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A mathematical model of cadmium transport in a biological system
~SYIKOSMENI-AI.
RESE~K<.H
13, X9-214
A Mathematical K. E.
SHANK,
(1977)
Model of Cadmium Biological System
Transport
in a
R. J. VETTEK, AND P. L. ZIEMER
Received
January
28. 1976
The research was undertaken to analyze the uptake and distribution of “Yd in the mouse following repeated administrations and to develop a mathematical model that would describe the transport of cadmium in an animal organism. The model used was a mammillary compartmental model with the compartments corresponding to selected organs and tissues analyzed for “Vd content. The nine-compartment model was described by first-order kinetics. From the research it was observed that no administration of cadmium was influenced by previous administrations, and this allowed the use of the same rate constants in the model for each of the multiple dosings. The linear mammillary compartmental model was able to approximate the observed laboratory values in the present investigation. The derived model was also shown to be valuable for predicting the retention ofcadmium inother species without adjusting any of the rate constants.
INTRODUCTION
Cadmium is a toxic heavy metal and an environmental pollutant (Schroeder and Balassa, 1961; Friberg et crl., 1971) which has sprung into prominence to satisfy the demands of new industrial processes. The distribution of cadmium in laboratory animals has become of prime interest recently due to the metal’s toxicity and danger to the public. The literature contains reports of single-administration studies (Decker et al., 1957; Berlin and Ullberg, 1963: Miller et nl., 1968; Gunn ef nl., 1968; Lucis et (II., 1969; Mann, 1973). and investigations of tissue retention values at one sacrifice-time following multiple dosings (Dyer, 1973; Shanbaky, 1973). However, the pattern of cadmium distribution after each administration in a multiple dosing study has not been reported. The present investigation was designed to study the distribution of cadmium at specific times following each injection of repeated administrations to provide information regarding the dynamic transport of cadmium and to develop a mode1 to describe this transport. MATHEMATICAL
MODEL
Cadmium accumulation in mouse organs has been described mathematically by Matsubara-Khan (1974) and Matsubara-Khan and Machida (1975) using onecompartment and two-compartment theoretical models. The purpose of the present study was to develop a mathematical model that would describe the observed uptake and distribution of cadmium in a number of compartments. This model is important to better understand the transport of cadmium in a mammalian animal that is repeatedly exposed to the heavy metal from the environment. Predictions based on this mode1 could be of importance in assessing the impact of cadmium exposure in a mammalian system. A multiple-compartment mammillary mode1 was used to describe the biological
210
SHANK,
VETTER,
L
AND
%IEMER
J
G----l Spleen
FIL.
1, Schematic
of model
used in this study.
system (Sheppard, 1962). The model assumes first-order kinetics which implies that the rate coefficients are not a function of time or level of cadmium in a compartment. The model is presented in Fig. 1, where the compartments correspond to the selected organs and tissues analyzed for losCd content. The model is shown in system dynamics notation, where a rectangle implies a compartment, and the valve figure represents a rate (Fort-ester, 1968). The system equations were derived by noting that the rate of change of cadmium present in a compartment was equal to the amount entering per unit time minus the amount leaving per unit time. The equations are based on an income-and-loss rationale dictated by the conservation principle in which all the cadmium transmitted must be accounted for. In Fig. 1 the abbreviation L is used for the liver, K for kidney, P for pancreas, S for spleen, G for gastrointestinal tract, T for testes, B for blood, C for residual carcass, and F for feces. The equation for the liver in differential form is i = RBL*B - RLB*L, where L is the time rate of change of cadmium in the liver, RBL*B is the rate of movement into the liver where RBL is the rate constant for the blood to the liver and B is the percentage blood retention, and RLB*L is the rate of movement out of the liver where RLB is the rate constant for the liver to the blood and L is the percentage liver retention. Similar equations can be written for the other compartments.
Cd
20
TRANSPORT
MODE:L
IN
BIOLOGICAL
SSSTEMS
211
GI -irOLl . .
.
. 8
12 0 E
3
6
9 1
FIG. 3. Cadmium retention in the various compartments. The computed model values are shown as curved lines while the means of laboratory data are shown as solid circles. Values on the ordinate represent dose units where each injection contained 100 units (0.0833 I.L~of cadmium with 0.5 &i of ‘“9Cd).
METHODS AND MATERIALS
Male adult mice of the SW/NIH strain weighing from 22 to 26 g each were purchased from Murphy Farms and were fed Wayne Lab Blox and water ad lib. Carrier-free logCd (C,H,O,), purchased from New England Nuclear had no radionuclidic impurities as determined by differential gamma-ray spectroscopy and compared to a secondary NBS standard. Carrier was added to adjust the specific activity of each injection to 0.5 /.LCi per 0.0833 pg Cd, and the quantity injected was 0.1 ml. Sixty animals were assigned randomly to three subgroups and received one, two, or three injections spaced at 48-hour intervals. Each subgroup contained 20 animals with 4 mice being sacrificed after each of five different time periods. Preliminary studies indicated that appropriate sacrifice times were 2 minutes, 10 minutes, 1 hour, 10 hours, and 48 hours after dosing.
212
SHANK,
VETTER,
AND
ZIEMER
Compartment
Rate constant
Liver
RBL RLB
0.66 0.0016
Kidney
RBK RKB
0.067 0.0014
Pancreas
RBP RPB
0.03 0.0020
Spleen
RBS RSB
0.003 0.0038
GI
RBG RGB
0.24 0.006,
Testes
RBT RTB
0.0025 0.0021
Carcass
RBC RCB
0.39 0.0042
Feces
RGF
0.00072,
(1 The second
value
should
be used after
Value
the second
(min-‘)
0.01”
0.00043”
injection.
Intravenous administration via the tail vein was selected for the study because it was considered as being closest to the ideal of an instantaneous pulse which aided the modeling procedures. Only those animals with perfect first-attempt injections were used in this study. The animals were sacrificed by decapitation and exsanguination. Organ and tissue samples taken from each mouse included the liver, kidneys, pancreas, spleen, gastrointestinal tract (including contents), testes, whole blood, and feces. All samples were analyzed for l OsCd content in a large Harshaw NaI (TI) well crystal connected to an Ortec single-channel analyzer calibrated at 1 keV per channel and set to count gammas from 60 to 100 keV. RESULTS
AND DISCUSSION
The experimental values obtained from the distribution study are plotted as solid circles in Fig. 2. Standard errors were small throughout the experiment averaging TABLE COMPARISON
OF WE
VALUES
LABORATORY
2
PREDICTED VALUES
Liver Species Dogs Goats Sheep Model values
BY THE
ot MANN
75.46 77.33 74.00 67.50
MODEL
TO THE
(1973). Kidneys
5.97 5.31 6.52 7.83
‘13
Organ Liver Kidneys Pancreas Spleen 01
Model
value
Laboratory
value
60.67 7.05
69.50 7.46
1.20 0.12
2.31
3.36
2.99
0.17
curves about 3%. The solid lines in Fig. 2 represent the predicted compartmental generated from the model. The compartmental retention values, which were obtained from the distribution study, were put into the model differential equations. The rate constants for the model were then determined by a linear regression scheme and were optimized by a least squares method. The final values are given in Table I. The rate constants were then utilized in a Gasp 1V simulation program (Pritsker. 1974) written to calculate the compartmental values over time. It should be noted that due to the independence of the injections, the same rate constants were used for all three injections, except RGB and RGF which were changed after the second injection to help the model values agree more accurately with the observed values. One purpose of developing the model was to provide a valuable tool in assessing the uptake and distribution of cadmium in all biological systems following exposure to the heavy metal. To test the model for other animals and to extend it beyond the present investigation, the model was used to predict the retention of cadmium in dogs, goats, and sheep without altering any rate constants. The predicted values were compared to the laboratory values of Mann (1973), who dosed each of the above species with one intravenous injection of cadmium acetate containing approximately 30 FCi. Comparison of the model and laboratory values for the liver and kidneys (the principal organs of retention) at 8 weeks after administration is presented in Table 2 in percentage of activity per organ. The laboratory values are the arithmetic means of three animals, and it can be noted that the predicted values were in good agreement with the observed values. The predictions of the model were tested against an experiment in which the intravenous route of administration was not used. Gunn et N/. (1968) injected mice subcutaneously with cadmium chloride. Two weeks after the administration they found 69.33% of the dose in the liver and 8.53% in the kidneys. The model values for the same time period were 67.51 and 7.83%, respectively, for the liver and kidneys. Finally, the model was tested for its validity dealing with multiple injections into another animal. Shanbaky (1973) injected cadmium acetate into rats at 48 hour intervals for a total of five times. He also administered zinc simultaneously with the cadmium at a Zn:Cd ratio of 10: 1 by weight. Comparison of the model values to the laboratory values in percentage of dose at 48 hours after the fifth injection is presented in Table 3. The laboratory values are arithmetic means of six animals. The model values were in good agreement with the observed values.
214
SHANK,
VETTER,
AND
ZIEMER
It has been shown, therefore, that the derived mathematical model is indeed a valuable tool for predicting the retention of cadmium in biological systems. The model was shown valid not only for mice as used in the present investigation, but for rats, dogs, sheep and goats. It has been shown to be valuable whether dealing with one or multiple dosings. It adequately approximated the cadmium retention values for different routes of administration, for different dose levels, for different chemical forms, and for additional zinc administration. Thus. it is suggested that the derived model has the potential for giving an approximation as to what can be expected for the transport of cadmium in all mammalian animals (including humans) that are exposed to the heavy metal. ACKNOWLEDGMENTS The authors research was Environmental
wish to thank Mrs. Kathryn Johnson for her valuable assistance supported in part by the United States Public Health Service, Health Sciences. under Training Grant 5-TOI-ESOO071.
in the laboratory. National Institute
This for
REFERENCES Berlin, M., and Ullberg. S. (1963). The fate of cadmium-109 in the mouse. Arch. Ent~iron. Hedrh 7, 686-693. Decker, C. F., Byerrum, R. U., and Hoppert. C. A. (1957). A study ofthe distribution and retention of cadmium-l 15 in the albino rat. Arch. Biochem. Biophsy. 66, 140-145. Dyer, R. D. (1973). “An Organ Distribution Study of Intraperitoneally Injected Cadmium-l l5m in Mice, Gerbils, Rats, Rabbits, and Chickens.” M. S. Thesis, Purdue University. Wright-Allen Press, Cambridge. Forrester. J. W. (1968). “Principles of Systems.” Friberg, L.. Piscator, M., and Norberg. G. (1971). “Cadmium in the Environment.” Chemical Rubber Co., Cleveland. Ohio. Gunn, S. A.. Gould, T. C.. and Anderson, W. A. D. (1968). Selectivity of organ response to cadmium injury and various protective measures. J. Prrtlrol. Bncrrriol. 96, 89-94. Lucis, 0. J., Lynk. M. E., and Lucis, R. (1969). Turnoverofcadmium-109inrats.,4uc/r. Erzr,iron. Heulth 18, 307-3 IO. Mann. S. J. (1973). “Whole Body Retention and Tissue Distribution of Intravenously Administered M. S. Thesis. Purdue University. lrsmCd in Goats, Sheep, and Dogs.” Matsubara-Khan. J. (1974). Compartmental analysis for the evaluation of biological half-lives of cadmium and mercury in mouse organs. Ent?ron. Rex. 7. 54-67. Matsubara-Khan. J., and Machida, K. (1975). Cadmium accumulation in mouse organs during the sequential injections of cadmium-109. Enr~iron. Res. IO, 29-38. Miller, W. J., Blackman, D. M., and Martin, Y. G. (1968). Cadmium-109 absorption, excretion. and tissue distribution following single oral and intravenous doses in young goats. J. Dairy Sci. 51, 1836-1839. Pritsker. A. A. B. (1974). “The Gasp IV Simulation Language.” Wiley, New York. Schroeder, H. A.. and Balassa, J. J. (1961). Abnormal trace metals in man: Cadmium.J. Chronic Dis. 14, 936-258. Shanbaky. M. M. (1973). “A Radiotracer Distribution Study of Repeated AdministrationofCadmium in the Albino Rat.” M. S. Thesis, Purdue University. Sheppard, C. W. (1962). “Basic Principles of the Tracer Method,” Wiley, New York.
RESE~K<.H
13, X9-214
A Mathematical K. E.
SHANK,
(1977)
Model of Cadmium Biological System
Transport
in a
R. J. VETTEK, AND P. L. ZIEMER
Received
January
28. 1976
The research was undertaken to analyze the uptake and distribution of “Yd in the mouse following repeated administrations and to develop a mathematical model that would describe the transport of cadmium in an animal organism. The model used was a mammillary compartmental model with the compartments corresponding to selected organs and tissues analyzed for “Vd content. The nine-compartment model was described by first-order kinetics. From the research it was observed that no administration of cadmium was influenced by previous administrations, and this allowed the use of the same rate constants in the model for each of the multiple dosings. The linear mammillary compartmental model was able to approximate the observed laboratory values in the present investigation. The derived model was also shown to be valuable for predicting the retention ofcadmium inother species without adjusting any of the rate constants.
INTRODUCTION
Cadmium is a toxic heavy metal and an environmental pollutant (Schroeder and Balassa, 1961; Friberg et crl., 1971) which has sprung into prominence to satisfy the demands of new industrial processes. The distribution of cadmium in laboratory animals has become of prime interest recently due to the metal’s toxicity and danger to the public. The literature contains reports of single-administration studies (Decker et al., 1957; Berlin and Ullberg, 1963: Miller et nl., 1968; Gunn ef nl., 1968; Lucis et (II., 1969; Mann, 1973). and investigations of tissue retention values at one sacrifice-time following multiple dosings (Dyer, 1973; Shanbaky, 1973). However, the pattern of cadmium distribution after each administration in a multiple dosing study has not been reported. The present investigation was designed to study the distribution of cadmium at specific times following each injection of repeated administrations to provide information regarding the dynamic transport of cadmium and to develop a mode1 to describe this transport. MATHEMATICAL
MODEL
Cadmium accumulation in mouse organs has been described mathematically by Matsubara-Khan (1974) and Matsubara-Khan and Machida (1975) using onecompartment and two-compartment theoretical models. The purpose of the present study was to develop a mathematical model that would describe the observed uptake and distribution of cadmium in a number of compartments. This model is important to better understand the transport of cadmium in a mammalian animal that is repeatedly exposed to the heavy metal from the environment. Predictions based on this mode1 could be of importance in assessing the impact of cadmium exposure in a mammalian system. A multiple-compartment mammillary mode1 was used to describe the biological
210
SHANK,
VETTER,
L
AND
%IEMER
J
G----l Spleen
FIL.
1, Schematic
of model
used in this study.
system (Sheppard, 1962). The model assumes first-order kinetics which implies that the rate coefficients are not a function of time or level of cadmium in a compartment. The model is presented in Fig. 1, where the compartments correspond to the selected organs and tissues analyzed for losCd content. The model is shown in system dynamics notation, where a rectangle implies a compartment, and the valve figure represents a rate (Fort-ester, 1968). The system equations were derived by noting that the rate of change of cadmium present in a compartment was equal to the amount entering per unit time minus the amount leaving per unit time. The equations are based on an income-and-loss rationale dictated by the conservation principle in which all the cadmium transmitted must be accounted for. In Fig. 1 the abbreviation L is used for the liver, K for kidney, P for pancreas, S for spleen, G for gastrointestinal tract, T for testes, B for blood, C for residual carcass, and F for feces. The equation for the liver in differential form is i = RBL*B - RLB*L, where L is the time rate of change of cadmium in the liver, RBL*B is the rate of movement into the liver where RBL is the rate constant for the blood to the liver and B is the percentage blood retention, and RLB*L is the rate of movement out of the liver where RLB is the rate constant for the liver to the blood and L is the percentage liver retention. Similar equations can be written for the other compartments.
Cd
20
TRANSPORT
MODE:L
IN
BIOLOGICAL
SSSTEMS
211
GI -irOLl . .
.
. 8
12 0 E
3
6
9 1
FIG. 3. Cadmium retention in the various compartments. The computed model values are shown as curved lines while the means of laboratory data are shown as solid circles. Values on the ordinate represent dose units where each injection contained 100 units (0.0833 I.L~of cadmium with 0.5 &i of ‘“9Cd).
METHODS AND MATERIALS
Male adult mice of the SW/NIH strain weighing from 22 to 26 g each were purchased from Murphy Farms and were fed Wayne Lab Blox and water ad lib. Carrier-free logCd (C,H,O,), purchased from New England Nuclear had no radionuclidic impurities as determined by differential gamma-ray spectroscopy and compared to a secondary NBS standard. Carrier was added to adjust the specific activity of each injection to 0.5 /.LCi per 0.0833 pg Cd, and the quantity injected was 0.1 ml. Sixty animals were assigned randomly to three subgroups and received one, two, or three injections spaced at 48-hour intervals. Each subgroup contained 20 animals with 4 mice being sacrificed after each of five different time periods. Preliminary studies indicated that appropriate sacrifice times were 2 minutes, 10 minutes, 1 hour, 10 hours, and 48 hours after dosing.
212
SHANK,
VETTER,
AND
ZIEMER
Compartment
Rate constant
Liver
RBL RLB
0.66 0.0016
Kidney
RBK RKB
0.067 0.0014
Pancreas
RBP RPB
0.03 0.0020
Spleen
RBS RSB
0.003 0.0038
GI
RBG RGB
0.24 0.006,
Testes
RBT RTB
0.0025 0.0021
Carcass
RBC RCB
0.39 0.0042
Feces
RGF
0.00072,
(1 The second
value
should
be used after
Value
the second
(min-‘)
0.01”
0.00043”
injection.
Intravenous administration via the tail vein was selected for the study because it was considered as being closest to the ideal of an instantaneous pulse which aided the modeling procedures. Only those animals with perfect first-attempt injections were used in this study. The animals were sacrificed by decapitation and exsanguination. Organ and tissue samples taken from each mouse included the liver, kidneys, pancreas, spleen, gastrointestinal tract (including contents), testes, whole blood, and feces. All samples were analyzed for l OsCd content in a large Harshaw NaI (TI) well crystal connected to an Ortec single-channel analyzer calibrated at 1 keV per channel and set to count gammas from 60 to 100 keV. RESULTS
AND DISCUSSION
The experimental values obtained from the distribution study are plotted as solid circles in Fig. 2. Standard errors were small throughout the experiment averaging TABLE COMPARISON
OF WE
VALUES
LABORATORY
2
PREDICTED VALUES
Liver Species Dogs Goats Sheep Model values
BY THE
ot MANN
75.46 77.33 74.00 67.50
MODEL
TO THE
(1973). Kidneys
5.97 5.31 6.52 7.83
‘13
Organ Liver Kidneys Pancreas Spleen 01
Model
value
Laboratory
value
60.67 7.05
69.50 7.46
1.20 0.12
2.31
3.36
2.99
0.17
curves about 3%. The solid lines in Fig. 2 represent the predicted compartmental generated from the model. The compartmental retention values, which were obtained from the distribution study, were put into the model differential equations. The rate constants for the model were then determined by a linear regression scheme and were optimized by a least squares method. The final values are given in Table I. The rate constants were then utilized in a Gasp 1V simulation program (Pritsker. 1974) written to calculate the compartmental values over time. It should be noted that due to the independence of the injections, the same rate constants were used for all three injections, except RGB and RGF which were changed after the second injection to help the model values agree more accurately with the observed values. One purpose of developing the model was to provide a valuable tool in assessing the uptake and distribution of cadmium in all biological systems following exposure to the heavy metal. To test the model for other animals and to extend it beyond the present investigation, the model was used to predict the retention of cadmium in dogs, goats, and sheep without altering any rate constants. The predicted values were compared to the laboratory values of Mann (1973), who dosed each of the above species with one intravenous injection of cadmium acetate containing approximately 30 FCi. Comparison of the model and laboratory values for the liver and kidneys (the principal organs of retention) at 8 weeks after administration is presented in Table 2 in percentage of activity per organ. The laboratory values are the arithmetic means of three animals, and it can be noted that the predicted values were in good agreement with the observed values. The predictions of the model were tested against an experiment in which the intravenous route of administration was not used. Gunn et N/. (1968) injected mice subcutaneously with cadmium chloride. Two weeks after the administration they found 69.33% of the dose in the liver and 8.53% in the kidneys. The model values for the same time period were 67.51 and 7.83%, respectively, for the liver and kidneys. Finally, the model was tested for its validity dealing with multiple injections into another animal. Shanbaky (1973) injected cadmium acetate into rats at 48 hour intervals for a total of five times. He also administered zinc simultaneously with the cadmium at a Zn:Cd ratio of 10: 1 by weight. Comparison of the model values to the laboratory values in percentage of dose at 48 hours after the fifth injection is presented in Table 3. The laboratory values are arithmetic means of six animals. The model values were in good agreement with the observed values.
214
SHANK,
VETTER,
AND
ZIEMER
It has been shown, therefore, that the derived mathematical model is indeed a valuable tool for predicting the retention of cadmium in biological systems. The model was shown valid not only for mice as used in the present investigation, but for rats, dogs, sheep and goats. It has been shown to be valuable whether dealing with one or multiple dosings. It adequately approximated the cadmium retention values for different routes of administration, for different dose levels, for different chemical forms, and for additional zinc administration. Thus. it is suggested that the derived model has the potential for giving an approximation as to what can be expected for the transport of cadmium in all mammalian animals (including humans) that are exposed to the heavy metal. ACKNOWLEDGMENTS The authors research was Environmental
wish to thank Mrs. Kathryn Johnson for her valuable assistance supported in part by the United States Public Health Service, Health Sciences. under Training Grant 5-TOI-ESOO071.
in the laboratory. National Institute
This for
REFERENCES Berlin, M., and Ullberg. S. (1963). The fate of cadmium-109 in the mouse. Arch. Ent~iron. Hedrh 7, 686-693. Decker, C. F., Byerrum, R. U., and Hoppert. C. A. (1957). A study ofthe distribution and retention of cadmium-l 15 in the albino rat. Arch. Biochem. Biophsy. 66, 140-145. Dyer, R. D. (1973). “An Organ Distribution Study of Intraperitoneally Injected Cadmium-l l5m in Mice, Gerbils, Rats, Rabbits, and Chickens.” M. S. Thesis, Purdue University. Wright-Allen Press, Cambridge. Forrester. J. W. (1968). “Principles of Systems.” Friberg, L.. Piscator, M., and Norberg. G. (1971). “Cadmium in the Environment.” Chemical Rubber Co., Cleveland. Ohio. Gunn, S. A.. Gould, T. C.. and Anderson, W. A. D. (1968). Selectivity of organ response to cadmium injury and various protective measures. J. Prrtlrol. Bncrrriol. 96, 89-94. Lucis, 0. J., Lynk. M. E., and Lucis, R. (1969). Turnoverofcadmium-109inrats.,4uc/r. Erzr,iron. Heulth 18, 307-3 IO. Mann. S. J. (1973). “Whole Body Retention and Tissue Distribution of Intravenously Administered M. S. Thesis. Purdue University. lrsmCd in Goats, Sheep, and Dogs.” Matsubara-Khan. J. (1974). Compartmental analysis for the evaluation of biological half-lives of cadmium and mercury in mouse organs. Ent?ron. Rex. 7. 54-67. Matsubara-Khan. J., and Machida, K. (1975). Cadmium accumulation in mouse organs during the sequential injections of cadmium-109. Enr~iron. Res. IO, 29-38. Miller, W. J., Blackman, D. M., and Martin, Y. G. (1968). Cadmium-109 absorption, excretion. and tissue distribution following single oral and intravenous doses in young goats. J. Dairy Sci. 51, 1836-1839. Pritsker. A. A. B. (1974). “The Gasp IV Simulation Language.” Wiley, New York. Schroeder, H. A.. and Balassa, J. J. (1961). Abnormal trace metals in man: Cadmium.J. Chronic Dis. 14, 936-258. Shanbaky. M. M. (1973). “A Radiotracer Distribution Study of Repeated AdministrationofCadmium in the Albino Rat.” M. S. Thesis, Purdue University. Sheppard, C. W. (1962). “Basic Principles of the Tracer Method,” Wiley, New York.