Advances in Pharmaceutical Technology

By Dr. S. C. Dinda

Advances in Pharmaceutical technology is designed to present a comprehensive coverage on the advanced area of Pharmaceutical Technology utilized towards designing of Drug delivery and Drug Targeting systems. It is a compilation of Pharmaceutical Biotechnology, Nanotechnology, Immunotechnology and such other Drug Delivery Technology, utilized for drug delivery and drug targeting through the oral as well as parenteral route. The book covers the basic scientific principles involved in designing approach of drug delivery systems and focuses on current area of biomedical applications as well as highlight on its ongoing research at present scenario, which may encourage the readers to develop their basic knowledge, update about the present area of research output and will create interest towards the research involvement.
Features
• Covers the basic scientific principles involved in designing approach of drug delivery systems
• Focuses on current area of biomedical applications as well as highlight on its ongoing research

• Language: English
• Publisher: BSP BOOKS
• Release date: Nov 5, 2019
• ISBN: 9789386211385

Book preview

Index

Chapter 1

Pharmacokinetic Approaches in Designing of Drug Delivery Systems

Introduction

Pharmacokinetics is the study of those rate process involved in the absorption, distribution, metabolism, and excretion of drugs and their relationship to the pharmacological, therapeutic, or toxic response in animals or humans. Pharmacokinetics technique attempts mathematically to define the time course of drug in tire body by assaying the drug and it's metabolite in readily accessible fluids such as blood and urine. The goal is to quantify the amount of drug available to the body fluid (bioavailability) from the time of administration to the time of total clearance.

Schematic representation of drug absorption, distribution and elimination.

The study of pharmacokinetics involves experimental approaches to develop the biologic sampling techniques; analytical method development for the measurement of the drugs and metabolites; data collection and manipulation; as well as theoretical aspects to develop pharmacokinetic models to predict the drug disposition after drug administration. The estimation and interpretation of data result in predicting dosage regimen for individual or group of patients.

Pharmacokinetic concepts are utilized at all stage of drug development. Clinical application of pharmacokinetics have resulted in the improvement of drug utilization and consequently, direct benefit to patients. Clinically, two of the important applications of pharmacokinetic principles are: design of an optimal dosage regimen and clinical management of individual patient and therapeutic drug monitoring.

1.1 Rate, Rate Constant, and Order

1.1.1 Rate of a Chemical Reaction

It is defined as the quantity of a reactant consumed or the quantity of a product formed in unit time in a chemical reaction.

Mathematically:

In other words,

1.1.2 Velocity of Reaction

Since the rate of reaction is not constant throughout the reaction, therefore, we cannot determine the uniform rate of reaction precisely. Thus velocity of reaction may be defined as the rate of reaction at a particular given moment i.e. at a specific time.

If we consider a very small interval of time dt, in which the change in concentration dx, which is taken to be nearly constant, then velocity of reaction is given by:

Rate expression and rate constant

Consider a general reaction:

Reactants -> products

According to the law of mass action, rate of reaction is directly proportional to the active mass, hence for the above reaction becomes:

Rate of reaction a [Reactants]

Rate of reaction = K [Reactants]

This expression is called rate expression and K is called rate constant or velocity constant.

Characteristics of rate constant

1.1.3 Order of Reaction

The order of reaction is defined as the sum of all the exponents of the reactants involved in the rate equation.

It should be noted down that all the molecules shown in a chemical equation do not determine the value of order of reaction but only those molecules whose concentrations are changed are included in the determination of the order of a reaction. In other w ords:

The number of reacting molecules whose concentration alters as a result of chemical reaction is termed as the order of reaction.

For example:

For a reaction, maximum order is three and the minimum is zero.

First order reactions

The reaction in which only one molecule undergoes a chemical change is called first order reactions. Example:

Second order reactions

The reaction in which two molecules undergo a chemical change is called second order reactions. Example:

Third order reactions

The reaction in which only three molecules undergo a chemical change is called third order reactions. Example:

In chemical kinetics, the order of reaction with respect to a certain reactant, is defined as the power to which its concentration term in the rate equation is raised.

For example, given a chemical reaction 2A + B —> C with a rate equation

The reaction order with respect to A would be 2 and with respect to B would be 1; the total reaction order would be 2 + 1 = 3. It is not necessary that the order of a reaction be a whole number i.e. zero and fractional values of order are possible, but they tend to be integers. Reaction orders can be determined only by experiment. Their knowledge allows conclusions about the reaction mechanism.

The reaction order may not necessarily be related to the stoichiometry of the reaction, unless the reaction is elementary. Complex reactions may or may not have reaction orders equal to their stoichiometric coefficients. For example:

•   The alkaline hydrolysis of ethyl acetate is:

It has the following rate equation: r = k [CH3COOC2H5] [OH]

•   The rate equation for imidazole catalyzed hydrolysis may be:

Although, there may not be presence of imidazole in the stoichiometric chemical equation.

•   In the reaction of aryldiazonium ions with nucleophiles in aqueous solution i.e., ArN2+ + X" —> ArX + N2> the rate equation may be:

Reactions can also have an undefined reaction order with respect to a reactant, for example one can not talk about reaction order in the rate equation, when dealing with a bimolecular reaction between adsorbed molecules, the rate equation will be:

If the concentration of one of the reactants remains constant (because of it is a catalyst or it is in great excess with respect to the other reactants) its concentration can be included in the rate constant, obtaining a pseudo constant: if B is the reactant, whose concentration is constant then the eq. becomes:

The second-order rate equation has been reduced to a pseudo-first-order rate equation. This makes the treatment to obtain an integrated rate equation much easier.

Zero-order reactions are often seen for thermal chemical decompositions where the reaction rate is independent of the concentration of the reactant (changing the concentration has no effect on the speed of the reaction).

Broken-order reactions

This order is a non-integer typical of reactions with a complex reaction mechanism. For example, the chemical decomposition of ethanol into methane and carbon monoxide proceeds with an order of 1.5 with respect to ethanol. The decomposition of phosgene to carbon monoxide and chlorine has order-1, with respect to phosgene itself and order 0.5 with respect to chlorine.

Mixed-order reaction

This order of a reaction changes in the course of a reaction as a result of changing variables such as pH. An example is the oxidation of an alcohol to a ketone by a ruthenate (RuO4² ) and a hexacyanoferrate, the latter serving as the sacrificial catalyst converting Ru(IV) back to Ru(VI): the disappearing-rate of the ferrate is zero-order with respect to the ferrate at the onset of the reaction (when its concentration is high and the ruthenium catalyst is quickly regenerated) but changes to first-order, when its concentration decreases.

Negative-order reactions are rare, for example, the conversion of ozone (order 2) to oxygen (order 1).

1.1.4 Rate Equation

The rate law or rate equation for a chemical reaction is an equation, which links the reaction rate with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial reaction orders). To determine the rate equation for a particular system one can combine the reaction rate with a mass balance for the system. For a generic reaction /nA + wB —> C’ with no intermediate steps involved in its reaction mechanism (that is, an elementary reaction), can be expressed as:

Where, [A | and [B] expresses the concentration of the species A and B, respectively [(usually in moles per liter (molarities)]; m and n are the respective stoichiometric coefficients of the imbalanced equation; they must be determined experimentally, k is the rate coefficient or rate constant of the reaction. The value of this coefficient k depends on conditions such as temperature, ionic strength and surface area of the adsorbent or light irradiation. For elementary reactions, the rate equation can be derived from first principles using collision theory. Again, m and n are not always derived from the balanced equation.

The rate equation of a reaction with a multi-step mechanism cannot, in general, be deduced from the stoichiometric coefficients of the overall reaction; it must be determined experimentally. The equation may involve fractional exponential coefficients, or it may depend on the concentration of an intermediate species.

The rate equation is a differential equation, and it can be integrated to obtain an integrated rate equation, which links concentrations of reactants or products with time.

If the concentration of one of the reactants remains constant (because it is a catalyst or it is in great excess with respect to the other reactants), its concentration can be grouped with the rate constant, obtaining a pseudo constant. If, B is the reactant, whose concentration is constant, then r = L[A][B] = k'[A]. The second order rate equation has been reduced to a pseudo first order rate equation. This makes the treatment to obtain an integrated rate equation much easier.

Zero-order reactions

Zero-order kinetics has a rate which is independent of the concentration of the reactant) s). Increasing the concentration of the reacting species will not speed up the rate of the reaction. Zero-order reactions are typically found when a material that is required for the reaction to proceed, such as a catalyst, which is saturated by the reactants. The rate law for a zeroorder reaction can be expressed as:

Where, r is the reaction rate, and k is the reaction rate coefficient with units of concentration/time. It occurs with a condition such as: 1) the reaction occurs in a closed system; 2) there is no net build-up of intermediates; and 3) there are no other reactions occurring simultaneously, which can be shown by solving a mass balance for the system using the following equation:

If, this differential equation is integrated it gives an equation, which is often called the integrated zero-order rate law as:

Where, represents the concentration of the chemical of interest at a particular time, and [Α]t represents the initial concentration.

A reaction is zero order, if concentration data are plotted versus time and the result is a straight line. The slope of this resulting line is the negative of the zero order rate constant k.

The half-life of a reaction describes the time need for half of the reactant to be depleted (same as the half-life involved in nuclear decay, which is a first-order reaction). For a zero-order reaction the half-life is given by

Example of a zero-order reaction is:

It should be noted that the order of a reaction cannot be deduced from the chemical equation of the reaction.

First-order reactions

A first-order reaction depends on the concentration of only one reactant (a unimolecular reaction). Other reactants can be present, but each will be zero-order. The rate law for an elementary reaction, which is first order with respect to a reactant ‘A’ is:

Where, k is the first order rate constant, which has units of 1/time.

The integrated first-order rate equation is

A plot of ln[A] Vs. time t gives a straight line with a slope of - k.

The half life of a first-order reaction is independent of the starting concentration and is given by

Examples of some reactions, which are first-order with respect to the reactant, are:

Further Properties of First-Order Reaction Kinetics

The integrated first-order rate equation, In [A] = - kt + In [A]o, may usually be written in the form of the exponential decay equation as:

A different (but equivalent) way of considering first order kinetics is as follows:

The exponential decay equation can be rewritten as:

where, Δtρ corresponds to a specific time period and n is an integer corresponding to the number of time periods. At the end of each time period, the fraction of the reactant population remaining relative to the amount present at the start of the time period,^, will be:

Such that after n time periods, the fraction of the original reactant population will be:

where, fBp corresponds to the fraction of the reactant population that will break down in each time period. This equation indicates that the fraction of the total amount of reactant population, which will break down in each time period is independent of the initial amount present. The chosen time period may corresponds to:

Where, the fraction of the population, which will break down in each time period will be exactly Ά the amount present i.e. the time period corresponds to the half-life of the first-order reaction).

The average rate of the reaction for the nth time period may be given by:

Therefore, the amount remaining at the end of each time period will be related to the average rate of that time period and the reactant population at the starting time period follows the equation:

Then the fraction of the reactant population, which will break down in each time period can be expressed as:

The amount of reactant, which break down in each time period in relation to the average rate over a time period may be expressed by:

Such that the amount that remains at the end of each time period w ill be related to the amount present at the starting time according to the equation as:

This equation is a precursor, allow ing the calculation of the amount present after any number of time periods, without need of the rate constant, provided that the average rate for each time period is know n.

Second-order reactions

A second-order reaction depends on the concentrations of one second-order reactant, or two first-order reactants.

For a second order reaction, its reaction rate is given by:

In several popular kinetic, the rate law for second-order reaction is defined as:

Conflating the 2 inside the constant for the first derivative form may make it the second integrated form (presented below). The option of keeping the 2 out of the constant in the derivative form is considered more correctly as it is almost always used in universally.

There by the integrated second-order rate equations may be written as:

[A]o and [B]o must be different to obtain that integrated equation.

The half-life equation for a second-order reaction may be:

It may be considered that for a second-order reaction, half-lives progressively double.

Another way to present the above rate equation by taking the log on both sides as:

Example of a Second-order reaction is:

Pseudo first order

Measuring a second order reaction rate with reactants A and B can be problematic. The concentrations of the two reactants must be followed simultaneously, which is more difficult; or measurement of one of them and calculate the other as a difference, which is less precise. A common solution forthat problem is the pseudo first order approximation

If either [A] or [B] remain constant as the reaction proceeds, then the reaction can be considered pseudo first order because of it only depends on the concentration of one reactant. For example, [B] remains constant then:

Where, k' = £[B]0 (k' or kobs with units s ') and an expression is identical to that of the first order expression above.

One way to obtain a pseudo first order reaction is to use a large excess of one of the reactants ([B]»[A]), so that, as the reaction progresses only a small amount of the reactant is consumed and its concentration can be considered to stay constant. By collecting k for many reactions with different (but excess) concentrations of [BJ; a plot of k versus [B] gives k (the regular second order rate constant) and the slope.

Reactions with order 3 (called ternary reactions) are rare and unlikely to occur.

Summary for reaction orders 0, 1, 2 and n

Where. M stands for concentration in molarity (mol · L ¹), t for time, and k for the reaction rate constant. Half life is often expressed as ti/2=0.693/k for a first order reaction (as In 2=0.693).

1.2 Pharmacokinetic Models

A model in pharmacokinetics is a hypothetical structure, which can be used to characterize with reproducibility, the behavior and fate of a drug in biological system when administered through certain route of administration and in a particular dosage form.

1.2.1 One-Compartment Model

Following drug administration, the body is depicted as a kinetically homogeneous unit (see Fig 1.1). This assumes that the drug achieves instantaneous distribution throughout the body and that the drug equilibrates instantaneously between tissues. Thus the drug concentration-time profile shows a monophasic response (i.e., it is monoexponential; Figure 1.1a). It is important to note that this does not imply that the dmg concentration in plasma (Cp) is equal to the drug concentration in the tissues. However, changes in the plasma concentration quantitatively reflect changes in the tissues. The relationship described in Fig 1.1(a) can be plotted on a log Cp Vs time graph (Fig. 1.1b) will show a linear relation, represents a one-compartment model.

Fig. 1.1 One-compartment model; Ka = absorption rate constant (h=1), K = elimination rate constant (h=1). (a) Plasma concentration (Cp) versus time profile of a drug showing a one-compartment model, (b) Time profile of a one-compartment model showing log Cp versus time.

1.2.2 Two-Compartment Model

The two-compartment model resolves the body into a central compartment and a peripheral compartment (see Fig 1.2). Although these compartments have no physiological or anatomical meaning, it is assumed that the central compartment comprises tissues that are highly perfused such as heart, lungs, kidneys, liver and brain. The peripheral compartment comprises less well-perfused tissues such as muscle, fat and skin. A two-compartment model assumes that, following drug administration into the central compartment, the drug distributes between the central compartment and the peripheral compartment. However, the drug does not achieve instantaneous distribution, i.e. equilibration, between the two compartments. The dmg concentration-time profde shows a curve (Fig. 1.2 a), but the log dmg concentration-time plot shows a biphasic response (Fig. 1.2 b)

Fig. 1.2 Two-compartment model; K12, K21 and K are first-order rate constants: K12 = rate of transfer from central to peripheral compartment; K21 = rate of transfer from peripheral to central compartment; K= rate of elimination from central compartment (a) Plasma concentration versus time profile of a drug showing a two compartment model, (b) Time profile of a two-compartment model showing log Cp versus time.

Fig 1.2(b) can be used to distinguish whether a drug shows a one- or two-compartment model. Fig 1.2(b) shows a profile in which initially there is a rapid decline in the drug concentration owing to elimination from the central compartment and distribution to the peripheral compartment. Hence during this rapid initial phase, the drug concentration will decline rapidly from the central compartment, rise to a maximum in the peripheral compartment. After a time interval (t), distribution equilibrium is achieved between the central and peripheral compartments, and elimination of the drug is assumed to occur from the central compartment as that of the one compartment model. All the rate processes may be described by first-order rate process.

1.2.3 Multi-compartment Model

In this model, the drug distributes into more than one compartment and the concentration-time profile shows more than one exponential (Fig 1.3a). Each exponential on the concentration-time profile describes a compartment. For example, gentamycin can be described by a three-compartment model following a single IV dose (see Fig 1.3b).

Fig. 1.3 (a) Plasma concentration versus time profile of a drug showing Multi-compartment model, (b) Time profile of a multi-compartment model showing log Cp versus time.

1.3 Pharmacokinetic Parameters in Dosage Regimen Fixation

For a majority of drugs subject to pharmacokinetic monitoring, the goal is to design individualized dosage regimen for patients, in order to keep the plasma concentrations of the drug within a preset minimum (Cmin) and maximum (Cmax) for multiple dosing regimens or at steady state (Css) for constant input regimens. For most drugs, the Cmin and Cmax values correspond to the minimum effective (MEC) and minimum toxic (MTC) concentrations. For example, the therapeutic range of digoxin for cardiac dysfunction is between 0.8 and 2ng/mL. This implies that the concentrations above 2ng/mL are more likely to be associated with toxicity and concentrations below 0.8ng/mL are more likely to produce little or no effect. Therefore, it is desired to fix a dosage regimen for digoxin to produce plasma concentrations within this therapeutic range.

The required data to design a dosage regimen depends on the kinetics of the drug in the patient and the reported therapeutic range of the drug. The kinetic parameters are derived from the plasma sample(s) taken from the patient (patient-specific values), adjustment of the reported average kinetic values with patient characteristics such as renal function (adjusted population values), or both the population and patient-specific data i.e. disease conditions, age, sex, body weight, etc.

1.3.1 Volume of Distribution

The volume of distribution (Vd) has no direct physiological meaning; it is not a ‘real’ volume and is usually referred to as the apparent volume of distribution. It is defined as the volume of plasma to which the total amount of drug in the body would be available to show the therapeutic effect. The body is not a homogeneous unit, even though a one-compartment model can be used to describe the plasma concentrationtime profile of a number of drugs. It is important to realize that the concentration of the drug (Cp) in plasma is not necessarily the same in the liver, kidneys or other tissues. Thus Cp in plasma does not equal Cp i.e. amount of drug (X) in the kidney or Cp i.e. amount of drug (X) in the liver or Cp i.e. amount of drug (X) in tissues. However, changes in the drug concentration in plasma (Cp) are proportional to changes in the amount of drug (X) in the tissues.

Since, Cp (plasma) a Cp (tissues) i.e. Cp (plasma) aX (tissues)

Then Cp (plasma) = Vd xX (tissues)

Where, Vd is the constant of proportionality and is referred to as the volume of distribution, which thus relates the total amount of drug in the body at any time to the corresponding plasma concentration.

And, Vd can be used to convert drug amount X to concentration.

This formula describes a mono-exponential decay (see Fig 1.2), where Cpi = plasma concentration at any time t.

The curve can be converted to a linear form (Fig 1.4) using natural logarithms (In):

Where, the slope = - k, the elimination rate constant and the y, is the intercept = In C° .

Then, at zero concentration (C° ), the amount administered is the dose, £>,

Fig. 1.4 In Cp Vs time profile.

If the drug has a large Fd, which does not equate to a real volume, e.g. total plasma volume, this suggests that the drug is highly distributed in tissues. On the other hand, if the Vd is similar to the total plasma volume, which will suggest that the total amount of drug is poorly distributed and is mainly in the plasma.

1.3.2 Elimination Rate Constant

Consider a single IV bolus injection of drug X (see Fig 1.2). As time proceeds, the amount of drug in the body is eliminated. Thus the rate of elimination can be described (assuming first-order elimination) as:

Where, X = amount of drug X, Xo = dose and k = first-order elimination rate constant.

1.3.3 Half-Life

The time required to reduce the plasma concentration to one half its initial values and is defined as the half-life (q 2)· Consider

Let Cp decay to Cp / 2 and solve fort = t1/2:

Hence

And

This parameter is very useful for estimating how long it will take to reduce the half of the original/initial concentration. It can also be used to estimate, how long a drug should be stopped, if a patient showing toxic manifestations, assuming the drug shows linear one-compartment pharmacokinetics.

1.3.4 Clearance

Drug clearance (CL) is defined as the volume of plasma in the vascular compartment cleared off drug per unit time by the processes of metabolism and/or excretion. Rate of clearance of a drug is constant, if the drug is eliminated following first-order kinetics. Drug can be cleared by renal excretion or through metabolic processes or following both. With respect to the kidney and liver, etc., clearances are additive, which can be determined as:

Mathematically, clearance is the product of the first-order elimination rate constant (k) and the apparent volume of distribution (ff).

Thus.

Hence, the clearance is the elimination rate constant (i.e. the fractional rate of drug loss) from the volume of distribution.

Clearance can be expressed in relation to half-life by the rate process following 1st order kinetics as:

If a drug has a CL of 2L/h, which tells that the 2 liters of the Fd is cleared off drug per hour. If the Cp is 10 mg/L, then 20 mg of drug is cleared per hour.

1.4 Relationship among Pharmacokinetic Parameters

Before discussing a modification of dosage regimens, it is necessary to realize, how the three major kinetic parameters (V, Cl, and ti/2 or K) are related to each other. This is important because of these kinetic parameters determine the shape of the plasma concentration-time profiles and therefore, affect the fluctuation between the Cαmax and Cαmin values. The mathematical relationship among these three parameters is demonstrated below:

If, two of these parameters are known, the third can easily be estimated from the above equation. However, the use of the above equation without an understanding of the underlying physiological relationship among these three parameters may result in erroneous conclusions. This is because of V and Cl, which are independent parameters, while the elimination half life (or rate constant) is a composite parameter dependent on both V and Cl as described below-.

Clearance is a measure of the efficiency of the organ(s) of elimination and is dependent on certain physiologic parameters in the organ (e.g., organ blood flow and drug intrinsic clearance and free fraction in the blood). For instance, for a drug eliminated by renal excretion, clearance is dependent on how well the kidneys can excrete the drug in urine. Therefore, in the elderly with reduced renal function, the clearance of renally excreted drugs would be reduced.

The extent of distribution of drugs, however, is independent of their clearance. Distribution is dependent on certain physiologic parameters such as perfusion and permeability of tissues to drugs and the level of binding proteins in the blood and tissues. Therefore, a reduction in the renal clearance of a drug in the elderly does not necessarily mean that the volume of distribution of the drug will also be different in this population. In other words, clearance and volume of distribution are independent of each other.

On the other hand, the elimination half life is dependent on both V and Cl. An increase in Cl (elimination efficiency) results in a reduction in t1/2 (or an increase in K). This is easy to understand that the more efficient elimination pathway may result in a faster decline in the plasma concentrations. An increase in V, however, results in prolongation of t1/2, because of the drug is distributed more extensively into the tissues (where it is safe to be eliminated). However, because the distribution is a reversible process, as the drug gets eliminated and plasma concentrations decline, the drug in the tissue will return to plasma, resulting in a more sustained level in plasma (increased half life). Therefore, the half life is dependent on both the clearance and volume of distribution, and a more appropriate presentation of the relationship among these three parameters is:

Or,

For example, if the volume of distribution changes, the half life (or rate constant) changes proportionally while clearance remains the same. To demonstrate this point, consider the following scenario. The volume of distribution and elimination rate constant of a drug in a patient are 35 L and 0.091 hr-1, respectively. While under treatment with this drug, the patient receives a second drug, which increases the drug V to 70 L. What is the clearance of the drug in the absence and presence of the interacting drug?

The mathematical relationship Cl = K * V may be used to estimate Cl in the absence of drug interaction:

Clearance is independent of V changes. Therefore, when V is increased to 70 L due to a drug interaction, Cl remains the same (3.2 L/hr). However, the above equation without an understanding of this fundamental concept may be misleading. This is because one may erroneously conclude from the equation Cl = K * V that doubling V would result in doubling Cl. This conclusion, however, is not valid as a two-fold increase in V would result in a two-fold decrease in K without any effect on Cl.

1.5 Steady State

Steady state occurs, when the amount of drug administered (in a given time period) is equal to the amount of drag eliminated in that same period. At steady state, the plasma concentrations of the drag (Cpss) at any time during any dosing interval, as well as the peak and trough, are similar. The time to reach steady-state concentrations is dependent on the half-life of the drag under consideration.

Effect of Dose on Steady state

The higher the dose, the higher the steady-state levels, but the time to achieve steady-state levels is independent of dose (see Fig 1.5). Note that the fluctuations in Cp max and Cp min are greatest with higher doses.

Fig. 1.5 Time profile of multiple IV doses - reaching steady state using different doses.

Time to reach steady state

For a drag with one-compartment characteristics, the time to reach steady state is independent of the dose, the number of doses administered, and the dosing interval, but it is directly proportional to the half-life.

Prior to steady state

As an example, estimate the plasma concentration at 12 h after therapy commences with a drag A given 500 mg three times a day.

Fig. 1.6 Multiple intravenous doses prior to steady state.

Consider each dose as independent and calculate the contribution of each dose to the plasma level at 12 h post dose (see Fig 1.6).

From the first dose:

From the second dose:

Thus, total Cpt at 12 h is

Remember that Cp = D / Vd

This method uses the principle of superposition. The following equation can be used to simplify the process of calculating the value of Cp at any time t after the nth dose:

Where, n = number of doses, T= dosing interval and t = time after the nth dose.

At steady state

To describe the plasma concentration (Cp) at any time (/) within a dosing interval (τ) at steady state (see Fig 1.7):

Fig. 1.7 Time profile at steady state and the maximum and minimum plasma concentration within a dosage interval.

Remember that Cp = D / Vd . Alternatively, for some drugs, it is important to consider the salt factor (S). Hence, if salt factor applicable, Cp =SD/Vand Cptwill be:

To describe the maximum plasma concentration at steady state (i.e. t =0 and exp (-kt) = 1):

To describe the minimum plasma concentration at steady state (i.e. t = τ):

To describe the average steady-state concentration. Cp

Steady state from first principles:

At steady state the rate of drug administration is equal to the rate of drug elimination. Mathematically the rate of drug administration can be stated in terms of the dose (D) and dosing interval (τ). It is always important to include the salt factor (.S') and the bioavailability (F). The rate of drug elimination will be the clearance of the plasma concentration at steady state:

Rate of drug elimination = CL x Cpss

At steady state:

Rearranging the equation:

In practice, steady state is assumed to be reached in 4-5 half-lives.

1.6 Dosage Regimen Fixation

1.6.1 Intravenous Infusion

Some drugs arc administered as an intravenous infusion rather than as an intravenous bolus. To describe the time course of the drug in the plasma during the infusion prior to steady state, one can use:

(If a salt form of the drug is given)

During the infusion, at steady state, as the rate in = rate out,

Where, R = D/τ = infusion rate (dose/h)

Loading dose

The time required to obtain steady-state plasma levels by IV infusion will be long, if the drug has a long half-life. It is, therefore, useful in such cases to administer an intravenous loading dose to attain the desired drug concentration immediately and then attempt to maintain this concentration by a continuous infusion.

To estimate the loading dose (LD), where Cpss is the final desired concentration, one can use to calculate LD by the equation:

If the patient has already received the drug, then the loading dose should be adjusted accordingly as:

or,

if, the salt form of the drug (salt factor S) is used.

Now consider the plasma concentration-time profile following a loading dose and maintenance infusion (see Fig 1.8).

Fig. 1.8 Profile following a loading dose and maintenance infusion.

The equation to describe the time course of the plasma drug concentrations following simultaneous administration of