IDIT - Quantitative Pharmacokinetics/Issar
What is a compartment?
> A compartment is not a real physiologic or anatomic region but is considered as a tissue or group of tissues that have similar blood flow and drug affinity. > Within each compartment, the drug is considered to be uniformly distributed. > Mixing of the drug within a compartment is rapid and homogeneous and is considered to be "well stirred," so that the drug concentration represents an average concentration, and each drug molecule has an equal probability of leaving the compartment.
Describe the one compartment model.
> Rate constants are used to represent the overall rate processes of drug entry into and exit from the compartment. > The model is an open system because drug can be eliminated from the system. Compartment models are based on linear assumptions using linear differential equations. One compartment linear model: According to this model we will consider the body to behave as a single well- mixed container. To use this model mathematically we need to make a number of assumptions. Ka = absorption rate constant and K = elimination rate constant What are the assumptions for this model? 1. Drug achieves instantaneous distribution throughout the body and that the drug equilibrates instantaneously between tissues. Thus the drug concentration-time profile shows monophasic response (i.e. monoexponential). It is important to note that this does not imply that the drug 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 can be plotted on a log Cp vs time graph and will then show a linear relation; this represents a one-compartment model. 2. We also need to assume that the drug is mixed instantaneously in blood or plasma. The actual time taken for mixing is usually very short, within a few of minutes, and in comparison with normal sampling times it is insignificant. We usually don't sample fast enough to see drug mixing in the blood. 3. We will assume that drug elimination follows first order kinetics. First order kinetics means that the rate of change of drug concentration by any process is directly proportional to the drug concentration remaining to undertake that process. Remember first order kinetics is an assumption of a linear model not a one compartment model. If we have a linear system and if we double the dose, the concentration will double at each time point.
Describe the SINGLE INTRAVENOUS BOLUS DOSE - A TWO COMPARTMENT MODEL.
A drug following two compartment model after a single bolus dose D = dose administered K (or Kelim)= elimination rate constant V1 = volume of the central compartment (blood) V2 = volume of the peripheral compartment (tissues) K12 = inter-compartmental rate transfer constant from central to peripheral (1 to 2) K21 = inter-compartmental rate transfer constant from peripheral to central (2 to 1) Shortly after the drug is injected into the central compartment most of the drug is in the compartment 1 as there has been insufficient time for any significant proportion of dose to distribute into compartment 2. There will be a marked disequilibrium between these compartments and the drug movement will be much greater from 1 to 2 than from 2 to 1. Hence net drug movement is going to happen from 1 to 2. Drug elimination will also occur from compartment 1 hence at this early time there would be two processes (a) distribution and (2) elimination that would removing the drug from the first compartment. Thus we will see a rapid drug fall in this compartment. Later, concentrations in compartment 1 would have fallen with a rise in the 2nd compartment and equilibrium would have reached. Thus we have only one process occurring in the central compartment (i.e. drug elimination) so the graph now fall more gently. At time point, continued drug elimination would have created another disequilibrium and the drug concentration in the 2nd compartment would be higher thus back movement of drug from 2 to 1 will occur. So we have two processes affecting the amount of drug in compartment 1. Elimination is removing the drug from the central compartment but requilibration is bringing the drug back from 2 to 1. Thus the rate of drug decline in compartment 1 is very slow since all we see is the balance of between the two processes (extent of elimination exceeds redistribution). The effect of elimination and re-distribution on the amount of drug in blood and the rest of the first compartment and their overall effect on drug concentration. [However, for a two compartmental model, the elimination phase half-life is often abbreviated t1/2 beta or (t1/2 β) to distinguish it from the distribution phase half-life (t1/2 alpha or t1/2 α).]
Describe how half-life contributes to the extent of fluctuation.
A major factor that controls the extent of fluctuation is the drug's half-life. Short half-life produces more fluctuations than the drugs that has a slower elimination. Very short half-life will bring in a risk of complications. Just like the long half-life drugs are problematic in achieving accumulation over an extended period (take too long to achieve SS). Thus both ends of the half-life spectrum are not desirable. If a drug is likely to suffer excessive fluctuations in concentrations there are two ways to moderate this - dose division and slow release formulations.
Describe accumulation in multiple dosing.
A similar relationship holds with multiple dosing compared to what was described in intravenous infusion. Long half-life elimination drug will take longer time to accumulate to steady state compared to a shorter half-life drug. Clinicians would be forced to use a loading dose to achieve therapeutic effect within a reasonable time-frame.
What are loading doses?
Addition of loading doses gives an opportunity to achieve steady state faster with an infusion rather without it. Giving a loading dose as an i.v. bolus, the Css drug levels are achieved rapidly without having the to wait for the drug to accumulate in the body. Using the same principle, D = Cp x VD, where Cp is the expected target concentration and VD the apparent volume of distribution for the drug would give us the dose required to meet the target concentration. Loading dose = target conc. x VD.
What is the accumulation period?
Although it is counterintuitive, there is a relationship between the drug's accumulation during the infusion and its elimination half-life. If the elimination t1/2 is T hrs then after infusing the drug for T hrs the drug levels will accumulate to half of projected Css. Ex. A drug with an elimination half-life of 3.4 hrs. T1 indicates a time equal to one half-life and the conc. has risen to ½ of Css. T2 marks where the equivalent of two half-lives (6.8 hrs) has passed and the conc. is ¾ of the Css. T3 - at this point 7/8 of the Css has been achieved and so on. For practical purposes we assume that the drug reaches Css in 3-4 half-lives. Thus, a drug with a short half-life will achieve Css rapidly compared to a drug that has a long half-life. Thus long half-life drugs would need a loading dose more often compared to drugs with shorter half-life.
Factors that Affect Half-Life
Any alteration of distribution, metabolism or excretion can affect the drug half-life (e.g., factors that affect plasma protein binding, liver metabolism, or renal excretion). Production of alkaline urine will decrease the half-life of a weak acid such as aspirin or phenobarbital. Age may markedly change the capacity for eliminating certain drugs, and therefore alter half-lives of drugs.
CALCULATION OF AREA UNDER THE CURVE (AUC)
Area under the curve can be calculated using the following example where drug X is given via i.v. administration to a patient and his blood samples were collected at the following time intervals. Time (hrs) Concentration in plasma (µg/mL) 0.5 - 38, 1.0 - 30, 2.0 - 18, 3.0 - 11, 4.0 - 8, 5.0 - 4 To calculate the area under the plasma conc-time curve for drug X between time 1 and 4 hours, the area between time intervals is an area of a trapezoid and can be calculated using the following equation. Ex AUC btwn 1 and 2 hrs AUC = [30+18]/2 x [2-1] =24 ig.hr/mL The total AUC between 1 and 4 hours is obtained by adding the three smaller AUC values together. i.e. 24+14.5+9.5 = 48 µg.hr/mL
What is the extraction ratio?
Assume blood flowing through the liver from left to right. The blood contains a drug that can be eliminated (extracted) by the liver. The concentration of drug entering the liver (Cin) is 10 mg/L but the time the blood exits the liver, this has been reduced to 4 mg/L (Cout). The extraction ratio (E) is the proportion of drug eliminated as blood passes through the liver. The amount removed is the difference between Cin and Cout (10-4 = 6 mg/L) and this is then expressed as a proportion of drug entering the liver which is 0.6 or 60%. NOTE: In terms of rate of drug elimination there is no difference in taking out all of the drug from 1L or 50% of the drug from 2L. The general equation for hepatic clearance is: CL=Qℎ×E hepatic blood flow x extraction ratio
Describe the RELATIONSHIP BETWEEN RINF (rate of infusion) AND CSS (concentration at steady state).
At steady state: Rinf = rate of elimination The rate of elimination at SS = mass of drug present at SS times the elimination rate constant (Mss x kelim) Kelim =Kinf = Mss×Kelim On rearranging Mss= Kinf⁄ Kelim Dividing by VD on both sides gives Mss⁄Vd= Rinf/(Kelim×Vd) Mss/VD = Css and kelim x VD is CL, thus Css= Rinf/CL
What is clearance?
Clearance is a measure of drug elimination from the body without identifying the mechanism or process. Clearance (drug clearance a.k.a systemic clearance a.k.a total body clearance, ClT) considers the entire body as a drug-eliminating system from which many elimination processes may occur. The mechanisms of drug elimination are complex, but collectively drug elimination from the body may be quantified using the concept of drug clearance. Drug clearance refers to the volume of plasma fluid that is cleared of drug per unit time. Clearance may also be considered as the fraction of drug removed per unit time. It is important to note that clearance does not indicate how much drug is being removed. The units of clearance are generally ml/min.
Describe dose division.
Ex: The overall rate of drug administration is 200 mg/day. However, it is modeled as being administered as a single dose (200 mg every 24 hrs) or in divided doses of 10 mg per 12 hrs or 50 mg per every 6 hrs. The most fluctuating trace: large doses are being given and we see prominent peak after each dose but we have to wait for a long time for the next dose over this extended period. Hence the fluctuations repeat in a similar fashion. In the central trace the individual doses are modest (50 mg) causing only small peaks and the dosage intervals are short (6hrs) allowing limited time to fall between doses. Thus we see a much smoother pattern. Extreme dose division gives a superior pattern of concentrations that stay comfortably within the desirable range in contrast to single daily doses where levels oscillate between toxicity and ineffectiveness. Frequent dosing encompasses - patient non-compliance hence a reasonable dosing interval would be 100 mg/12 hr where the drug fluctuations are still smoother compared to the 200 mg/day doses.
Describe first-order reactions/processes.
First-order reaction If the amount of drug A is decreasing at a rate that is proportional to the amount of drug A remaining, then the rate of disappearance of drug A is expressed as the differential equation: dA/dt=−kA where k* = the first-order rate constant. The reaction proceeds at a rate that is dependent on the concentration of drug A present in the body. It is assumed that the processes of ADME follow first-order reactions and most drugs are eliminated in this manner. However, there are notable exceptions, for example phenytoin and high-dose salicylates don't show first-order elimination. For drugs that show a first-order elimination process, as the amount of drug administered increases, the body is able to eliminate the drug accordingly and accumulation will not occur. However, if you continue to increase the amount of drug administered then all drugs will change from showing a first-order process to a zero-order process, for example in an overdose situation.
What are practical ways to restrict fluctuation?
For any given drug there is very little we can do about its half-life. However, we have the ability to control the dosage interval (dose division) and dosage form. If the drug has a rather short half-life and if we want to control the fluctuations we could do it in two ways. One, give the drug at shorter intervals (e.g. 4 times a day) or the other approach could be to use a sustained release formulation that is only given twice a day. The two methods will achieve very similar envelope of drug concentrations but for the patient's standpoint more easier to comply with.
Describe half-life and time-course of drug action
For many drugs the time-course of the plasma concentration of the drug and the time-course of the biological or pharmacodynamic effect run parallel. However, this is not always the case.
Describe the relationship between half-life, volume of distribution and clearance.
Half-life = 0.693/Kelim Substitution of Kelim in above equation by Cl/VD would rearrange equation as below: Half-life = 0.693.x Vd / Cl Half-life is shortened by a decrease in volume of distribution or an increase in clearance. Reduced clearance and increased volume of distribution tend to lengthen half-life. An increased volume of distribution reflects greater amounts of the drug distributed in the tissues, therefore becoming less available for elimination.
What is a half-life?
Half-life refers to the time it takes for the plasma concentration of the drug to fall by 50% regardless of the initial value. For each interval of elapsed time equal to a half-life, the concentration falls to one-half the value at the beginning of that interval. The terms plasma half-life, elimination half-life and biological half-life have been used interchangeably to refer to this concept. Strictly speaking, the term biological half-life should be reserved for the time it takes to eliminate half of the total body content of the drug. Note that the amount of drug eliminated during each half-life interval decreases as the concentration decreases. However, the fraction of drug eliminated during each interval is constant, i.e., 50%. When the half-life of a drug is referred to, it is assumed that the reference is being made to the elimination phase rather than the distribution phase, unless otherwise specified. The elimination phase half-life for a 1-compartmental model would be represented by t1/2. After four half-life intervals, more than 90% of the drug has been eliminated. If one drug has a longer half-life than another drug, then the time required for elimination of 90% of the first drug will be longer than the time required to eliminate 90% of the second drug. How do we calculate half-life? half-life = 0.693/kelim
Describe the extent of fluctuation in drug concentrations
If we give discrete doses of drugs, there will inevitably be a fluctuating pattern of drug conc. with a peak after each dose and a decline to a trough prior to the next dose. The three factors that influence extent of fluctuation are half-life, dose division and dosage form.
Describe mixed-order reactions/processes.
Mixed-Order or Non-Linear Kinetics The elimination of several important drugs converts from first-order to zero-order kinetics at clinically relevant concentrations. These drugs demonstrate mixed-order kinetics, sometimes referred to as nonlinear or dose-dependent kinetics. This occurs when the rate-limiting step or enzyme becomes saturable at concentrations approximating the range of therapeutic concentrations. At lower concentrations, the rate will depend upon the drug concentration (first-order). At higher concentrations within the clinical range, the system becomes saturated and elimination proceeds at a constant rate regardless of the drug concentration. Examples of mixed order kinetics: Phenytoin (Dilantin) (an anti epileptic drug), which switches kinetics within the therapeutic range, and salicylic acid (active metabolite of acetylsalicylic acid or aspirin), which exhibits first-order kinetics at concentrations that may be effective against fever and mild pain but switches to zero-order kinetics at larger doses and in the concentration range required for the treatment of rheumatoid arthritis.
What are pharmacokinetic models?
Pharmacokinetic models are hypothetical structures that are used to describe the fate of a drug in a biological system following its administration. Because of the vast complexity of the body, drug kinetics in the body are frequently simplified to be represented by one or more tanks, or compartments, that communicate reversibly with each other.
Loading dose and target concentrations.
The accumulation of certain drugs may be very slow and loading doses have to be used to speed up the achievement of adequate concentrations. The same applies here similar to infusions. Loading dose = target conc. x VD/F
In what situation is the concept of a half-life applicable to drug elimination?
The concept of half-life is meaningful only when the drug is eliminated according to first-order kinetics. If a drug were eliminated according to zero-order kinetics, then a constant amount rather than constant fraction would be eliminated per unit time.
Describe the concentrations at steady state.
The term peaks and trough are commonly used to describe the highest and lowest concentrations at steady state. We will use the terms "Css, max" and "Css, min". Css,avg is also used to represent average concentrations at steady state. Css (C bar SS) is also used to represent Css,avg. How do you calculate the average concentration at SS - Css,Avg? Css,avg = F.DCl.τ F = bioavailability, D = dose Cl = clearance τ = dosage interval (if doses given 12 hr then tau =12hrs.) This equation is independent of whether a drug follows 1-CM or 2-CM. for oral doses we have to multiply the F with dose. Where F is the bioavailability for a drug. EXAMPLE: A drug is intended to be given to a patient with a target average plasma concentration at steady state of 1.5mg/L. The clearance of the drug is 1.6 L/hr and the drug is to be administered orally twice daily. The oral bioavailability is 90% and the drug is to be administered as a salt containing 75% of the parent drug. What dose of the drug salt should this patient administer to meet the target plasma drug concentration? rearranging the equation D=Css,avg.CL.τF.S D=1.5 ×1.6×12 / 0.9×.75 = 42.7
Describe the two-compartment model (2-CM).
The two-compartment model resolves the body into a central compartment and a peripheral compartment. 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. What are the assumptions for the 2-CM? 1. 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. 2. The peripheral compartment comprises less well-perfused tissues such as muscle, fat and skin. 3. A two-compartment model assumes that, following drug administration into the central compartment, the drug distributes between that compartment and the peripheral compartment. However, the drug does not achieve instantaneous distribution, i.e. equilibration, between the two compartments. The drug concentration-time profile shows a curve, but the log drug concentration-time plot shows a biphasic response and can be used to distinguish whether a drug shows 1-CM or 2-CM. 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, and then decline. 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 with the one compartment model, all the rate processes are described by first-order reactions.
What are the different rates of reactions/processes possible for ADME?
To consider the processes of ADME, the rates of these processes have to be considered; they can be characterized by two basic underlying concepts. The rate of a reaction or process is defined as the velocity at which it proceeds and can be described as either zero-order or first-order.
Describe intravenous infusion.
To understand the concentration versus time for intravenous infusion we always need to consider the balance between input and output. If these are constant then the total body load (or the blood concentration) will remain constant. But if they are unequal the blood conc. will rise or fall. When very little drug has been delivered: low body load of drug implies a low rate of elimination. Therefore, the rate elimination <<< rate of infusion. The load has risen considerably and the rate of elimination will also be increased. The blood levels are still rising thus the elimination rate is still < rate of infusion. The drug concentrations (drug load) have risen sufficiently for the elimination rate to match the infusion rate of the drug. At this point the input and the output balance out and the drug concentration in the blood settle down. This condition is called steady state and the associated blood concentrations is conc. at steady state (Css). Steady State is the condition when the rate of drug entry into the body is balanced by the rate of elimination and no further drug accumulation occurs.
Describe multiple dosing.
Two types of patterns emerge during multiple dosing. The lower trace: the drug is eliminated rapidly (k = 0.5/hr) and by the next time the dose is due, the entire dose has been eliminated. Thus the second dose just repeats the first with no accumulation occurring with subsequent dosing. However in the upper trace, the elimination rate is slower (k =0.05/hr) hence only a fraction of the dose is eliminated in 8 hrs. So the next dose is additional to a large residue still present in the body and the second peak is much higher than the first. This is known as pharmacokinetic accumulation. Accumulation does not happen indefinitely. The amount of drug eliminated depends upon the drug load in the body. At early point (A) the drug load in the body is still very low so that the amount eliminated in that time interval is much less than the dose that enters and there is rapid accumulation. By the second marked interval (B) the body load and the amount eliminated are higher but the elimination rate still does not quite match the dose entering and there is continued accumulation. By the late interval (C) the body load is high enough to drive the rate of elimination to match the incoming dose and there is no further accumulation. This late period is referred to as the steady state. In constant infusion these levels very constant but in this situation the blood levels fluctuate as the individual doses are given but is considered SS because these is no further net accumulation.
What is the apparent volume of distribution (VD)?
Under drug distribution we have studied that VD is the hypothetical volume into which the drug is distributed under equilibrium conditions. We can also calculate the plasma concentration at any time when we know kel and Cp0 (concentrarion in the plasma at onset?). However, usually we don't know Cp0 ahead of time, but we do know the dose. A dose in mass units, may be in mg. To calculate Cp0 we need to know the volume that the drug is distributed into. That is, the apparent volume of the mixing container, the body. This apparent volume of distribution is not a physiological volume. It won't be lower than blood or plasma volume but it can be much larger than body volume for some drugs. It is a mathematical 'fudge' factor relating the amount of drug in the body and the concentration of drug in the measured compartment, usually plasma. Vd = amount of drug in the body/concentration in plasma On intravenous administration the amount of drug in the body would be the dose. Thus, Vd = dose/Cp0 It would not be possible to calculate Cp0 unless we know the VD of the drug or the first observed drug concentration is extrapolated to the Y-Axis to obtain Cp0 from the conc.-time graph. Or extrapolated graphically VD can also be calculated using a model independent approach where Vd=dose/kelim[AUC∞] AUC can be calculated from the concentration time plot using the trapezoidal rule. ----------------------- Example: The plasma concentration of drug X after a dose of 200 mg (assuming a 70 kg human) was 0.05 mg/mL. Assuming 100% bioavailability and measurement of plasma concentration after distribution is complete but before any elimination has taken place, what is the volume of distribution of the drug? Vd=200 mg/0.05=4000 mL ----------------------- Using the volume of distribution to calculate a dose based on target concentration to be achieved in a patient. For clinical purpose, the main use of the VD is to allow the calculation of a dose that will result in an appropriate blood concentration of a drug in a patient. As an example we will calculate the dose of aminophylline: The target concentration for theophylline is between 10-20 mg/L, so an average of 15 mg/L would be acceptable. How do we get the VD for theophylline - population mean expressed form studies conducted during drug development process. The generally accepted mean for theophylline is about 0.5L/kg Vd=0.5x 70kg= 35 L dose = Cp x Vd=15x35=525 mg If the patient was dosed as an aminophylline salt, where the salt factor would be 0.75 then: Correction for salt factor is: = Dose/Salt = 525/0.75 = 700 mg of aminophylline
Describe slow release formulations.
With the fast release we get the same high, early peak as with the single dosing and then once absorption is largely completed it follows a fairly long period of elimination followed by a large drop in concentrations. The slow release form gives peaks that are later and lower and there is now a shorter period of decline. The low troughs are eliminated. Thus slow release can very effectively decrease the fluctuations in concentrations.
Describe zero-order reactions/processes.
Zero-order reaction Consider the rate of elimination of drug A from the body. If the amount of the drug, A, is decreasing at a constant rate, then the rate of elimination of A can be described as: dA/dt=−k∗ where k* = the zero-order rate constant. The reaction proceeds at a constant rate and is independent of the concentration of drug A present in the body. An example is the elimination of alcohol. Drugs that show this type of elimination will show accumulation of plasma levels of the drug and hence nonlinear pharmacokinetics.