Principles of Pharmacokinetics

Pharmacokinetics is also very useful in fundamental research of the delivery of drugs in animals and humans, with the use of PET screening, location emission tomography, where you can directly imagine the binding of drugs to the action quote. The dose-response interaction is now the core tenet of pharmacology. We closely research medication exposure-response interactions to find the correct dosage for a given clinical indication. Now, of course, dose reaction refers to both medication effectiveness and toxicity. It is important to understand the range of doses that are useful therapeutically and the range of doses and resulting plasma concentrations that may lead to toxicity with the use of this drug. Now, there are a number of pharmacokinetic/pharmacodynamic modelling approaches that have been used to define these drug exposure-response relationships, and you will deal with that in subsequent sessions of the course. Now, linked to this notion of the dose-response relationship is the target concentration strategy that has been very useful clinically for a number of drugs.

We already addressed the concern with individual variation in drug exposure when drugs are used in standard doses, as we saw with pioglitazone or metformin. So, this approach, the target concentration strategy, attempts to individualize therapy when therapeutic and toxic ranges of drug concentrations in plasma have been established. This is important to define a useful therapeutic range and then to target therapy to that range of therapeutic concentrations. The ultimate objective is to maximize effectiveness and mitigate toxicity. Now, the first explanation of the medicinal drug control that we have documented is that of Dr. Wuth, using bromide and setting ranges of therapeutic doses of bromide as a sedative. Of necessity, this technique is now used by a variety of medications and, for example, lithium carbonate in bipolar disorder is prescribed with very close regard to the resultant lithium plasma levels so that you maximize potency and prevent any potentially dangerous toxicity by using this agent. Now, what medications are candidates for clinical drug surveillance? Generally, medications with a low clinical index, which means that we will speed up from concentrations that are therapeutic into ranges of concentrations that can cause toxicity.

Pharmacokinetics Of Drug Action

The example of lithium is a very good example of a drug with a low therapeutic index, but there are many others, like digoxin and some antibiotics. But in any event, that category of drugs is a good category of agents for therapeutic drug monitoring. You may also be faced with a clinical condition where you do not have, whether you do, physiological endpoints that you can observe on a continuous basis, or biomarkers to direct your dose. You may be concerned with people with seizure disorder—epilepsy—where seizures are rare and, of course, unacceptable. So, if you want to use the spectrum of therapeutic concentrations as your biomarker to guide your dose and hope that this will lead to a substantial decrease in the number of seizures. We have already noted that the pharmacokinetics differ significantly between people, even if you have a target concentration, you should change the dose on an individual basis. We may sometimes use plasma-drug concentration measurements to track conformity, but there are also problems with this technique. But, let’s see, schematically, what happens when the target focus technique is used. We have an estimated initial dose that we administer with a target level in mind. Some drugs need a loading dose to establish a therapeutic concentration quickly, followed by a maintenance dose. Other drugs we begin simply with a maintenance dose. Therapy is initiated, and then we have to evaluate the patient. We need to see the reaction in the patient, and we may also calculate the amount of the medication. On the basis of this estimation, we can then refine the dosage level, change the dose, and then proceed on an iterative basis to maximize the spectrum of concentrations that we want to achieve during therapy.

Now, how are we going to pick a target level? Ok, there is an empiric method in terms of determining the levels of doses are medicinal and where you have limited to no toxicity. So, we’ll have an example of digoxin to answer this issue of how to identify a clinical spectrum of concentrations.

Drug Infusion: Case Study

This was a study conducted in Boston by Dr Smith and Haber in patients that were being treated with digoxin because they had either congestive heart failure or atrial fibrillation requiring rate control, and what they found in a sample of patients categorized as toxic or non-toxic, depending on clinical characteristics and electrocardiographic characteristics, without understanding of the resulting digoxin levels. This is a histogram of the distribution of digoxin concentrations in non-toxic patients, and higher concentrations of digoxin are then determined in patients that were clinically toxic. Thus, on the basis of these observational findings, a clinical goal range is proposed; in this case, 0.8 to 1.6 nanograms per mL of plasma. It was considered that levels in the range of 1.6 to 3.0 per mL were potentially harmful and that patients had levels of toxicity. of 3.0 nanograms per mL or greater were probably already having digoxin toxicity. But once again, based on further evaluation of the effects of digoxin not only on function in patients with congestive heart failure, but now in terms of survival after long-term treatment with digoxin.

This study was published in the early 2000s looking at patients on therapy for congestive heart failure and receiving digoxin throughout this period of observation that lasted 48 months, and then looking at survival on the basis of the observed levels of digoxin. Now, there was a placebo group here that you see with the continuous line.

Digoxin: A Case Study

These patients were receiving treatment for congestive heart failure, but were not receiving digoxin as part of their regimen. And then patients that were receiving digoxin but now stratified based on their digoxin levels: Low levels of 0.5 to 0.8; intermediate levels of 0.9 to 1.1; and high levels greater than 1.2 nanograms per mL. Now, you see that survival changed based on the digoxin levels and the range of digoxin levels that were measured actually at one month into the trial — one month into the trial. The better survival is actually in patients that have low digoxin levels in plasma, and there is a disadvantage in terms of survival for patients that continue digoxin and maintain the level or at least had a level at one month after beginning the trial that exceeded 1.2 nanograms per mL. So, of course, the question is, what were the digoxin levels well into the trial? We don’t have that data, but based on this survival analysis for the use of digoxin in patients with congestive heart failure, there is a new therapeutic range that has been proposed; namely, 0.5 to 0.9 nanograms per mL, much lower than what was usual in clinical practice. And the benefits may result from inhibition of the sympathetic nervous system rather than improved inotropy or improved contractility of the myocardia. There are limitations for this study — we already pointed that out — that no digoxin levels were done after one month in the study, and considering that the observations lasted for 48 months. So, that’s how we estimate a target level. And then, in the case of drugs that require a loading dose — and that was the practice, actually, with digoxin — we need to estimate the loading dose based on the concept of distribution volume. Distribution volume or apparent distribution volume, the main pharmacokinetic parameter.

So, let’s use the digoxin example again. Here, we map the concentration of digoxin in plasma—this is in logarithmic scale— versus time in a linear scale, and we demonstrate the plasma concentration versus time curve for digoxin after intravenous administration of three-quarters of a milligram; a single dose.
It’s a filling dosage. And now we see that plasma concentration versus time, semi-logarithmically plotted, is declining bio-exponentially. This is referred to as the delivery process, and then this terminal phase is referred to as the removal phase.

Now, the modelling here is plotting the tissue concentrations of digoxin over time, and we see that those tissue concentrations of digoxin rise as the plasma concentrations of digoxin are declining. Now, in order to estimate the apparent volume of distribution for digoxin, one approach is that of the extrapolation method; namely, extrapolating from the terminal phase of this curve back times zero and estimated this C subzero [spelt phonetically] or initial concentration of the drug. Now, that is, again, one approach to estimating the apparent volume of distribution, and we are using what we call a single compartmental model of drug distribution and elimination. We administered the dose; in our example, we gave this dose intravenously. Then we have this single body compartment, a hypothetical compartment, where the drug is distributed, and then we are showing here the parameter of elimination clearance. And basically, what we’re doing in this example — the volume of distribution by extrapolation — is estimated as the ratio of the dose over that extrapolated initial concentration. The assumption, of course, is that instantaneous distribution occurs. We saw that that is not the case, but once again, this is one approach that has been useful in terms of estimating the apparent volume of distribution.

 

There are other approaches that we will discuss later in the course, the volume of distribution by area and the volume of distribution at steady state. So, the example of digoxin.

Initial digitization—this is the term corresponding to the loading dosage of digoxin—a quarter of a milligram is delivered and distributed in a single compartment, resulting in an initial concentration of 1.4 nanograms per mL. You see, here we are doing our proper dimensional analysis of the dosage that was given.

The measure concentration in plasma in terms of nanograms per mL, and then applying that principle — the delusion [spelled phonetically] principle, if you will — we have now our dose in nanograms per mL, our concentration in nanograms per mL, and we have this rather large volume, apparent volume of distribution of 536 liters for digoxin. Of course, this does not agree with the reality of physiological body fluid compartments, but nevertheless, the apparent volume of distribution is a critical and very important pharmacokinetic parameter to determine.

The process of drug distribution

We saw that distribution, in fact, was not instantaneous, and that has an impact on the action of the drug — in this case, the chronotropic action of digoxin — in that digoxin slows the heart rate. Here, we’re looking at ventricular rate in a group of patients with atrial fibrillation with rapid ventricular response, and we have both oral and intravenous administration. This is from the classic work of Harry Gold and his coworkers in the early 1950s. And what we’re seeing here is a significant reduction in heart rate after the intravenous administration of digoxin, but you see that the effect is not instantaneous. The maximal effect, in fact, requires six hours before we can observe that significant slowing of the heart rate in patients with atrial fibrillation. So, drug distribution may, in fact, impact the onset of drug action. That is, the rate of drug distribution may impact the onset of drug action. So, now, if we want to continue treatment, we have to select the maintenance dose.

 

So, what are the principles that apply here? Now, in order to estimate the maintenance dose, we need to understand the concept of elimination half-life and elimination clearance, clearance being the other primary pharmacokinetic parameter we referred to a moment ago. So, simple definition.

Elimination half-life:

The time taken for the plasma concentration or total body storage of the drug to decrease to half of the concentration or volume present at a previous time. It’s a very simple term, but again, half-life refers exclusively to drugs that obey first-order or exponential elimination kinetics, and we’ll get back to them in a second. So, let’s look at some basic equations that apply to half-life here. Again, assuming first-order reduction kinetics. And the half-life can then be calculated as a function of a normal logarithm of two times the apparent amount of delivery, divided by the removal clearance for that compound.  The first-order elimination rate constant can be estimated as the ratio of the natural logarithm of two over the observed half-life.

And finally, the elimination clearance can be calculated as the product of k times the apparent volume of distribution, but in fact, k does not determine clearance.  This is one way to measure removal clearance, but clearance really decides both the half-life and the first-order rate constant.

Now, maintenance treatment for digoxin. Now, how much do we need to offer to sustain the medicinal standard we’ve been searching for? 1.4 nanograms per mL in this case. Ok, we need to guess how much of the medication is missing over time. In this scenario, it has been reported that one third of the total body stocks of digoxin are lost everyday.

In the case of digoxin, the drug is eliminated primarily via the kidneys. So, one third of the total body stores times zero; namely, a quarter — or rather, three quarters of a milligram. One third of that is a quarter of a milligram, so that is the daily loss, and that is the loss that has to be replaced on a regular basis. So, that’s how you establish what your maintenance dose should be.

Now, you of initiate therapy without giving a loading dosage, because this is a blunt force example of the fact that drug accumulation would take place—will take place—within time before you enter or near the plateau. After seven doses, in this case, you are fairly similar to the total body store of 0.75 milligrams that was determined by providing a loading dose.

 

As a result, compound aggregation can take place exponentially because you have a continuous maintenance dose and have first-order removal kinetics for the drug. Now, of course, there is another way to estimating the degree of substance concentration using this cumulation element seen here.

This parameter Tao [spelled phonetically] is the dosing interval — the dosing interval. I mean, in the case of our example, it was 24 hours, or one day. And then, of course, you need to know or have an estimation of the elimination rate constant, the first-order elimination rate constant, for that drug.

Now, you will find the derivation [spelled phonetically] of this and other equations in your textbook, and again, the elimination rate constant that we showed as in the accumulation factor equation, calculated as the normal logarithm of two separated by elimination half-life. Now, let’s see graphically what’s going on in three separate situations:

The first is that no digitizing dosage, no loading dose, was applied, and the drug accumulates exponentially before it hits the plateau.

The solid line here would be a situation where a loading dose was administered to establish a therapeutic level quickly, and then the optimal maintenance dose was administered over a period of time. Actually, the maintenance dose here is the same as the maintenance dose here. Now, let’s say that you gave a higher loading dose, twice the loading dose you gave before, but then administered the same maintenance dose that was used here and here.

Over a period of time, the concentration that will be achieved at the plateau or when a steady state is achieved is the same. So, this illustrates the fact that the loading dose that’s not determined what the concentration is going to be at steady state, and now we’re illustrating another useful estimation; namely, that 90 per cent of the steady-state level with continuous drug administration will be achieved in approximately 3.3 half-lives for that particular drug. Now, practically, think about an individual with normal renal function that is receiving a quarter of a milligram of digoxin for maintenance and approaches the plateau concentration in approximately seven days, as we saw in our example. Now, think of an individual with uremia, impaired renal function, and consequently, impaired elimination of digoxin.

The drug will accumulate, again, using the same maintenance dose, and you will anticipate that the plateau concentration is going to be double if the clearance of elimination, say, is reduced by 50 percent. But the other thing that is critical is to understand that in a patient with compromised renal function, you will not hit the peak until later. This is natural renal activity, the normal half-life of digoxin. This is reduced renal function and a longer half-life of digoxin. As a result, this steady state concentration will not be achieved until later; in this case, in this scenario, until 14 days of dosing have taken place.

Clearance as a primary parameter in pharmacokinetics.

We need to understand clearance in the context of drug evaluation and use in clinical medicine. Now, this is a traditional creatinine clearance equation that you learn in your physiology courses that describes the clearance of creatinine — this is an endogenous product that can be measured in plasma — and the clearance of creatinine being used as an index of renal function. And we have this relationship here that says that U times V over P determines what the creatinine clearance is in that context. So, U refers to the urine concentration of the drug, or rather, of creatinine, in this case. V is the urine volume produced over a period of time.

Typically, the creatinine clearance requires a 24-hour urine collection, so this is really a urine formation rate. And then, P standing for plasma concentration of creatinine. Now, let’s look at this again and think about the appearance of creatinine in the urine, the rate of appearance of creatinine in the urine: dE — think about excretion of creatinine — dE over dt. And now this is equal to the clearance for creatinine and the plasma concentration at that time. So, again, that equation that we had before is really a differential equation in disguise. Now, let’s think about the rate of change of creatinine in the body, X being the creatinine in the body. So, we have dX over dt now being equal to I, I being the rate of creatinine synthesis — this is an endogenous product — minus the clearance of creatinine, times the plasma concentration.

This would be the creatinine excretion rate. At steady state, we can of course discard this term, dX over dt, such that the plasma concentration now is equal to the rate of creatinine synthesis or is directly proportional to the rate of creatinine synthesis and inversely proportional to the rate of creatinine clearance. And let’s look at these steady state equations, because these are truly some of the most useful equations you’re going to use in pharmacokinetics. So, if we look at continuous synthesis of creatinine, the steady state plasma concentration of creatinine equals the endogenous rate of production of creatinine over the clearance. And if you think about a drug that is being given continuously — say, by intravenous infusion — the steady state concentration is going to be equal to the infusion rate over the elimination clearance for that drug. So, again, one of the most useful equations for you to keep in mind in addressing what are the determinants of the steady state concentration of the drug.

 

Now, we don’t often do creatinine clearance determinations and collect urine for 24 hours, and a number of equations have been developed over the years to estimate the clearance of creatinine in the case of the Cockcroft and Gault equation that has been in use since the 1970s. And you have these parameters here that consider age, that consider weight of the individual, and of course, the serum creatinine concentration in milligrams per deciliter. Now, this estimate, based on the Cockcroft and Gault equation, has to be reduced by 15 percent for women because, generally, they have a smaller body mass — specifically, skeletal muscle mass — and that leads to a reduced estimate for women when using this approach. Now, in this equation — or rather, in this slide — what you see is that the terms that are shown in red are actually estimating the creatinine synthesis rate that we had in our basic equation previously.

An example of the importance of relying on the estimated clearance of creatinine, as opposed to simply measuring a serum concentration of creatinine is illustrated in the work by Piergies and colleagues in the early ’90s. They had a group of individuals that were clinically toxic due to the use of digoxin, and what they were trying to see is what was the clearance of creatinine in these patients, as opposed to the serum concentration of creatinine. And they grouped their patients into individuals that had creatinine and serum of 1.7 milligrams per deciliter or less, or individuals that had greater than 1.7 milligrams per deciliter of creatinine. And these are their estimated clearances of creatinine, using the Cockcroft and Gault equation.

What you find here is that in a population of people with a low serum creatinine concentration, a comparatively low serum creatinine concentration, 19 individuals out of 23 reportedly had an average creatinine clearance of less than 50. On the other hand, the majority of people with serum creatinine greater than 1.7 have creatinine clearance less than 50. So, again, it is necessary to estimate the clearance of creatinine.

Now, another approach to estimating renal function is based on this equation, the MDRD equation—many versions—which directly measure the glomerular filtration rate.

Not the creatinine clearance, but the glomerular filtration rate normalized to body surface area. Now, you’re going to have more discussions of this equation in addressing pharmacokinetics alternations in patients with renal disease. A more modern equation is the CKD-EPI collaboration equation that is more accurate than the MDRD equation in estimating the glomerular filtration rate, and actually has less bias if the GFR is greater than 60 millilitres per meeting — per minute, rather. Once again, normalized to body surface area. So, back to our steady-state equations.

If you have a continuous drug infusion, the steady-state concentration is a function of the infusion rate and the clearance of elimination for the drug. If you’re using intermittent dosing, say, giving the drug once a day or twice a day or whatever the case may be, this is the estimated mean serum concentration over that dosing interval, now being equal to the dose over the dosing interval, and again, over the clearance of elimination for the drug. So, the steady-state concentration. Let us emphasize it is not determined by the loading dose. Now, once again, some drugs require the administration of a loading dose to establish a therapeutic concentration rapidly, but the loading dose does not determine what the steady-state concentration will be with continuous administration of the drug.

The mean steady-state concentration of irregular drug administration is not determined by the amount of delivery. But, on the other hand, we need to pay attention to peak and trough levels because they will be affected by the apparent volume of distribution, and this is shown in this example, where the volume of distribution is either large or small and the same dose is administered over a dosing interval. And you can see variations in peaks and troughs, but the estimated mean concentration over the dosing interval is the same and, of course, corresponds to the dosing rate and the elimination clearance, and an important element to highlight is that changes in maintenance dose for most drugs, when we’re dealing with first-order kinetics of elimination, result indirectly proportional changes in the steady-state concentration.

Once again, for most drug that follow first-order kinetics of elimination. And we are re-emphasizing our steady state equations because, truly, these are equations you should remember because of their conceptual and practical use. But some drugs are not eliminated by first-order kinetics, and I’m giving you three examples here. Phenytoin, ethyl alcohol, and aspirin — acetylsalicylic acid. These are drugs that deviate from the general pattern of first-order kinetics of elimination. And let’s focus on phenytoin; phenytoin undergoes metabolism in the liver via this main pathway of cytochrome CYP 2C9, and we have this parahydroxylated metabolite that is generated through this pathway. And here we have an example, actually from Dr. Arthur J. Atkinson, Junior, who had the opportunity to study a patient with phenytoin toxicity.

Very high levels of phenytoin observed in this patient upon admission. And which signs of toxicity — just to give you a reference, the therapeutic range for phenytoin is typically 10 to 20 micrograms per mill. We’re near 60 micrograms per mL when this patient was admitted with signs of toxicity. And what they did in this very elegant analysis was that they followed the plasma amounts of the substance over time, and at the same time began gathering urine to test the appearance of this parahydroxylated phenytoin metabolite. And you see here, day after day, that the amount of this phenytoin metabolite recovered in the urine remained remarkably stable.

Plasma amounts are decreasing with time, as you can see here, but for a period of time, the volume of the metabolite that occurs in the urine is stable. And then we reach a point when the plasma concentrations begin to decline more rapidly, and also, we see that the amount of metabolite recovered in the urine also diminishes. What this is indicating — and of course, they’re measuring urine creatinine to validate their urine collections, if you will, over time — and over here they started re-administering phenytoin once again, and they see the increase in the level, and then the subsequent decline of the level. What this indicates is that the metabolic pathway, CYP2C9, that generates this hydroxylated metabolite of phenytoin is saturated over a significant period of time because of these very high concentrations of phenytoin that that metabolic pathway cannot handle, if you will.

Phenytoin kinetics actually follow the pattern of Michaelis-Menten kinetics. Again, concentration over time; in this case, intravenously giving the drug. This is the rate of change of the plasma concentration of phenytoin, which does not obey first-order kinetics, expressed here—or rather, calculated by the V-max—that is, the maximal potential of the metabolic pathway—the Michaelis constant, and here again the phenytoin concentration terms. So, this is a departure from the first-order elimination kinetics.

Steady-state equations.

So, let’s look again at our steady-state equations. We’re giving the drug — a drug, if you will — at intervals orally, and this is the equation we described before for drugs that follow first-order kinetics: Clearance of elimination times the mean steady-state concentration. In the case of drugs that follow Michaelis-Menten kinetics, like phenytoin and ethyl alcohol and aspirin, you need to apply this equation and this term, if you will, in lieu of this clearance of elimination term. A very important issue, because when you follow this type of kinetics you lose the element of dose proportionality. Here, the patient receiving 300 milligrams per day of phenytoin has a concentration of 10 micrograms per mL in plasma; again, the mean steady-state concentration. We go up to 400 milligrams and the concentration already doubles. We go to 500 milligrams per day; we have triple the concentration of phenytoin. So, we do not have dose proportionality with drugs that follow Michaelis-Menten kinetics of elimination. And again, this is another example of a patient that became toxic on a phenytoin dose of 300 milligrams per day. A typical dose, if you will, but excessive in the case of this individual with a slower rate of metabolism.

Defining, then, the therapeutic dose for this patient should really be 200 milligrams per day. And one thing, of course, that arises as a question is, well, there is a large number of drugs that are metabolized in the liver, so an enzymatic pathway is involved, and yet we do not see Michaelis-Menten kinetics for those drugs. So, we have apparent first-order kinetics of elimination, and what we can see here is that in situations where the KM, the Michaelis-Menten constant for that particular drug and that enzyme, is much greater than the plasma concentrations that we will need or observe in a therapeutic context in the clinic.

If the Km is much greater than C, then we can neglect this term here in the denominator, such that now we have V-max over Km becoming a pseudo-first-order rate constant of elimination. So, the ratio of two constants, of course, is a constant, so now this becomes the equivalent term, if you will, under conditions in which the KM for that drug and that enzymatic pathway is much greater than the concentrations that we need to obtain for a therapeutic response.

I hope I have provided you with an overview of Pharmacokinetics in the context of therapeutic drug utilization. Thank you very much.

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