### Definition/Introduction

Pharmacokinetics (PK) describes the mathematical relationship between the dose of the drug administered and its measured concentration at an easily accessible site of the body. PK, in particular, is a study of what the body does to a drug, deals with the processes of absorption, distribution, metabolism, and elimination (acronym ADME). Within the PK, the *steady-state* is a concept of fundamental importance in pharmacology. It describes a dynamic equilibrium in which drug concentrations consistently stay within therapeutic limits for long, potentially indefinite, periods. The concentration around which the drug concentration consistently stays is known as the *steady-state concentration*.

The meaning of steady-state, and its clinical value, can only be understood after the necessary acquisition of some concepts of PK. In the context of clinical pharmacology and PK, mathematically, the kinetics of absorption and elimination represent complex processes, although they are subject to basic rules that can be schematized and easily applied to different aspects of drug therapy.

The half-life or t1/2 is a key concept of PK. Following repeated administration of a drug, a steady-state is reached when the quantity of drug eliminated in the unit of time equals the quantity of the drug that reaches the systemic circulation in the unit of time. Consequently, the half-life represents the time required to reduce the plasma concentration of the drug reached in steady-state by 50%. The half-life can be calculated with the following formula:

**t 1/2 = 0,693•Vd/CL**

Where Vd is the volume of distribution at the steady-state, and CL is the clearance. Although approximate, from a clinical point of view, this formula relates t1/2, Vd, elimination (CL), and steady-state, which represent the basic PK parameters. The Vd is defined as the theoretical volume necessary to contain the total quantity of drug present in the organism at the same concentration as that present in the plasma. Thus, the so-called single-compartment model (or one-compartment open model) was designed assuming that the human body is one compartment consisting of a single well-stirred compartment with a Vd; this latter parameter can be calculated with the following equation:

**Vd = Q/[P]**

Q is the amount of drug present in the body; [P] is the drug's plasma concentration. Of note, unlike other PK parameters (e.g., the rates of absorption, biotransformation, and elimination of the drug from the plasma in the unit of time), Vd is not subjected to first-order kinetics. Indeed, it does not depend on [P] but on the chemical-physical features of the drug. However, because Vd is an apparent volume, it is not a physical space but a hypothetical space if the drug could distribute uniformly in the body. If the drug concentrations are known, Vd can also be calculated by the formula:

**Vd = D/C0**

Where D is the dose and C0 the concentration at time 0.

The other variable of the t1/2 is the clearance (CL). It represents the volume of plasma that is cleaned of the drug in the unit of time and is better defined as the relationship between the rate of elimination of the drug and the concentration of the same in the plasma:

**CL = drug elimination rate/[P]**

Although the concentrations of pharmacological agents continuously decrease in the body via metabolism and elimination, periodic administrations of the drug, known as maintenance doses, are necessary to maintain therapeutic concentrations by balancing the amount of drug leaving the body. Initially, a relatively large dose of the drug can also be administered to reach a therapeutic or steady-state concentration more rapidly; this is known as a *loading dose* and represents a more consistent initial dose of the drug to saturate the binding sites.

Half-life and Vd formulae are also single-dose PK equations. In the case of multiple infusions, the mathematical approach becomes more complicated. The concept of bioavailability comes into play, which is the fraction of an administered dose of the unmodified drug that reaches the systemic circulation. For a drug given intravenously, the bioavailability is 100%. When the bioavailability of a drug, the Vd, and the body’s CL of a drug are known, the loading dose and maintenance doses after multiple administrations can be calculated by the following multiple-dose (or infusion rate) equations:

**LD = SSC•Vd/B**

In the above equation, LD is the loading dose.

**MD = SSC•CL•DI/B**

MD is the maintenance dose, SSC the desired steady-state concentration of the drug, B is the drug's bioavailability, DI is the dosing interval.

Note that the formulae can become more complex when taking into account the salt fractions of drugs and non-continuous infusions with varying durations between the administration of maintenance doses. Another multiple-dose formula allows calculating the infusion rate k0:

**K0 = CL•SSC**

*First-order and zero-order kinetics*

A fundamental aspect that requires attention is to evaluate the plasma concentrations of drugs after multiple administrations relate to the not-saturable and saturable mechanisms. In clinical terms, the clinical differences are striking. In particular, if the absorption and elimination systems are not saturable for small dosing increments, the increase in the concentration of the drug is proportionate. In this case, the drugs follow first-order kinetics. According to this model, a constant fraction of the drug is eliminated in the unit of time, and, in turn, the kinetics is exponential. The rate of elimination is directly proportional to drug concentration that, in turn, decays exponentially.

On the other hand, if the absorption and elimination systems are saturable, a zero-order (or saturation) kinetics is followed. In this model, the kinetics is not exponential but initially linear as drug removal occurs at a constant speed, originally independent from the plasma concentration. After saturation, the relationship between the administered dose and the steady-state plasma concentration is unpredictable, not following the rule of proportionality. When the system is saturated, indeed, increases in the administered dose will not correspond to increases in plasma concentrations, and small changes in dose may induce a significant change in plasma concentrations; interestingly, while the absorption of most drugs follows first-order kinetics, several drugs such as ethanol, phenytoin, and aspirin exhibit the saturation kinetics, especially at toxic concentrations.

*Accumulation *

Drug accumulation occurs with repeated dosing (dosing interval), and there is a delay in time to eliminate the drug from the body. This phenomenon usually manifests when the dosing interval is shorter than four half-lives. An index of accumulation is the accumulation ratio (AR):

**AR = 1 dose/FD**

Where FD represents the fraction of drug eliminated after one dosing interval. This formula's first-order kinetics of the drug can predict the AR after many different dosing regimens by inserting different dosing intervals. Nevertheless, as this mathematical approach is highly dependent upon the estimate for the elimination rate, its inaccuracies represent a potential bias.