Numerical simulations of the infusion of adriamycin using a physiological flow model

Comput. Math. Applic. Vol. 14, No. 9-12, pp. 861-868, 1987 Printed in Great Britain. All rights reserved

0097-4943/87 $3.00+0.00 Copyright © 1987 Pergamon Journals Ltd

NUMERICAL SIMULATIONS OF THE INFUSION OF ADRIAMYCIN USING A PHYSIOLOGICAL FLOW

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W. D. WOSILAIT, 1 R. H. LUECKE 2 and M. P. RYAN j ~Dcpartments of Pharmacology and 2Chemical Engineering, University of Missouri, Columbia, MO 65212, U.S.A. A ~ t r a e t - - T h e antineoplastic drug Adriamycin is administered by short term i.v. infusion. The drug is noted for its acute and chronic cardiotoxieity in addition to its toxicity on the bone marrow and gut. A physiological flow model and computer program, which contains a subprogram for continuous infusion of a drug, was used to simulate the distribution of Adriamycin during infusion in heart, bone marrow, gut, kidney, muscle, skin, liver and bile. Simulations were carried out for 1, 2, 5, 10 and 20 min of infusion. Computations were made for the concentration of drug in these organs as a function of time over a total period of 48 min. Simulations showed the heart and kidney would contain high concentrations of drug during infusion which rapidly declined upon cessation of infusion. Bone marrow and gut showed a less rapid accumulation and decline. Muscle and skin showed even a slower accumulation and decline. Simulations also showed the effect of altering the rate of infusion on the concentration in the liver and excretion in the bile.

1. I N T R O D U C T I O N

Adriamycin is an antineoplastic drug which is noted for its cardiotoxicity in addition to its toxicity on the bone marrow and the gut. There can be an acute cardiotoxicity of arrythmias and a delayed toxicity similar to congestive heart failure., Because of its toxicity, Adriamycin is administered not by bolus injections, but by short term infusion. Tissue concentrations of the drug vary depending on the rate of blood flow to the organ and the tissue-to-plasma distribution ratio, as well as the concentration of the drug in the plasma. The concentration in the plasma is a dynamic function of the infusion rate and differential rates of uptake by tissues. In the past, the kinetics of infusion have been described by polyexponential equations and abstract compartments [1]. In view of the potential problem with toxicity, it would be very useful to have some means to determine the changing concentrations of drug in crucial tissue regions during and after its administration. To aid in this determination, a set of computer programs has been developed which are based on physiological function and flow rates, and which simulate the distribution of Adriamycin to various tissue regions and its elimination by excretion in the bile. These programs were adapted from a physiological flow model that was developed for the distribution and elimination of warfarin in the bile of the rat [2]. The simulation was extended and validated for Adriamycin in the rat [3]. In addition to bolus injections, the program can handle continuous infusion of a drug. This is important since Adriamycin is usually administered by short term infusion. The purpose of this report is to describe the results of numerical simulations of the effect of varying the rate of infusion of Adriamycin on the calculated concentration of the drug in various organs in the rat using a physiological flow model. 2. METHODS

2.1. Mathematical model A mathematical model was developed using a physiological flow model which included organ mass or volume, blood flow to the organs and tissue to plasma partition ratios [3]. Organ volumes or mass were computed using allometric equations of the following form: organ volume Y = a M b, t A preliminary report of this research was presented at the Federation of American Societies of Experimental Biology Meetings, St. Louis (April 1986). Fed. Abstr. 45(3), 207 (1986). 861

W.D. WOSILAITet al.

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in which Y is an anatomical variable, M is the mass of the body and a and b are constants which have been reported for different organs. The following volumes (ml) were calculated for a 234 g rat: plasma, 10.45; muscle, 117.0; gut, 12.5; kidneys, 2.15; liver, 9.6; skin, 38.8; and marrow, 2.5. Allometric equations are useful in scaling up experimental studies for animals of different weights, and from laboratory animals to humans [4, 5]. Blood flows to the organs were determined also using allometric equations of the same form. The computed flows (ml/min) used in the model for a 234 g rat were: muscle, 3.45; gut, 6.08; kidney, 5.75; liver, 7.46; skin, 0.28 and bone marrow, 0.64. The concentration of drug within each organ or tissue region was assumed to be uniform as in a stirred tank reactor. Separate differential equations are used for computing the concentration of drug in each organ as a function of time; an algebraic equation was used to compute the concentration of the drug in the heart. The differential equations in verbal equivalents can be represented as follows: [ t h e rate of ] I rate of inflow I uptake of = of drug to a drug an organ

Irate of outflow ] of drug from the organ

[- rate of metabolism ] [ and/or elimination l_by that organ

2.2. Animals studies

The basic model has been verified in the rat. In brief, the procedure used ~4C-Adriamycin which was obtained from the National Cancer Institute. The drug was diluted with non-radioactive Adriamycin for i.v. injection over a period of 2 min into anesthetized male Sprague-Dawley rats. Adriamycin is excreted primarily in the bile. The bile flow rate was 25 #1/min. Bile samples were collected from the cannulated duct for 120-150 min to measure the rate of excretion of the drug. These values were used in testing the model. Blood samples were obtained from the snipped tail at frequent intervals. These values also were used in validating the model. Tissue samples were obtained at the end of the experiment for the determinations of tissue to plasma partition ratios for the model parameters used in the present report: muscle, 4.27; gut, 17.87; kidney, 25.86; liver, 36.60; skin, 5.75; marrow, 47.74; and heart, 32.17. The kinetic parameters for the elimination rates were determined from the concentrations of drug in bile and plasma using a weighted least squares regression analysis. The values used were: Vmax= 10.25nmole x min -j g-~ and Km= 125 nmole x g-~. 2.3. Computer program

The simulation was implemented in FORTRAN to compute the concentration of drug in each organ in the model as a function of time [3]. The program can handle a single bolus injection of a drug, repeated (up to 10) bolus injections, as well as continuous infusion. Simulated continuous infusions (of 1, 2, 5, 10 and 20 min) were used exclusively for the present report for a total time of analysis of 48 min. It is possible in the programs to combine infusion with injections. The computed data for each organ and for each time of infusion (1, 2, 5, 10 and 20 min) were merged using SAS (statistical analysis systems) procedures and stored as merged data in a SAS Systems Library. The merged data were printed using SAS PROC PRINT and plotted on a CalComp 1077 precision plotter using SAS PROC GPLOT. 3. RESULTS 3.1. Concentration of Adriamycin in plasma

In the example shown (Fig. 1) the dose of Adriamycin was (10 mg/kg) for an animal weighing 0.234 kg. The computed volume of plasma (Vp) was 10.44 ml; thus, the calculated concentration of Adriamycin in the plasma at zero time, following a bolus injection, would be 224.1 nmole ml-~. The calculated concentrations in the plasma at the end of infusion for 1, 2, 5, 10 and 20 min of infusion are presented in Table 1 along with the percentage of the calculated concentration for uniform distribution in the plasma at zero time. Note that for infusion times greater than 2 min, the maximum plasma concentration is almost exactly proportional to the rate of infusion. Adriamycin is noted for its rapid disappearance from the plasma and its rapid distribution to tissues. This behavior is reflected in Fig. 1, which shows the computed concentration of Adriamycin

Numerical simulations o f the infusion of Adriamycin

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Fig. 1. Simulated concentration of Adriamycin in plasma during continuous infusion. The concentration of Adriamycin in the different organs in the model were computed for 1, 2, 5, 10 and 20 min ofi.v, infusion and stored on disk. The computed concentrations (nmole/ml - t ) in plasma for the different durations of infusion were merged and plotted using SAS P R O C G P L O T using the splin¢ option to provide the solid lines. For clarity the spline option was not used for the concentrations for 2 rain (<>) o f infusion.

in the plasma after infusions of the same total amount over periods of I, 2, 5, 10 and 20 min duration. The concentration of Adriamycin in the plasma increases throughout the course of infusion. The rate of increase in concentration depends on the rate of infusion. At the lowest rate (20 min duration), a near steady state condition is reached in which the rate of loss from the plasma is nearly equal to the rate of infusion. Following the infusion, there is a rapid decrease in the concentration of drug in the plasma which is a result, in part, of the low degree of binding by plasma proteins and the high degree of binding by tissue components. 3.2. Concentration in heart and kidney The concentration profiles of drugs in various tissues is a function of a number of factors, such as the concentration of drug in the plasma, the rate of blood flow to the tissues, and the tissue-to-plasma partition ratio. The shape of the profiles of the concentrations of Adriamycin in the heart (Fig. 2) and kidney (not shown) are very similar to that of the plasma but differ markedly from the profiles in other organs. These two organs with very high blood perfusion rates have a Table 1. The calculated concentration of Adriamycin in the plasma at the end of infusion for different times Time of infusion (min)

Conc. in plasma (nmole/ml-t )

Percent of 0 min' (%)

(P I 2 5 10 20

224.1 142.4 101.7 40.7 20.3 10.2

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Fig. 2. Simulations of the concentrations of Adriamycin in the heart, marrow and gut. The concentration of Adriamycin was computed for each organ in the model for 1, 2, 5, 10 and 20 min of infusion and stored on disk. The concentrations in each organ were merged and plotted using SAS PRO(3 GPLOT using the spline option (--) for 1, 5, 10 and 20 rain of infusion. For clarity, the sDline option was not used for 2 rain of infusion (<>).

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Table 2. Comparison of peak concentrations and time at different rates of infusion of Adriamycin

Tissue

Tissue a

Duration of

infusion (min)

Plasma

Muscle

Kidney

Liver

Bile

Gut

Marrow

Heart

Skin

1 2 5 10 20

142 102 41 20 10

8.7 (15)b 8.7 (16) 8.7 (18) 8.7 (21) 8.6 (27)

461(2) 455 (3) 391 (5) 309 (10) 204 (20)

35.2(6) 35.2 (7) 34.8 (9) 33.6 (12) 30.0 (20)

1030(12) 1031 (13) 1022 (15) 993 (18) 910 (26)

109(4) 109 (5) 108 (7) 105 (11) 93 (20)

79 ( 1 9 ) 79 (20) 79 (22) 79 (22) 78 (29)

4581(I) 3272 (1) 1764 (4) 1038 (9) 601 (19)

2.8(>48) 2.8 (> 48) 2.8 (> 48) 2.8 (>48) 2.8 (> 48)

"Concentration (nmole/ml- s). bThe numbers in parentheses indicate the time of the peak (rain) as determined from the computer printout of the data.

distinct concentration profile which is characterized by a peak which is reached during infusion, and then followed by a sharp drop in concentration when infusion is discontinued. As for the plasma, halving the rate of infusion halves the peak concentration reached in the heart. When infusion stops, Adriamycin is quickly transported out of the heart and kidney and into regions with higher binding ratios and slower rates of blood flow. In contrast, for example, the concentration in the gut (Fig. 2) and bone marrow (Fig. 2) reach a peak with a moderate decline when infusion is discontinued. 3.3. Concentration in gut and bone marrow

The toxicity of a drug in an organ can be related to various chemical and/or biochemical factors which differ for different organs depending upon their specialized functions. The gut and bone marrow are both noted for their toxic reactions to a variety of anti-neoplastic agents. In both of these tissues, the Adriamycin infusion causes a gradual increase in concentration followed by a gradual decline (Fig. 2). The peak concentrations occur well after the infusion has ceased. This behavior reflects the redistribution of the drug stored in other areas during the infusion. Note, unlike the plasma and heart peaks, that the peak concentrations in the gut and bone marrow are not greatly affected by the infusion rates studied here. The infusion rate would have to have been reduced by an order of magnitude to affect the concentrations in the bone marrow. The peak concentrations in the gut were attained earlier than in the marrow (Table 2), primarily because of the more rapid blood flow to the gut. The decrease in concentration in the gut was also faster than in the bone marrow for the same reason. 3.4. Concentration in muscle and skin

The muscle and skin are not special sites of toxicity for Adriamycin but, since they constitute a rather large fraction of the body mass, they store a significant fraction of the drug and are therefore included in the model. Table 2 shows the peak concentrations in these tissues and the time of the peak. The concentration profiles of the muscle resembles that of the bile (Fig. 3). The skin is unusual with a slow build-up that has not peaked at 48 min, presumably because the slow rate of blood flow to this tissue delays the attainment of the peak concentration. It represents a reservoir of Adriamycin that is returned to the plasma over an extended period. 3.5. Excretion of Adriamycin

Adriamycin is taken up by the liver, from which it is excreted in the bile. Figure 3 shows the time course of concentration in the liver. Note the difference in pattern from that shown for the heart. The profile in the liver more nearly resembles that of the gut which has a similar ratio of binding capacity to the blood perfusion rate. The profile for the concentration in the bile peaks somewhat later than that of the liver (Fig. 3), but note that the bile concentrations are 30 times higher than the average for the liver. This makes it clear that the drug is not distributed uniformly on a microscale in the liver. 4. DISCUSSION Drug distribution and elimination has been described by polyexponential models [1] and physiological flow models [2-4]. The polyexponential models use rate constants for the distribution of a drug in abstract compartments. On the other hand, the physiological flow model describes

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Fig. 3. Simulations of the excretion of Adriamycin in liver and bile. Computations were made, stored and plotted as for Figs 1 and 2.

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the kinetics of distribution of a drug into anatomical compartments using anatomical, physiological, and pharmacological parameters. A previous report has shown some relationships between these two models [6]. The physiological flow model can be reduced to a polyexponential model but not vice versa without providing additional information which in essence converts it to a physiological flow model. Formulae have been presented for continuous infusion of a drug using polyexponential models [1]. In the present work, continuous infusion is shown for a physiological flow model. Adriamycin is usually administered to humans in not less than 3-5 min. Our simulations included more rapid (1 and 2 min) and less rapid (10 and 20 min) infusions which provide some theoretical information which can be of practical value. The simulations show different concentration profiles. The heart and kidney showed sharp concentration peaks. The present simulations show that reducing the rate of infusion for the same total dose reduces the concentration, particularly in rapidly perfused tissues such as the heart and the kidney. One could conclude this intuitively, but simulations using anatomic and physiologic parameters provides a more quantitative picture of the magnitude and the kinetics of changes which would occur with different rates of infusion. Furthermore, one might not as easily predict that peak concentrations in slowly perfused regions such as bone marrow, are not greatly affected by drug infusion rates. Adriamycin is a drug noted for its reactivity forming free radicals which can react with DNA [7], and other cell fractions [8-10], and for its conspicuous organotoxicity. Elevated concentrations of the drug in a tissue could result in toxicity resulting from the higher concentration of drug and its highly reactive products. The model [3] also includes a compartment for almost irreversibly bound drug which can incorporate the covalent bonding. The present analysis is primarily kinetic rather than mechanistic. The physiological flow model has additional advantages [5, 11] beyond the ability to predict tissue concentrations. The interlinking of the equations in the model permits one to integrate anatomical, biochemical, physiological and organ pathology into a comprehensive picture which can be of value to the theoretical pharmacologist, the clinical pharmacologist and the oncologist. For example, the model and program can be adapted to cope with organ pathology such as liver disease, which is especially important for drugs metabolized in the liver and excreted in the bile [12]. Although the present report does not include the effects of liver damage on the kinetics and distribution of Adriamycin, the feasibility of including such effects has been described previously [ll, 12]. Although physiological flow models may appear excessively complex, they are really just a combination of relatively simple parts. Current computers and numerical methods can cope with these mathematical combinations. In.cancer chemotherapy, such models can help present a clearer picture of what actually occurs in the tissues and organs. Physiological flow models also can be scaled up from experimental animals to man [5]. Lee et al. [13] have reported a number of examples in which models have been successfully applied to human therapy. The combined understanding of anatomy, physiology and pathology incorporated into a comprehensive pharmacokinetic model could enhance the safety of cancer chemotherapy with toxic drugs.

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