Computer Methods and Programs in Biomedicine 26 (1988) 137-144 137 Elsevier

CPB 0087,1

Sect ion II. Systems and p rog rams

Microcomputer program for interactions in drug elimination in the rat

J o h n C. L e w i s 1, R i c h a r d H. L u e c k e 2 a n d W a l t e r W o s i l a i t 3

1 Department of Chemical Engineerin~ 2 Department of Chemical Engineering and 3 Department of Pharmacology University of Missouri, Columbia, MO, U.S.A.

In modern medicine patients often require multiple drug therapy. Such therapy can be modeled by a physiological flow model using interlinked differential equations to represent the differential rates of delivery and uptake of a drug by organs. Included in the model are the sites of action, toxicity, elimination and drug binding. A menu-driven computer program for this model was developed in FORTRAN 77 for execution on a microcomputer. The program computes the rate of uptake and concentration of drug in key tissues as a function of time. Drug administration can be modeled for bolus injection with up to 10 repeats and/or for continuous infusion of the drug. The drugs warfarin and adriamycin are used as illustrative examples. The program can handle multifactorial problems such as acute and chronic competitive elimination interactions, liver damage, and features of aging such as reduced drug binding and organ function. The computed results can be printed numerically or displayed graphically, either on a screen terminal or as hard copy. The results of several simulations may be cross-plotted for studies involving parametric changes such as would occur in multiple pathology and aging.

Drug elimination; Rat; Microcomputer; Physiological flow model; FORTRAN

1. In t roduct ion A ma themat i ca l mode l can s imula te the t ime

vs. concen t ra t ion behav ior of drugs in vi ta l t issues A ma jo r goal of medic ine and p h a r m a c o l o g y is the which are the sites of ac t ion o r toxicity. Simula- safe and effective use of drugs to a t ta in a des i red t ions pe rmi t an invest igator , or a cl inician, to the rapeu t ic effect. Drugs are po ten t i a l ly danger- s tudy the effects of var ious d rug regimens, of ous, especial ly those wi th a na r row marg in of a l te red phys io logica l s tates of the pat ient , or of safety. Mul t ip le d rug the rapy is c o m m o n l y em- drug in terac t ions occurr ing in mul t ip le drug ther- p loyed which presents the risk of e l imina t ion in- apy. A good mode l also can give in fo rma t ion teract ions. The admin i s t r a t ion of mul t ip le combi - abou t var ious combina t ions of these factors which na t ions of drugs is po ten t i a l ly more hazardous might occur s imul taneously . than that of a single drug [1,2]. Whi le drug in ter - Bischoff [3] and others deve loped a mode l form ac t ions are not necessar i ly all harmful , not know- k n o w n as the phys io logica l f low model which is ing they are occurr ing, or not p rope r ly evalua t ing based upon organ weights or volumes (V) , ra tes of them, can be harmful . Quan t i t a t ive in fo rma t ion b lood flow to the organs (Q) , and the ra t io of concern ing an e l imina t ion in te rac t ion can be the b ind ing affinit ies of the, drug in var ious tissues basis for a change to a safer drug schedule, c o m p a r e d to the concen t ra t ion in the p l a sma (R) .

The dyna mic flux of drugs through the b o d y and tissues is represented by a set of d i f ferent ia l equa- t ions deve loped f rom uns t eady state mass bal- Correspondence: R.H. Luecke, Department of Chemical En-

gineering, University of Missouri, Columbia, MO65211, U.S.A. ances. The descr ib ing equat ions are solved using

0169-2607/88/$03.50 © 1988 Elsevier Science Publishers B.V. (Biomedical Division)

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numerical methods on digital computers. Drug processing. No direct programming ability is re- concentrations are calculated at the sites of action, quired. The output can be presented in the form elimination and toxicity, of tables of drug concentrations in tissues vs. time,

We have developed a computer program for or as graphs of these quantities. The computed this type of model using a main-frame computer, output can be displayed either on the computer It was verified experimentally in the rat with the screen or as printed tables. anticoagulant warfarin [4,5], which is involved in These programs are intended to be useful for numerous drug-drug interactions. We have dem- the medical practitioner and in teaching or re- onstrated its effectiveness in simulation of search. elimination interactions between warfarin and BSP [5]. BSP is especially useful as an interactive com- petitor because of its rapid clearance from the 2. Hardware and software specifications

animal which permits a study of the onset, peak The mathematical model consists of a set of FOR- effect and duration of an interaction. The interac-

TRAN-77 programs that compute drug concentra- tions are modeled using kinetic parameters de- tions as a function of time for the plasma and termined for each of the single drugs, various organs and tissue regions. These programs

The anticoagulant drug model was expanded for the antineoplastic drug, adriamycin [6,7]. The were modified, where necessary, to compile and

execute using the MicroSoft FORTRAN version modifications consisted of changes for the values 3.31 Compiler on an IBM microcomputer. Except for various parameters, the addition of new com- for the built-in graphics output, which is compati- partments to represent the distribution of ble only with IBM Color graphics, there is no adriamycin and for evaluation at the sites of ac-

reason why these programs should not operate on tion or toxicity, i.e., bone marrow and heart. This other PCs with a FORTRAN-77 compiler. The model was also verified experimentally in the rat programs also operate on mainframe computers. [7]. These simulations provide a vivid means of The source listing is about 70K, the object deck illustrating the changes in tissue concentrations about 120K, and the executive run module is which are not readily apparent from general con- siderations, about 160K. A microcomputer with a memory size

In this paper, the two models have been merged of 200K or larger is necessary. For the IBM XT type of microcomputer, an

into a single set of programs so that only changes 8087 math coprocessor is a practical necessity for of parameters are required to simulate one system the use of these programs. With the math copro- or the other. With suitable changes in the parame- cessor, the computer run-time is about 80 seconds ters, many other drug systems also may be simu- for a case study for one injection of one drug lated. The simulations may be run with a single studied for a two-hour period. For a more com- drug or with two interacting drugs. The method of

plex case of two interacting drugs, with a single numerical solution to the differential equations bolus injection of one drug, and two bolus injec- was also changed to eliminate certain numerical tions plus part-time infusion of the other, simula- problems that could occur, tion of a two-hour period requires about 4 minutes

The most important change, however, was that of computer time. A version linked to operate only the programs were adapted for use on personal on PCs with the 8087 coprocessor is twice as fast. microcomputers for those who do not have ready Without the math coprocessor, computer simula- access to mainframe computers. The parameter tion time is increased about tenfold. information is input to the program, not in a data file, but instead in the 'user-friendly' method of answering questions displayed on the computer 3. Program structure screen.

These programs may now be used by anyone The program set consists of a main program and having basic skills in operating a PC, such as word subprograms. The mathematical models have been

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described in some detail elsewhere [4-7]. In this entered. The values of various parameters are paper, most attention will be given to the PC echoed to an output file, after which the compu- interface, tations begin.

3.1. Input 3.2. Program operation

The program is menu driven and the user may The equations are integrated numerically using a specify either of two basic drug systems for de- new differential equation solver which is in the fault values of the various parameters. One of the public domain developed by M.E. Davis (personal systems described is the anticoagulant, warfarin, communication). This method uses a fourth-order as it interacts with the drug BSP [4,5] and the Runge-Kutta-Fehlberg routine unless the equa- other is the antineoplastic drug, adriamycin [6,7]. tions become 'stiff'. This is evidenced when the A prototype interacting drug may also be specified internally adjusted step sizes become very small. for adriamycin, although experimental parameter When that occurs, the method switches to the data is not yet available. Any or all of the parame- semi-implicit Runge-Kutta method of Michelson ters for the drugs, or for the allometric equations [8]. None of the examples shown exhibited enough of the rat, may be changed from the default to stiffness to require the Michelson method. If the simulate physiological changes resulting from pa- Michelson semi-implicit method is used, computer thology or aging, run time increased 3 to 10 times.

The model consists of a set of 13 simultaneous Separate initial starts are provided for each new first-order differential equations for each drug, i.e. bolus injection of drugs. Multiple injections cause 26 in total. For warfarin, the model computes longer run times because of the steeper gradients drug concentrations in the plasma, muscle, skin, and because of the adaptation of step size. kidney, liver, gut, bile duct and bile. For The programs are written so that the integra- adriamycin, a 'tight' compartment was added to tion procedure is called as subroutine DGEAR account for the almost irreversible binding of a with the same argument list as the commercial portion of the drug, and compartments were added IMSL [9] library subroutine. If the user has a for the bone marrow and heart. The bone marrow, version of the IMSL library available, it may be gut, and heart are principal sites of toxicity for linked with the drug program. If the stiff method this antineoplastic drug. is required, with DGEAR the run times are in-

Input specifications are entered in response to creased only 20-40% over the times required for prompts on the computer screen. Default values the nonstiff equations. are shown for each prompt; changes may be entered to reflect abnormal physiological function. 3.3. Output After specifying the animal weight (rat), changes may be made in the basic allometric parameters. Parameter input is echoed to a file (PARAM.OUT) The type of drug system is entered, after which for later reference if desired. This reference file one specifies whether a single drug or two inter- also contains other computational information, acting drugs are to be simulated, such as the number of steps required for various

Prompts then appear for any changes desired portions of the total time interval. for drug binding or kinetic elimination parameters The time vs. concentration data is output to a ( K m and Vm~x). Finally the input dosage and working file (DRUG.OUT). When the solution to treatment regimens are specified. For each drug, the equation is complete, the user may print or up to 10 bolus intravenous injections may be plot the results. Any or all of the tissue or plasma specified of varying doses and time spacing, concentrations (up to six at a time) may be selected and/or an intravenous infusion may be specified for printing; 100 points are printed over the entire for part or all of the simulation interval. The time interval. Injection and infusion times are simulation time interval length is the last variable extended to the nearest output time. Plots of the

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data may be requested in addition to, or in place Michaelis-Menten kinetic rate coefficients for drug of, printing. A plotting menu allows the user to i and drug j respectively. plot any combination of concentrations of either BSP has a second pathway for elimination that or both drugs, with up to seven lines on the graph, is not shared with warfarin. An unlimited number of graphs and tables may be produced, r = Vm" CB

All concentrations are plotted dimensionless (C B + Km) 2 from zero to 1 with the data scaled to the maxi- mum of the first curve to be plotted. There is an The program contains the first rate equation option, however, for scaling each curve to its own for both drugs, and the second for BSP only. Any maximum value. To aid in composing the plots, values may be entered for the kinetic parameters, they are all displayed on the screen before print- V~ and K m. The values may be obtained by direct ing. The graphics are produced using the MEF experiment, or by fitting the parameters to NOLIMIT subroutine package for Microsoft elimination data[11]. FORTRAN [10]. The previously described results of warfarin-

Several computer runs using different condi- BSP interaction [5,12] can be obtained by using all tions may also be cross-plotted. This is only slightly of the parameter defaults for warfarin. The pro- more complicated than plotting data from a single gram also handles repeated administration of a run. A separate program, CROSSP, reads data drug, which is a common practice. The concentra- from multiple files (as prompted) and produces a tions of warfarin in the plasma and bile with single file for plotting, injections of BSP at 20, 80, and 140 minutes are

All plots are printed on a dot-matrix printer, shown in Fig. 1 [5]. These plots were made using The figures shown in this article were printed on the built-in graphics package. The labels on the an Okidata Microline 192 printer. The quality of curves were added manually; the graphics package the plots depends, to some extent, on the quality does not have the capability to label curves. The of the hardware, rate of elimination of warfarin is greatly reduced

The concentration vs. time data may also be plotted with other packages that contain graphics such as Lotus 123 or VP Planner. ~ ..... i BsP

~" I i -'~ n p osma 3.3.1. Sample results ~ Warfarin in p lasma ~i

An example of an interactive drug system is the ~ ~ ~ 1 ' combination of the anticbagulant warfarin and BSP. BSP is especially useful for modeling because I I I~i

Wo or / it is rapidly cleared from the body, permitting x~ [ ~Jn bile f ~/ /.--"$, '~. ~-- study of the onset, the peak effect, and duration of E ] ] ~ j II~/ / / /,ll ! / / an inhibiting interaction with warfarin. These g ] I I \ / [ l ~ ' / ~ / . drugs share a common route of elimination which u / can be described by the competitive inhibition ~ / I ~'v\ Ik~k ]k~',~ form of the Michaelis-Menten equation [12]:

v.,,. C, ri = ( C, + Kmi + Cj x Kmi//Kmj) warfarin; BSP injections C 20,80,140; 200 min

Fig. 1. The effect of repeated injections of BSP on the elimina- Here, r~ is the rate of elimination of drug i p e r tion of warfarin. A single bolus injection of warfarin (300 g) is

followed by bolus injections of BSP (15 mg) at 20, 80, and 140 unit volume of liver, C; is the concentration of minutes. The reduced concentrations of warfarin in the bile drug i in the liver, Cj is the concentration of drug indicates the suppression of the clearance of warfarin by j in the liver, and Vmi, Kmi and Kmj are the competition of the BSP.

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after each injection of BSP, as indicated by the Many drugs are administered by infusion. reduced concentration of warfarin in the bile; the Adriamycin is administered by short-term infu- rate increases again as the BSP clears from the sion. The effect of the length of infusion on the system, concentration of adriamycin in plasma and mar-

Many drugs are metabolized in the liver and row can be analyzed using this model. The longer excreted in the bile. Liver damage can affect both infusion times greatly reduce peak concentrations of these processes. Curves showing the combined in the heart, which is a site of toxicity for effects of interactions of warfarin with injections adriamycin, but has a lesser effect on the con- of BSP and reduced liver function are shown in centrations in marrow, a site of action. Fig. 2. This graph was made by cross-plotting data The elimination of adriamycin also can be af- obtained from separate simulation runs; the curves fected by reduced liver function and/or competi- are scaled to a common maximum. The concentra- tive inhibition by another drug such as BSP or tions of warfarin in bile are shown for normal rifamycin. Parameters for competitive inhibition liver function without an interacting drug. Also of adriamycin by any specific second drug are not shown are the concentrations in bile when the yet available, but a prototype drug can be hy- liver function is reduced to 50% of normal and pothesized with the same parameters as BSP with when BSP is injected. For this demonstration ex- warfarin. ample, the reduction in liver function is assumed The effect that such a drug would have on the to be (a) in capacity of the organ to store the drug elimination of adriamycin is illustrated in Figs. 3 and (b) in the rate of eliminating the drug. Of and 4. With a five-minute infusion of adriamycin, course, for some special situations the liver dys- multiple bolus injections of drug 'B' retard function may be modeled by different parameters, elimination of adriamycin. Fig. 3 shows the re- For the case shown, the rate of clearance of duced elimination rate as evidenced by low bile warfarin is greatly reduced with the lower liver concentrations of adriamycin with injections of function and the interaction with BSP. The in- 'B'; Fig. 4 shows the higher concentration of creased retention of the drug could be vitally adriamycin in the marrow with the injections of important in a clinical situation. 'B'. The concentration in the marrow is somewhat

,,'--~---____.__.__ NO BSP, 100% liver function / \

7 ,"

i / /t"/ " I f~ / .,'-, t ' / / ,

" i i, 'BSP, 100% t ~ " / I ,' liver function , /" ,.,' BSP, 100% liver function , / I I ~

~'ti: f !11 U ~ / __..~.. 7 /~ ~ eE ~" ~ . ~ i BS,,I 100~,o" hver function " 0_ ~ ~_ / °

, BSP, 50% liver function BS i"~'"q"'l"" I'"I"'I'"F'"r"'I'W" T" T"I"" F"l'"l''l'" F'T "'1 50;~o li - - --

Warfarin; W &W/O BSP & liver damoge .... I r r " l r r " l ' i " l ' " ' ! ' " ' f r " r l " ' [ ' " ' J r " ' [ ' " ' i ' i " ] " I . . . . . . . . . , , : - - ~ - t . . - . , ~

Adriomycin- W & W/O BSP & liver domoge Fig. 2. The effect of liver damage on the elimination interac- tion between warfarin and BSP. As in Fig. 1, an injection of Fig. 3. Simulation of the effects of multiple injections of BSP warfarin is followed by three injections of BSP. Concentrations and reduced liver function on the clearance of adriamycin. An of warfarin in the bile are shown also for no BSP injections, infusion of adriamycin (585 #g/min for 4 min) is followed by and for the BSP injections with a 50% reduction in liver bolus injections (15 nag) of BSP at 20, 80, and 140 rain. The function. The concentrations in the bile are proportional to the lowered concentrations in the bile show reduced elimination

clearance rates, rates.

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~--'-'~sp, ~ suits presented in easy to understand tabular form / ~ ~ ~ . l i v e r f u n c t i o n ~ or vivid graphical displays. Simulations are an

elegant and effective way to demonstrate the con- sequences of altering one or more physiological or

,// - - ~ [ o liver function drug-scheduling parameter(s). Although warfarin

and adriamycin were used as illustrative examples, xa No BSP,'lOO%liver funct~n ~. .~ -~-~ ._ the program may be adapted to many other drug E ~ systems by appropriate changes of parameters. 'E The program described here provides vivid

graphic examples for normal drug elimination as 0_ well as for elimination occurring in patients with

liver damage. Simulations permit the pharmacolo- gist or clinician to get a better quantitative ap-

i~.~r.~i~.~r'~i~.~.~.~'r.~'1.~.l~'~r''i~'~I~..I~'.'F~1..~r.~r"~1 preciation of drug distribution under normal and Adriamycin- W &W/O BSP & liver damage pathological conditions.

Fig. 4. The effect of liver damage on the concentration of Future plans include extending the models to adriamycin in bone marrow. As in Fig. 3, an infusion of adriamycin is followed by bolus injections of BSP. Concentra- allow scale-up from the laboratory rat to man [13]. dons of adriamycin remain higher as a result of multiple The ultimate goal is to make multiple drug ther- injections of BSP, and remain high for a much longer period of apy in complex medical situations safer and more

time if there is also reduced liver function, effective. Suitable models, with the new ready availability of powerful microcomputers, make this a realistic goal.

higher in the presence of the competitive drug, and is much higher when there are both a compe- titive drug and liver damage. This effect could, of References course, be significant in drug treatment, resulting in higher concentrations of the drug in the tissue. [1] J. Hull, W.J. Murray, H.S. Brown, B.a. Williams, S. Chi

and G.G. Koch, Clin. Pharmacol. Ther. 234 (1978), 644-649.

4. Discussion [21 F.E. May, R.B. Stewart and L.E. Cluff, Clin. Pharmacol. Ther. 22 (1977) 322-328.

[3] K.B. Bischoff, R.L. Dedrick and J.A. Langstreth, J. Pharm. Physiological flow modeling permits remarkable sci. 60 (1971)1128-1133. analysis of drug distribution among tissues which [41 R.H. Luecke and W.D. Wosilait, Comput. Prog. Biomed. are the sites of action, toxicity, and elimination. It 8 (1978) 35-43. is based upon anatomical, physiological, biochem- [51 R.H. Luecke, L.E. Thomason and W.D. Wosilait, Camp.

Prog. Biomed. 11 (1980) 88-98. ical and pharmacological processes which are [61 R.H. Luecke, M.P. Ryan and W.D. Wosilait, Camp. Meth. fundamental to drug therapy. The model is flexi- Prog. Biomed. 20 (!985). 23-31. ble, in that parameters can be altered in accor- [7] W.D. Wosilalt, R.H. Luecke and M.P. Ryan, (1987) in: dance with changes resulting from multiple pa- Advances in Mathematics and Computers in Medicine,

ed. M. Whitten, (Pergammon Press, New York, 1987). thology and multiple drug therapy. While the [81 M.L. Michelson, AIChEJ. 22 (1976)594-598. abstractness of many interlinked differential equa- [91 IMSL Library, Edition 8 (IMSL, Houston TX, 1980). tions makes it difficult to visualize the conse- [10] NOLIMIT FORTRAN Subroutines Version 2 (MEF En- quences of changing one or more parameters to vironmental Inc., Austin TX, 1985). reflect pathological changes, readily available [11] R.H. Luecke and W.D. Wosilait J. Pharmacokin. Bio- powerful new microcomputers can help alleviate pharm. 14 (1986)65-78. this problem. If suitable programs have been [12] R.H. Luecke and W.D. Wosilait J. Pharmacokin. Bio-

pharm. 7 (1979) 629-641. developed with appropriate parameters, these [13] Y.S. Lee, K.K. Chan, P.A. Harris and J.L. Cohen, Cancer equations may be solved numerically and the re- 45 (9)2231-2239.

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Appendix: file management and execution or plotting. They may also be imported into SAS. The output working data files, PARAM.OUT

The execution files are: and DRUG.OUT, may be printed (or edited) if (1) PCDRUG. Uses input data files WARF.DAT desired. A previously calculated case may be re- or ADRI.DAT; integrates the differential equa- plotted or printed if the DRUG.OUT file still tions; produces a parameter file, PARAM.OUT; exists (or if it exists as a different file that may be produces a working output file, DRUG.OUT; can renamed DRUG.OUT). Type in "PRINTIT" to produce a data file for printing PRNT.OUT, can prepare PRNT.OUT from DRUG.OUT; type in produce plots. "PLOTIT" to plot the data in the DRUG.OUT (2) PRINTIT. Uses DRUG.OUT to produce print file. In all of the plotting routines, the "?" will files PRNT.OUT. bring up the main menu at any time. Note that (3) PLOTIT. Uses DRUG.OUT to produce plots. "7" exchanges the method of scaling. (4) CROSSP. Uses filenames supplied by the user To cross-plot data from several runs, type in to produce a working file DRUG.OUT. "CROSSP" and follow the instructions given. A (5) PLOTCP. Uses DRUG.OUT to produce cross DRUG.OUT file will be produced that contains plots, up to 4 concentrations from each of up to 6 files.

To use the program run the main program by These may be plotted by typing in "PLOTCP". typing in "PCDRUG" and then following the The plotting is slightly more complicated than instructions that appear on the screen. Plots may with "PLOTIT" since you must calculate the in- be produced and print files created. The print files dex position. For example, to plot the first con- may be imported into spread sheets such as centration of each of six runs, you must specify LOTUS 123 or VP-PLANNER for manipulation plot indices 1, 5, 9, 13, 17, and 21.