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6 Mathematics and pharmacokineticsy = 2x + 2
264y = − 0.5x + 6xy
AB(^0246)
Figure 6.1.The linear function.Line A has a positive gradient of 2 and an intercept on
the y axis of 2; line B has a negative gradient of 0.5 with an intercept on the y axis of 6. The
equation for line B can also be written 2y= 12 −xbymultiplying through by 2 and
rearranging the terms on the right side of the equation.
Foraone-compartment model the function is:
C=C 0 e−kt.
Plasma concentration, C, depends on time, so t is the independent variable and
plasma concentration, which is measured at a series of time-points, is the dependent
variable. The relationship in this case is described by an exponential function.
The linear function
The equation for a straight line with a gradient of m is given by:
y=mx+c.
The constant, c, tells us the intercept on the y-axis and allows us to position the
straight line in relation to the axes. If we knew only the gradient, we couldn’t draw
the line; we need at least one point to fix the exact place to draw it. Thus we need two
pieces of information to draw a particular straight line: its gradient and its intercept
on the y-axis. If m is negative the slope of the line is downward, if m is positive then the
slope is upward (Figure6.1). In a later section we meet differentiation; for a straight
line the differential equation simply gives a constant, the value of the gradient, m. We
meet a straight line in pharmacokinetics when taking a semi-logarithmic plot of the
concentration–time curve for a simple one-compartmental model. The expression
may look more complicated than the one above:
ln(C)=ln(C 0 )−kt.
Inthis case we think of the y-axis as being ln(C) and the x-axis as being t. If we then
compare this expression with y=mx+citshould be clear that−kislikemand
represents the gradient and ln(C 0 )represents the intercept on the y-axis.