Physiological flow model for drug elimination interactions in the rat

Cor.tputer Programs in Biomedicine I 1 (1980) 88-98 © Elsevier/NorthAloUand Biomedical Press

PHYSIOLOGICAL

FLOW MODEL FOR DRUG ELIMINATION

INTERACTIONS

IN THE RAT

Richard H. LUECKE and Lynn E. THOMASON DetJartment of Chemical Engineering, University of Missouri and Walter D. WOSILAIT Department of Pharmacology. University of Missouri, Columbia. Mt~r¢212. USA

Drug elimination interactions in the rat are modelled based on phys~,lmogicalblood flow rates and organ weights. A previous model has been substantially improved by the addition of a compartment representing the skin and the interactions are computed using Michaelis-Menten kinetics for competitive inhibition in the shared pathways. Furthermore, the results of repetitive dosing may also be simulated. The programs, which are extensively annotated and user oriented, are illustrated on the results of an acute warfarin-BSP interaction experiment in rats. Drug interactions

Physiological models

Warfarin

Evaluation o f interactive effects is very important and should modulate drug treatment. One type o f interaction that often occurs is the rate at which drugs are eliminated from the b o d y . Knowledge of this type o f interaction can be very important so that prope., adjustments in the, magnitude and frequency o f dosage can be made without exceeding toxicity limits or going into a sub-effective range. A computer program for calculation o f unsteady state concentrations o f interacting drugs is described in this paper. in classical pharmacokinetics, drug concentrations are calculated on the basis o f an empirical model with 2 or 3 compartments. As has been pointed out [ 1], these compartments rarely have anatomic or physiological significance and the form o f the model is determined on a statistical basis as ~he simplest that can adequately represent experimental data. These models are limited in their ability to extrapolate to conditions different from those under which the data were collected or to predict drug concentrations in specific tissues or organs.

1. Introduction

Modern drug therapy is marked b y the use o f multiple combinations o f drugs to achieve several different effects simultaneously. In these circumstances, many interactions can occur in addition to the main effects of drugs. Numerous such interactions have been reported in recent years and consideration o f these interactions is a vital element o f drug treatment. Furthermore, the frequency o f occurrence o f such interactions increases with the number o f drugs used.

Nometwlature : B = sub~trate conce~.tration in rate equation (specifically BSP in this Taper) (nmol/ml); C -~drug concentration in body regions ~nmol[ml);g(t) = function of time for drug input (nmol/min); k K = rate constant for elimination in kidney (min -l ); Km = rate equation parameter (nmol/ml); Q = blood flow rates (ml]min);r = general elimination rate in liver (nmol/min);r s = elimination rate for warfarin (nmol/min); r B = elimination rate for BSP (nmol/min);ri = t:ansport through bile duct (nmoi); R -- tissue binding ratio (dimensionless); S = substrate concentration in rate equation ($pecificially warfarin in this paper) (nmol/ml); V =-volume (ml); Vmax = rate equation parameter (nmol/min); r = delay time in bile duct (min)

Subscripts for tissues: -K = kidney, -L = liver, -M = muscle; - p = plasma; -S = skin 88

R.H. Luecke et al., Flow model for drug elimination imeractions

Pharmacokinetic models can also be developed by utilizing physiologk;al quantities such as volumes of organs and tissues, and blood flow rates. Although a large number of parameters and variables are involved in these models, many may be obtained from the literature. Some, such as blood flow rates are specific for the animal, while others such as tissue-to-plasma binding ratios must be derived from experiments using the drug in question. Elimination rate experiments usually are focused only on those parameters which model the reactions of the drugs with the enzymes that lead to the formation of metabolites. A physiological flow model can approximate drug concentrations in various tissues and organs throughout the body. it is predictive of some internal concentrations that are impossible to monitor. Because the parameters have identifiable anatomical significance, the flow models can even serve to some extent as a monitor of experimental data. Another important advantage of the physiological tlow model becomes evident with interacting drug systems or disease processes. Elimination interactions often result from competition between drugs for enzymes in rate-limiting steps in the elimination involving drug metabolism. This leads to the competitive inhibition equation for the elimination rate equatmns for drugs S and B of the following form [21: q -

VmaxS

Kms +

IBI [Sl

KmB

VmaxB rB = 1 ÷ KmB+ KmB" ISl

[B!

(1)

89

analysis on several of the rate parameters. A physiological flow model of interactive drug elimination in the rat is presented here. The particular interacting drugs that will be used to illustrate the computer program are BSP and the anticoagulant warfarin; this combination was also considered in an earlier paper [3], In the case of anticoagulants, interactions can occur at various points in the elimination pathway. The interaction between warfarin and BSP is particularly useful for kinetic analysis because the onset of action, the peak effect and the duration of action can be studied over a relatively shert period of time. This is quite important in the collection of experimental data to verify the model since the laboratory techniques involved with the animals are difficult and tedious. The model developed is quite general and is applicable to a wide variety of interacting drug systems if the retevam parameters are available. The present program is based upon many features of a previous one but has the following new features: (1) It is user oriented for teaching and research; (2) It has specific terms for competitive enzymatic reactions at the site of drug metabolism; (3) Au additional compartment representing the skin has been added which substantially improves the model fit; (4) The model includes repetitive dosing, up to 10 injections of each drug of arbitrary magnitudes and timing.

2. Hardware and software specifications (2)

[BI gins

In these equations the parameters Vmax, VmaxB (nmol/min), Km and KmB (nmol/ml) refer to maximum elimination rates and K-values of the respective drugs in single drug systems without interactions; rs and r B are the elimination rates (nmol/min) in the interacting systems. The interacting drug can be modelled using the single drug parameters; it is not necessary to introduce a separate interaction parameter. In principle, the interaction model :an be develo~d without any interaction experiments althe, ugh, as will be shown here, model fit may be considerably improved by a least squares

The mathematical model is basically a set of FORTRAN programs that compute drug concentrations as a function of time in plasma and in various tissues and organs. These results may be printed out in tabular form and/or plotted on the line printer. The programs were developed at the University of Missouri-Columbia on an Amdah1470/V7 computer system and they are compatible with FORTRAN G, H and WATFIV Compilers. In WATFIV the object and array area is about 60 k bytes and a simulation of 50 time periods uses about 0.3 min. CPU time (at a cost of about $3.00). The programs are extensively annotated and are intehded to be user oriented. We intend that the results will be useful in teaching as well as research.

R.H. Luecke et al.. Flow model/or druF elimination interactions

90 Scheme 1

FLOW CHARTOF PROG~ i l

Input-output contro] variables, No. o f ~ drugs, binding factors, rate parameters~k Drug recovery and molecular weight ~k Drug dosages---magnitude and timing Measured plasma and bile concentrations ~k Parameters for equation solver ~ k Weight of rat. ~ k

1

Compute parameters for rat: Blood and bile flow Organ weights

{

I

1

.

JSolve differential Equations (4th order Runge Kutta)

I

I

yes Ist 20 intervals fter injection _ Interval : 1/I0 specified

nO

[ DERV: Set up for d.'ug. #I or #2 )m~

l

1

,

DRV: Compute derivatives of concentrationsJ•

1

~BI

EM: Infusion input ND: Nonconstant binding coefficients I RATE: Elimination rate in l i v e r ii

l

no

PLOT: 1. Tabular output 2. Interpolate concentration at data times 3. Calculate sum of squares error 4. Plot on line printer

R.fI. Luecke et aL, blow model for drug elimination interactions

drug elimi,lation and concentration studies [3,4]. Various organs and tissues are mathematically lumped into a number of compartments representing more or less physiologically meaningful body regions (fig. 1). The properties of each of the body regions which are interconnected by blood flows, are considered to be homogeneous. The concentrations of drugs in the blood plasma and in tissues of the various regions are governed by unsteady state mass balance utilizing tissue-to-plasma binding ratios and the chemical kinetics of the rate limiting elimination steps. For the drug system used to demonstrate the programs, eight simultaneous ordinary differential equations describe

3. Program structure The program set consists of a main program and 14 subprograms. System parameters and initial conditions are read in using an INPUT subprogram and physiological parameters of the rat are computed in subprogram RAT. The results of up to 10 injections of each of 2 drugs can be modeled; the dosage of each injection and the time between injections must he specified and may vary. Infusion (intravenous) of each drug is also permitted via arbitrary function subprograms EM 1 and EM2 (see flowchart). The model follows along similar lines to previous SCHEME

EOUATIONS

gill

1 1.

Plasma. VP dCP. g (ll +

dt

I om

2.

Cs/RsJ_s CL/eL~e"

3.

Skin Vs dCs= QS ICp - CS) ~

V dC = oK 4Cp 5.

eL

(QL + QK + QM ÷ QsICp

VMdCM= QM(Cp'CM) dt RM

4.

CKJRK Kidney I , QK

QL CL+ QKCK + QMCM ÷ QS CS " RL RK RM

Muscle

Muscle j :

cK VK

Liver V dCL =o L I C P - C L I - r V L L.~R-'L

6.

Bile duct 1"(Irl ~ r - r|

dt Bile duct

r-~Z"

rl " rz r2

91

r3

Fig. 1. Blockdiagram of model showing8 compartments and blood flows and concentrations.

R.H, Luecke et al., Flow model for dnlg elimination interactions

92

the response of each drug. For drugs that are reabsorbed from the bile in the gut or gut lumen, 5 more equations are needed with reabsorption kinetics. Although not needed in our example these additional 5 equations are operative in the programs. The set of up to 26 simultaneous differential equations is solved nmnerically using a fixed interval, fourth-order Runge-Kutta method. Because of the discontinuities caused by the intravenous injection of drugs, the equation-solver is re-initiafized at each injection poi~it. The initial 2 intervals following any injection are integrated using 1/10th the step size of the following intervals since the most rapid changes occur immediately after an injection. Automatic variable interval methods and methods designed for 'stiff' differential equations have proved to be not necessary so far in this work. The integration subprogram calls other subprograms tbr calculations of the derivatives. The entire set of derivatives are computed for the first drug and then the same equations with different parameters are u~ilized for the second drug. Elimm:tion ra',,es and s.:ne non-constant binding coefficients ~.~ecomputed il~ separate subprograms. Calculated values of the concentrations are interFola:ed at the times for which experimental data is

available and the differences between computed and observed data are determined, The weighted sum of squares of these deviations are computed. The weighting factors may be varied but the usual weighting factor is the observed value of the concentration. This choice implies that the error in the data has been assumed proportional to the square root o f its magnitude. In that case, the minimum of the we ~ t e d sam squares represents a maximum of the likelihood function. Thus, this value may be minimized to obtain a minimum variance estimate for parameters used in the model. The programs contain coding which permits determination of parameters at this minimfim.

The actual minimization can be achieved maaually an,! interactively with time shared operation: An interactive input (unit 11) is set up for the terminal and the value of I 1 is read in for the parameter INPUT. in this mode, the program halts at a statement requesting input; values for the parameters to be fitted are entered at the TSO terminal, in the present work, these parameters were Fmax s, Kms and Vmax B- .ddl other parameters were assumed to be known from literature values or from studies with the earlier model. Four weighted sum-of-squares of residuals are cal-

30

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3

i 0

I 30

i

I 60

i

I 90

,

I I00

MINUTES

Fig. 2. The clearance of warfarin. [ 14CiWarfarin (l.0 mg/kg; 2 t~Ci/kg) was injected intravenously at zero time and BSP (50 rng/kg) at 60 min. The experimental points for 3 rats are presented with SEM. The curves were computed using the simulation model. The solid curve shows the effect o f too large a choice of reaction rate parameter K m.

R.H. Luecke et al., Flow model for drug elimination interactions

culated: one for each drug in the plasma and in the bile. The objective is to fmd values for the parameters that simultaneously minimize all 4 sets of residuals. It is found that the residuals for bile concentrations are particularly sensitive to changes in Vmax. The parameter Kms is then adjusted to improve the overall fit of all of the experimental data but particulady for the extent of the decrease in bile concentration of warfarin following the injection of BSP (fig. 2). This sequence of parameter adjustments is repeated until a minimum of the total sum-of-squares of the residuals is obtained. The search process could of course, be automated but the decoupling of variables obtained by manual consideration of the residuals gives very rapid convergence. The plasma concentrations are only secondarily influenced by Vmax and Kin. With values giving the smallest bile residuals, a significant change in the fit for the plasma concentrations could only be obtained by changes to other parameters, notably the binding ratios. These values were assumed here to be the .:he as found in previous studies [3]. This interactive mode may be used not only to fit parameters to data but also to modify certain model values to observe the response in drug concentrations. 3.1. Input

Input data are read from cards in a separate subprogram. The amount of output that is desired to be printed can be specified. The physiological parameters for the rat are obtained by empirical correlations of the total weight with the weight of various organs, the blood flow rate and bile flow rate [4]. Bile or blood flow may be modified if experimental conditions had noted or caused such changes. Tissue-to-plasma binding ratios are dependent upon both the animals and the drug and must be read in as data. Values for the physiological parameters for the rat and drug binding ratios used in the sample problem were listed in an earlier paper [3]. Percent of the total drug recovery in the bile, the molecular weight of the drugs and the elimination rate parameters are also part of the input data. The programs permit multiple (up to 10) injections of each drug. A set of inputs specify the numbel of injections of each drug and the magaitude and the exact time of each injection.

93

Finally integration parameters are entered: the numerical integration interval size, the number of intervals between each printout and the final time of the last computed concentration. Experimental concentration data may also be entered - up to 50 points each for plasma and bile concentrations for each drug. These values are compared to computed values to aid in estimation of model parameters. 3.2. Output

All of the input data, both read-in data and computed parameters of the rat, are printed out to confirm that the desired values of these parameters have been obtained. Similarly, the dosage schedule for each drug and all e'~perimental data points are printed out. The com~,uted concentrations from the model are printed out in tabular format similar to that shown for a previous program [3]. The concentrations are also plotted on a line printer. The plot also contains the experimental data points and the standard error of their mean for visual comparison. A table of the experimentally observed and computed concentrations is printed and finally the weighted sum of squares of the error in these values.is printed. Portions of this output may be suppressed by specification of a parameter. A zero value for INPUT gives all of the output; a value of 4 suppresses all output except the confirmation of input parameters and the sum of squares output. 3.3. Sample problem

An example of an interactive drug system is the warfarin and BSP combination. In the series of experiments to be simulated here, 300/zg (1 mg/kg) of [14C] warfarin were injected intravenously into 300 g Sprague Dawley rats. One hour later 15 mg (50 mg/kg) BSP were injected (also intravenously). Plasma concentrations were measured and bile samples were collected for a 2 h period. Three experiments on different rats are represented. The output listing of data used in this .,,imulation is shown in fig. 4. All of the input data is printed for verification purposes along with some computed

94

R.H. Luecke et al.. Flow model for drug elimination interactt'ons

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Fig. 3. The clearance of BSP in the interaction experiment. BSP (50 mg/kg) was injected intravenously 60 min after warfarin. Plasma and bile samples were collected and assayed at the indicated points -*5EM. The simulation using Clarenburg and Kao's

rarameters [6 ] for BSP is shown by the solid line; the dashed line shows the fit for an increase in Vrnax for BSP of 30%.

parameters (organ volumes and blood flow rates). The injection of BSP reduced the concentration of warfarin in the bile indicating the shared elimination pathway. The measured values of the concentrations of both drugs in bile and in plasma along with the standard error of the mean are shown in fig. 2, 3. Two sets of computed concentrations are shown. The solid lines represent concentrations that were computed using rate constants which were derived solely from single drug systems. The data fit for this a priori data is reasonable considering the diverse sources of the data. Certainly the shapes of the predicted curves are generally correct. The fit of this data with the experimental data can be improved by a regression analysis on just a few of the parameters. Tile dashed curves were obtained when the values for the rate constants Vmax for BSP were increased about 30% and when the values for the values of Km for warfarin was decreased by a factor of 4. These values were found to minimize the sum of the weighted least squares errors for the bile concentration data for the 2 drugs. It should be noted that because of large intersubject variability ~n these experiments, the formal statistical significance level of the difference in fits is low.

Simulations of concentrations for 1 injection of 300/ag warfarin in a 300 g rat followed by 4 injections of 15 mg BSP at 60 min intervals are shown in fig. 5, 6. Superimposed on the sawtooth profiles in bile concentration are small increases in the overall level of BSP and decreases in the level of warfarin. The program can also simulate injections at irregular time intervals and continuous infusion. 3.4. Program and mode o f availability

A print out of the program is available on request.

4. Discussion

Elimination of anticoagulants occurs by a series o f enzyme-catalyzed reactions which convert the drugs to a form that can pass througi~ membranes in the liver. The metabolites are removed in the bile. A limiting rate step occurs in which the substrate and enzyme combine to form a complex which decomposes to the metabolite and regenerated enzyme. This classic biochemical scheme lea,l,s to the well-known

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VP= 1 3 . 3 5 9 8 8 V~I= 1 5 0 . 0 0 0 0 0 VG= 1 5 . 8 0 1 2 0 ~tt= 4.28930

TOTAL NEIGHT OF RAT =

DRUG 1 DRUG 2

RL

VH2 = EKH2 = RCOV2=

2.66309 15.80120 7.1~883

RH

0.0350 0.1742 0.8500

VM3 = 0.0200 EKt;3 = 0.06~0 ~1 = 3~8.0000

0.02500 7.56730

0.2780 3.0000

0.2400 0.6000

BILE: G3=

RS RG

11.92819 ~5. 8G 956 9.2760~

0.0600 0.1000

VL= VS= QL=

0 30000 KS

0.4800 1.2000

RK

BZNDZKG FACTORS

K = 25 H = 0.10000 TFXHAL = 1 2 0 . 0 0 0 0 0 ~ N ~ NOTE: T F Z N A L / ( K ~ H ) . L E . 4 8 ~.~

DOSE

TZME

NO. OF I V ZH~ECTZDHS OF DRUG 2 =

0.000000

TIME

NO, OF I V INJECTIONS OF DRUG 1 =

70.0000 BO.O000 90.0000

TZHE

3 DATA POZHTS

0.34372

EXFN = 2.0000 ~2 = 838.0000

qS=

Fig. 4. O u t p u t listing o f data used in the program. Organ volumes VP, VM, VS, VK, VG and blood flow rates (QM, QL, QK, QS) are computed from rat weight.

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16.1000 16.7000 15.9000 15.1000 15.1000 14.0000 14.0000 13.4000 10.2000 5.5000 6.3000 8.5000 10.0000 12.0000 12.4000 12.9000 1~.7000 12.6000 13.0000

13 DATA POZNTS

57.0000 70.0000 60.0000 90.0000 120.0000

31.0000

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31.0000 36.0000 ~1.00~0 46.0000 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000 90.0000 9~.0000 100.0000 105.0000 110.0000 115.0000 120.0000

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96

R.H. Luecke et al.. Flow model for drug elimination interactions

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Fig. 5. Concentrations of warafin in plasma and bile. Warfarin (300 ug) was injected at t -- O; 15 tag BSP was injected at 60 min intervals starting at 20 rain.

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R.H. Luecke et al., Flow model for drug elimination interactions

Michaelis-Menten rate equation: r --

Vmax l + Km/ISl

(3)

when r is the rate, [S] is the substrate concentration and Vr,ax and Km are parameters. Interactions occur when a second drug shares an enzyme or a group of enzymes for a rate limiting step. The same sort of reaction scheme then leads to a modified form of the Michaelis-Menten equation (eq. (1) and eq. (2)). These two symmetric equations represent the only linkage - and hence all of the interaction - between the systems of equations for the two drugs. Parameters for these equations can be obtained from experiments with the single drugs. Data for warfarin was taken from some earlier work by Wosilait [5]; BSP rate parameters were presented by Clarenburg and Kao [6]. In the latter work, a secondary pathway for BSP was also observed. When the primary elimination route was partially blocked by a competing drug, a large fraction of the BSP was eliminated via the secondary pathway in which the rate was proportional to approximately the square of the concentration. This secondary BSP elimination pathway was included in the model. It was noted (fig. 2, 3) that the use of these values from the literature for the elimination rate parameters gave a moderately good fit to the data. A subsequent least squares analysis with just 2 degrees of freedom (variation of Kin for warfarin and simultaneous variation of Vmax for both routes with BSP) gave a very close fit to the data. Such results suggest that the form of the model is valid and that the extensive structure which is implemented by the physiological flow modelling procedure is useful in extrapolation of data beyond experimental conditions. Two features of the model equations deserve further comment. The elimination reaction in the liver is computed as a linear function of liver volume or weight (eq. (5) in fig. 1). This procedure is in accordance with the practice with chemical reactor methodology and provides a nominal correction to elimination rate kinetics that are attributable to liver weight differences. Simple variation of liver weight may result from such factors as animal size or sex dif-

97

ferences or differences resulting from disease treatment affecting the liver. However, such extrapolations must be considered cautiously in biological situations since there are a number of implied assumptions that may be unwarranted such as similar concentrations of elimination enzymes per unit mass of liver and identical elimination pathways. In particular, extrapolation between species is risky and experimental verification is essential. The second feature to be noted is the skin compartment. In an earlier model [2,3] this was not included and some difficulties were experienced with the fit of the data during the initial period and with the overall material balance, i.e., recovery. These discrepancies triggered considerable searching for sources of the differences. It was reported [7] that the skin had considerable storage capacity in another drug system and subsequent work showed that the skin also had high storage capacity for warfarin. Since the perfusion rate for the skin is low, the effective result is a relatively long time constant. Inclusion of this compartment substantially improved the fit of the data and yielded a material balance near 100%. Drug interactions are potentially hazardous, especially when multiple combinations of drugs are used. While the mathematical a~,alysis of multiple interactions is complex using the physiological flow model, it does provide a functional basis for simulating interactions. The programs can be used for research purposes in the area which can be designated the pharmacokinetics of multiple drug interactions. It can also be used for teaching graduate students, pharmacists, clinicians and allied health personnel who are concerned with some of the interactive components of multiple drug interactions by simulating conditions in which other drugs are used. Furthermore, the simu!ations can include the effect of diseases. For example if the drugs are metabolized and excreted in either the liver or the kidney or both the input parameters for these organs can be changed to reflect the seslerity of the disease. Thus, the flexibility of the physiological flow model, as applied to multiple interactions, can aid in our understanding of such complex situations. The present report and program should provide the basis for analyzing and testing even more complex combinations of drugs which could provide insights concerning multiple combinations of drugs making their use safer and more effective.

98

R.H. Lueeke et ai.. Flow model for drug elimination interactions

Acknowledgement This work was funded in part, by grant HL 15698 from USPHS.

References [ 11 II.S.G. Chen and J.F. Gross, Cancer Chemother. Pharmacoi. 2 (1979) 85-94.

[2] R.H. Laecke and W.D. Wosilait, J. Pharmacokin. Biophatm. (1980) in press. [3] R.H. Lueckc and W.D. ~'osilait, Comput. Prog. Biomed. 8 (1978) 35-43. 14l K.B. Bischoff, R.L. Ded~i~kand J.A. Longstr~.th,J. Pharm. Sci. (1971) 1128-1133. {5] W.D. Wosilait,Gen. i'!;.,annacol.8 (1977) 349-353. [6] R. Clarenburg and C.C. Kao, Am. J. Physiol. 25 (1973) 192-200. [7] R.L. Dedrick, D.S. Zaharko and RJ. Lutz, J. Pharm. Sci. 62 (1973) 882-890.