AMERICAN
JOURNAL
OF PHYSIOLOGY Printed
Vat. 229, No. 2, August 1975.
in U.S.A.
Interaction of 0, diffusion and 0, metabolism in cat urinary bladder tissue HUGH D. VAN LIEW AND PAUL-YUAN CHEN Department of Physiology, State Uniuersi~ of New York School 03’ Medicine,
VAN LIEW, Hucw D., AND PAUL-YUAN CWEN. Interaction of Q2 d@usiun and 02 metabolism in cut urinary bladder tissue. Am. J. Physiol. 1975.-We measured oxygen exchange across 229(Z) : 444-448. the inside surface of excised urinary bladders that were inflated with gas mixtures. By using a range- of PO 2 differences between the
two sides of the tissue and relatively
simple mathematical
models,
we could infer the exchanges across the outside surface total 02 economy of the tissue so that we could evaluate action between diffusion and metabolic 02 consumption. mate of the Gogh diffusion constant or permeation
and the the interThe esti-
with this preparation
coefficient avoids errors due to 02 consumption and
unstirred layers. At 37”C, the value of bladder tissue 02 consumpand the value of the Gogh constant tion was 4,4 X IO-” min-l for 02 was 2.22 X low5 cm2 min-l atm? diffusion oxygen
coefficient;
consumption;
excised Warburg
tissue; Gogh respirometer
diffusion
constant;
MANY CLASSICAL TREATMENTS of diffusion focus on one process, the dispersion of a material throughout some limited space (e.g., 3). H owcver, in biological applications, there are almost always at least two time-requiring processes occurring at once. In oxygen utilization of living tissue, even in vitro, diffusion occurs simultaneously with oxygen consumption. The situation is even more complicated in vivo, as in the Krogh model of a capillary domain (1 I), because the additional time-requiring process of blood perfusion is coupled to the diffusion and metabolism processes. ‘The strategy that has been used to understand interactions between two or more ongoing processes is to set up a differential equation based on the principle of conservation of mass on the molecular, or differential scale, and then generalize to a macroscopic, observable scale by mathematical integration. Greven (6) took advantage of this approach to provide estimates of diffusion coefficients by allowing 02 consumption of the tissue to establish the concentration gradient for diffusion. By treating the interaction of 02 diffusion and 02 reaction with hemoglobin as a two-process system, Kreuzer and Hoofd (9) have provided an analysis of facilitated O2 diffusion that is more satisfying than earlier explanations The differential equation approach was used by Ganfield, Nair, and Whalen (5) to explain their findings with microelectrode analyses of Pop versus distance in excised brain tissue and by Van Liew (16) to deal with diffusion and blood perfusion in gas exit from subcutaneous gas pockets in rats. The aims of the present communication a,re: a) to explore h verge simple two-process situation-02 diffusion and 02
Bufalo,
New York I4214
consumption are studied in a mass of excised tissue in various gaseous 02 environments-and 6) to measure directly the Krogh diffusion constant for 02 through tissue at 37°C without the errors that would be incurred if metabolism were not taken into account. METHODS
Our experimental preparation has the advantage that gases are measured directly and that there are no bathing liquids that could lead to unstirred layers next to the tissue (2, 17). Male cats (1.3-5.1 kg) were killed by an overdose of sodium pentobarbital. The urinary bladders were excised, washed with saline, inflated with 12-24 ml of various gas mixtures, and suspended inside a moist chamber at 37”C, with either room air or pure O2 outside the bladder. We took no special precautions to rid the tissue of blood. Gas was added or removed from the bladder with syringes by way of a stopcock connected to a multiholed tube tied into the bladder cavity through the urethra. Gas samples were analyzed with a Scholander apparatus (12) and volumes were measured with a calibrated syringe. Measurements were started at least 10 min after introduction of a new gas mixture to assure a steady state of diffusion. The amount of Oa contained in the bladder (frbm 02 concentration times total gas volume) was measured before and after a timed period. Difference in O2 amount during the period divided by time was the estimate of exchange rate, v, either into or out of the bladder. Surfqce area, A, for each exchange rate measurement was estimated from V, the gas volume contained in the bladder, by the empirical formula (17) : A = 4.84 V213 + 4. Thickness of the bladder for each measurement was estimated by dividing weight of the tissue by area times density. Timed periods were l-2 h. Average 02 partial pressure inside during a timed period was taken as the arithmetic average of the initial and final values. Volumes were reduced to standard temperature and pressure dry, and partial pressures were expressed as fractions of a standard atmosphere, 760 Torr. Average values of bladder parameters are presented in Table 1, A. RESULTS
Figure 1 shows data obtained when the bladders were suspended in air. Each exit-rate value is normalized by multiplication by the quantity (thickness/area) for that particular measurement. The abscissa is Paz inside the
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Oa DIFFUSION
VS.
METABOLISM
IN
EXCISED
bladder. There is a significant O2 exit from the bladder when Paz inside is 0.2 atm, the approximate POT outside in the air. Figure 2 shows data from two types of experiment. a) The group of points to the right (from APO* of -0.2 to +0.6 atm) are the Fig. 1 data replotted, but with the abscissa given as Paz difference between inside and outside. Three data points from the extreme left of Fig. 1 were deleted on the grounds that at low Po2 it is likely that part of the tissue may be anoxic (see DISCUSSION). b) The points at the left in which (from APoz of -0.9 to -0.4) are from experiments pure O2 was outside the bladder. These latter points appear as negative APoz values, because inside Paz was less than outside and with one exception as negative 02 exit since 02 was entering the bladder cavity. DISCUSSION
Our estimates of the Krogh diffusion constant, the O2 consumption of the tissue, and depth of penetration of O2 from air into the metabolizing bladder tissue appear in Table 1, B. The rationale and method by which they were calculated are in the sections that follow. Our Table 1 value of the Krogh constant for 02, 2.22 X 10m5 cm2 min-l atm-l is greater than most of the values from various sources in the handbook compilation of Bartels (1) and is almost twice that calculated by Ganfield, Nair, and Whalen (5) for cat cerebral cortex at 37°C. Our value
A) Average values of measured parameters Weight of bladder, g Surface area of inflated bladder, cm2 Thickness of bJadder wall, cm B) Estimates based on exchange data (Figs. 1 and 2) and measured parameters (above) 02 consumption of tissue, min+ Gogh diffusion constant (K), cm2 min-’ atm-l Depth of penetration of 02 from room air (PO2 = 147 torr), cm
445
TISSUE
2.45 35.7 0.066
4.37 2.22
x
m-3
x
10-s
,045
A
2.0
A 1.5 cm* mtn-l 1.0
0.5
FIG. 1. Exit rate of 02 from bladders suspended in air vs. PQ inside. Exit rate is expressed as volume (STPD) multiplied by thickness and divided by area. Use of this normalization factor, X/A, serves two purposes: it decreases variability due to differences in size and volume of bladders and, as shown in DISCUSSION, slope of data plotted this way can be used to estimate Krogh diffusion constant.
l
l
x 1o-5 l l*,.& rp
1.5
!
-.8
-.6
l 42
l 5’. +e%a
l
cm2 min-’
g,;
l
/
l o
l
l
-.
2
.6
d PO, , atm
FIG. 2. Data from experiments with bladders suspended in air (right side of diagram) plotted together with bladders suspended in pure 02 (left side). Ordinate: same as in Fig. 1; abscissa: 02 partial pressure difference, inside minus outside.
is a little less than that measured by Grote (7) for rat lung tissue at 37”C, 2.5 X W5, and is also less than the 2.45 X 10m5 and 2.7 X 10d5 for heart and brain tissue, respectively, recalculated from earlier measurements (8, 15) on the basis of better values for O2 diffusion in water and temperature dependence for 02 diffusion in lung tissue (7) m Digusion and metabolism combined. Consider a surface that is parallel to the inside and outside surfaces of the bladder and at a variable distance, x, from the inside. Our three basic assumptions are that the rate of gas diffusion across this surface equals the rate of diffusion across the inside surface minus the amount that has been consumed by metabolism, that diffusion across the surface at x follows Fick’s second law, and that there is no net diffusion in directions parallel to the inside surface. In the equations that follow, dV/dt is volume of gas per minute across the plane at x; v is volume per minute across the inside surface; 4 is metabolic rate per unit of tissue volume; A is surface area (assumed to be the same at any x); K is the Krogh diffusion constant; P is partial pressure at depth x; and Pi is the 02 partial pressure inside the bladder cavity. The basic statement is: dV/dt
w
2.0x 1o-5
ox
= J? -
PAX
= -
KA(dP/dx)
(4
The two right-hand parts of equation I can be integrated to give vx - ;iAx2/2 = - KAP + C. The constant of integration C can be eliminated by the condition that when x = 0, P = Pi, so that: VX
-
qAx2/2
= + KA(Pi
-
P)
(2)
If the tissue metabolism were zero (4 = 0), equation 2 would be the usual form of Fick’s law that is used with permeation of gases through inert membranes or of inert gases through tissues (1, 2). When q is not zero, three situations must be considered. First, 02 may not penetrate to all regions of the tissue, so that Paz is zero beyond some depth and the amount of 02 that diffuses out of the cavity is subsequently completely consumed by the inside layers of the bladder. Boundary conditions in this case are that when x = x’, P = 0 where x’ is some depth beyond which there is no more Ozl and
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446 v = PAX’.
H.
Substitution
of these into equation2 gives: (3)
Equation 3 for this case of an anoxic core in the tissue has been used by Greven (6) in study of tissue slices in respirometers and by Van Liew (16) in study of 02 diffusion out of subcutaneous gas pockets of freshly killed rats. It is apparent from equation 3 that flux, ii/A, is dependent on the square root of Pi. There are no terms for partial pressure outside and thickness of the bladder, since the inside is independent of the outside because of the anoxic zone of variable size inside the tissue. The second possibility is that PO:! inside the bladder is so high that none of the 02 consumption of the tissue comes from the outside environment. Boundary conditions are that when x = X, P = P,, where X is total thickness of the bladder and P, is partial pressure in the outside environment. This condition yields : vX/A
= K(Pi
-
PO> + 4x2/2
(4)
D. VAN
LLEW
AND
P.-Y.
CHEN
suggests that the assumptions are warranted at least as first approximations. The slope of the least-squares line in Fig. 2 is the desired K value for Table 1, B. Standard error of the slope is 0.07 X 10m5 cm2 min-l atm? According to equation 4, they intercept is 4X2/2, from which the ;1 value of Table 1, B was calculated with the use of an average thickness of 0.066 cm (Table 1, A). Total O2 economy of excised bladder. Our data, although exhibiting considerable scatter, are compatible with the mathematic models of equations 3 and 4. We will develop the models further to illustrate the total 02 economy of a bladder when the bladder is suspended in air and inflated with gas mixtures having different concentrations of 02* Results of such an experiment are shown in Fig. 1. Figure 3 shows schematically the 02 exchanges of a bladder that has characteristics of our average data in Table 1. The only curve in Fig. 3 starts at the origin (0) and then gives way to a dashed curve. It represents the O2 exit from the bladder when there is an anoxic core; the equation for this curve is obtained by multiplying both sides of equation 3 by X. The upper diagonal heavy line segment ad represents 02 exit when there is no anoxic core by equation 4. At point a, when Paz is about 0.05 atm, the curve and the diagonal line meet. Therefore at P&s above that at point a, there is no anoxic core. The exit of O2 from the gas-filled bladder is always caused by a diffusion gradient in the tissue, dP/dx. Since the curve Oa and line segment ad rise with increasing POT, the dP/dx must also increase with PO*. At the left, O2 consumption of the tissue was the only factor causing the dP/dx, whereas to
In equation 4 there are terms for the outside partial pressure and total thickness because the whole of the tissue is involved. If all 02 consumption is from outside, rather than the inside, the equation is the same, except that i and o can be reversed. Equation 4 describes a straight line with the Krogh constant K for slope and 4X2/2 as intercept. The third possibility is that part of the metabolized 0, comes from nside and part from outside. Equation 4 applies in this case also, as can be seen by adding together two equations that are similar to equation 4, one for 02 diffusion from inside the bladder and one for diffusion from the outside. In such a treatment, one assumes that there is a demarcation plane at XII, where both dV/dt and dP/dx are zero; a variable P” at x” is the boundary condition for diffusion in each direction, and v from inside plus ii’ from outside add to give ;IAX, the total 02 consumption of the whole tissue. The data in Fig. 2 are plotted in the form of equation 4. In the experiments in which pure 02 was outside (at the left of Fig. Z), any 02 appearing in the bladder cavity must have penetrated all the way through, so there is no question of an anoxic core. The Oz-outside points follow the same trend as the points in which the bladder was in air (at the right), indicating that only in the three very low Pi points that were deleted for Fig. 2 (3 points at extreme left of Fig. 1) was there any possibility of an anoxic core. Many assumptions are involved in application of the mathematical treatment above to our data: that bladder tissue is homogeneous and isotropic with regard to both diffusion and O2 consumption, that 02 consumption is independent of PoZ except that q is zero when PoZ is zero, FIG. 3. Schematic representation of 02 economy of an “average” excised bladder suspended in air. Plot is same as in Fig. 1 and upper that diffusion can be regarded as a one-dimensional process heavy oblique Iine segment ad and solid curve Oa represent the 02 that even though the bladder is essentially spherical, that an Xeaves the bladder cavity (data of Fig. 1). Lower oblique line segment inflated bladder has uni .form thickness, and that ou .r measbe and short horizontal line segment represent amount of 02 that uremen ts approximate steady states even though various penetrates through to the air outside the bladder. When Paz inside is parameters are changing slowly. These assumptions are below 0.62 atm, this is negative since 02 enters the tissue from the air. Distance between the 2 heavy lines is proportional to 02 consumption surely not all valid in an absolute sense. For example, tissue of tissue, which is constant except at low Paz, where there is an anoxic is not homogeneous with respect to O2 consumption, which core. The 2 light line segments illustrate errors in estimate of K that is known to be localized in mitochondria. However, the could be incurred if metabohsm of tissue were not accounted for (see agreement of the data with the expectations from the theory text).
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O2 DIFFUSION
VS. METABOLISM
IN EXCISED
447
TISSUE
the right of point a, the gradient due to O2 consumption was modified when inside Paz was great enough to cause the 02 penetrating from inside to meet that from outside. The dashed continuation of the Oa curve shows what the 02 exit rate would have been if 02 from the outside had not had an effect to the right of point a (as could have been the case if the outside were Nz rather than air). The upper solid line segment and curve Oad corresponds to the data of Fig. 1. The second heavy diagonal line segment, be, joined to the horizontal line segment at the bottom of the graph are drawn to show O2 exit from the bladder tissue into the outside air. We could have obtained such data the inflated bladder in an air-filled vessel so bY enclosing tha t we could measure changes of O2 n the ou tside environment. According to line be, when Paz inside the bladder is above 0.62 atm, the bladder gives off O2 to the air because 02 is permeating all the way through. When Paz inside is low, there is movement of 02 from the air into the bladder. At Paz of 0.62 atm, there is no net entrance or exit of 02 from the outside; 02 from the inside is sufficient to furnish the 02 for metabolism of the entire tissue. When Pea = 0.2, so that Pea is equal inside and outside, half the metabolism of the tissue is supplied by the inside and half by the outside, as shown by the equidistance of lines ad and be from the axis. Point a, the critical Pea for development of an anoxic core, is below 0.2 atm, so there is a finite Pea at all points through the tissue when there is air both inside and outside. The total 02 consumed by the bladder tissue is proportional to the vertical distance between lines ad and be in Fig. 3. When Paz is below the Paz at points a and 6, the Oa consumption decreases because of the anoxic core. On the horizontal line to the left of b, 02 entering the tissue from outside reaches its maximum depth. Although the size of the anoxic core increases as inside Paz decreases, the increase is all due to decreasing O2 penetration from the inside. To the left of b, penetration from outside is constant because Pea outside is constant. One can calculate the depth of penetration of 02 from air with vX/A from b to the axis and 6 The result is 0.045 cm, shown in Table 1. This indicates that excised tissue, or nonperfused tissue open to the air during surgery, can obtain O2 from the air for a depth of about half a millimeter, provided the diffusion and metabolic characteristics are the same as in these bladders. If the temperature were lower, the upper heavy curve and line segment Oad would be changed in four ways: a) The curve Oa would rise less rapidly because of change in equation 3 due to the effect of lower 4 at lower temperature and to a minor extent, the temperature effect on K. b) The slope of ad would be slightly changed due to temperature effect on K. c) The vertical displacement of the ad away from point c would decrease because of the lower value of the intercept 4X2/2 of equation 4. d) Point a would move downward and to the left until the metabolic rate was so low that influx from outside air could nourish the whole tissue. In that situation, the line ad would pass through the origin or through the abscissa; there would be no curved portion Oa and no anoxic core. Error due to neglect of metabolism. The light line segment cd
in Fig. 3 shows the error tha t might be incurred if metabolic O2 consumption were not accounted for. If 02 exit rate were to be measured by putting pure O2 on one side of a membrane and air on the other side, one might divide the normalized exit rate at d by the presumed difference, P, This would markedly overestimate the value of K, Pd as shown by the difference of slope cd versus the correct slope of ad. On the other hand, if only the appearance of O2 on the outside were measured, one would find an exchange rate shown as point e, and the slope ce would underestimate the proper K by a factor of about 2. Data for an inert membrane having the same K as the live tissue would fall on a line that would be parallel to the two heavy diagonal lines but would pass through point c. Since data for metabolizing tissue are displaced from the inert membrane line by the quantity 4X2/2, the error in K due to neglect of metabolic rate increases as metabolism increases and increases as the square of the thickness. through tissues by Krogh ( 10) studied 02 penetration measuring O2 appearance in deoxygenated hemoglobin and therefore apparently made an error such as that illustrated by line ce+ However, his error was probably small because he used thin tissue pieces and worked at low temperatures that would keep metabolism low. Takahashi and Fatt (14) avoided error due to 02 consumption in measurements of diffusion in the cornea of the eye by inhibiting metabolism by cold and by dehydration and rehydration of the tissue. Thews (15) and Grote (7, 8), in measuring 02 diffusion through tissue slices held in place by plastic membranes, apparently did not incur an error even though their mathematical treatment does not account for 02 consumption+ They found no change in their results when the tissue was poisoned by cyanide. Probably they disrupted the metabolic machinery by their routine procedure of freezing the tissue to facilitate slicing. Our previous work (17) showed that when cat bladders were frozen and rethawed CO2 production was greatly inhibited, but COz diffusion through the tissue was unchanged. The preceding paragraphs discussed the possibility of error i n estimation of diffusion constants due to failure to account for O2 consumption. Conversely, there is a danger that inadequate diffusion to all regions of a tissue will confound the attempt to measure 02 consumption in vitro (4, 13). The Warburg equation for deciding on thickness of tissue slices for respirometer measurements is the same as equation 3 but with the substitution of QAx’ for v, where x’ is depth of penetration of 02. Depth of penetration is proportional to the quantity dKP/q and thus is weakly dependent on true 4, K, and Po2. Inadequate oxygenation would underestimate 11 and would be especially misleading in comparisons of 4 of a particular tissue in different situations. When there is an anoxic core, an experimental treatment that doubled the true would increase the observed 02 consumption by only 40 because depth of penetration would decrease to 70 % of its original value. The finding by us and by Grote (7, 8) and Thews (15) of K values that are about twice those of earlier workers (1) might be thought to show that the possibility of insufficient oxygenation is less significant because penetration of 02 is greater with high K values. However, depth of penetration l
9 a/0
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448
H‘. D.
is only weakly tied to K by the square-root function. Furthermore, the earlier low K values were probably appropriate for estimates of penetration in respirometer measurements because both the K values and the respirometer 4 values were obtained in fluids that had unstirred layers of similar magnitude adjacent to the tissue.
VAN
LIEW
AND
P.-Y.
CHEN
This study was aided by Contract NOOO14-68-A-0216 (NR 101-722) between the Office of Naval Research and the State University of New York at Buffalo and by National Institutes of Health Grant 5
POI HL 1414 .
Received
for publication
27 November
1974.
REFERENCES 1. BARTELS, Circulation,
2.
3. 4.
5.
6.
7.
8,
H. Tables of diffusion coefficients. In: Respiration and edited by P. L. Ahnan and D, S. Dittmer. Washington, DC.: Fed. Am Sot. Exptl. Biol., 1971, p. 21-24. CHEN, P.-Y., AND H. D. VAN LIEW. Krogh constants for diffusion of nitrogen and carbon monoxide in bladder tissue. Resp. Physiol. In press. CRANK, J. The Mathematics of Di$usion. London : Oxford Univ. Press, 1967. FARR, D. A., AND F. A. FUHRMAN. Role of diffusion of oxygen in respiration of tissues at different temperatures. J. @pL. physiol. 20: 637-646, 1965. GANFIELD, R. A., P. NAIR, AND W. J. WHALEN. Mass transfer, storage, and utilization of 02 in cat cerebral cortex. Am. J. Physiol. 219: 814-821, 1970. GREVEN, IS. uber die Bestimmung des Sauerstoffdiffusionskoeffizienten aus der Atmungsgrosse von Geweben in der Warburg-apparatur. PJluegers Arch. 269 : 38-54, 1959. GROTE, J. Die Sauerstoffdiffusionskonstanten in Lungengewebe und Wasser und ihre Temperaturabhgngigkeit. P’uegers Arch. 295 : 245-254, 1967. GROTE, J., AND G. THEWS. Die Bedingungen fiir die Sauerstoffversorgung des Herzmuskelgewebes. P’uegers Arch. 276 : 142-l 65, 1962.
9. KREUZER,
F., AND L. J. C. HOOFD. Factors influencing facilitated of oxygen in the presence of hemoglobin and myoglobin. Resp. Physiol. 15 : 104-l 24, 1972. KROGH, A. The rate of diflusion of gases through animal tissues, with some remarks on the coefficient of invasion. J. Physiol., London 52: 391-408, 1919. KROGH, A. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissues. J. Physiol., London 52 : 409-415, 1919. SCHOLANDER, P. F. Analyzer for accurate estimation of respiratory gases in one-half cubic centimeter samples. J. Viol. Chem. 167 : 235-250, 1946. STEVENS, N. L., AND C. M. SKOOE. Tracheal smooth muscle and rate of oxygen uptake. Am. J. Physiol. 226: 1462-1467, 1974. TAKAHASHI, G. H., AND I. FATT. The diffusion of oxygen in the cornea. Exptl. Eye Res. 4 : 4- 12, 1965. THEWS, G. Ein Verfahren zur Bestimmung des Q-Diffusionskoeffizienten, der Qz-Leitfahigkeit und des Q-L&lichkeitskoeffizienten im Gehirngewebe. PJuegers Arch. 27 1 : 227-244, 1960. VAN LIEW, H. D. Coupling of diffusion and perfusion in gas exit from subcutaneous pockets in rats. Am. J. Physiol. 214: 1176-l 185, 1968. VAN LIEW, H. D. Diffusion constant for CO2 through urinary bIadders of cats. Resp. Physiol. 13 : 372-377, 1971. diffusion
10.
11.
12.
13. 14. 15. 16 l
17.
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JOURNAL
OF PHYSIOLOGY Printed
Vat. 229, No. 2, August 1975.
in U.S.A.
Interaction of 0, diffusion and 0, metabolism in cat urinary bladder tissue HUGH D. VAN LIEW AND PAUL-YUAN CHEN Department of Physiology, State Uniuersi~ of New York School 03’ Medicine,
VAN LIEW, Hucw D., AND PAUL-YUAN CWEN. Interaction of Q2 d@usiun and 02 metabolism in cut urinary bladder tissue. Am. J. Physiol. 1975.-We measured oxygen exchange across 229(Z) : 444-448. the inside surface of excised urinary bladders that were inflated with gas mixtures. By using a range- of PO 2 differences between the
two sides of the tissue and relatively
simple mathematical
models,
we could infer the exchanges across the outside surface total 02 economy of the tissue so that we could evaluate action between diffusion and metabolic 02 consumption. mate of the Gogh diffusion constant or permeation
and the the interThe esti-
with this preparation
coefficient avoids errors due to 02 consumption and
unstirred layers. At 37”C, the value of bladder tissue 02 consumpand the value of the Gogh constant tion was 4,4 X IO-” min-l for 02 was 2.22 X low5 cm2 min-l atm? diffusion oxygen
coefficient;
consumption;
excised Warburg
tissue; Gogh respirometer
diffusion
constant;
MANY CLASSICAL TREATMENTS of diffusion focus on one process, the dispersion of a material throughout some limited space (e.g., 3). H owcver, in biological applications, there are almost always at least two time-requiring processes occurring at once. In oxygen utilization of living tissue, even in vitro, diffusion occurs simultaneously with oxygen consumption. The situation is even more complicated in vivo, as in the Krogh model of a capillary domain (1 I), because the additional time-requiring process of blood perfusion is coupled to the diffusion and metabolism processes. ‘The strategy that has been used to understand interactions between two or more ongoing processes is to set up a differential equation based on the principle of conservation of mass on the molecular, or differential scale, and then generalize to a macroscopic, observable scale by mathematical integration. Greven (6) took advantage of this approach to provide estimates of diffusion coefficients by allowing 02 consumption of the tissue to establish the concentration gradient for diffusion. By treating the interaction of 02 diffusion and 02 reaction with hemoglobin as a two-process system, Kreuzer and Hoofd (9) have provided an analysis of facilitated O2 diffusion that is more satisfying than earlier explanations The differential equation approach was used by Ganfield, Nair, and Whalen (5) to explain their findings with microelectrode analyses of Pop versus distance in excised brain tissue and by Van Liew (16) to deal with diffusion and blood perfusion in gas exit from subcutaneous gas pockets in rats. The aims of the present communication a,re: a) to explore h verge simple two-process situation-02 diffusion and 02
Bufalo,
New York I4214
consumption are studied in a mass of excised tissue in various gaseous 02 environments-and 6) to measure directly the Krogh diffusion constant for 02 through tissue at 37°C without the errors that would be incurred if metabolism were not taken into account. METHODS
Our experimental preparation has the advantage that gases are measured directly and that there are no bathing liquids that could lead to unstirred layers next to the tissue (2, 17). Male cats (1.3-5.1 kg) were killed by an overdose of sodium pentobarbital. The urinary bladders were excised, washed with saline, inflated with 12-24 ml of various gas mixtures, and suspended inside a moist chamber at 37”C, with either room air or pure O2 outside the bladder. We took no special precautions to rid the tissue of blood. Gas was added or removed from the bladder with syringes by way of a stopcock connected to a multiholed tube tied into the bladder cavity through the urethra. Gas samples were analyzed with a Scholander apparatus (12) and volumes were measured with a calibrated syringe. Measurements were started at least 10 min after introduction of a new gas mixture to assure a steady state of diffusion. The amount of Oa contained in the bladder (frbm 02 concentration times total gas volume) was measured before and after a timed period. Difference in O2 amount during the period divided by time was the estimate of exchange rate, v, either into or out of the bladder. Surfqce area, A, for each exchange rate measurement was estimated from V, the gas volume contained in the bladder, by the empirical formula (17) : A = 4.84 V213 + 4. Thickness of the bladder for each measurement was estimated by dividing weight of the tissue by area times density. Timed periods were l-2 h. Average 02 partial pressure inside during a timed period was taken as the arithmetic average of the initial and final values. Volumes were reduced to standard temperature and pressure dry, and partial pressures were expressed as fractions of a standard atmosphere, 760 Torr. Average values of bladder parameters are presented in Table 1, A. RESULTS
Figure 1 shows data obtained when the bladders were suspended in air. Each exit-rate value is normalized by multiplication by the quantity (thickness/area) for that particular measurement. The abscissa is Paz inside the
Downloaded from www.physiology.org/journal/ajplegacy by ${individualUser.givenNames} ${individualUser.surname} (129.186.138.035) on January 16, 2019.
Oa DIFFUSION
VS.
METABOLISM
IN
EXCISED
bladder. There is a significant O2 exit from the bladder when Paz inside is 0.2 atm, the approximate POT outside in the air. Figure 2 shows data from two types of experiment. a) The group of points to the right (from APO* of -0.2 to +0.6 atm) are the Fig. 1 data replotted, but with the abscissa given as Paz difference between inside and outside. Three data points from the extreme left of Fig. 1 were deleted on the grounds that at low Po2 it is likely that part of the tissue may be anoxic (see DISCUSSION). b) The points at the left in which (from APoz of -0.9 to -0.4) are from experiments pure O2 was outside the bladder. These latter points appear as negative APoz values, because inside Paz was less than outside and with one exception as negative 02 exit since 02 was entering the bladder cavity. DISCUSSION
Our estimates of the Krogh diffusion constant, the O2 consumption of the tissue, and depth of penetration of O2 from air into the metabolizing bladder tissue appear in Table 1, B. The rationale and method by which they were calculated are in the sections that follow. Our Table 1 value of the Krogh constant for 02, 2.22 X 10m5 cm2 min-l atm-l is greater than most of the values from various sources in the handbook compilation of Bartels (1) and is almost twice that calculated by Ganfield, Nair, and Whalen (5) for cat cerebral cortex at 37°C. Our value
A) Average values of measured parameters Weight of bladder, g Surface area of inflated bladder, cm2 Thickness of bJadder wall, cm B) Estimates based on exchange data (Figs. 1 and 2) and measured parameters (above) 02 consumption of tissue, min+ Gogh diffusion constant (K), cm2 min-’ atm-l Depth of penetration of 02 from room air (PO2 = 147 torr), cm
445
TISSUE
2.45 35.7 0.066
4.37 2.22
x
m-3
x
10-s
,045
A
2.0
A 1.5 cm* mtn-l 1.0
0.5
FIG. 1. Exit rate of 02 from bladders suspended in air vs. PQ inside. Exit rate is expressed as volume (STPD) multiplied by thickness and divided by area. Use of this normalization factor, X/A, serves two purposes: it decreases variability due to differences in size and volume of bladders and, as shown in DISCUSSION, slope of data plotted this way can be used to estimate Krogh diffusion constant.
l
l
x 1o-5 l l*,.& rp
1.5
!
-.8
-.6
l 42
l 5’. +e%a
l
cm2 min-’
g,;
l
/
l o
l
l
-.
2
.6
d PO, , atm
FIG. 2. Data from experiments with bladders suspended in air (right side of diagram) plotted together with bladders suspended in pure 02 (left side). Ordinate: same as in Fig. 1; abscissa: 02 partial pressure difference, inside minus outside.
is a little less than that measured by Grote (7) for rat lung tissue at 37”C, 2.5 X W5, and is also less than the 2.45 X 10m5 and 2.7 X 10d5 for heart and brain tissue, respectively, recalculated from earlier measurements (8, 15) on the basis of better values for O2 diffusion in water and temperature dependence for 02 diffusion in lung tissue (7) m Digusion and metabolism combined. Consider a surface that is parallel to the inside and outside surfaces of the bladder and at a variable distance, x, from the inside. Our three basic assumptions are that the rate of gas diffusion across this surface equals the rate of diffusion across the inside surface minus the amount that has been consumed by metabolism, that diffusion across the surface at x follows Fick’s second law, and that there is no net diffusion in directions parallel to the inside surface. In the equations that follow, dV/dt is volume of gas per minute across the plane at x; v is volume per minute across the inside surface; 4 is metabolic rate per unit of tissue volume; A is surface area (assumed to be the same at any x); K is the Krogh diffusion constant; P is partial pressure at depth x; and Pi is the 02 partial pressure inside the bladder cavity. The basic statement is: dV/dt
w
2.0x 1o-5
ox
= J? -
PAX
= -
KA(dP/dx)
(4
The two right-hand parts of equation I can be integrated to give vx - ;iAx2/2 = - KAP + C. The constant of integration C can be eliminated by the condition that when x = 0, P = Pi, so that: VX
-
qAx2/2
= + KA(Pi
-
P)
(2)
If the tissue metabolism were zero (4 = 0), equation 2 would be the usual form of Fick’s law that is used with permeation of gases through inert membranes or of inert gases through tissues (1, 2). When q is not zero, three situations must be considered. First, 02 may not penetrate to all regions of the tissue, so that Paz is zero beyond some depth and the amount of 02 that diffuses out of the cavity is subsequently completely consumed by the inside layers of the bladder. Boundary conditions in this case are that when x = x’, P = 0 where x’ is some depth beyond which there is no more Ozl and
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446 v = PAX’.
H.
Substitution
of these into equation2 gives: (3)
Equation 3 for this case of an anoxic core in the tissue has been used by Greven (6) in study of tissue slices in respirometers and by Van Liew (16) in study of 02 diffusion out of subcutaneous gas pockets of freshly killed rats. It is apparent from equation 3 that flux, ii/A, is dependent on the square root of Pi. There are no terms for partial pressure outside and thickness of the bladder, since the inside is independent of the outside because of the anoxic zone of variable size inside the tissue. The second possibility is that PO:! inside the bladder is so high that none of the 02 consumption of the tissue comes from the outside environment. Boundary conditions are that when x = X, P = P,, where X is total thickness of the bladder and P, is partial pressure in the outside environment. This condition yields : vX/A
= K(Pi
-
PO> + 4x2/2
(4)
D. VAN
LLEW
AND
P.-Y.
CHEN
suggests that the assumptions are warranted at least as first approximations. The slope of the least-squares line in Fig. 2 is the desired K value for Table 1, B. Standard error of the slope is 0.07 X 10m5 cm2 min-l atm? According to equation 4, they intercept is 4X2/2, from which the ;1 value of Table 1, B was calculated with the use of an average thickness of 0.066 cm (Table 1, A). Total O2 economy of excised bladder. Our data, although exhibiting considerable scatter, are compatible with the mathematic models of equations 3 and 4. We will develop the models further to illustrate the total 02 economy of a bladder when the bladder is suspended in air and inflated with gas mixtures having different concentrations of 02* Results of such an experiment are shown in Fig. 1. Figure 3 shows schematically the 02 exchanges of a bladder that has characteristics of our average data in Table 1. The only curve in Fig. 3 starts at the origin (0) and then gives way to a dashed curve. It represents the O2 exit from the bladder when there is an anoxic core; the equation for this curve is obtained by multiplying both sides of equation 3 by X. The upper diagonal heavy line segment ad represents 02 exit when there is no anoxic core by equation 4. At point a, when Paz is about 0.05 atm, the curve and the diagonal line meet. Therefore at P&s above that at point a, there is no anoxic core. The exit of O2 from the gas-filled bladder is always caused by a diffusion gradient in the tissue, dP/dx. Since the curve Oa and line segment ad rise with increasing POT, the dP/dx must also increase with PO*. At the left, O2 consumption of the tissue was the only factor causing the dP/dx, whereas to
In equation 4 there are terms for the outside partial pressure and total thickness because the whole of the tissue is involved. If all 02 consumption is from outside, rather than the inside, the equation is the same, except that i and o can be reversed. Equation 4 describes a straight line with the Krogh constant K for slope and 4X2/2 as intercept. The third possibility is that part of the metabolized 0, comes from nside and part from outside. Equation 4 applies in this case also, as can be seen by adding together two equations that are similar to equation 4, one for 02 diffusion from inside the bladder and one for diffusion from the outside. In such a treatment, one assumes that there is a demarcation plane at XII, where both dV/dt and dP/dx are zero; a variable P” at x” is the boundary condition for diffusion in each direction, and v from inside plus ii’ from outside add to give ;IAX, the total 02 consumption of the whole tissue. The data in Fig. 2 are plotted in the form of equation 4. In the experiments in which pure 02 was outside (at the left of Fig. Z), any 02 appearing in the bladder cavity must have penetrated all the way through, so there is no question of an anoxic core. The Oz-outside points follow the same trend as the points in which the bladder was in air (at the right), indicating that only in the three very low Pi points that were deleted for Fig. 2 (3 points at extreme left of Fig. 1) was there any possibility of an anoxic core. Many assumptions are involved in application of the mathematical treatment above to our data: that bladder tissue is homogeneous and isotropic with regard to both diffusion and O2 consumption, that 02 consumption is independent of PoZ except that q is zero when PoZ is zero, FIG. 3. Schematic representation of 02 economy of an “average” excised bladder suspended in air. Plot is same as in Fig. 1 and upper that diffusion can be regarded as a one-dimensional process heavy oblique Iine segment ad and solid curve Oa represent the 02 that even though the bladder is essentially spherical, that an Xeaves the bladder cavity (data of Fig. 1). Lower oblique line segment inflated bladder has uni .form thickness, and that ou .r measbe and short horizontal line segment represent amount of 02 that uremen ts approximate steady states even though various penetrates through to the air outside the bladder. When Paz inside is parameters are changing slowly. These assumptions are below 0.62 atm, this is negative since 02 enters the tissue from the air. Distance between the 2 heavy lines is proportional to 02 consumption surely not all valid in an absolute sense. For example, tissue of tissue, which is constant except at low Paz, where there is an anoxic is not homogeneous with respect to O2 consumption, which core. The 2 light line segments illustrate errors in estimate of K that is known to be localized in mitochondria. However, the could be incurred if metabohsm of tissue were not accounted for (see agreement of the data with the expectations from the theory text).
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O2 DIFFUSION
VS. METABOLISM
IN EXCISED
447
TISSUE
the right of point a, the gradient due to O2 consumption was modified when inside Paz was great enough to cause the 02 penetrating from inside to meet that from outside. The dashed continuation of the Oa curve shows what the 02 exit rate would have been if 02 from the outside had not had an effect to the right of point a (as could have been the case if the outside were Nz rather than air). The upper solid line segment and curve Oad corresponds to the data of Fig. 1. The second heavy diagonal line segment, be, joined to the horizontal line segment at the bottom of the graph are drawn to show O2 exit from the bladder tissue into the outside air. We could have obtained such data the inflated bladder in an air-filled vessel so bY enclosing tha t we could measure changes of O2 n the ou tside environment. According to line be, when Paz inside the bladder is above 0.62 atm, the bladder gives off O2 to the air because 02 is permeating all the way through. When Paz inside is low, there is movement of 02 from the air into the bladder. At Paz of 0.62 atm, there is no net entrance or exit of 02 from the outside; 02 from the inside is sufficient to furnish the 02 for metabolism of the entire tissue. When Pea = 0.2, so that Pea is equal inside and outside, half the metabolism of the tissue is supplied by the inside and half by the outside, as shown by the equidistance of lines ad and be from the axis. Point a, the critical Pea for development of an anoxic core, is below 0.2 atm, so there is a finite Pea at all points through the tissue when there is air both inside and outside. The total 02 consumed by the bladder tissue is proportional to the vertical distance between lines ad and be in Fig. 3. When Paz is below the Paz at points a and 6, the Oa consumption decreases because of the anoxic core. On the horizontal line to the left of b, 02 entering the tissue from outside reaches its maximum depth. Although the size of the anoxic core increases as inside Paz decreases, the increase is all due to decreasing O2 penetration from the inside. To the left of b, penetration from outside is constant because Pea outside is constant. One can calculate the depth of penetration of 02 from air with vX/A from b to the axis and 6 The result is 0.045 cm, shown in Table 1. This indicates that excised tissue, or nonperfused tissue open to the air during surgery, can obtain O2 from the air for a depth of about half a millimeter, provided the diffusion and metabolic characteristics are the same as in these bladders. If the temperature were lower, the upper heavy curve and line segment Oad would be changed in four ways: a) The curve Oa would rise less rapidly because of change in equation 3 due to the effect of lower 4 at lower temperature and to a minor extent, the temperature effect on K. b) The slope of ad would be slightly changed due to temperature effect on K. c) The vertical displacement of the ad away from point c would decrease because of the lower value of the intercept 4X2/2 of equation 4. d) Point a would move downward and to the left until the metabolic rate was so low that influx from outside air could nourish the whole tissue. In that situation, the line ad would pass through the origin or through the abscissa; there would be no curved portion Oa and no anoxic core. Error due to neglect of metabolism. The light line segment cd
in Fig. 3 shows the error tha t might be incurred if metabolic O2 consumption were not accounted for. If 02 exit rate were to be measured by putting pure O2 on one side of a membrane and air on the other side, one might divide the normalized exit rate at d by the presumed difference, P, This would markedly overestimate the value of K, Pd as shown by the difference of slope cd versus the correct slope of ad. On the other hand, if only the appearance of O2 on the outside were measured, one would find an exchange rate shown as point e, and the slope ce would underestimate the proper K by a factor of about 2. Data for an inert membrane having the same K as the live tissue would fall on a line that would be parallel to the two heavy diagonal lines but would pass through point c. Since data for metabolizing tissue are displaced from the inert membrane line by the quantity 4X2/2, the error in K due to neglect of metabolic rate increases as metabolism increases and increases as the square of the thickness. through tissues by Krogh ( 10) studied 02 penetration measuring O2 appearance in deoxygenated hemoglobin and therefore apparently made an error such as that illustrated by line ce+ However, his error was probably small because he used thin tissue pieces and worked at low temperatures that would keep metabolism low. Takahashi and Fatt (14) avoided error due to 02 consumption in measurements of diffusion in the cornea of the eye by inhibiting metabolism by cold and by dehydration and rehydration of the tissue. Thews (15) and Grote (7, 8), in measuring 02 diffusion through tissue slices held in place by plastic membranes, apparently did not incur an error even though their mathematical treatment does not account for 02 consumption+ They found no change in their results when the tissue was poisoned by cyanide. Probably they disrupted the metabolic machinery by their routine procedure of freezing the tissue to facilitate slicing. Our previous work (17) showed that when cat bladders were frozen and rethawed CO2 production was greatly inhibited, but COz diffusion through the tissue was unchanged. The preceding paragraphs discussed the possibility of error i n estimation of diffusion constants due to failure to account for O2 consumption. Conversely, there is a danger that inadequate diffusion to all regions of a tissue will confound the attempt to measure 02 consumption in vitro (4, 13). The Warburg equation for deciding on thickness of tissue slices for respirometer measurements is the same as equation 3 but with the substitution of QAx’ for v, where x’ is depth of penetration of 02. Depth of penetration is proportional to the quantity dKP/q and thus is weakly dependent on true 4, K, and Po2. Inadequate oxygenation would underestimate 11 and would be especially misleading in comparisons of 4 of a particular tissue in different situations. When there is an anoxic core, an experimental treatment that doubled the true would increase the observed 02 consumption by only 40 because depth of penetration would decrease to 70 % of its original value. The finding by us and by Grote (7, 8) and Thews (15) of K values that are about twice those of earlier workers (1) might be thought to show that the possibility of insufficient oxygenation is less significant because penetration of 02 is greater with high K values. However, depth of penetration l
9 a/0
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448
H‘. D.
is only weakly tied to K by the square-root function. Furthermore, the earlier low K values were probably appropriate for estimates of penetration in respirometer measurements because both the K values and the respirometer 4 values were obtained in fluids that had unstirred layers of similar magnitude adjacent to the tissue.
VAN
LIEW
AND
P.-Y.
CHEN
This study was aided by Contract NOOO14-68-A-0216 (NR 101-722) between the Office of Naval Research and the State University of New York at Buffalo and by National Institutes of Health Grant 5
POI HL 1414 .
Received
for publication
27 November
1974.
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2.
3. 4.
5.
6.
7.
8,
H. Tables of diffusion coefficients. In: Respiration and edited by P. L. Ahnan and D, S. Dittmer. Washington, DC.: Fed. Am Sot. Exptl. Biol., 1971, p. 21-24. CHEN, P.-Y., AND H. D. VAN LIEW. Krogh constants for diffusion of nitrogen and carbon monoxide in bladder tissue. Resp. Physiol. In press. CRANK, J. The Mathematics of Di$usion. London : Oxford Univ. Press, 1967. FARR, D. A., AND F. A. FUHRMAN. Role of diffusion of oxygen in respiration of tissues at different temperatures. J. @pL. physiol. 20: 637-646, 1965. GANFIELD, R. A., P. NAIR, AND W. J. WHALEN. Mass transfer, storage, and utilization of 02 in cat cerebral cortex. Am. J. Physiol. 219: 814-821, 1970. GREVEN, IS. uber die Bestimmung des Sauerstoffdiffusionskoeffizienten aus der Atmungsgrosse von Geweben in der Warburg-apparatur. PJluegers Arch. 269 : 38-54, 1959. GROTE, J. Die Sauerstoffdiffusionskonstanten in Lungengewebe und Wasser und ihre Temperaturabhgngigkeit. P’uegers Arch. 295 : 245-254, 1967. GROTE, J., AND G. THEWS. Die Bedingungen fiir die Sauerstoffversorgung des Herzmuskelgewebes. P’uegers Arch. 276 : 142-l 65, 1962.
9. KREUZER,
F., AND L. J. C. HOOFD. Factors influencing facilitated of oxygen in the presence of hemoglobin and myoglobin. Resp. Physiol. 15 : 104-l 24, 1972. KROGH, A. The rate of diflusion of gases through animal tissues, with some remarks on the coefficient of invasion. J. Physiol., London 52: 391-408, 1919. KROGH, A. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissues. J. Physiol., London 52 : 409-415, 1919. SCHOLANDER, P. F. Analyzer for accurate estimation of respiratory gases in one-half cubic centimeter samples. J. Viol. Chem. 167 : 235-250, 1946. STEVENS, N. L., AND C. M. SKOOE. Tracheal smooth muscle and rate of oxygen uptake. Am. J. Physiol. 226: 1462-1467, 1974. TAKAHASHI, G. H., AND I. FATT. The diffusion of oxygen in the cornea. Exptl. Eye Res. 4 : 4- 12, 1965. THEWS, G. Ein Verfahren zur Bestimmung des Q-Diffusionskoeffizienten, der Qz-Leitfahigkeit und des Q-L&lichkeitskoeffizienten im Gehirngewebe. PJuegers Arch. 27 1 : 227-244, 1960. VAN LIEW, H. D. Coupling of diffusion and perfusion in gas exit from subcutaneous pockets in rats. Am. J. Physiol. 214: 1176-l 185, 1968. VAN LIEW, H. D. Diffusion constant for CO2 through urinary bIadders of cats. Resp. Physiol. 13 : 372-377, 1971. diffusion
10.
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17.
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