Respiration Physiology (1975) 24, 147-158;
ABSORPTION
North-Holland
OF IN VIVO INERT
Publishing
Company,
Amsterdam
GAS BUBBLES’
M. P. HLASTALA and H. D. VAN LIEW Deparrmentsof 98195
Medicine
and Department
and of Physiology and Biophysics, of Physiology,
State
University
University of
New
of
Washington,
York at Buffalo,
Seattle, Buffalo,
Washington, New
York,
14214. U.S.A.
Abstract. The resolution rates of gas bubble-s in perfused tissue are examined using a mathematical model which is consistent with transient-state and steady-state data from large gas pocket experiments. The tissue surrounding the bubble is assumed to have an inlinite number of infinitesimally small capillaries (distributed sink). The time for complete disappearance of a bubble depends on initial bubble size. blood perfusion rate of the tissue, physical characteristics of the inert gas and the fraction (1- Pa/Pg), where Pa and Pg are inert gas partial pressures in the surrounding tissue and inside the bubble, respectively. Calculated time for disappearance of a nitrogen bubble of 1 mm radius in an O,-breathing man is 100-250 min, depending on blood perfusion. Breathing of air increases the time of persistence of the bubble about IO-fold. The ‘transient state’, when gas is dissolving in the immediate surroundings of the bubble, is most influential when there is no perfusion and the partial pressure gradient gas volume is the largest. However, our major conclusion is that the initial transient state has little effect on total lifetime of the bubble in any case. Bubbles Decompression sickness Diffusion Helium
Hydrogen Neon Nitrogen Transient state
Large changes in environmental gas pressure, as in deep-sea diving entail a risk of formation of gas bubbles in the blood and tissues. Diving tables have been designed with the philosophy that proper elimination of inert gas by the lungs will minimize the supersaturation of inert gas in tissue, thus preventing the initial formation of bubbles, but in practice, the philosophical considerations have been overridden by empirical corrections to prevent symptoms. Spencer and Clarke (1972) and Evans Accepted
for
publication
8 March
1975.
studies were supported in part by United States Public Health Service Grant HL-12174 and Contract NC@O14-68-A-0216,(NRIOl-722) between the Office of Naval Research, Department of the Navy, and the State University of New York at Buffalo. ’ These
148
M. P. HLASTALA AND H. D. VAN LIEW
et al. (1972) have demonstrated the presence of venous gas emboli in divers following accepted decompression tables, so it seems that in decompression sickness, the avoidance of bubble formation is not the crucial factor. Presumably, bubbles form wherever dissolved gas supersaturation exceeds some prescribed level (bubble nucleation is of major interest but is not within the scope of the present paper). Intravascular bubbles may be a very common occurrence in routine hyperbaric excursions, and it is probable that extravascular bubbles occur also, although extravascular bubbles have not yet been demonstrated following accepted decompression. Because study of decompression bubbles by direct observation is possible only to a limited degree, it is necessary to characterize the processes of bubble growth or decay by mathematical models. A very important aspect of the characterization of any particular bubble is the convection field in the surroundings. By virtue of the fact that there is a supersaturation, it can be inferred that bubbles form only in tissue where convection occurs by perfusion of blood, or at some location within a reasonable diffusion distance of a gas phase. If the bubble is extravascular, it will grow and resolve in place without moving. A large-sized analog of extravascular decompression bubbles has been investigated by Van Liew (1968a,b) using subcutaneous gas pockets in rats. The gas pocket preparation allows study of diffusion in one direction from a flat surface because the distance of penetration of gas into the tissue is very small compared to the radius of curvature of the gas-tissue interface. The concepts developed from gas pocket experiments were later used to predict the behavior of small spherical bubbles in perfused tissue under steady-state conditions (Van Liew and Hlastala, 1969). For the prediction, the gas is assumed to diffuse through the tissue but also to disappear in an infinite number of inhnitesimally small capillaries (uniformly distributed sink). If the bubble is intravascular, it may grow and resolve in situ, in which case it will be subject to the same boundary conditions as an extravascular bubble. If it is swept away into the blood stream, it will be subject to a forced convection where the diffusion of gas between bubble and blood is assisted by the convective movement of blood past the bubble. In this case the bubble dynamics will follow the predictions of mass transfer between a gas phase and a moving liquid phase (Hlastala and Farhi, 1973; Yang et al.. 1972). If the bubble lodges in a small vessel, any surface adjacent to a moving stream ofblood will follow a forced convection pattern, while the other surfaces will follow a pattern similar to the extravascular bubble. For the previous predictions concerning a bubble in tissue (Van Liew and Hlastala, 1969) it was assumed that the gas content of the immediate surroundings ofthe bubble changed only if bubble size changes. The purpose of the present paper is to extend the theory to include the initial or ‘unsteady-state’ behavior for the simple case of a spherical bubble that suddenly comes into contact with tissue. Recent experimental evidence (Hlastala, 1974) suggests that the model for perfused tissue developed by Van Liew also holds under this transient state condition where gas is going into or out of solution in the tissue near the bubble.
INERT GAS BUBBLES
149
Theory The unique property of a bubble in perfused tissue is that gas diffusing from the bubble through the tissue meets capillaries at various depths. At each capillary, the gas can either continue to diffuse or go into solution in the blood in proportion to its partial pressure and be carried away by perfusion. Thus, the dissolved gas concentration decreases with distance from the gas-tissue interface because of two factors: (a) blood leaving capillaries removes gas from the system (mass consumption) and (b) divergence of flux lines around the bubble (mass diffusion). Our present model could apply to a bubble which is moved intravascularly until it lodges in a vessel surrounded by homogeneous, perfused tissue. The bubble is assumed to be spherical and, in addition to inert gas, contains CO2 and O2 in equilibrium with tissue Pco, and Po2, respectively. Only one inert gas is involved. The subject is breathing the same inert gas as that in the bubble and dissolved in the tissue. At time zero, the inert gas partial pressure in the bubble is higher than that in the tissue. The ‘transient state’ or ‘unsteady state’ describes the initial rapid effiux of gas during the development of the ‘steady-state’ partial pressure profile in the surrounding tissue. According to the principle of mass conservation, the difference between the rates of mass diffusion and mass consumption is equal to the time-rate of change in the amount of locally dissolved gas. The governing equation has been derived elsewhere (Van Liew, 1967) for the steady state. For the unsteady state, the diffusion equation is:
(1)
VZp’-lZp’
1 dP’ = Bat
In eq. (1) P’ is the difference between P, partial pressure of inert gas, at any place in tissue, and Pa, partial pressure of inert gas in the incoming arterial blood, (P’=P-Pa), mm Hg; V2 is the divergence of the gradient; D is the diffusivity of the gas in tissue, cm2/min; rb and sl, are the solubilities of gas in blood and tissue respectively, ml gas/( ml *mm Hg); and the product (kQ) is effective blood perfusion, ml blood/(ml tissue.min) (Van Liew, 1968a). Eq. (1) can be rewritten in spherical coordinates with the assumption of spherical symmetry: (2)
:rJ’ ; f ‘Cj-
jb2p'
=
With the transformations P’=u/r, spherical bubble, eq. (2) becomes: (3)
‘$
-
ju2U
=
A$
$‘C? and t= r- R, where R is the radius of the
M. P.
150
HLASTALA
AND H. D. VAN LIEW
with the boundary conditions: u(
b) If tissue perfusion is zero, the bubble behaves exactly as if it were in an unstirred medium. Eq. (8) becomes the same as that derived by Epstein and Plesset (1950): (11)
$=
-a,DPs(l
- E)(i+
&)
c) If the bubble size is very large so that the surface is essentially flat, the diffusion is like that from a subcutaneous gas pocket. Eq. (8) becomes the same as one previously used (Hlastala, 1974) to describe transient state diffusion of diethyl ether into rat subcutaneous tissue after volume is resubstituted for bubble radius: (12)
g=
-a,DAPs(l
- E)(nerf[J(Dt)*]
+ s)
In a real situation, the resolution dynamics of a decompression sickness bubble will depend on the temporal pattern of the boundary conditions. During inert gas washin or washout caused by changes in breathing mixture or ambient pressure, tissue inert gas will be changing, creating non-constant boundary conditions. We have dealt with the simpler situations of non-changing tissue gas tension to assess the relative importance of the ‘transient state’ and other factors. References Altman, P. L. and D. S. Dittmer, ed. (1971). Respiration and Circulation. Federation of American Societies for Experimental Biology. Bethesda, Md. Crank, J. (1970). The Mathematics of Diffusion. London, Oxford University Press, p. 130. Epstein, P. S. and M. S. Plesset (1950). On the stability of gas bubbles in liquid-gas solutions. .I. Chem Phys. 18: 1505-1509. Evans, A., E. E. P. Barnard and D. N. Walder (1972). Detection of gas bubbles in man at decompression. Aerosp. Med. 43 : 1095-1096. Gertx, K. H. and H. H. Loeschcke (1954). Bestimmung der Diffusions-KoetEzienten von Hz, Os, N, und He in Wasser und Blutserum bei konstant gehaltener Konvektion. Z. Nururforsch. 9b: l-9. Hlastala, M. P. and L. E. Farhi (1973). Absorption of gas bubbles in flowing blood. J. Appl. Physiol. 35: 311-316. Hlastala, M. P. (1974). Transient state diffusion in rat subcutaneous tissue. Aerosp. Med. 45: 269-273. Spencer, M. P. and H. F. Clarke (1972). Pre-cordial monitoring of pulmonary gas embolism and decompression bubbles. Aerosp. Med. 43: 762-767.
158
M. P. HLASTALA AND H. D. VAN LIBW
Van Liew, H. D. (1967). Factors in the resolution of tissue gas bubbles. In: Underwater Physiology, edited by C. J. Lambertsen, Baltimore, Williams & Wilkins Co., pp. 191-204. Van Liew, H. D. and M. Paike (1%7). Permeation of neon, nitrogen and sulfur hexatluoride through walls of subcutaneous gas pockets in rats. Aerosp. Med. 38: 829-831. Van Liew, H. D. (1968a). Coupling of diffusion and perfusion in gas exit from subcutaneous pockets in rats. Am. J. Physiol. 214: 1176-1185. Van Liew, H. D. (1968b). Interaction of CO and 0s with hemoglobin in perfused tissue adjacent to gas pockets. Respir. Physiol. 5: 202-210. Van Liew, H. D. and M. P. Hlastala (1969). Influence of bubble size and blood perfusion on absorption of gas bubbles in tissue. Respir. Physiol. 7: 111-121. Yang, W. J., R. Echigo, D. R. Wotton and J. B. Hwang (1971). Experimental studies of the dissolution of gas bubbles in whole blood and plasma. I. Stationary bubbles. J. Biomech. 4: 275-281. Yang, W. J., R. Echigo, D. R. Wotton, J. W. Ou and J. B. Hwang (1972). Mass transfer from gas bubbles to impinging flow of biological fluids with chemical reaction. Biophys. .I. 12: 1391-1404.
ABSORPTION
North-Holland
OF IN VIVO INERT
Publishing
Company,
Amsterdam
GAS BUBBLES’
M. P. HLASTALA and H. D. VAN LIEW Deparrmentsof 98195
Medicine
and Department
and of Physiology and Biophysics, of Physiology,
State
University
University of
New
of
Washington,
York at Buffalo,
Seattle, Buffalo,
Washington, New
York,
14214. U.S.A.
Abstract. The resolution rates of gas bubble-s in perfused tissue are examined using a mathematical model which is consistent with transient-state and steady-state data from large gas pocket experiments. The tissue surrounding the bubble is assumed to have an inlinite number of infinitesimally small capillaries (distributed sink). The time for complete disappearance of a bubble depends on initial bubble size. blood perfusion rate of the tissue, physical characteristics of the inert gas and the fraction (1- Pa/Pg), where Pa and Pg are inert gas partial pressures in the surrounding tissue and inside the bubble, respectively. Calculated time for disappearance of a nitrogen bubble of 1 mm radius in an O,-breathing man is 100-250 min, depending on blood perfusion. Breathing of air increases the time of persistence of the bubble about IO-fold. The ‘transient state’, when gas is dissolving in the immediate surroundings of the bubble, is most influential when there is no perfusion and the partial pressure gradient gas volume is the largest. However, our major conclusion is that the initial transient state has little effect on total lifetime of the bubble in any case. Bubbles Decompression sickness Diffusion Helium
Hydrogen Neon Nitrogen Transient state
Large changes in environmental gas pressure, as in deep-sea diving entail a risk of formation of gas bubbles in the blood and tissues. Diving tables have been designed with the philosophy that proper elimination of inert gas by the lungs will minimize the supersaturation of inert gas in tissue, thus preventing the initial formation of bubbles, but in practice, the philosophical considerations have been overridden by empirical corrections to prevent symptoms. Spencer and Clarke (1972) and Evans Accepted
for
publication
8 March
1975.
studies were supported in part by United States Public Health Service Grant HL-12174 and Contract NC@O14-68-A-0216,(NRIOl-722) between the Office of Naval Research, Department of the Navy, and the State University of New York at Buffalo. ’ These
148
M. P. HLASTALA AND H. D. VAN LIEW
et al. (1972) have demonstrated the presence of venous gas emboli in divers following accepted decompression tables, so it seems that in decompression sickness, the avoidance of bubble formation is not the crucial factor. Presumably, bubbles form wherever dissolved gas supersaturation exceeds some prescribed level (bubble nucleation is of major interest but is not within the scope of the present paper). Intravascular bubbles may be a very common occurrence in routine hyperbaric excursions, and it is probable that extravascular bubbles occur also, although extravascular bubbles have not yet been demonstrated following accepted decompression. Because study of decompression bubbles by direct observation is possible only to a limited degree, it is necessary to characterize the processes of bubble growth or decay by mathematical models. A very important aspect of the characterization of any particular bubble is the convection field in the surroundings. By virtue of the fact that there is a supersaturation, it can be inferred that bubbles form only in tissue where convection occurs by perfusion of blood, or at some location within a reasonable diffusion distance of a gas phase. If the bubble is extravascular, it will grow and resolve in place without moving. A large-sized analog of extravascular decompression bubbles has been investigated by Van Liew (1968a,b) using subcutaneous gas pockets in rats. The gas pocket preparation allows study of diffusion in one direction from a flat surface because the distance of penetration of gas into the tissue is very small compared to the radius of curvature of the gas-tissue interface. The concepts developed from gas pocket experiments were later used to predict the behavior of small spherical bubbles in perfused tissue under steady-state conditions (Van Liew and Hlastala, 1969). For the prediction, the gas is assumed to diffuse through the tissue but also to disappear in an infinite number of inhnitesimally small capillaries (uniformly distributed sink). If the bubble is intravascular, it may grow and resolve in situ, in which case it will be subject to the same boundary conditions as an extravascular bubble. If it is swept away into the blood stream, it will be subject to a forced convection where the diffusion of gas between bubble and blood is assisted by the convective movement of blood past the bubble. In this case the bubble dynamics will follow the predictions of mass transfer between a gas phase and a moving liquid phase (Hlastala and Farhi, 1973; Yang et al.. 1972). If the bubble lodges in a small vessel, any surface adjacent to a moving stream ofblood will follow a forced convection pattern, while the other surfaces will follow a pattern similar to the extravascular bubble. For the previous predictions concerning a bubble in tissue (Van Liew and Hlastala, 1969) it was assumed that the gas content of the immediate surroundings ofthe bubble changed only if bubble size changes. The purpose of the present paper is to extend the theory to include the initial or ‘unsteady-state’ behavior for the simple case of a spherical bubble that suddenly comes into contact with tissue. Recent experimental evidence (Hlastala, 1974) suggests that the model for perfused tissue developed by Van Liew also holds under this transient state condition where gas is going into or out of solution in the tissue near the bubble.
INERT GAS BUBBLES
149
Theory The unique property of a bubble in perfused tissue is that gas diffusing from the bubble through the tissue meets capillaries at various depths. At each capillary, the gas can either continue to diffuse or go into solution in the blood in proportion to its partial pressure and be carried away by perfusion. Thus, the dissolved gas concentration decreases with distance from the gas-tissue interface because of two factors: (a) blood leaving capillaries removes gas from the system (mass consumption) and (b) divergence of flux lines around the bubble (mass diffusion). Our present model could apply to a bubble which is moved intravascularly until it lodges in a vessel surrounded by homogeneous, perfused tissue. The bubble is assumed to be spherical and, in addition to inert gas, contains CO2 and O2 in equilibrium with tissue Pco, and Po2, respectively. Only one inert gas is involved. The subject is breathing the same inert gas as that in the bubble and dissolved in the tissue. At time zero, the inert gas partial pressure in the bubble is higher than that in the tissue. The ‘transient state’ or ‘unsteady state’ describes the initial rapid effiux of gas during the development of the ‘steady-state’ partial pressure profile in the surrounding tissue. According to the principle of mass conservation, the difference between the rates of mass diffusion and mass consumption is equal to the time-rate of change in the amount of locally dissolved gas. The governing equation has been derived elsewhere (Van Liew, 1967) for the steady state. For the unsteady state, the diffusion equation is:
(1)
VZp’-lZp’
1 dP’ = Bat
In eq. (1) P’ is the difference between P, partial pressure of inert gas, at any place in tissue, and Pa, partial pressure of inert gas in the incoming arterial blood, (P’=P-Pa), mm Hg; V2 is the divergence of the gradient; D is the diffusivity of the gas in tissue, cm2/min; rb and sl, are the solubilities of gas in blood and tissue respectively, ml gas/( ml *mm Hg); and the product (kQ) is effective blood perfusion, ml blood/(ml tissue.min) (Van Liew, 1968a). Eq. (1) can be rewritten in spherical coordinates with the assumption of spherical symmetry: (2)
:rJ’ ; f ‘Cj-
jb2p'
=
With the transformations P’=u/r, spherical bubble, eq. (2) becomes: (3)
‘$
-
ju2U
=
A$
$‘C? and t= r- R, where R is the radius of the
M. P.
150
HLASTALA
AND H. D. VAN LIEW
with the boundary conditions: u(
b) If tissue perfusion is zero, the bubble behaves exactly as if it were in an unstirred medium. Eq. (8) becomes the same as that derived by Epstein and Plesset (1950): (11)
$=
-a,DPs(l
- E)(i+
&)
c) If the bubble size is very large so that the surface is essentially flat, the diffusion is like that from a subcutaneous gas pocket. Eq. (8) becomes the same as one previously used (Hlastala, 1974) to describe transient state diffusion of diethyl ether into rat subcutaneous tissue after volume is resubstituted for bubble radius: (12)
g=
-a,DAPs(l
- E)(nerf[J(Dt)*]
+ s)
In a real situation, the resolution dynamics of a decompression sickness bubble will depend on the temporal pattern of the boundary conditions. During inert gas washin or washout caused by changes in breathing mixture or ambient pressure, tissue inert gas will be changing, creating non-constant boundary conditions. We have dealt with the simpler situations of non-changing tissue gas tension to assess the relative importance of the ‘transient state’ and other factors. References Altman, P. L. and D. S. Dittmer, ed. (1971). Respiration and Circulation. Federation of American Societies for Experimental Biology. Bethesda, Md. Crank, J. (1970). The Mathematics of Diffusion. London, Oxford University Press, p. 130. Epstein, P. S. and M. S. Plesset (1950). On the stability of gas bubbles in liquid-gas solutions. .I. Chem Phys. 18: 1505-1509. Evans, A., E. E. P. Barnard and D. N. Walder (1972). Detection of gas bubbles in man at decompression. Aerosp. Med. 43 : 1095-1096. Gertx, K. H. and H. H. Loeschcke (1954). Bestimmung der Diffusions-KoetEzienten von Hz, Os, N, und He in Wasser und Blutserum bei konstant gehaltener Konvektion. Z. Nururforsch. 9b: l-9. Hlastala, M. P. and L. E. Farhi (1973). Absorption of gas bubbles in flowing blood. J. Appl. Physiol. 35: 311-316. Hlastala, M. P. (1974). Transient state diffusion in rat subcutaneous tissue. Aerosp. Med. 45: 269-273. Spencer, M. P. and H. F. Clarke (1972). Pre-cordial monitoring of pulmonary gas embolism and decompression bubbles. Aerosp. Med. 43: 762-767.
158
M. P. HLASTALA AND H. D. VAN LIBW
Van Liew, H. D. (1967). Factors in the resolution of tissue gas bubbles. In: Underwater Physiology, edited by C. J. Lambertsen, Baltimore, Williams & Wilkins Co., pp. 191-204. Van Liew, H. D. and M. Paike (1%7). Permeation of neon, nitrogen and sulfur hexatluoride through walls of subcutaneous gas pockets in rats. Aerosp. Med. 38: 829-831. Van Liew, H. D. (1968a). Coupling of diffusion and perfusion in gas exit from subcutaneous pockets in rats. Am. J. Physiol. 214: 1176-1185. Van Liew, H. D. (1968b). Interaction of CO and 0s with hemoglobin in perfused tissue adjacent to gas pockets. Respir. Physiol. 5: 202-210. Van Liew, H. D. and M. P. Hlastala (1969). Influence of bubble size and blood perfusion on absorption of gas bubbles in tissue. Respir. Physiol. 7: 111-121. Yang, W. J., R. Echigo, D. R. Wotton and J. B. Hwang (1971). Experimental studies of the dissolution of gas bubbles in whole blood and plasma. I. Stationary bubbles. J. Biomech. 4: 275-281. Yang, W. J., R. Echigo, D. R. Wotton, J. W. Ou and J. B. Hwang (1972). Mass transfer from gas bubbles to impinging flow of biological fluids with chemical reaction. Biophys. .I. 12: 1391-1404.
