BIOPHARMACEUTICS & DRUG DISPOSITION, VOL. 2. 11S121 (1981)

THEORETICAL CONSIDERATIONS IN

TIMES FOLLOWING ORAL DOSES, ILLUSTRATED WITH MODEL DATA

CALCULATION OF TERMINAL PHASE HALF-

STEPHEN H. CURRY College of Pharmacy. Universiry 01' Florida, Gainesville.

Florida 32610. U .S .A .

ABSTRACT Concentrations of drugs in plasma following oral doses, when plotted as a function of time, invariably display a growth and decay curve. The terminal phase of this curve is commonly used for calculation of a half-time (sometimes designated the 'half-life') of the drug concentration in plasma. However, difficulty is often experienced in assigning a suitable time point as that of commencement of the terminal phase. This paper examines theoretically and with model data the issue of whether there is a useful systematic relation between this time and other identifiable times such as that of maximum concentration. The conclusion is reached that no rule of thumb can be applied. It is stressed that computer simulation. statistical testing for linearity, and comparison with data obtained following intravenous doses are all necessary before a conclusion can be reached that a particular half-time has been validly assessed.

KEY WORDS Half-time Half-life Growth and decay curve Compartments Models Oral doses

INTRODUCTION

A common requirement in pharmacokinetic analysis of data for drug concen- trations in plasma is calculation of a half-time ( I t ) of decay of the concentration in the terminal phase of the growth and decay graph invariably seen following administration of an oral dose into a one-compartment or two-compartment system. This half-time is often assigned biological significance and is commonly designated as the 'half-life' of the drug. However, assessment of a half-time requires a prior decision concerning the time-point at which the terminal phase can be considered to start. This decision is not made easily, and it has been suggested that calculation of the half-time using data from twice the time of maximum concentration onwards is a valid procedure. This paper assesses the concept that there is a useful systematic relation between the time when the

0142-2782/81/020115-07$01 .OO @ 1981 by John Wiley & Sons, Ltd.

Received 7 July 1980

116 S. H. CURRY

terminal phase becomes dominant in the data. and other identifiable points such as the time of maximum concentration. Theoretical considerations, and model data, are used.

THEORY

Four cases are considered: (i) one-compartment kinetics following intravenous doses; ( i i ) two-compartment kinetics following intravenous doses; ( i i i ) one- compartment kinetics following oral doses; and ( iv) two-compartment kinetics following oral doses.

In case (i). a graph of logarithm of concentration against time is a straight line, and

Cp, = Cp,e-" ( 1 )

where Cp, is the concentration in plasma at time f ; Cp, is the concentration in plasma at time 0, obtained from the graph of log Cp, wnus I by extrapolation to f = 0; and k is the rate constant of decay. In this case the graph is monoexponen- tial, and no decisions on assignment of different parts of the graph as representative of different phases of decay are needed.

In case (ii), there are two phases of decay. the earlier one being the faster. The equation for the graph is the sum of two exponentials. and

(2) where A and Bare the theoretical concentrations at time 0, obtained from the two exponentials by extrapolation. applying the principles described for case (i) above. r and pare the rate constants of the faster and slower phases respectively.

Adequate graphical and computer techniques exist for the derivation of A , B . r , and p, and provided these techniques are correctly applied, correct assignment of rate constants to the two phases of decay poses relatively few problems.'

The problem defined in the introduction to this paper is best illustrated by case (iii). In this case, except for the special situation of k = y,? the equation for the graph is

where y is the rate constant of the growth phase. The concentration maximum occurs when dCp/dt = 0. I t has been repeatedly shown (see, for example, Reference 3) that the time of this maximum (zmax) is given by

Cp, = A e-"+ Be-'"

Cp, = Cpo(e-"-e- I ) (3)

Since the growth and decay curve implicit in equation (3) is the sum of two exponentials, it does not exclusively show the kinetics of decay from fmax onwards. Rather, it increasingly represents the decay kinetics as 1 increases, approaching exclusive representation of decay only when f approaches z. The

TERMINAL PHASE HALF-TIMES 117

problem can usefully be considered through an approximation, such that dominance by the decay phase is deemed to occur when

where r is a figure indicating the proportion of Cp, accounted for by the decay exponential. If r = 0-9, Cp, is 90 per cent accounted for in this way. The time (tdef) at which the curve is defined as in the decay phase to the criterion represented by r is then given by

Cp, e-& 3 rCp, ( 5 )

In r - In ( r - 1) 'def = .,-k

I

Unfortunately, this equation involves the logarithm of a negative number and so it cannot easily be used to determine tdef. This is inevitable as r < 1 (in this application). However, it can be seen that the ratio tdcf/tmax will involve r, 7 and k. and this ratio will therefore vary over a wide range.

One approach is the use of a computer. Calculations with model data have shown that, if 7 = 1, r = 0.9, and k varies from 0.01 to 0-4, tdef and t,,, show an inverse relation and the ratio varies from 2.6 1 to 0.52 (i.e. for low values the graph is 90 per cent decay from a point before the peak). Similarly, if k = 0.1, t = 0.9. and 7 varies from 0.25 to 2, the two times show a direct relation and the ratio varies from 0.80 to 2.62. Thus any ideas that the ratio is approximately 2 are erroneous. Also, it should be noted that although the highest ratio in these model numbers in not much greater than 2, this arises from use of a conservative value of r of 0.9. Higher values of r will raise the ratio. However, as with case (ii), adequate graphical and computer techniques exist for more valid approaches in this case (see, for example, Reference 4).

The greatest problem arises with case (iv). In this case, the simplest equation for the growth and decay curve is

CpI = Ae-"+Be-l"-(A+B)e- ' ( 7 )

As with case (iii), the maximum concentration occurs when dCp/dt = 0, which can be shown by differentiation to occur when

7 ( A + B ) = 0 (8)

However, as in the earlier derivation, this leads to an equation which cannot be solved, in this case for 1, without bringing negative logarithms into the solution.

A e(.-x)I+/jBel, - / ! ) I -

The equation for tdef, in this case, is derived from the relation:

Be-'" = rCp, (9)

and it is

1 r (7 - r ) A z - p ( l - r ) ( y - p ) B fdcf = -In

118 S . H . CURRY

Generally, however, y%a, and yafl, and a simplified relation is

1 rA tdcr = -1n-

a - p (1-r)B

In this case the equation can be solved with r c 1, but it is obvious that definition of a ratio of the type indicated as useful for case (iii) would be extremely complex, even without the difficulty of solving equation (8).

MODEL DATA

Case (iv) requires a computer solution. A suitable iterative program has been used to produce the model data shown in Figures 14, in which r = 0.9 (in the figures this is shown as R = 90), and a, fl, 7, A, and B are varied as shown in the

.70 -

.60

- n

TIHE

Figure 1. Model concentration-time curves fitting equation (7). with A = 0.7. B = 0.3. /j = 0.1.

points at which 90 per cent of Cp, is accounted for by the term Be-"' .. , - - 1, and z varied from 0-2 to 0.7. The dotted lines link the values at I,,,. The dashed lines link the

legends. In these figures, a dotted line defines t,,,, and a dashed line defines the point at which the slower phase of decay accounts for 90 per cent of the curve. With changes in a, p, A, and B, the two points of interest vary inversely. The implications for assignment of data to particular phases are obvious.

The point tdef is not affected in this case by changes in 7 , as demonstrated by equation (1 l), and in 'Figure 3, the concentration at tdef is almost a constant. so there is no dashed'line. However, there is still variation in f,,, to affect the ratio tdefltrnax.

TERMINAL PHASE HALF-TIMES 119

.70

.60

- CHANGING VALUE OF BETA.R=90-

-

.50 i

P

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 V. V"

TI* Figure2. Modelconcentration-timecurvesfittingequation(7). withA = 0.7. B = 0.3.1 = 0.5.;. = I.

and p varied from 0.01 to 0.4. Dotted and dashed lines as in Figure I

.70

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.so z E! I- 2 . 4 0 + z w u 5 .30 u

.20

.I0

0.00

TIME

Figure 3 . Model concentration-time curves fitting equation ( 7 ) . with A = 0.7, B = 0.3. I = 0.2. p = 0.1, and 7 varied from 0.5 to 2.0. Dotted lines as in Figure I

120

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. .60

.so z 0 - c 2 . 4 0 c z Lu U

u d .30

.20

. l o

S . H. C U R R Y

C H A N G I N G VALUES OF A A N D B SIMULTANEOUSLY.R=90-

00

T I M E

Figure 4. Model concentration-time curves fitting equation ( 7 ) . with 1 = 0.5. /{ = 0.1. ;' = 1. A + B = 1. and A varied from 0.5 to 0.9. Dotted and dashed lines as in Figure I

Notwithstanding the fact mentioned earlier, that. generally 7 $2. a further problem with case (iv) can still arise when z and 7 approach each other. I f z = 7 , equation (7) reduces to

Cp, = B(e-'"-e- ') (12)

and the data will then suggest that there is a one-compartment system in operation.

It is thus apparent that extreme care is needed with assignment of rate constants to particular phases of growth and decay curves for drugs apparently distributed through a two-compartment system. Some essential guidelines would appear to be: (i) use a computer simulation where possible; (ii) never apply any rule of thumb relating to peak time (or indeed peak concentration); ( i i i ) examine any apparently straight line by appropriate statistical tests for linearity, and only calculate a half-time if this test is favourable. Additionally, it is incautious to conclude that a particular half-time indicates the kinetics of a biological process without additional evidence. This will generally mean that conclusions can only be drawn if reference intravenous data are available when an apparent two- compartment system is studied.

ACKNOWLEDGEMENTS

This work was partly conducted at The London Hospital Medical College (University of London, England). I am grateful to various members of the

TERMINAL PHASE HALF-TIMES 121

computing unit of The London Hospital Medical College, Mr. Stephen Evans, Mr. Ashley Hyams, and Miss Fiona Borth, for their assistance in the mathematics and computer work involved in this report.

REFERENCES I . D. S. Riggs. The Marhemarical Approach 10 Physiological Problems, MIT Press. Cambridge

(Massachusetts), 1970. 2. M. Bialer. Pharmacokinrr Biopharmacrur.. 8. I I 1 (1980). 3. M. Gibaldi and D. Perrier, Pharmocokinrtics. Marcel Dekker, New York. 1975. 4. L. Saunders. The Absorption and Distribution of Drugs, Bailliere Tindall. London, 1974.