the R = co curve by the factor R/r. Profiles are illustrated in fig. 3, which shows 3 sets of curves for bubbles of differenl sizes, and each set shows three different perfusion rates. The three curves for a 10 micron bubble (furthest to the left) illustrate the relative unimportance of blood flow for small bubbles. Consider the three curves for a blood flow of .l ml,/mlu,/min, a reasonable value for well-perfused human tissues. As bubble radius increases, partial pressure at any given distance increases until the limit described by eq. (17) is approached when R becomes very large. Conversely, it is seen that the pressure falls much more rapidly for the smaller bubbles. This is because of the greater divergence of flux lines at low bubble radii. We have indicated above that because perfusion makes little difference when radius is small, the present model, which includes capillary perfusion, approaches the unstirred case of EPSTEIN and PLE~SET (1950). Figure 3 suggests that the present model

118

HUGH

D. VAN LIEW AND MICHAELP. HLASTALA

1

260

’

4&o

’

Distance from Bubble f r - R f

do , microns

Fig. 3. Relative partial pressure of dissolved gas as a function of distance from bubble. Curves for three different bubble sizes and three different perfusion rates are shown. It is assumed that cq, = at, k = 1, and D = 1.3 x 1O-3cm2/min.

also approaches the shell model (VAN LIEW, 1967) for small bubbles. The profiles for bubbles of 10 ~1could be approximated by a straight line which would intersect the axis near the 30 microns that was estimated for the nonstirred shell by workers with in vitro bubbles (HERTZ and LOESCHCKE, 1954; WYMAN et al., 1952). An important aspect of fig. 3 is the demonstration of the distance that the gradient penetrates out into the tissue. Gradient for the smallest bubble shown (radius of 10 p) penetrates out to about 100 microns, which is about 3 intercapillary distances (assume capillaries are 30 p apart), before P is down to 10 % of its initial value. Gas from larger bubbles penetrates farther. When R= 100, the gradient reaches out about 500 ,u, or on the order of 20 intercapillary distances, to bring P down to 10% of its bubble value. These dimensions show that capillaries in a fairly large region will be involved in gas uptake from decompression bubbles. FACTORSWHICH ARE NEGLECTEDIN THE DERIVATION

The demonstration that perfusion is unimportant in absorption or growth of small bubbles suggests that eq. (14) will be a useful approximation formula for prediction of behavior of small decompression sickness bubbles. However, in the derivations we have assumed that the bubbles were in a steady state with regard to a) total pressure in the bubble due to surface tension and tissue elastic pressure, b) partial pressures of the inert gas in the bubble and in the blood and tissues, and c) amount of gas in reversible solution in the gradient. We argue that elastic pressure which can be exerted by most tissues is negligible. Thus bubbles in soft tissues will separate the tissue rather than build up pressure. In the extreme case of bone, a growing bubble can let off excess gas through blood vessel channels. The dissection, swelling and distortion caused by gas bubbles can give rise

EFFECTS OF SIZEANDPERFUSION ON BUBBLEBESOLUTION

119

to pain and symptoms, but it seems unlikely that enough pressure can build up to have an important effect on gas diffusion. The neglect of surface tension may be a more serious oversimplification. Bubbles of 10 micron radius may have excess pressure of 100 mm Hg due to surface tension, and at lower radii, the excess pressure increases rapidly. From an equation which includes surface tension, EPSTEINand PLESSET(19SO)compute that the effect is usually unimpor~nt. In any case, use of eq. (14) will always be conse~ative from the point of prevention or cure of decompression sickness; surface tension will always increase rate of absorption and decrease rate of growth of bubbles. The neglect of several unsteady-state phenomena seems warranted. a) Changes of gradient as bubble radius changes. According to calculations of EPSTEINand PLESSET (1950), effect of a time-dependent term which accounts for the amount of gas which goes reversibly into the tissue immediately around the bubble is negligible so long as the bubble remains in one place and there are no stirring effects other than perfusion through capillaries. b) Changes of the inert gas composition within the bubble due to changes in the other components. Data with large subcutaneous gas pockets in rats showed a transient adjustment period after a change of total pressure which lasted about 60 min (VANLIEW et al., 1965). With bubbles of 1 mm diameter or less a greater area-to-volume ratio should decrease the time of the adjustment period 10 or lOOfold. c) Changes of diffusivity or surface tension due to films of foreign substance that become con~ntrated at the interface between gas and fluid. LIEB~~NN (1957) obtained only a small effect that could be attributed to such films. Whether lipoprotein or other surface active agents could be important in uivo remains to be determined. d) Changes of the gas partial pressure within the tissues brought on by wash-in or washout of gas via the blood. The equations of this paper were developed on the assumption that average partial pressure of inert gas beyond the influence of the bubble was constant and equal to arterial partial pressure, P,. A useful approximation to the case where average tissue partial pressure was changing due to wash-in or washout of gas could be made by replacing P, of eq. 114) with a function of time. Thus in a simple washout situation, the approximate equation for small bubbles analogous to eq. (14) would be:

(18)

dRjdt = (a,DP,) ( 1 -3$.?)(k)

where PO is original gas partial pressure in the tissue far from the bubble, the time constant for washout of the inert gas from the tissue. Note that in text, blood perfusion has an indirect effect on small bubbles, for perfusion a strong influence on the time constant for washout or wash-in. Bubbles may be cylindrical due to their location in blood vessels. The partial pressure profile near the cylindrical surface is: (19)

p=p

a

+ (Pg-PX&r) R&R)

r>R

and 8 is this conrate has resulting

120

HUGH D. VAN LIEW AND MICHAEL P. HLASTALA

where K, (Jr) is the Bessel function of zero order of the second kind. Equation (19) corresponds to the profile equation for spherical bubbles, eq. (11). According to eq. (19) the partial pressure at any radius is slightly greater for the cylindrical bubble than for a spherical bubble because there is only two-dimensional divergence as opposed to three-dimensional divergence for the spherical bubble. Therefore gas efflux per unit of interface area will be less. Further analysis of a cylindrical bubble would require special assumptions because of the complexity of diffusion near the ends of the bubble where homogeneity of tissue cannot be assumed. Finally, it must be pointed out that eq. (14) can be used only for growing or shrinking bubbles in a fixed location. It can be shown by integration of profiles (as in fig. 3), that for small bubbles the amount of gas in solution in the gradient is large compared to the amount in the bubble; therefore, in generation of a bubble where none existed before, in movement of a bubble from one location to another, or in a bubble which has a flow of blood past it, unsteady state effects or stirring effects will have to be considered. Acknowledgement We wish to acknowledge very helpful conversations with Dr. RICHARD G. BUCKLES. References BATEMAN,J. B. (1951). Review of data on value of preoxygenation in prevention of decompression sickness. In: Decompression Sickness, edited by J. F. Fulton. Philadelphia, W. B. Saunders Co., pp. 242-277. BUCKLES, R. G. (1968). The physics of bubble formation and growth. Aerospace Med. 39: 10621069. EPSTEIN, P. S. and M. S. PLESSET(1950). On the stability of gas bubbles in liquid-gas solutions. J. Chem. Physics 18 : 1505-l 509. G~RTZ, K. H. and H. H. L~E~CHCKE(1954). Bestimmung der Diffusion Koeffizienten von HZ, 02, NZ und He in Wasser und Blutserum bei konstantgehaltener Konvection. Nuturforschung 96: l-9. HARVEY,E. N. (1951). Physical factors in bubble formation. In: Decompression Sickness, edited by J. F. Fulton. Philadelphia, W. B. Saunders Co., pp. 90-114. HILLS, B. A. (1966). A thermodynamic and kinetic approach to decompression sickness. Thesis, University of Adelaide, Libraries Board of South Australia. LIEBERMANN,L. (1957). Air bubbles in water. J. Appl. Physics 28: 205-211. NIMMS, L. F. (1951). A physical theory of decompression sickness. In: Decompression Sickness, edited by J. F. Fulton, Philadelphia, W. B. Saunders Co., pp. 192-222. PIIPER, J., R. E. CANFIELDand H. RAHN (1962). Absorption of various inert gases from subcutaneous gas pockets in rats. J. Appl. Physiol. 17: 268-274. PIIPER, J., H. T. HUMPHREYand H. RAHN (1962). Gas composition of pressurized, perfused gas pockets and the fish swim bladder. J. Appl. Physiol. 17: 275-282. RUFF, S. and K. G. M~SLLER(1966). Theorie der Druckfallbeschwerden und ihre Anwendung auf Tauchtabellen. Znt. 2. Angew. Physiol. einschl. Arbeitsphysiol. 23 : 251-292. VAN LIEW, H. D., B. BISHOP, P. WALDER D. and H. RAHN (1965). Effects of compression on composition and absorption of tissue gas pockets. J. Appl. Physiol. 20: 927-933. VAN LIEW, H. D. (1967). Factors in the resolution of tissue gas bubbles. In: Underwater Physiology, edited by C. J. Lambertsen. Baltimore, Williams and Wilkins Co., pp. 191-204.

BFFBCTS OF SIZE AND PBBFTJSION ON BUBBLE BBSOLUTlON VAN Lraw, H. D. and M. PASSKE(1967). Permeation

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of neon, nitrogen and sulfur hexafluoride through walls of subcutaneous gas pockets in rats. Aerospace Med. 38: 829-831. VAN Lrew, H. D. (1968a). Coupling of diffusion and perfusion in gas exit from subcutaneous pocket in rats. Am. J. Physiol. 214: 1176-1185. VAN LIEW, H. D. (1968b). Interaction of CO and OS with hemoglobin in perfused tissue adjacent to gas pockets. Respir. Physiol. 5: 202-210. VAN LIEW, H. D., W. H. SCHOEN~~CH and A. J. OLSZOWKA (1968/1969). Exchanges of Nx between a gas pocket and tissue in a hyperbaric environment. Respir. Physiol. 6: 23-28. WYMAN,J., JR., P. F. SCHOLANDER,G. A. EDWARDS and L. IRVING(1952). On the stability of gas bubbles in sea water. Sears Foundation J. Marine Research 11: 47-62.