12

Pharmacokinetic simulations with hand- held programmable calculators We have recently taken an interest in teaching principles of simu- lation exploiting the small, pro- grammable calculators commonly possessed by our students. The calculators concerned are typified by the TI-55-II (Texas Instru- ments), the EL-512 (Sharp) and the HP-33C (Hewlett-Packard), and all are in the $30-35 price range. Cal- culators such as these usually have between five and seven user accessible memories, and 55-120 programming step capability, with 3-8 of the programming steps used for each memory. Cal- culators of a somewhat larger type have already made a significant contribution to the teaching and practice of pharmacokinetics in the clinical setting 1-12, but the papers concerned have mostly focussed attention on dosage ad- justment following the collection of limited pharmacokinetic data, using pharmacokinetic para- meters (population data) for refer- ence. Our interest is in learning through simulation, and we pre- sent here details of a group of programs which provide serial solutions to some representative equations. It is our intention that these programs should be used as models by other investigators interested in analogous problems.

The first program (Table I) simulates one compartment kine- tics. It is presented in the lan- guage used by all three calculators

which we have studied, together with details of the TI-55-II display and a commentary. The program considers an i.v. dose (5 g) ad- ministered into an apparent volume of distribution of 50 I. The half-life of the drug is I h. Prelim- inary calculations indicate that Cpo and kel are 1 0 0 m g l -z and 0.693 -z respectively in the equa- tion:

Cpt = Cpo exp (-l%lt)

where Cpt is the concentration in plasma at time t, Cpo is the theor- etical concentration in plasma at time 0, and 1% I is the rate constant of first order plasma concentration decay. The program gives a series of solutions of the equation for one-hour time intervals.

Three other programs are pre- sented in TI-55-II language only (Table II). The one compartment growth and decay program uses information on a dose (D) (100 mg), an apparent volume of distribution (Vd) (501), fraction absorbed (F,1) z, an absorption rate constant (ka)(1 h -z) and an elimin- ation rate constant (ket)(0.3 h -z) in generating a series of concentra- t ion-t ime data points from the equation:

k a

Cpt - ka - kel

FD Vd [exp(--k~lt)-- exp(-- kat)]

TIPS - January 1985

The protein binding program uses information on the number of binding sites per protein molecule (n,2)(Ref. 2), the protein concentra- tion (Pt)(6 X 10 -4 M), and the equi- librium constant of b inding (K)(1 x 104) to relate fraction bound (6) to concentration un- bound (D/)(starting at 1 x 10 -s M), from the equation:

1 1 + Dl/nPt + 1/nKPt

A series of D/ and [3 values is generated, together with a series of Dt (total drug concentration) values. The i.v. infusion simula- tion program uses an infusion rate (ko)(100 mg h-Z), a volume of dis- tribution (Vd)(501) and a first order elimination rate constant (kel)(0.5 h -z) to generate a series of concentration-time data pairs using the equation:

ko [1 - exp(-kel)t] Cpt =

The reader is invited to study these programs with a view to enhancing his or her own ability, which can be tested by attempting to change the constants in our solutions, and by writing pro- grams for other pharmacokinetic equations, for example for area under the curve using the trape- zoidal rule or for two-compart- ment decay of plasma concentra- tion following i.v. doses. It should be noted that the calculators examined vary in memory capa- city and order and nomenclature of key strokes, but not in applic- ability to those problems. They do have one other useful function - built-in linear regression pro-

TABLE II. TI-55-11 keystrokes for a series of simple pharmacokinetic simulations.

Simulations Keystrokes

One compartment growth ON/C and decay curves X

[CP] RCL

.5 ON/C

Protein binding ON/C +

LRN 4 1 5

STO Intravenous infusion with first ON/C order elimination X

100 [Pause]

1

ON/C 2nd [Part] 3 1 + ( 1 - .3 ) = 1 = X 100 = + 50 = STO 0 LRN 2nd .3 + / - X RCL 1 = INV Lnx = STO 2 1 + / - X 1 = INV I_nx = STO - 2 RCL 1 2rid [Pause]

STO + 1 RCL 2 X RCL 0 = RST LRN RST ON/C STO 1 P,/S (Press R/S repeatedly to generate pairs of time and concentration figures) ON/C 2nd CSR 2 X 10,000 = X 0.0006 = 2nd 1/x

1 = STO 0 0.00001 STO 1 ON/C ON/C 2nd [Part] 3 2nd [CP] RCL 1 2nd [Pause] = 2 = 6 EE + / -

+ RCL 0 = STO 2 2~1 1/x 2nd [Pause] + / - + = 2nd 1/x X RCL 1 = 2nd [Pause] 1 EE + / - + RCL 1 = STO 1 RST LRN RST ON/C ON/C 0.00001 t R/S (Press R/S repeatedly to obtain a series of trios of D~, [~ and Dt figures)

ON/C 2nd [Part] 2 LRN 2nd [CP] 1 STO 0 .5 + / - RCL 1 = INV Lnx = STO - 0 RCL 0 X

= = 50 = = .5 = STO 0 RCL 1 2rid .25 STO + 1 RCL 0 RST LRN RST ON/C ON/C STO R/S (Press R/S repeatedly to generate a series of pairs of time and concentration figures)

1985, Elsevier Science Pub l i she r s B.V., A m s t e r d a m 0165 - 6147/85/$02.00

TIPS - January 1985

o

0

O. -v

o

~d

t~

W

=,

t--

8

-v.~3

"u

cq

i i

co rr

z ~ ~ . . a rr~

w

9 0

c~

o.

z 0

z 0

>

==

E

t---

~ ~.~_

o

8

o

0. r r

..a r r o

O

~ x rv ft.

O Z

8 8

8 ~

ft.

~, ~ o

~o

~ 5 ._~

_ "~

O × ~,

8 -

X oa

8 8

8

X

li!-, ~ . ~

~ +

8 8

~lt 'N

o

l ii

i ~ 1 ~

~ z

kO

i ~.~_

8 8 ~ 8 ~

13

g r a m s . W e h a v e f o u n d t h e s e e spe - cial ly u se fu l in s i m u l a t i n g p h e n y - t o i n dos ing . T h i s fo l lows M i c h a e l i s - M e n t e n k ine t ics . In t h a t Cps~, t h e m e a n c o n c e n t r a t i o n at s t e a d y s ta te re la tes to v a l u e s for V m ~ a n d Km as in t he e q u a t i o n :

R - Rma x X Cp~ K m + Cps~

w h e r e R is t h e da i l y d o s e a n d Rmax is e q u i v a l e n t to V m a x a n d r e p r e - s e n t s a t heo re t i c a l m a x i m u m poss - ib le t o l e r ab l e i n p u t / o u t p u t ra te w i t h m a i n t e n a n c e of s t eady- s t a t e . R e a r r a n g e m e n t g ives :

R R = Rmax - Km Cpss

w h i c h d e s c r i b e s a s t r a i g h t l i ne re- l a t i o n s h i p f r o m w h i c h p r e d i c t i o n of Cpss i n d u c e d b y a n y R is pos s - ib le p r o v i d e d c a l i b r a t i o n da t a exists . The n e e d is to p r e d i c t a n e w R/Cpss d a t a pa i r . The l i n e a r r e g r e s s i o n p r o g r a m can b e u s e d to ca l ib ra t e w i t h t w o or m o r e k n o w n R/Cpss da t a pa i r s , a n d to ca lcula te R . . . . Km, a n d Cpss for a n e w R va lue , or to ca lcu la te t he n e e d e d R v a l u e for a c h o s e n Cpss. Deta i l s of t h i s are a v a i l a b l e to a n y o n e w h o cares to w r i t e to us for t h e m .

S T E P H E N H. C U R R Y A N D

K A M L E S H M. T H A K K E R

Division of Clinical Pharmacokinetics, College of Pharmacy, University of Florida, Gaines- ville, FL 32610 USA

R e f e r e n c e s

1 Bowman, J. D. and Mungall, D. (1983) in Applied Clinical Pharmacokinetics (Mun- gall, D., ed.), pp. 389-434, New York, Raven Press

2 Polk, R. E., Moskowitz, S. M. and May- hall, C.G. (1981) Drug lntell. Clin. Pharm. 15, 751-757

3 Chennavasin, P. and Brater, D. C. (1982) Eur. J. Clin. Pharmacol. 22, 91-94

4 Gee, J. P. and Jim, L. K. (1982) Am. J. Hosp. Pharm. 39, 1508-1510

5 Wilson, J. V. (1982) Pharm. ]. 229, 105 6 Bennett, S. W. and Scott, C. A. (1980)

Am. J. Hosp. Pharm. 37, 523-529 7 Crow, J. W. and Levy, G. (1978) Am. J.

Hosp. Pharm. 35, 1075-1077 8 Foster, T. S. and Bourne, W. A. (1977)

Am. J. Hosp. Pharm. 34, 22-24 9 Hepler, C. D. and Prince, R.A. (1980)

Am. J. Hosp. Pharm. 37, 1631-1635 10 King, W. and Kaul, A. F. (1980) Drug

lntell. Clin. Pharm. 14, 686--693 11 Madsen, B. W., Tarala, R. A. and Pater-

son, J. W. (1980) Eur. J. Clin. PharmacoL 17, 393-399

12 Manion, C. V. (1978) Am. J. Hosp. Pharm. 35, 947-951