Computer Programs in Biomedicine 7 (1977) 41-44 © Elsevier/North-Holland Biomedical Press
APPLICATION OF A PROGRAMMABLE CALCULATOR IN DATA FITTING ACCORDING TO ONE AND TWO COMPARTMENT OPEN MODELS IN CLINICAL PHARMACOKINETICS S. NIAZI Department o f Pharmacy, College of Pharmacy University of lllinois at the Medical Center, 833 South Wood Street, Chicago, IlL 60612, USA
The plasma or blood concentration profiles are fitted by single or two compartment open models using log-linear regression analyses. For two compartment models, "feathering" is performed at 95% equilibration time obtained from raw pharmacokinetic data. The equations have been programmed onto a Texas Instrument SR 52 pocket calculator and recorded on 8.5 × 1.7 cm magnetic strips, facilitating drug dosage regimen calculations through individual patient titrations in a clinical setting. Programmable Calculator Feathering Technique
Clinical Pharmacokinetics SR-52
One- and Two-Compartment Models Log-Linear Regression.
1. Introduction The blood concentration profiles o f drugs following intravenous administration have frequently been fitted
b y single compartment or two compartment models ( I ) as shown in fig. 1. In the past, the pharmacokinetic parameters in sample populations from these models have been used to calculate dosage regimens that would provide most effective drug therapy. However, in recent years, emphasis has been placed on individually titrating the patients in view o f the multiplicity of factors including disease states that can affect the pharma-
Dose
C°
=
= klo
Vdo
A
B i
~o~omow oo
Tissue
(
k12
Central Dose
k21
~2)
-~
cO =
*°°mRoe °°mowm
k10 )
Vdo
(1)
Fig. 1. One and two compartment open models for drug disposition.
time
Fig. 2. Two-compartment "feathering" plot.
S. Niazi, Application of a programmable calculator
42
cokinetic profile. This approach is especially recommended for drugs with low therapeutix index, such as cardiac glycosides and antocoagulants. Generally, in a clinical setting, the pharmacokinetic parameters can be calculated by plotting plasma concentration (C) on a semi-logarithmic paper to obtain the intercept(s) and slope(s) as shown in fig. 2 to give: One compartment:
C = A e -at
(1)
Two compartments:
C = Ae -at + Be -3t
(2)
The volume of the central compartment, Vclo, is approximated by One compartment: Two compartments:
Vdo = Dose/A Vdo = Dose/(A + B)
(3)
Vdeq
~A+B~
(5)
and various rate constants as shown in fig. 1 are expressed (1): klo
_
a/~
k21
A +B
k12 = ot +/3 - k21 - kl0
The blood concentration versus time data is fitted according to log-linear relationship:
In Y = b ' x + i n a
(9)
Ex i lnyi - (l/n) EX i Y.hTyi b=
(10)
Zx~ - (1/n)(Zxi) 2 a = exp[~l:yi - b ~-~]
(11)
and the goodness of fit obtained from coefficient of determination, r2 =
(12)
[~,x2
(~i)21i~(lnyi) 2
(~yi)2]
(6)
_ A 3 + Ba k21
2. Description of the mathematical model
(4)
and in two compartment models, the final volume of distribution following steady state (2) is expressed as: Dose
solute kinetics in a single compartment model. The purpose of this paper is to report the use of Texas Instrument SR-52 calculator in fitting one and two compartment open model data and to calculate various derived parameters.
(7) (8)
The accuracy of these derived pharmacokinetic parameters depends on the accuracy of observed parameters (A, B,/3, a) read from the log-linear graph paper. The observed parameters are subject to human errors which can be very significant especially if a is very large. Although several elaborate computer programs such as AUTOAN (3), NONLIN (3), MIMED (1) and CSTRIP (4) are available, the feasibility of their use is rather limited in a clinical setting. Recently, two programmable hand held calculators have been made available by Hewlett-Packard and Texas Instrument that can possibly be used for such data fitting. The use of a Hewlett-Packard calculator was recently reported in this journal (5) in calculating
where n = number of data points. An average coefficient of determination for the two exponential phases can be used in calculating the overall fit. The two exponents are separated by "feathering" the terminal exponent:
Be #t = C - A e ~t .
(13)
The exponent substraction is performed only up to the times such that gat is not approaching zero; but as the body reaches an equilibrium, the blood concentration is essentially described by a monoexponential equation. An equilibrium state can be defined when the ratio of the amounts of drug in the tissue compartment, X 2, and in the central compartment, X1, becomes constant: X2 --
X1
k12(~-3t _ ~-o.t) -
= constant (k21 -/3)e ~ + (or - k21)~-at
(14)
S. Niazi, Application of a programmable calculator
Table 1 Plasma concentration of warfarin following administration of 200 mg intravenous dose.
As time approaches, a limit such that e S ~ _ k12
(X2/X1)t-eq - (k12 _ fl)
(15)
at all other times, (X2/X1 ) = Feq (X2/X1 )t-eq
(16)
where Feq is the equilibrium fraction. Although mathematically, time required for Feq = 1 is infinity, 95% equilibration time, t0.95, can be used with small error in feathering subroutine initiation. F r o m eqs. (14) and (15):
t0-95 -
k21 - / 3 + (or - / 3 )
43
(17)
3. Program description The program consists o f two segments: (i) log-linear regression o f C versus time data with subroutines for hybrid parameter calculation and feathering initiation (ii) calculation of derived pharmacokinetic parameters, k12 , k21 , k l 0 , t0.95 , C(t) and average r 2. The first step involves the estimation o f t0.95, which is calculated b y entering a few terminal (at least two) and a few initial C versias time points and calculating the raw pharmacokinetic parameters to give t0.95. Complete data is then entered with a call for subroutine at time below t0.95. In the case o f single compartment model, t0.95 will approach zero.
4. Typical sample run Table 1 contains typical plasma concentrations o f warfarin following administration o f a 200 mg dose. Program number 1 is loaded and last three C and time values entered. At this time a feathering initiation subroutine is called and the first three data point entered. The program number 2 is then loaded and t0.95 obtained. The above procedure is then repeated entering all data points and calling the feathering subroutine at time less than t0.95. Table 2 reports actual program function.
t (h)
C Ozg/ml)
0.25 0.50 0.75 1.00 3.00 6.00 8.50 12.50 24.00 37.00 48.00 72.00 90.00 117.00 145.00 168.00 192.00
41.3 33.8 30.2 28.4 26.2 24.0 25.0 23.0 19,0 15,6 13.0 9.0 7.0 4.5 2.9 2.0 1.4
Table 2 Typical sample run of data in table 1. Step
Description
Input
Key
Print out
1. 2. 3. 4.
Load Program No. 1 Initialize time (n) C (n)
192 1.4
E A RUN
5. 6.
time ( n - l ) C (n-l)
168 2.0
A RUN
7. 8.
time (n-2) C (n-2)
145 2.9
A RUN
9.
Calculate r2
(skip) 192 1.4 (skip) 168 2.0 (skip) 145 2.9 (skip) 0.999 (skip) (skip) 0.75 30.2 (skip) 0.5 33.8 0.25 41.3 (skip) 0.999 (skip)
B
10. 11. 12.
Initiate subroutine time (n-14) C (n-14)
0.75 30.2
C A RUN
13. 14. 15. 16.
time (n-15) c (n-15) time (n-16) C (n-16)
0.5 33.8 0.25 41.3
A RUN A RUN
17.
Calculate r2
B
18. 19. 20.
Load Program No. 2 Initialize to.95
E A
1.25 (skip)
44
S. Niazi, Application of a programmable calculator
Table 2 (continued) Step Description 21. 22. 23. 24.
5. Hardware and software specifications Input
Key
Load Program No. 1 Initialize time (n) C (n) At time < 1.25
E A RUN B
ti Ci
c A RUN
Print output
time (n) C (n) 0.999 (skip)
The " c o m p u t e r " used for this program is the Texas Instrument Model SR52 programmable calculator. Some o f the disadvantages in using these kind o f calculators (5) are minimum in SR52 which has 20 memories, 223 programmable steps and print out. The small size and simplicity of operation make this machine valuable for spot clinical situations.
6. Mode of availability Calculate r2
B
Load Program No. 2 Initialize Parameters A
E B
c~ B
k21 klo k12 r2 for C (t)
time
C
e.g.
7.00
C
0.999 (skip)
31.63 (skip) 3.22 (skip) 27.60 (skip) 0.0155 (skip) 1.51 (skip) 0.033 (skip) 1.69 (skip) 0.999 (skip) t C (skip) 7.00 24.76 (skip)
Program listing and reprints can be obtained from: Dr. S. Niazi, Department o f Pharmacy, College o f Pharmacy, University o f Illinois at the Medical Center, 833 South Wood Street, Chicago, Ill. 60612, USA.
References [ 1 ] M. Gibaldi and D. Perrier, Pharmacokinetics (Marcel Dekker Publications, New York, 1975). [2] S. Niazi, J. Pharm. Sci. 65 (1976) 452. [3] J.G. Wagner, Clinical Pharmacokinetics, Drug Intelligence Publications, Hamilton, II1., 1975. [4] A.J. Sedman and J.G. Wagner, J. Pharm. Sci. 65 (1976) 1006. [5] W.E. Walker, D.A. Hall, M.L. Sanfelippo and R.S. Swenson, Comp. Prog. Bio. Med. 5 (1975) 99.
APPLICATION OF A PROGRAMMABLE CALCULATOR IN DATA FITTING ACCORDING TO ONE AND TWO COMPARTMENT OPEN MODELS IN CLINICAL PHARMACOKINETICS S. NIAZI Department o f Pharmacy, College of Pharmacy University of lllinois at the Medical Center, 833 South Wood Street, Chicago, IlL 60612, USA
The plasma or blood concentration profiles are fitted by single or two compartment open models using log-linear regression analyses. For two compartment models, "feathering" is performed at 95% equilibration time obtained from raw pharmacokinetic data. The equations have been programmed onto a Texas Instrument SR 52 pocket calculator and recorded on 8.5 × 1.7 cm magnetic strips, facilitating drug dosage regimen calculations through individual patient titrations in a clinical setting. Programmable Calculator Feathering Technique
Clinical Pharmacokinetics SR-52
One- and Two-Compartment Models Log-Linear Regression.
1. Introduction The blood concentration profiles o f drugs following intravenous administration have frequently been fitted
b y single compartment or two compartment models ( I ) as shown in fig. 1. In the past, the pharmacokinetic parameters in sample populations from these models have been used to calculate dosage regimens that would provide most effective drug therapy. However, in recent years, emphasis has been placed on individually titrating the patients in view o f the multiplicity of factors including disease states that can affect the pharma-
Dose
C°
=
= klo
Vdo
A
B i
~o~omow oo
Tissue
(
k12
Central Dose
k21
~2)
-~
cO =
*°°mRoe °°mowm
k10 )
Vdo
(1)
Fig. 1. One and two compartment open models for drug disposition.
time
Fig. 2. Two-compartment "feathering" plot.
S. Niazi, Application of a programmable calculator
42
cokinetic profile. This approach is especially recommended for drugs with low therapeutix index, such as cardiac glycosides and antocoagulants. Generally, in a clinical setting, the pharmacokinetic parameters can be calculated by plotting plasma concentration (C) on a semi-logarithmic paper to obtain the intercept(s) and slope(s) as shown in fig. 2 to give: One compartment:
C = A e -at
(1)
Two compartments:
C = Ae -at + Be -3t
(2)
The volume of the central compartment, Vclo, is approximated by One compartment: Two compartments:
Vdo = Dose/A Vdo = Dose/(A + B)
(3)
Vdeq
~A+B~
(5)
and various rate constants as shown in fig. 1 are expressed (1): klo
_
a/~
k21
A +B
k12 = ot +/3 - k21 - kl0
The blood concentration versus time data is fitted according to log-linear relationship:
In Y = b ' x + i n a
(9)
Ex i lnyi - (l/n) EX i Y.hTyi b=
(10)
Zx~ - (1/n)(Zxi) 2 a = exp[~l:yi - b ~-~]
(11)
and the goodness of fit obtained from coefficient of determination, r2 =
(12)
[~,x2
(~i)21i~(lnyi) 2
(~yi)2]
(6)
_ A 3 + Ba k21
2. Description of the mathematical model
(4)
and in two compartment models, the final volume of distribution following steady state (2) is expressed as: Dose
solute kinetics in a single compartment model. The purpose of this paper is to report the use of Texas Instrument SR-52 calculator in fitting one and two compartment open model data and to calculate various derived parameters.
(7) (8)
The accuracy of these derived pharmacokinetic parameters depends on the accuracy of observed parameters (A, B,/3, a) read from the log-linear graph paper. The observed parameters are subject to human errors which can be very significant especially if a is very large. Although several elaborate computer programs such as AUTOAN (3), NONLIN (3), MIMED (1) and CSTRIP (4) are available, the feasibility of their use is rather limited in a clinical setting. Recently, two programmable hand held calculators have been made available by Hewlett-Packard and Texas Instrument that can possibly be used for such data fitting. The use of a Hewlett-Packard calculator was recently reported in this journal (5) in calculating
where n = number of data points. An average coefficient of determination for the two exponential phases can be used in calculating the overall fit. The two exponents are separated by "feathering" the terminal exponent:
Be #t = C - A e ~t .
(13)
The exponent substraction is performed only up to the times such that gat is not approaching zero; but as the body reaches an equilibrium, the blood concentration is essentially described by a monoexponential equation. An equilibrium state can be defined when the ratio of the amounts of drug in the tissue compartment, X 2, and in the central compartment, X1, becomes constant: X2 --
X1
k12(~-3t _ ~-o.t) -
= constant (k21 -/3)e ~ + (or - k21)~-at
(14)
S. Niazi, Application of a programmable calculator
Table 1 Plasma concentration of warfarin following administration of 200 mg intravenous dose.
As time approaches, a limit such that e S ~ _ k12
(X2/X1)t-eq - (k12 _ fl)
(15)
at all other times, (X2/X1 ) = Feq (X2/X1 )t-eq
(16)
where Feq is the equilibrium fraction. Although mathematically, time required for Feq = 1 is infinity, 95% equilibration time, t0.95, can be used with small error in feathering subroutine initiation. F r o m eqs. (14) and (15):
t0-95 -
k21 - / 3 + (or - / 3 )
43
(17)
3. Program description The program consists o f two segments: (i) log-linear regression o f C versus time data with subroutines for hybrid parameter calculation and feathering initiation (ii) calculation of derived pharmacokinetic parameters, k12 , k21 , k l 0 , t0.95 , C(t) and average r 2. The first step involves the estimation o f t0.95, which is calculated b y entering a few terminal (at least two) and a few initial C versias time points and calculating the raw pharmacokinetic parameters to give t0.95. Complete data is then entered with a call for subroutine at time below t0.95. In the case o f single compartment model, t0.95 will approach zero.
4. Typical sample run Table 1 contains typical plasma concentrations o f warfarin following administration o f a 200 mg dose. Program number 1 is loaded and last three C and time values entered. At this time a feathering initiation subroutine is called and the first three data point entered. The program number 2 is then loaded and t0.95 obtained. The above procedure is then repeated entering all data points and calling the feathering subroutine at time less than t0.95. Table 2 reports actual program function.
t (h)
C Ozg/ml)
0.25 0.50 0.75 1.00 3.00 6.00 8.50 12.50 24.00 37.00 48.00 72.00 90.00 117.00 145.00 168.00 192.00
41.3 33.8 30.2 28.4 26.2 24.0 25.0 23.0 19,0 15,6 13.0 9.0 7.0 4.5 2.9 2.0 1.4
Table 2 Typical sample run of data in table 1. Step
Description
Input
Key
Print out
1. 2. 3. 4.
Load Program No. 1 Initialize time (n) C (n)
192 1.4
E A RUN
5. 6.
time ( n - l ) C (n-l)
168 2.0
A RUN
7. 8.
time (n-2) C (n-2)
145 2.9
A RUN
9.
Calculate r2
(skip) 192 1.4 (skip) 168 2.0 (skip) 145 2.9 (skip) 0.999 (skip) (skip) 0.75 30.2 (skip) 0.5 33.8 0.25 41.3 (skip) 0.999 (skip)
B
10. 11. 12.
Initiate subroutine time (n-14) C (n-14)
0.75 30.2
C A RUN
13. 14. 15. 16.
time (n-15) c (n-15) time (n-16) C (n-16)
0.5 33.8 0.25 41.3
A RUN A RUN
17.
Calculate r2
B
18. 19. 20.
Load Program No. 2 Initialize to.95
E A
1.25 (skip)
44
S. Niazi, Application of a programmable calculator
Table 2 (continued) Step Description 21. 22. 23. 24.
5. Hardware and software specifications Input
Key
Load Program No. 1 Initialize time (n) C (n) At time < 1.25
E A RUN B
ti Ci
c A RUN
Print output
time (n) C (n) 0.999 (skip)
The " c o m p u t e r " used for this program is the Texas Instrument Model SR52 programmable calculator. Some o f the disadvantages in using these kind o f calculators (5) are minimum in SR52 which has 20 memories, 223 programmable steps and print out. The small size and simplicity of operation make this machine valuable for spot clinical situations.
6. Mode of availability Calculate r2
B
Load Program No. 2 Initialize Parameters A
E B
c~ B
k21 klo k12 r2 for C (t)
time
C
e.g.
7.00
C
0.999 (skip)
31.63 (skip) 3.22 (skip) 27.60 (skip) 0.0155 (skip) 1.51 (skip) 0.033 (skip) 1.69 (skip) 0.999 (skip) t C (skip) 7.00 24.76 (skip)
Program listing and reprints can be obtained from: Dr. S. Niazi, Department o f Pharmacy, College o f Pharmacy, University o f Illinois at the Medical Center, 833 South Wood Street, Chicago, Ill. 60612, USA.
References [ 1 ] M. Gibaldi and D. Perrier, Pharmacokinetics (Marcel Dekker Publications, New York, 1975). [2] S. Niazi, J. Pharm. Sci. 65 (1976) 452. [3] J.G. Wagner, Clinical Pharmacokinetics, Drug Intelligence Publications, Hamilton, II1., 1975. [4] A.J. Sedman and J.G. Wagner, J. Pharm. Sci. 65 (1976) 1006. [5] W.E. Walker, D.A. Hall, M.L. Sanfelippo and R.S. Swenson, Comp. Prog. Bio. Med. 5 (1975) 99.
