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Can diffusion coefficients be estimated from plasma clearance curves of intact animals?
J. theor. Biol. (1982) 95, 399-407
Can Diffusion Coefficients be Estimated from Plasma Clearance Curves Of Intact Animals? KENNETH
H. NORWICH
Department of Physiology and Institute of Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada h45.S lA4 (Received
8 December
1980, and in revised form 29 October 1981)
Certain substances, when injected intravenously, are removed from the plasma in accordance with the simple power law, C =AtP. C is the concentration in plasma;A and a are constkmts.It is suggestedthat when a is of the order of 1.5, this power law may result from diffusion of the injected substanceunder conditions of sphericalsymmetry. It is shown for two such substances,that their diffusion coefficient in plasma,D, is given approximately by the expressionD =&T (I’&/A)~‘~, where ME is the effective diffusing mass.
1. Introduction It has been shown (Norwich & Siu, 1980) that when certain substances are injected intravenously as a bolus, the plasma disappearance or clearance curve can be described quite precisely by a function of the form C = At-”
(1)
where C is concentration and A and a are constants. These substances may be radioactively labelled metabolites, drugs or dyes. In the cases of 3H-glycerol, 3H-lactate, indocyanine green and bromsulphalein (BSP) the value of the constant, a, as determined from a least squares curves fit, ranged from about 1.3 to 1.5. It was hypothesized that the above equation had its basis in a convection-diffusion process with spherical symmetry. It can be shown (as we have done in the Appendix) that, to a degree of approximation, such a process can give rise to equation (1) with a equal to 312. Suppose that a mass, ME, of material is injected suddenly at zero time, and is carried radially within an infinite medium by convective and diffusive
processes. It is shown in the Appendix that when diffusion is the predominant process, the following equation will provide an approximation for the value of the concentration, C, at the point of injection (center of 399
0022-5193/82/060399+09$03.00/0
01982
Academic
Press
Inc. (London)Ltd.
400
K.
H.
NORWICH
the sphere) at any time, t, after the injection: ME ’
=
(4*0)3/z
-3/2 t
*
D is the coefficient of diffusion, which, for dilute solutions, we may take as constant. Equations (1) and (2) are of the same form. All the substances whose clearance was governed by power laws of the form of equation (l), which were not bone-seeking elements, were found to be removed from the circulation in large part by the liver. Many were found to have flow-limited rates of clearance and high extraction ratios. Therefore, it was further hypothesized that the diffusion process was occurring within the liver acinus, the spheroidal cluster of cells which is found centred about the terminal portal venule, and which Dr Aron Rappaport and others have shown to be the basic unit of liver structure. It will now be shown that from the value of the constant, A, obtained by curve-fitting function (1) to the experimental data, one can calculate the value of the diffusion coefficient, D, for the substance in plasma. 2. Method
for Calculating
D
The injected substance will diffuse through all of the acini within the liver. That is, the process of diffusion occurs in parallel at many loci. Removal of the substance occurs in one of the peripheral zones of the liver acinus (concentric spherical shells) shown by Rappaport and others (Schiff, 1975) to contain the necessary enzymes. The site of removal of the substance from the blood is taken to be remote from the centre of the acinus (terminal portal venule). Portal blood is well intermixed with systematic blood in this instance; no appreciable difference in concentration of the injected substance between portal and systematic blood is expected. The problem is to estimate the value of ME which is the effective mass diffusing through each of the acini. Suppose that the original bolus of material injected intravenously contained a mass M of the diffusing substance. This mass, M, is divided up among the N acini contained within liver. Therefore, a mass M/N diffuses through each acinus. If all the diffusing material were confined to the blood within the liver, M/N would then equal ME, the mass proceeding through each acinus. But the diffusing material is distributed throughout a much larger distribution volume, VT, which is specific for each substance-extracellular water, vascular space, etc. Therefore, the total mass injected, M, must be divided by a further factor, VT/V=, where VL is the volume of blood contained by the liver.
CLEARANCE
AS
A PROCESS
OF
DIFFUSION
The effective mass diffusing through each acinus is M ME=N(V&)=
M(VL/VT) N
’
That is, the effect of a large intermixed pool of fluid is to reduce the actual diffusing mass per acinus, M/N, to an effective diffusing mass per acinus (M/N) (VJ VT). This is an approximation. It must not be supposed that the acini can handle the increased load of diffusing material contained in VT without a perturbation of the point source solution given in equation (2). Substituting the effective mass, ME, from equation (3) into equation (2) we obtain M ’ = N( VT/ V,) (~TD)~‘~
By comparing
-3/2
’
’
(4)
equations (1) and (4) we can write A=
M N( V,l V,) (47rD)3’2’
(5)
Solving for the diffusion coefficient, D,
(The dimensions of A are [M. Lp3 . T3’2].) This is the result we sought. All the qualities on the right-hand side of equation (6) are either known or can be estimated for two substances, 3H-glycerol and bromsulphalein. M is the mass of material injected. We shall normalize it to give mass injected per kg body weight. VL, the volume of blood contained within the liver, has been estimated by various investigators. It will be taken here to be 35% of the liver volume, a little higher than the average figure reported by Greenway & Stark (1971). Since liver comprises about 2.5% of total body weight, Vr can be estimated as about 35% of 2.5% = 0.9% of body volume. VT, the total volume of distribution of the injected substance, will, of course, vary with the substance. Glycerol is believed to be distributed in total body water, or about 70% of body volume. BSP, which is bound to plasma albumin, should be confined to the plasma volume or about 4.5% of body volume. A has been evaluated by curve-fitting the clearance curve to the function (1). A value for N, the number of acini per kg body weight, has not, to my knowledge, been published. Upon my request, Dr Aron Rapport and his
3H-glycerol ‘H-glycerol BSP
(dog 29)
Substance
3.914 x lo6 4.502 x lo6 0.7927
Value of A [C = At-“] mass ml-’ mm 3/2
(5)
20 000
BSP
20 000
20 000
(dog 29)
‘H-glycerol
3H-glycerol
Substance
Number of acini/kg body weight N
(1)
2.942 3.041 2.212
x 1om6 x 1om6 x 1o-6
(7)
(6)
1.765 x 1O-4 1.835 x lo-‘+ 1.327 x 1O-4
4.5
70
70
Total distribution volume % body volume VT
(3)
0.9
0.9
0.9
Volume of blood in the liver % body volume V,
(2)
7.094 x 1om6 7.094x lomh 0.5187 x lo-’
D in vitro cm’ set 1
(8)
63.6 x 10’ dpm 76.9x 10’ dpm 5.4 mg
Mass of substance injected per kg A4
(4)
Norwich
Norwich Norwich
data measured
.
& Gram
Kallai-Sanfacon, Kallai-Sanfacon, Winkler & Gram
Clearance
Winkler
Kallai-Sanfacon,
Norwich
data measured
Kallai-Sanfacon,
Clearance
& Steiner & Steiner
by
& Steiner
& Steiner
by
Summary of the data used to calculate the diffusion coefficient, D, from plasma clearance curves. The value of D reported in column 7 is obtained by substituting the numbers given in columns 2-6 in right-hand side of equation (6).
TABLET
2 i; Lt
2
1
;r:
iz
CLEARANCE
AS
A
PROCESS
OF
403
DIFFUSION
colleagues sectioned a number of human livers in various planes, and arrived at an average density of one acinus per 1.26 mm3 of liver volume. Thus we expect about 1000/1*26 or 794 acini per cubic centimetre. Since there are about 25 g of liver per kg body weight, we expect 25 x 794 or about 20,000. acini per kg body weight. The canine liver was taken to be similar to the human liver. All these numbers have been summarized in Table 1. 3. Calculation (A)
DIFFUSION
COEFFICIENT
The data of Kallai-Sanfacon, dog 29, the in viuo calculation
FOR
of D “H-GLYCEROL
IN
THE
DOG
Norwich & Steiner (1978) were used. For of D, using equation (6), is
D = 2.242 x lop6 cm2 set-’ D = 3.041 x 1O-6 cm2 set-’ for dog 30 The value for the diffusion coefficient for glycerol measured in vitro (Reid, Prausnitz & Sherwood 1977), corrected approximately for temperature (Weast, 1974), and corrected approximately for the viscosity of plasma (Daniels & Alberty, 1966; McDonald, 1974) is 7.094 x lop6 cm2 set-‘. (B)
DIFFUSION
COEFFICIENT
FOR
BSP
IN
THE
HUMAN
The data of Winkler & Gram (1961) were used. For reasons discussed below, low doses of BSP are expected to give the best results. For the lowest injected dose of BSP in control subjects, an in uivo measurement using equation (6) gives D = 2.212 x 10m6 cm2 set-‘. The value for the diffusion coefficient for bovine albumin measured in vitro (Van Holde, 1971), corrected approximately for temperature and plasma viscosity as before is 0.5 187 x lop6 cm2 set-’ The above results are summarized in Table 1. 4. Discussion
The in viuo estimate (from clearance curves) of D for glycerol is in error by a factor of 2. The in viva estimate of D for BSP is in error by a factor
404
K.
H.
NORWICH
of about 4. These errors are well within the range that would be expected from the very simple mathematical model of the clearance process which was used. The clearance of BSP is known to be dose-dependent. As the injected dose becomes smaller, the data progressively straighten on full logarithmic paper, and the value of a (see equation (1)) becomes progressively closer to 3. Therefore, the clearance curve produced by the lowest dose is best. In the data of Winkler and Gram, the clearance curve produced by the lowest dose consisted of an average of four experiments, and the concentrations reported are averages over a 5 min interval. It has been assumed that BSP remains bound to plasma albumin throughout its excursion through the liver. The title of this paper is in the form of a question, rather than an assertion. This has been done because of the difficulty in estimating the effective diffusing mass, ME. If the clearance curves of drugs and labelled metabolites are to be viewed as solutions to the point-source diffusion equation, then a reduced mass given by an equation like equation (3) must be used. Certainly the entire injected mass of material does not remain confined to the hepatic vasculature. Other models of the process give rise to variations of equation (3). It should be appreciated that the calculation of diffusion coefficients from plasma clearance data is not being advocated here as an alternative to standard, in vitro methods for determining this coefficient, which are incomparably better. Rather the ability of the clearance calculation to approximate the value of the diffusion coefficient is taken as evidence that the kinetics of the process is more aptly described by a random walk or diffusive process than by an exchange of material between well-mixed pools. A similar model was suggested (Marshall & Onkelinx, 1968) for diffusion of alkaline earth radioisotopes in adult bone. Our view that the speed of diffusion is greater than that of convection in the transport of certain substances through the liver is in keeping with the view of Goresky (1964). He regarded the diffusive equilibration of BSP between vascular and extravascular spaces as a nearly instantaneous process. The picture of material rapidly diffusing through the liver acinus provides a qualitive explanation for the shape of plasma clearance curves which are of the labelled glycerol type. These curves fall precipitously during the first few moments after the substance is injected, containing as much as 50% of the total area beneath the curve in the region t = 0 to t = 2 min. This steep fall is attributed to diffusion into an “empty” liver, with high concentration gradients favouring exit from the bloodstream. During this phase the liver is being loaded with metabolite, and metabolite is being excreted.
CLEARANCE
AS
A PROCESS
OF
DIFFUSION
405
During later phases of removal, the clearance curve, governed by tP with larger c,falls more slowly than it would if it were governed by an exponential function. At this time the liver is already loaded with metabolite. The rate of removal from the bloodstream may be even less than the rate of excretion of metabolite. Finally, it may be observed that when both convective and diffusive transport are taken into account, equation (1) should be replaced by equation (A5) of the Appendix, which has the form C = At-” e-bta This equation
has sometimes been used empirically
(7) to fit clearance data.
This work has been supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada. I am grateful to my colleagues, Dr George Steiner and Dr Mary-Ann Kallai-SanfaGon, without whom the 3H-glycerol data would not have been obtained. And I am most indebted to Dr Aron Rappaport and his colleagues at the Liver Laboratory of Sunnybrook Hospital, Toronto, for their co-operation in determining the acinar density. Thanks also to my colleague, Professor Diran Basmadjran of the Department of Chemical Engineering, University of Toronto, for his help in locating and interpreting the diffusion coefficients measured in vitro. REFERENCES DANIELS, F. & ALBERTY, R. A. (1966). Physical Chemistry. 3rd. ed. New York: Wiley & Sons. p. 406. GORESKY, C. A. (1964). Am. J. Physiol. 207,13. GREENWAY, C. V. & STARK, R. D. (1971). Physioi. Rev. 51,23. KALLAI-SANFACON, M. A., NORWICH, K. H. & STEINER, G. (1978). Can. 1. Physiol. Pharmacol. 56,934. MARSHALL, J. H. & ONKELINX, C. (1968). Nature, Land. 217,742. MCDONALD, D. A. (1974). Blood Flow in Arteries. London: Edward Arnold. NORWICH, K. H. & ZELIN, S. (1970). Bull. Math. Biophys. 32, 25. NORWICH, K. H. & SIU, S. (1980). J. Theor. Biol. 95, 000. REID, R. C., PRAUSNITZ, J. M. & SHERWOOD, T. K. (1977). The Properties of Gases and Liquids. New York: McGraw Hill. p. 577. SCHIFF, L. Editor. (1975). Diseases of the Liver. 4th edition . Philadelphia: J. P. Lippincott. VAN HOLDE, K. E. (1971). Physical Biochemistry. Englewood Cliff, New Jersey: PrenticeHall. p. 105. WEAST, R. C. Editor. (1974). Handbook of Chemistry and Physics. 55th edition. Cleveland: CRC Press. WINKLER, K. & GRAM, C. (1961). Acta Med. Sand. 169,263.
APPENDIX
The equation of convective diffusion (governing transport by convection of fluid and diffusion within the fluid) may be written
~=DV2C-V *(CC)
(Al)
406
K.
H.
IWRWICH
where C is the concentration of some drug or labelled metabolite, D is a constant coefficient of diffusion, and v’ is the fluid velocity (blood, interstitial fluid, etc.) at some point. For the idealization of purely radial transport with constant radial velocity, V, within a spherical liver acinus, equation (Al) becomes A21 An approximate solution to equation (A2) for a point source within an infinite sphere can be found as follows. The solution to the corresponding one-dimensional problem is (Norwich & Zelin, 1970)
where ME is the mass of material injected at the origin at t = 0, and u is here the constant convective or drift velocity along the x axis. The corresponding three dimensional case with constant drift velocities VI, L’~, L‘? along the x, y and z axes respectively is
To approximate the case of radial symmetry in three dimensions, let us consider an “eight-point star”, where &of the injected molecules experience drift with velocity (v,v,v) (directed into one octant), & of the injected molecules experience drift with velocity (-v,v,u), 6 experience (v,-tl,v 1, etc. In each of the eight cases concentration is governed by an equation of the form (A4). The superposition of the eight functions for C(x,y,~,t) provides an approximation to the solution of the purely radial differential equation (A2). This was verified for the two-dimensional case by calculating values for C(x,y,t) from (the two-dimensional version of) equation (A4), and then confirming by a finite difference method that these values satisfy (the cylindrical symmetrical version of) the differential equation (A2). In all cases tested, the superposition of rectangular solutions (A4) satisfied the symmetrical differential equation (A2) quite adequately. We are interested in the case where sampling of concentration takes place at the origin, corresponding to sampling within the bloodstream. Setting X, y and z all equal to zero in equation (A4) gives the superposition of the eight solutions:
ME C(O,O,O,t~ = (47rD)“’
CLEARANCE
When u*t is substantially (A5) becomes
AS
A
PROCESS
OF
DIFFUSION
407
smaller than the diffusion coefficient, D, equation
which is the same as equation (2).
Can Diffusion Coefficients be Estimated from Plasma Clearance Curves Of Intact Animals? KENNETH
H. NORWICH
Department of Physiology and Institute of Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada h45.S lA4 (Received
8 December
1980, and in revised form 29 October 1981)
Certain substances, when injected intravenously, are removed from the plasma in accordance with the simple power law, C =AtP. C is the concentration in plasma;A and a are constkmts.It is suggestedthat when a is of the order of 1.5, this power law may result from diffusion of the injected substanceunder conditions of sphericalsymmetry. It is shown for two such substances,that their diffusion coefficient in plasma,D, is given approximately by the expressionD =&T (I’&/A)~‘~, where ME is the effective diffusing mass.
1. Introduction It has been shown (Norwich & Siu, 1980) that when certain substances are injected intravenously as a bolus, the plasma disappearance or clearance curve can be described quite precisely by a function of the form C = At-”
(1)
where C is concentration and A and a are constants. These substances may be radioactively labelled metabolites, drugs or dyes. In the cases of 3H-glycerol, 3H-lactate, indocyanine green and bromsulphalein (BSP) the value of the constant, a, as determined from a least squares curves fit, ranged from about 1.3 to 1.5. It was hypothesized that the above equation had its basis in a convection-diffusion process with spherical symmetry. It can be shown (as we have done in the Appendix) that, to a degree of approximation, such a process can give rise to equation (1) with a equal to 312. Suppose that a mass, ME, of material is injected suddenly at zero time, and is carried radially within an infinite medium by convective and diffusive
processes. It is shown in the Appendix that when diffusion is the predominant process, the following equation will provide an approximation for the value of the concentration, C, at the point of injection (center of 399
0022-5193/82/060399+09$03.00/0
01982
Academic
Press
Inc. (London)Ltd.
400
K.
H.
NORWICH
the sphere) at any time, t, after the injection: ME ’
=
(4*0)3/z
-3/2 t
*
D is the coefficient of diffusion, which, for dilute solutions, we may take as constant. Equations (1) and (2) are of the same form. All the substances whose clearance was governed by power laws of the form of equation (l), which were not bone-seeking elements, were found to be removed from the circulation in large part by the liver. Many were found to have flow-limited rates of clearance and high extraction ratios. Therefore, it was further hypothesized that the diffusion process was occurring within the liver acinus, the spheroidal cluster of cells which is found centred about the terminal portal venule, and which Dr Aron Rappaport and others have shown to be the basic unit of liver structure. It will now be shown that from the value of the constant, A, obtained by curve-fitting function (1) to the experimental data, one can calculate the value of the diffusion coefficient, D, for the substance in plasma. 2. Method
for Calculating
D
The injected substance will diffuse through all of the acini within the liver. That is, the process of diffusion occurs in parallel at many loci. Removal of the substance occurs in one of the peripheral zones of the liver acinus (concentric spherical shells) shown by Rappaport and others (Schiff, 1975) to contain the necessary enzymes. The site of removal of the substance from the blood is taken to be remote from the centre of the acinus (terminal portal venule). Portal blood is well intermixed with systematic blood in this instance; no appreciable difference in concentration of the injected substance between portal and systematic blood is expected. The problem is to estimate the value of ME which is the effective mass diffusing through each of the acini. Suppose that the original bolus of material injected intravenously contained a mass M of the diffusing substance. This mass, M, is divided up among the N acini contained within liver. Therefore, a mass M/N diffuses through each acinus. If all the diffusing material were confined to the blood within the liver, M/N would then equal ME, the mass proceeding through each acinus. But the diffusing material is distributed throughout a much larger distribution volume, VT, which is specific for each substance-extracellular water, vascular space, etc. Therefore, the total mass injected, M, must be divided by a further factor, VT/V=, where VL is the volume of blood contained by the liver.
CLEARANCE
AS
A PROCESS
OF
DIFFUSION
The effective mass diffusing through each acinus is M ME=N(V&)=
M(VL/VT) N
’
That is, the effect of a large intermixed pool of fluid is to reduce the actual diffusing mass per acinus, M/N, to an effective diffusing mass per acinus (M/N) (VJ VT). This is an approximation. It must not be supposed that the acini can handle the increased load of diffusing material contained in VT without a perturbation of the point source solution given in equation (2). Substituting the effective mass, ME, from equation (3) into equation (2) we obtain M ’ = N( VT/ V,) (~TD)~‘~
By comparing
-3/2
’
’
(4)
equations (1) and (4) we can write A=
M N( V,l V,) (47rD)3’2’
(5)
Solving for the diffusion coefficient, D,
(The dimensions of A are [M. Lp3 . T3’2].) This is the result we sought. All the qualities on the right-hand side of equation (6) are either known or can be estimated for two substances, 3H-glycerol and bromsulphalein. M is the mass of material injected. We shall normalize it to give mass injected per kg body weight. VL, the volume of blood contained within the liver, has been estimated by various investigators. It will be taken here to be 35% of the liver volume, a little higher than the average figure reported by Greenway & Stark (1971). Since liver comprises about 2.5% of total body weight, Vr can be estimated as about 35% of 2.5% = 0.9% of body volume. VT, the total volume of distribution of the injected substance, will, of course, vary with the substance. Glycerol is believed to be distributed in total body water, or about 70% of body volume. BSP, which is bound to plasma albumin, should be confined to the plasma volume or about 4.5% of body volume. A has been evaluated by curve-fitting the clearance curve to the function (1). A value for N, the number of acini per kg body weight, has not, to my knowledge, been published. Upon my request, Dr Aron Rapport and his
3H-glycerol ‘H-glycerol BSP
(dog 29)
Substance
3.914 x lo6 4.502 x lo6 0.7927
Value of A [C = At-“] mass ml-’ mm 3/2
(5)
20 000
BSP
20 000
20 000
(dog 29)
‘H-glycerol
3H-glycerol
Substance
Number of acini/kg body weight N
(1)
2.942 3.041 2.212
x 1om6 x 1om6 x 1o-6
(7)
(6)
1.765 x 1O-4 1.835 x lo-‘+ 1.327 x 1O-4
4.5
70
70
Total distribution volume % body volume VT
(3)
0.9
0.9
0.9
Volume of blood in the liver % body volume V,
(2)
7.094 x 1om6 7.094x lomh 0.5187 x lo-’
D in vitro cm’ set 1
(8)
63.6 x 10’ dpm 76.9x 10’ dpm 5.4 mg
Mass of substance injected per kg A4
(4)
Norwich
Norwich Norwich
data measured
.
& Gram
Kallai-Sanfacon, Kallai-Sanfacon, Winkler & Gram
Clearance
Winkler
Kallai-Sanfacon,
Norwich
data measured
Kallai-Sanfacon,
Clearance
& Steiner & Steiner
by
& Steiner
& Steiner
by
Summary of the data used to calculate the diffusion coefficient, D, from plasma clearance curves. The value of D reported in column 7 is obtained by substituting the numbers given in columns 2-6 in right-hand side of equation (6).
TABLET
2 i; Lt
2
1
;r:
iz
CLEARANCE
AS
A
PROCESS
OF
403
DIFFUSION
colleagues sectioned a number of human livers in various planes, and arrived at an average density of one acinus per 1.26 mm3 of liver volume. Thus we expect about 1000/1*26 or 794 acini per cubic centimetre. Since there are about 25 g of liver per kg body weight, we expect 25 x 794 or about 20,000. acini per kg body weight. The canine liver was taken to be similar to the human liver. All these numbers have been summarized in Table 1. 3. Calculation (A)
DIFFUSION
COEFFICIENT
The data of Kallai-Sanfacon, dog 29, the in viuo calculation
FOR
of D “H-GLYCEROL
IN
THE
DOG
Norwich & Steiner (1978) were used. For of D, using equation (6), is
D = 2.242 x lop6 cm2 set-’ D = 3.041 x 1O-6 cm2 set-’ for dog 30 The value for the diffusion coefficient for glycerol measured in vitro (Reid, Prausnitz & Sherwood 1977), corrected approximately for temperature (Weast, 1974), and corrected approximately for the viscosity of plasma (Daniels & Alberty, 1966; McDonald, 1974) is 7.094 x lop6 cm2 set-‘. (B)
DIFFUSION
COEFFICIENT
FOR
BSP
IN
THE
HUMAN
The data of Winkler & Gram (1961) were used. For reasons discussed below, low doses of BSP are expected to give the best results. For the lowest injected dose of BSP in control subjects, an in uivo measurement using equation (6) gives D = 2.212 x 10m6 cm2 set-‘. The value for the diffusion coefficient for bovine albumin measured in vitro (Van Holde, 1971), corrected approximately for temperature and plasma viscosity as before is 0.5 187 x lop6 cm2 set-’ The above results are summarized in Table 1. 4. Discussion
The in viuo estimate (from clearance curves) of D for glycerol is in error by a factor of 2. The in viva estimate of D for BSP is in error by a factor
404
K.
H.
NORWICH
of about 4. These errors are well within the range that would be expected from the very simple mathematical model of the clearance process which was used. The clearance of BSP is known to be dose-dependent. As the injected dose becomes smaller, the data progressively straighten on full logarithmic paper, and the value of a (see equation (1)) becomes progressively closer to 3. Therefore, the clearance curve produced by the lowest dose is best. In the data of Winkler and Gram, the clearance curve produced by the lowest dose consisted of an average of four experiments, and the concentrations reported are averages over a 5 min interval. It has been assumed that BSP remains bound to plasma albumin throughout its excursion through the liver. The title of this paper is in the form of a question, rather than an assertion. This has been done because of the difficulty in estimating the effective diffusing mass, ME. If the clearance curves of drugs and labelled metabolites are to be viewed as solutions to the point-source diffusion equation, then a reduced mass given by an equation like equation (3) must be used. Certainly the entire injected mass of material does not remain confined to the hepatic vasculature. Other models of the process give rise to variations of equation (3). It should be appreciated that the calculation of diffusion coefficients from plasma clearance data is not being advocated here as an alternative to standard, in vitro methods for determining this coefficient, which are incomparably better. Rather the ability of the clearance calculation to approximate the value of the diffusion coefficient is taken as evidence that the kinetics of the process is more aptly described by a random walk or diffusive process than by an exchange of material between well-mixed pools. A similar model was suggested (Marshall & Onkelinx, 1968) for diffusion of alkaline earth radioisotopes in adult bone. Our view that the speed of diffusion is greater than that of convection in the transport of certain substances through the liver is in keeping with the view of Goresky (1964). He regarded the diffusive equilibration of BSP between vascular and extravascular spaces as a nearly instantaneous process. The picture of material rapidly diffusing through the liver acinus provides a qualitive explanation for the shape of plasma clearance curves which are of the labelled glycerol type. These curves fall precipitously during the first few moments after the substance is injected, containing as much as 50% of the total area beneath the curve in the region t = 0 to t = 2 min. This steep fall is attributed to diffusion into an “empty” liver, with high concentration gradients favouring exit from the bloodstream. During this phase the liver is being loaded with metabolite, and metabolite is being excreted.
CLEARANCE
AS
A PROCESS
OF
DIFFUSION
405
During later phases of removal, the clearance curve, governed by tP with larger c,falls more slowly than it would if it were governed by an exponential function. At this time the liver is already loaded with metabolite. The rate of removal from the bloodstream may be even less than the rate of excretion of metabolite. Finally, it may be observed that when both convective and diffusive transport are taken into account, equation (1) should be replaced by equation (A5) of the Appendix, which has the form C = At-” e-bta This equation
has sometimes been used empirically
(7) to fit clearance data.
This work has been supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada. I am grateful to my colleagues, Dr George Steiner and Dr Mary-Ann Kallai-SanfaGon, without whom the 3H-glycerol data would not have been obtained. And I am most indebted to Dr Aron Rappaport and his colleagues at the Liver Laboratory of Sunnybrook Hospital, Toronto, for their co-operation in determining the acinar density. Thanks also to my colleague, Professor Diran Basmadjran of the Department of Chemical Engineering, University of Toronto, for his help in locating and interpreting the diffusion coefficients measured in vitro. REFERENCES DANIELS, F. & ALBERTY, R. A. (1966). Physical Chemistry. 3rd. ed. New York: Wiley & Sons. p. 406. GORESKY, C. A. (1964). Am. J. Physiol. 207,13. GREENWAY, C. V. & STARK, R. D. (1971). Physioi. Rev. 51,23. KALLAI-SANFACON, M. A., NORWICH, K. H. & STEINER, G. (1978). Can. 1. Physiol. Pharmacol. 56,934. MARSHALL, J. H. & ONKELINX, C. (1968). Nature, Land. 217,742. MCDONALD, D. A. (1974). Blood Flow in Arteries. London: Edward Arnold. NORWICH, K. H. & ZELIN, S. (1970). Bull. Math. Biophys. 32, 25. NORWICH, K. H. & SIU, S. (1980). J. Theor. Biol. 95, 000. REID, R. C., PRAUSNITZ, J. M. & SHERWOOD, T. K. (1977). The Properties of Gases and Liquids. New York: McGraw Hill. p. 577. SCHIFF, L. Editor. (1975). Diseases of the Liver. 4th edition . Philadelphia: J. P. Lippincott. VAN HOLDE, K. E. (1971). Physical Biochemistry. Englewood Cliff, New Jersey: PrenticeHall. p. 105. WEAST, R. C. Editor. (1974). Handbook of Chemistry and Physics. 55th edition. Cleveland: CRC Press. WINKLER, K. & GRAM, C. (1961). Acta Med. Sand. 169,263.
APPENDIX
The equation of convective diffusion (governing transport by convection of fluid and diffusion within the fluid) may be written
~=DV2C-V *(CC)
(Al)
406
K.
H.
IWRWICH
where C is the concentration of some drug or labelled metabolite, D is a constant coefficient of diffusion, and v’ is the fluid velocity (blood, interstitial fluid, etc.) at some point. For the idealization of purely radial transport with constant radial velocity, V, within a spherical liver acinus, equation (Al) becomes A21 An approximate solution to equation (A2) for a point source within an infinite sphere can be found as follows. The solution to the corresponding one-dimensional problem is (Norwich & Zelin, 1970)
where ME is the mass of material injected at the origin at t = 0, and u is here the constant convective or drift velocity along the x axis. The corresponding three dimensional case with constant drift velocities VI, L’~, L‘? along the x, y and z axes respectively is
To approximate the case of radial symmetry in three dimensions, let us consider an “eight-point star”, where &of the injected molecules experience drift with velocity (v,v,v) (directed into one octant), & of the injected molecules experience drift with velocity (-v,v,u), 6 experience (v,-tl,v 1, etc. In each of the eight cases concentration is governed by an equation of the form (A4). The superposition of the eight functions for C(x,y,~,t) provides an approximation to the solution of the purely radial differential equation (A2). This was verified for the two-dimensional case by calculating values for C(x,y,t) from (the two-dimensional version of) equation (A4), and then confirming by a finite difference method that these values satisfy (the cylindrical symmetrical version of) the differential equation (A2). In all cases tested, the superposition of rectangular solutions (A4) satisfied the symmetrical differential equation (A2) quite adequately. We are interested in the case where sampling of concentration takes place at the origin, corresponding to sampling within the bloodstream. Setting X, y and z all equal to zero in equation (A4) gives the superposition of the eight solutions:
ME C(O,O,O,t~ = (47rD)“’
CLEARANCE
When u*t is substantially (A5) becomes
AS
A
PROCESS
OF
DIFFUSION
407
smaller than the diffusion coefficient, D, equation
which is the same as equation (2).