T C

T 50,2 + Curine kfiltration + ksecretion kreabsorption (1.16)

The minus sign and prime on kreabsorption indicate a different driving force concentration than for filtration and secretion and drug transport in the opposite direction. Elimination is the generic term given to the first-order rate constant K, or sometimes P, describing the parent drug lost by both metabolism km and excretion ke (Eq. 1.17):

Factors analogous to those affecting gut absorption also can affect drug distribution and excretion. Any transporters or metabolizing enzymes can be taxed to capacity—which clearly would make the kinetic process nonlinear (see "Linear versus Nonlinear Pharmacokinetics"). In order to have linear pharmacokinetics, all components (distribution, metabolism, filtration, active secretion, and active reabsorption) must be reasonably approximated by first-order kinetics for the valid design of controlled release delivery systems.

1.5.3 Convolution of input and disposition

To obtain a complete drug concentration profile, both the input and disposition kinetics must be known or assumed. If the input is an intravenous bolus, zero or first order, and disposition is first order, then the input and disposition can be combined mathematically through the convolution operation, represented by the * symbol. Mathematically, this is represented as rt

Cp (t) = input (t) * disposition (t) = J input(r) x disposition (t - t) dr

If we know input(t) and Cp(t), we can extract disposition(t). The easiest way to accomplish this deconvolution (extraction) is to give an intravenous bolus dose and measure Cp(t), which will exactly mirror the underlying disposition(t). Once disposition(t) is known (and assumed not to change for the same drug), Cp(t) can be predicted for any input(t), or more important for controlled release of delivery systems, the input(t) needed to produce a specific Cp(t) profile can be determined easily. Pharmacokinetics is simply the convolution of an input(t) with the drug/patient's disposition(t). Putting the whole package together, pharmacokinetics includes all kinetic aspects of input (liberation, absorption) and disposition (distribution, metabolism, and excretion).

1.6 Compartmental Pharmacokinetic Modeling

1.6.1 Single-dose input systems

Compartmental modeling is by far the most commonly used pharma-cokinetics modeling technique. In compartmental modeling, tissues having similar kinetic drug concentration profiles are lumped together into a compartment. For example, a common three-compartment model (Fig. 1.8) may have one compartment representing the blood/plasma and other tissues that reach their steady-state concentration very rapidly (< 3 hours) for a given dose (i.e., kinetically similar to the blood). This compartment is commonly called the central compartment and usually contains such organs as the blood/plasma, kidney, lungs, liver, and most other large internal organs. The second compartment in this three-compartment model could be called shallow tissues; these tissues do not reach their steady-state concentration as rapidly as the central compartment but still reach steady-state somewhat quickly (3 to 8 hours). Examples of the shallow compartment might be organs such as muscle, eyes, and other smaller internal organs, as well as sometimes the skin. The third compartment consists of tissues that reach their steady-state concentration slowly; examples of the deep compartment are adipose tissue, brain, and sometimes skin tissues (particularly when the drug

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