For most drugs, the time to reach steady state is four to five half-lives if the drug is given at regular intervals—no matter the number of doses, the dose size, or the dosing interval. A half-life is how long it takes for half of the drug to be eliminated from the body.
If a single dose is given every half-life, half of the first dose will be cleared from the body before the next dose. So, after the second dose, there will be 1. Half of that is eliminated and then the next dose is given, meaning there are now 1. At dose 5 after five half-lives , there will be close to two doses in the body, which means one entire dose is eliminated each dosing interval. If we continue dosing at the same frequency, the amount we dose will be eliminated during each dosing interval.
As a result, drug concentrations in the body remain constant steady. Another way to think about steady state:. But a very simple way to remember it is that the average C ss is the total exposure AUC over one dosing interval divided by the duration of the dosing interval.
For a drug with a short half-life, steady state is achieved pretty quickly. Samples of urine and blood were removed periodically and assayed for parent drug. The following data were obtained:. The sigma-minus method , or the amount of drug remaining to be excreted method , is an alternative method for calculation of the elimination rate constant k from urinary excretion data is.
This method is sometimes preferred over the previous method since minimum fluctuations in the rate of elimination is obtained by this method. It was observed that the plasma-level time curve for some drugs following its rapid IV injection does not decline linearly as a single, first-order rate process. This nonlinear plasma-level time curve is attributed to distribution of these drugs at various rates into different tissue groups.
So, these multicompartment models were developed to explain and also to predict the plasma and tissue concentrations of these drugs. Again, drug distribution in the body depends mainly on plasma protein binding, tissue affinity, and drug lipo- or hydrophilicity. It must be mentioned that the kinetic description of the multicompartment process assumes that the rate of drug transfer between central and tissue compartments is first-order.
It is noteworthy to mention that the body can be divided into organs with high blood perfusion and those with slow blood perfusion. The central compartment consists of the plasma, extracellular fluids and highly perfused tissues in which drug equilibrate rapidly. The kidney and liver, which are the tissues for drug elimination, are considered integral parts of the central compartment.
The plasma-level time curve for a drug that follows a two-compartment model shows that the plasma drug concentration declines biexponentially as the sum of two first-order processes—distribution and elimination. This is attributed to distribution of the administered drug between the central compartment and tissue peripheral compartment. Figure shows distribution of the administered drug between central and peripheral compartment. Construction of the plasma-level time curve for a drug that follows a two-compartment model indicates that the curve may be divided into two phases: a distribution phase and an elimination phase.
Later, the drug starts to enter the tissue peripheral compartment and when the drug reaches the maximum tissue concentrations, an equilibrium is established and the rate of drug entry into the tissue equals the rate of drug exit from the tissue. At this stage, the drug concentrations in the central and tissue compartments will decline in a parallel and slower manner when compared to the distribution phase.
This phase is the elimination phase and the decline is a first-order process. For most two-compartment models the elimination occurs from the central compartment model unless other information about the drug is known since the major sites of drug elimination renal excretion and hepatic metabolism occur from organs such as the kidney and liver, which are highly perfused with blood [ 6 , 7 ].
If k 12 and k 21 are first-order rate constants that govern the rate of drug change in and out of the tissues, then the change in drug concentration in the tissue with time could be calculated from the following equation:.
The constants a and b represent the rate constants for the two phases, distribution phase and elimination phase, respectively. The constants A and B are intercepts on the y axis obtained from the plasma-level time curve after IV bolus, which exhibit two compartments. These values may be obtained graphically by the method of residuals or by computer.
Semilog plot of plasma-level versus time for a two-compartment IV bolus model. This method is used for fitting curve into the experimental data when the drug does not follow a one-compartment model. The method is sometimes called Feathering or Peeling method. A kg patient was administered a drug by rapid IV injection in a dose of mg. Blood samples were taken periodically after the administration of drug, and the plasma samples were assayed for the drug concentration. If you plotted the provided data on semilogarithmic graph paper, a curved line is observed which indicates that the drug is distributed in more than one compartment.
From the data, the constants may be obtained either by the computer or by the method of residuals, in which the equation that describes the process is:. Plotting of the data indicates that the curve is biexponential: the first segment for the distribution phase rapid phase , while the second for the elimination phase.
The rapid distribution phase is confirmed when comparing the values for a and b , the constant a being larger than the rate constant b. Therefore, at some later time, the term Ae —at will approach zero, while Be —bt will still have a value. At this later time, the two-compartment IV bolus equation will reduce to:. To obtain the constant a, the values for the drug plasma concentrations at the extrapolated line are subtracted from the original experimental data point to get the residual plasma concentration.
Plotting the values of the residual plasma concentrations versus time will yield a straight line that represents the rapid distribution a phase. Semilogplot of Cp versus time showing residual line. A number of pharmacokinetic parameters may be derived by proper substitution of rate constants a , b , and y intercepts A and B into the following equations:.
As mentioned previously, drugs with extravascular distribution such as those with high peripheral tissue binding contribute to a large apparent volume of distribution, while drugs that are polar with low lipid solubility or which highly bound to plasma protein account for small apparent V D. In multiple-compartment kinetics, such as the two-compartment model, several volumes of distribution can be calculated. For many polar drugs, an initial volume of 7—10 L may be interpreted as rapid distribution of the administered drug within the plasma and some body extracellular fluids.
For example, the V p of the antibiotic moxalactam ranges from 0. If this is true, then the volume of distribution will equal 3 L and if not, then distribution of drug may also occur outside the vascular pool. When the rate of drug entering the peripheral tissue compartment from the central compartment is equal to the rate of drug exit from the tissue compartment into the central compartment, this condition is achieved at steady-state and the rates of drug transfer between the two compartments are described by the following expressions:.
Since the amount of drug in the central compartment, D p , is equal to V p C p ,. Assuming a steady-state condition is reached, therefore the apparent volume of drug distribution at steady-state V D ss may be determined by dividing the total amount of drug in the body by the concentration of drug in the central compartment at steady-state:. The volume of distribution by area, also known as V D area , or simply V D , is obtained by a method similar to that used to find V p , except that the rate constant b is used instead of the overall elimination rate constant k.
This is achieved through dividing the total body clearance by b and is influenced by drug elimination in the beta, or b , phase. So, the reduction in drug clearance from the body may increase the area under the curve AUC, which is reflected on the value of V D that is either reduced or unchanged depending on the value of b. Here the volume of distribution is related to the body clearance, and the body clearance usually occurs during the elimination phase.
The first two columns of the provided table represent the time and plasma concentration, which may be collected after IV bolus administration of mg of drug. If these data are plotted, the following figure 7 is obtained:.
C extrapolated values at early times are shown in column 3 and the residual in column 4. The concept for drug clearance that follows a two-compartment model is similar to that of the one-compartment.
Clearance may be calculated without consideration of the compartment model. Clearance is calculated by dividing the IV bolus dose by the area under the plasma-level time curve from zero to infinity. The last equation is simple and gives more accurate results than using the trapezoidal rule to obtain area. Such drugs are infused slowly through a vein into the blood at a constant rate zero order input which allows precise control of plasma drug concentrations.
The following figure represents the plasma-level time curve for a drug given by constant IV infusion. At time zero, no drug was present in the body after which the drug level gradually increases until it becomes constant plateau or steady-state. Once the drug has reached the steady-state, the rate of drug leaving the body is equal to the rate of drug entering the body.
Plasma-concentration time curve during the infusion of the administereddrug at constant rate. The mathematical expression or the pharmacokinetic equation for drug administered by infusion will depend on whether the drug follows the one- or two-compartment model.
Drugs administered by constant IV infusion show a zero-order input process, during which the drug is introduced into the bloodstream while the elimination process for most drugs is first-order.
Plasma-concentration time curve after IV Infusion. If D B is the amount of the drug in the body, R is the rate of drug input infusion rate and k is the elimination rate constant. The expression that best describes the process is:. At steady-state, the rate of drug input R is equal to the rate of drug output k D P ,.
The average half-life of theophylline is about 4 hr and the apparent volume of distribution is about 25 liter. What is the necessary infusion rate? During administration of a drug by IV infusion, the plasma drug concentration starts to increase and the rate of drug elimination will also increase since the latter is concentration-dependent.
C p keeps increasing until a steady-state condition is reached at which the rate of drug input IV infusion rate equals the rate of drug output elimination rate. At this stage, a steady-state C SS is reached and the resulting plasma drug concentration is directly related to the rate of infusion and inversely related to the body clearance of the drug.
For drug administered by IV infusion, the therapeutic activity is observed when the concentration of the drug is close to the desired plasma concentration, which is usually the required steady-state drug concentration. The time to reach steady-state could be determined by knowing the time to reach half the steady-state which can be derived:. It must be noticed that the time to reach half the steady-state has the same value for the elimination half-life and is dependent on the elimination process not the infusion rate while the value of C ss is controlled by the infusion rate.
An increase in the infusion rate will not shorten the time to reach the steady-state drug concentration. From the pharmacokinetic point of view, for drugs administered by IV infusion, the clinical effect of the drug activity is observed when the drug concentration in the plasma is close to the desired plasma drug concentration, which is the desired steady-state drug concentration. A drug belonging to the cephalosporins antibiotics has a volume of distribution of 10 L and an elimination rate constant k value of 0.
What is the infusion rate required to maintain this concentration. Calculate the total body clearance Cl T for this drug. An infinitely long period of time is needed to reach a steady-state level of a certain drug. Calculate a time required to reach steady-state relative to the half-life of this drug. As repeated doses of a drug are administered its plasma concentration builds up and reaches what is known as a steady state. This is when the amount of drug in the plasma has built up to a concentration level that is therapeutically effective and as long as regular doses are administered to balance the amount of drug being cleared the drug will continue to be active.
Using the eggs represents the variety of clearance mechanisms that eliminate drug molecules from circulation. And the replenishment of eggs represent taking new doses of medication. There are several special PK parameters associated with steady state kinetics. These parameters are not necessarily more important; however, they are useful because of the unique situation when drug input rate and elimination rate are equivalent.
The first is the average plasma concentration at steady state, or C ss. This parameter can be calculated based on the steady state definition where the rate of input is equal to the rate of elimination. Rate of elimination:. By solving for C p , you get the following:. As a further simplification, we know that there is a relationship between dose, clearance, and bioavailability shown by the following equation:. Thus, the average concentration at steady state is simply the total exposure over 1 dosing interval divided by the time of the dosing interval.
So while concentrations rise and fall during a dosing interval at steady state, the average concentration does not change. Furthermore, the only factors that control C ss are the dose, the dosing interval, and the clearance.
Assuming clearance cannot be altered by a clinician, the steady state levels of drug can be modulated using the dose and the dosing interval. Lower doses and longer intervals will result in lower C ss values, while higher doses and shorter intervals will give higher C ss values.
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