The Exponential Function

A function defines a unique value for the dependent variable, y, given a value for the independent variable, x. We write this as:

y = f(x)

For pharmacokinetics, we are interested in drug concentration (C) as a function of time (t). We can write:

C = f(t)

Observation tells us that after a single dose, plasma concentration falls with time and the rate at which that concentration falls is proportional to the concentration itself. This describes an exponential relationship, the simplest of which involves a single exponential. The general form of an exponential function is:

y = Anax

In this relationship n is the base and x the exponent; A and a are constants. Although it is possible to use any base for our exponential function, the natural number e is chosen for its mathematical properties. The exponential function, y = e^x, is the only function that integrates and differentiates to itself, making manipulation of relationships involving exponentials much easier than if another base were chosen. The transcendental number e is irrational, it cannot be expressed as a fraction, and takes the value 2.716 … where there is an infinite number of digits following the decimal point.

The rate at which a function changes is represented graphically by the tangent to the graph of y against x. Exponentials are positive if the rate at which y changes increases as x increases or negative if the rate at which y changes decreases as x increases (see Figure 6.1). The function y = Ae^ax is a positive exponential. An example of a positive exponential relationship is bacterial cell growth – as time goes on, the number of bacteria increases exponentially. Compound interest relating to the growth of an investment with time is a further example of a positive exponential relationship.

Figure 6.1 The exponential function. (a) Negative exponential function. Curve A is a simple wash-out curve for drug elimination, for example after a single bolus dose; the equation is C = C₀e^−kt, the asymptote is zero and the starting point on the concentration-axis is defined as C₀. Curve B is a simple wash-in curve such as is seen when a constant rate infusion is used. The starting point on both axes is now zero and the asymptote is the concentration at steady-state C_ss. The equation is C_ss(1 − e^−kt). In both cases the rate constant for elimination is k. (b) Positive exponential function. This represents the exponential growth in a bacterial colony starting from a single organism. This organism divided; the two resultant organisms both divide and so on. There is a regular doubling of the number of bacteria so the equation is N = 2^t/d, where N is the number of organisms at time t and d is the time between consecutive cell divisions.

After a single bolus, plasma concentration of drug falls at a rate that decreases with time and is dependent on the concentration itself. This describes a negative exponential relationship (Figure 6.1a). The simplest mathematical model that describes this is a single negative exponential function – it describes a single-compartment model:

C = C0e-kt

In this relationship the independent variable is time (t) and the dependent variable drug concentration (C). C₀, the concentration at time t = 0, and k, the rate constant for elimination, represent the constants that define the exact equation for a given drug. This relationship is commonly referred to as a wash-out curve for the drug. The starting point on the concentration axis is C₀ and the ‘steepness’ of the curve depends on the rate constant, k.

Mathematically we talk about the time constant (τ) for an exponential process – this is the time it takes the concentration to fall by a factor of e and is the inverse of the rate constant for elimination (τ = 1/k). A consequence of this is that concentration falls by the same proportion over equal time periods. In one time constant the process is approximately 63% complete (concentration has fallen from C to C/e) so after three time constants the process is virtually completed. Similarly it takes one half-life (t_1/2) for concentration to halve (concentration has fallen from C to C/2) and the process will be virtually complete in five half-lives. Note that the time constant is longer than the half-life (because e is greater than 2).

The way in which plasma concentration increases during a constant rate infusion, a wash-in curve, is also described by a negative exponential; although plasma concentration increases with time, the rate at which it approaches its maximum value is decreasing with time, making it a negative exponential (see Figure 6.1).

Logarithms

Any number can be expressed as a power of 10, for example 1000 can be written 10³ and 5 as 10^0.699. We call the power the exponent and 10 the base. Any number can be written in this way so that for any positive value of x:

x = 10y

We define the value y as the logarithm to the base 10 of x, so the logarithmic function is (see Figure 6.2):

y = log x

Thus any number can be written as:

x = 10logx

The exponent in 10^y is therefore equivalent to the logarithm (log) to the base 10 of x. Thus log1000 is 3 and log5 is 0.699. Similarly log10 is 1 (10¹ = 10) and log1 is 0 (because 10⁰ is 1).

Figure 6.2 The logarithmic function. This shows the function y = log(x) for logarithms to the base 10. When x = 1y = 0, which is a point common to all logarithmic relationships. Notice that unlike the exponential function, there is no asymptote corresponding to a maximum value for y. There is an asymptote on the x-axis, since the logarithm approaches negative infinity as x gets smaller and smaller and approaches 0; there is no such thing as a logarithm for a negative number.

So far we have described the familiar situation where the base is 10 and the exponent is the logarithm to the base 10. We can actually write a number in terms of any base, not just 10, with a corresponding exponent as the logarithm to that base.

In pharmacokinetics we are concerned with relationships involving the exponential function where the base is the natural number e, so we use e as the base for all logarithmic transformations.

By convention the logarithm of x to the base e is written as ln x, whereas log x is reserved for logarithms to base 10 and other logarithms to other bases are written as log_n x, where n is the base.

Logarithms to base e are known as natural logarithms, e.g. 2 can be written e^0.693, so the natural logarithm of 2, ln2, is 0.693 (this is a useful value to remember, as it is the factor that relates time constant to half-life).

Manipulating equations using logarithmic and exponential functions is described in more detail in the Appendix (page 62).

Calculus: Differentiation

Clinically we know that the rate of decline of plasma concentration with time in a wash-out curve is dependent on the plasma concentration. For a simple single-compartment model the constant of proportionality is k, the rate constant for elimination. Mathematically the rate at which concentration changes with time is expressed as the tangent to the curve. The tangent to any curve is described by its differential equation, so in the case of plasma concentration we can write:

dC/dt ∝ C or dC/dt = –kC

This simple expression describes a first-order relationship since the tangent depends on C raised to the power 1, also described as linear kinetics.

In mathematical terms we could start with the negative exponential describing the fall in plasma concentration and work out its differential:

d(C0e-kt)/dt

The exponential function differentiates to itself but we also need to take into account the factor −k so:

d(C0e-kt)/dt = –C0e-kt

Which gives us the relationship above, dC/dt = −kC

We also know clinically that for certain drugs, when elimination processes are saturated, drug concentration declines at a constant rate and the graph of concentration against time is a straight line rather than an exponential curve. When this happens the differential equation becomes:

dC/dt = –k

There is no dependence at all on concentration itself, so we describe this as zero-order kinetics since the tangent depends on C raised to the power zero (C⁰ = 1).

Calculus: Integration

In differential calculus we use mathematics to calculate the gradient to a curve described by a mathematical function. In integral calculus mathematics is used to calculate the area under the curve of a function. It is possible to calculate the area between any two limits given by values on the x-axis. In pharmacokinetics we are interested in the entire area under the curve between time zero (t = 0) and infinity (t = ∞). Integration is the opposite of differentiation – if we integrate the function described by the differential equation given above, dC/dt = −kC, we can get back to the original equation, C = C₀e^−kt, but only by knowing that at infinite time the plasma concentration must be zero.

This integral is written:

∫ C0e−kt dt, with limits from 0 to ∞

The ‘dt’ is used to show we are integrating with respect to time.

It is beyond the remit of this book to discuss integration in further detail.

Pharmacokinetic Models

Modelling involves fitting a mathematical equation to experimental observations of plasma concentration following drug administration to a group of volunteers or patients. These models can then be used to predict plasma concentration under a variety of conditions and, because for many drugs there is a close relationship between plasma concentration and drug activity, pharmacodynamic effects can also be predicted.

A number of models can be used:

• 
  

  compartmental models

  
  
• 
  

  physiological models

  
  
• 
  

  non-compartmental models.

We will concentrate on compartmental models.

The Single-compartment Model

The simplest model is that of a single, well-stirred, homogenous compartment. If a single dose of drug is given, then the model assumes that it instantaneously disperses evenly throughout this compartment and is eliminated in an exponential fashion with a single rate constant for elimination (see Figure 6.3a). Although such a model rarely applies to drugs used in clinical practice, it is important to understand because it introduces the concepts that are further developed in more complex compartmental models. We have seen that this model is described by an equation with a single negative exponential term:

C = C0e-kt

where C₀ is the concentration at time t = 0 and k is the rate constant for elimination. The volume of the single compartment is the volume of distribution, Vd. The rate constant for elimination, k, is the fraction of the volume of distribution from which drug is removed in unit time. The actual volume from which drug is removed in unit time is known as the clearance (Cl) of drug and has units of ml.min⁻¹ and is the product of volume of distribution and rate constant for elimination:

Cl = k.Vd

As described above, the time constant τ is the inverse of k so clearance also can be expressed as the ratio of the volume of distribution to the time constant:

Cl = Vd/τ

For any particular model k and Vd are constant, so clearance must also be constant. Because clearance is a ratio it is possible for drugs with very different values for Vd and τ to have the same clearance as long as the ratio of these two parameters is the same.

Figure 6.3 Compartmental models. (a) Single-compartment model, volume of distribution Vd, rate constant for elimination k. (b) Two-compartment model, central compartment has volume V₁ and peripheral compartment has volume V₂. Rate constants for transfer between compartments are described in the text. The rate constant for elimination is k₁₀. (c) Three-compartment model. This is similar to the two-compartment model but with the addition of a second peripheral compartment, volume V₃, with slower kinetics.

The values of the two parameters describing the relationship C = C₀e^−kt can be found by converting this equation to its logarithmic equivalent using natural logarithms (see Appendix (page 62) for the intermediate steps).

lnC = ln(C0e-kt)

lnC = lnC0 – kt

This gives the equation of a straight line with a gradient −k and lnC₀ the intercept on the lnC axis (see Figure 6.4). Since we know the dose of drug given (X) and C₀, the concentration at time zero, the volume of distribution will be:

Vd = X/C0

When a single bolus dose, X mg, of drug is given the concentration at time zero is C₀ or X/Vd mg.ml⁻¹. If we follow the amount of drug remaining in the body (X_t) it declines in a negative exponential manner towards zero in the same way as concentration:

Xt = Xe-kt

The rate at which the drug is eliminated (in mg per minute) is the tangent to this curve, described by the differential expression:

dXt/dt = -kXt

We know that clearance is the product of k and Vd, so k is the ratio of clearance (Cl) and Vd. If we put this into the expression above we can see that the rate of elimination is:

dXt/dt = –(Cl/Vd)Xt = –Cl(Xt/Vd) = –Cl.C

At a plasma concentration C, the drug is eliminated at a rate determined by the product of clearance and the plasma concentration with units mg.min⁻¹, the negative sign indicating that the amount of drug falls with time. Note that this is the rate of elimination, quite different from the rate constant for elimination.

Figure 6.4 Semilogarithmic transformation of the exponential wash-out curve. (a) An exponential decrease in plasma concentration of drug against time after a bolus dose of a drug displaying single-compartment kinetics. (b) A natural logarithmic scale on the y-axis produces a straight line (C₀ is the plasma concentration at time, t = 0).

So far we have considered how the plasma concentration falls after a single bolus dose. If we give multiple doses, each one given before the concentration has fallen to zero, then the plasma concentration will build up in a saw-tooth fashion. The clearance of the drug determines how rapidly the required plasma level builds up. A smoother rise in plasma concentration will be achieved if a fixed-rate infusion is used. The curve of concentration against time is a negative exponential (see Figure 6.1b) with the steady-state concentration (Css) determined by clearance (Cl), the concentration of the solution infused (Ci) and infusion rate (I).

At steady-state the input of drug must be equal to output. This is a very important concept in kinetics. At steady state:

Input is the amount of drug given in unit time – the product of drug concentration in the syringe and infusion rate: Ci.I

Output is the rate of elimination of drug at steady state concentration, which is the product of steady-state concentration and clearance: Css.Cl

Input = output

Ci.I = Css.Cl

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