## Mean residence time (MRT): Understanding how long drug molecules stay in the body

When I first began learning about pharmacokinetics, I was often confused by the mean residence time (MRT) parameter. I wasn’t really sure what it meant, how to interpret the value, and why it would ever be important. After many years of working with pharmacokinetic analysis, I still do not use MRT very often, but I now have a better appreciate of what it is telling me so that I can use it properly, if needed.

To discuss MRT, we need to change the conversation from the concentration of drug in the body at a given time to the residence time of individual molecules in the body. The idea behind calculating the mean residence time is that each molecule spends a different amount of time in the body, with some molecules lasting a very short amount of time and others lasting longer. You can plot the relative frequency of the residence time in the body, and it looks like a concentration-time curve.

Residence Time

Another way to look at MRT is to use a though experiment. Imagine we could inject exactly 10 molecules of a drug into the blood stream, and then could measure when each molecule left the body. The data is presented below:

Molecule #   Time in the body (min)
1 8.1
2 18.2
3 30.9
4 43.2
5 60.0
6 79.8
7 107.1
8 139.2
9 171.3
10 198.2

Note: Data from Parameters for Compartment-free Pharmacokinetics, Willi Cawello (Editor), 1999.

The half-life of the drug is the time to eliminate 50% of the drug in the body. In this case, the 5th molecule is eliminated at 60.0 minutes. The MRT is calculated by summing the total time in the body and dividing by the number of molecules, which is turns out to be 85.6 minutes. Thus MRT represents the average time a molecule stays in the body.

The generalized equation for an intravenous bolus injection is as follows:

$MRT=\frac{\sum {N_i*t_i}}{\sum {N_i}}$
where ti is residence time and Ni is the number of molecules with a given residence time (or all molecules for the denominator).

We can assume that every molecule that enters the body will also leave the body. So we can substitute Dose for Ni using the following relationship:

$\sum {N_i} = \int_0^{Dose} dA(t) = Dose$

Finally, for drugs with linear kinetics, the amount in the body is proportional to the concentration in plasma at all time points. By making these substitutions, we can arrive at the following for MRT calculations:

$MRT= \frac{\int_0^{\infty} tC(t)dt}{\int_0^{\infty} C(t)dt} = \frac{AUMC}{AUC}$

where AUMC is the area under the first moment curve or the curve of concentration*time versus time.

The calculation is more complex if you have a route of administration other than IV bolus because you have to account for the time required for the drug to enter the body. This is often called the mean input time (MIT). In this situation, the following equation would be used:

$MRT = \frac{AUMC}{AUC} - MIT$

There are many methods for estimating MIT depending on the type of dose administration.

So, why is MRT important? It can be used to estimate the average time a drug molecule spends in the body. It can also be used to help interpret the duration of effect for direct-acting molecules (e.g. blood pressure lowering agents). It should be noted that MRT is highly influenced by the measurements in the terminal phase. If there are inadequate samples to accurately estimate the terminal elimination rate constant, MRT estimates will be unreliable.

Drug therapy in chronic disease situations requires systemic drug levels to reach target steady-state levels for maximum safety and efficacy. The time it takes for a drug to reach steady-state is a function of the elimination half-life of the drug. The following table illustrates how long it will take to achieve steady-state relative to the half-life:

# of half-lives    % of Steady-State
1 50%
2 75%
3 87.5%
4 93.8%
5 96.9%

To achieve steady-state, you need approximately 5-7 half-lives of the drug. For drugs with rapid elimination and short half-life values, this is not a problem; however drugs with slow elimination could require days or weeks to achieve steady-state. If therapeutic effects are needed quickly, and the drug has a long half-life, one can use a loading dose to achieve therapeutic levels on the first dose. The loading dose rapidly achieves the therapeutic response and subsequent doses maintain the response.

$\textrm{Loading dose}=\frac{\textrm{Maintenance dose}}{(1-e^{-k\tau})}$

where τ is the dosing interval for the maintenance dose, and k is the terminal elimination rate constant.

## Saturable drug absorption

Drug absorption is the process by which a drug molecule moves from the site of administration to the systemic circulation. Following intravenous administration, there is no absorption process since the drug is directly introduced into the blood stream. However, for oral, intramuscular, subcutaneous, sublingual, buccal, transdermal, (and a many other routes), there will be an absorption process that occurs. Since there are so many different variants of extravascular administration, let’s focus on oral administration to explain the concept of saturable drug absorption. Following ingestion of a drug capsule or tablet, the drug enters the gastrointestinal (GI) tract and begins to dissolve in the GI fluids and is absorbed into the body via passive and active transport systems. The drug then passes through the portal vein to the liver and then into the systemic circulation. If any of these processes becomes saturated, then increases in the administered dose will not correspond to increases in amount of drug absorbed into the body. This is called saturable drug absorption.

Dissolution
If the drug is poorly soluble in the GI fluid, then you may encounter saturation of the dissolution process. At some point, no more drug can dissolve into the GI fluid to be available for absorption. This results in a plateau in drug exposure where increasing the dose will not increase the exposure. Sometimes, dissolution can be aided by adding additives that assist in the dissolution process, or change the pH of the GI fluid to aid in dissolution.

Transport-mediated uptake
Some drugs are absorbed through transport-mediated uptake mechanisms. For example, valacyclovir is absorbed by active transport by the peptide transporter PEPT1. Drug transporters can be saturated if the substrate (drug) levels are sufficiently high. Once the transporter is saturated, no additional drug can be absorbed from the GI tract, even if it is available for transport. Similar to the example of dissolution, increases in the dose will not increase the exposure. These transport-mediated limitations are difficult to modulate without re-desigining the drug molecule to take advantage of multiple transport systems.

Passive uptake
Most drugs are absorbed through passive mechanisms that are generally not saturable. These absorption mechanisms depend only on the available surface area of the GI tract and the strength of the concentration difference between the lumen of the GI tract and the portal vein. Thus, saturation of passive uptake mechanisms is very rare. On occassion, with very large doses, absorption can slow due to the large amount of drug present in a confined space. In general, saturable absorption does not occur in situations with passive uptake.

If you encounter saturable drug absorption, there is generally no harm because higher doses will not result in higher drug exposure.  Understanding the type of saturation (transporter or dissolution) will be useful information to make a decision on they approach to address the issue. If the saturation is in the dissolution, it may be possible to address it with reformulation and addition of specific materials to help the drug dissolve in the GI fluid. If the saturation is due to transport-mediated uptake, there is not much you can do to change the outcome.

When you have saturable absorption, drug exposure will plateau at higher doses. When you have saturable elimination (or clearance), the total exposure will increase faster than dose at higher doses. By plotting the exposure (area under the curve or AUC) on the y-axis and dose on the x-axis, you can determine if absorption and/or elimination are being saturated across the dose range studied.

## Why Cmax is a continuous variable and tmax is a categorical variable

Example PK Curve with Cmax and tmax labeled

The maximum observed concentration (Cmax) and the time of Cmax (tmax) are both obtained directly from the concentration-time data. In this post, I will review how to determine both of these parameters, and how to interpret information from the values. These two parameters are simple, but they pack some important information if you know how to extract it!

Cmax
This parameter is defined as the highest observed concentration in a concentration-time profile. In the image, the Cmax is represented by the value of the black horizontal line, just below 25 ng/mL. If two peaks of identical height are observed, the first peak is considered the Cmax. The Cmax can be determined in most statistical software packages by using the MAX function which selects the maximum value of a set of data. The Cmax is determined within the dosing interval. Thus if 2 doses are administered, you will have a Cmax value following each dose administration (assuming blood draws were taken after both).
tmax
This parameter is defined as the time of the sample identified as Cmax. In the image, the tmax is represented by the green vertical line. If two peaks of identical height are observed, the time of the first peak is considered tmax. Since each concentration is associated with a specific time in a pairwise fashion (time, concentration), the value for tmax is easily determined from the pair after the Cmax is selected.

Both of these parameters are obtained from observational data. Neither is calculated from a mathematical model, and thus each reported value must exist in the original dataset. For example, if you only drew blood samples at 0, 1, 2, 4, 6, and 8 hours, you could not have a tmax value of 3 hours. The actual time of blood sample collection is used rather than nominal (or planned) collection time.

The sampling scheme will have an enormous impact on the values determined for each of these parameters. As a general rule, the more frequently you sample around the expected time of the maximum concentration, the more accurate the value of Cmax and tmax will be relative to the “true” values. In contrast, if only 1 sample is taken around the expected time of the maximum concentration, you will tend to see low variation in your tmax variable, and high variation in your Cmax variable. This is because it is very unlikely that all subjects will have identical PK profiles, thus you are selecting a single time at which you make a measurement, and all variability is attributed to the Cmax value.

This discussion now brings us to a critical point. The variable Cmax is a continuous variable that can take any real number from 0 to infinity (∞). Thus when using statistics to summarize Cmax information, you can use normal probability theory. This means you can use averages and standard deviations to report summary statistics. I generally use geometric mean as the best representation of the population mean Cmax value because these values are log-normally distributed (tailed distribution biased away from zer0). But arithmetic means are also acceptable to many. When comparing Cmax values, you can use standard statistical tests based on normal distribution theory.

In contrast to Cmax, the tmax variable is a categorical variable that can only take values based on the planned sampling scheme. Thus if your planned sample schedule includes 0, 1, 2, 4, 6, and 8 hours, you cannot have a sample at 1.5 or 10 hours (unless there was a deviation). Thus the expected values for tmax are confined to some pre-selected categories. Thus when you summarize tmax results, you should NOT use averages and standard deviations, as they do not accurately describe the distribution of values. Instead, you should use medians and ranges. The median is the “middle” of the data, and the range represents the “extremes” of the data. This also applies to any statistical testing of tmax where you should use a non-parametric test like the Rank-Sum Test to compare tmax values from 2 different treatments (e.g. formulations).

So when designing your studies, make sure you take frequent samples around the time of the expected tmax, and always remember that Cmax is a continuous variable, but tmax is a categorical variable.

## What is quantification anyway?

Bioanalytical analysis is a fundamental tool for the pharmacokineticist. The results of a bioanalysis are the source data for all pharmacokinetic work. Thus a clear understanding of the methodologies and challenges associated with the bioanalytical analysis science is of great benefit to a pharmacokineticist. As an example, I was recently working on the development of a new compound. Several clinical trials had been run before I became involved in the project, and I was asked to take the molecule from its current state to registration (e.g. NDA with the FDA). As I began to review the pharmacokinetic data from the previous 4 clinical trials, I started to ask questions about the bioanalytical method being used to measure plasma concentrations. I discovered, that the drug was not stable in plasma, and there was rapid degradation after just a few minutes of time at room temperature. I also discovered that our preclinical group had begun to use a stabilizer in plasma collection tubes to prevent this degradation. However, that same stabilizer was not being used in the clinical setting. Furthermore, all of the projections of drug exposure for future studies were based off analyses of plasma samples without the stabilizer.

Clearly, the human exposure data was compromised, and potentially unreliable. We decided to conduct a few simple in vitro experiments and determined that the stabilizer was needed for human plasma samples also. We then used that stabilizer in an upcoming clinical pharmacology study and the results showed that the previous exposure estimates (without stabilizer) were off by as much as 50%! Furthermore, a projected difference in exposure between patients and healthy volunteers from the older studies turned out to be an artifact due to the poor bioanalysis technique used. Once we corrected the bioanalysis methodology, we were able to obtain high quality data with lower variability and more accuracy.

Limit of quantification (LOQ)

The limit of quantification is defines as the lowest concentration which can be determined with a given analytical assay with the required precision and accuracy. In most cases the precision and accuracy must be ± 20%, but sometimes it is ±15%. This means that the actual value can be within 15-20% of the reported value. If the LOQ is reported as 1.00 ng/mL, then the actual concentration can be between 0.80 ng/mL and 1.20 ng/mL (assumes ±20%). Values below the limit of quantification (abbreviated as BLQ or BQL) are not reported as number in most bioanalysis datasets. The bioanalyst does not report the value because he/she does not have adequate confidence in the accuracy of the number to report it. (Some pharmacokineticists would like this information and the associated variability reported … but that is a separate discussion.)

Limit of detection (LOD)

The limit of detection is the lowest concentration which can be measured analytically using prespecified criteria. Often the limit of detection is a bioanalytical response that is 5 times the background response in the assay. Below this limit, the bioanalytical scientist believes he/she cannot separate background noise from an actual analyte measurement.

Upper limit of quantification (ULOQ or ULQ)

The upper limit of quantification is the highest concentration in the calibration curve which can be determined with a given analytical assay with the required precision and accuracy. In most cases the precision and accuracy must be ±15% at the high end of the calibration curve. The highest concentration is selected arbitrarily by the bioanalytical scientist. If concentrations are observed above the ULOQ, the samples can be diluted and tested within the calibration range. Often the bioanalytical scientist will try to cover at least 3 orders of magnitude (e.g. 1 to 1000 ng/mL) in the analytical range to avoid having to do dilution work.

Precision

Precision is a measure of reproducibility or repeatability of a measurement. If the same sample is measured multiple times, the analytical assay may provide slightly different values (i.e. concentrations). Validated assays have an expected precision less than or equal to ±15% at all concentrations except the LOQ where precision of ±20% is acceptable. The precision can be determined by making at least 3 measurements of the same sample. The coefficient of variation (%CV) is calculated by dividing the standard deviation of the 3 measurements by the mean of the 3 measurements and multiplying by 100. Thus if the standard deviation is 4.37 ng/mL, and the mean is 49.6 ng/mL, the %CV would be

$\textrm{\%CV}={\frac{4.37}{49.6}}*100=8.76\%$

Accuracy

Accuracy is a measure of how close a measured value is to the actual (true) value. A known concentration is measured multiple times, and the analytical assay may report different values (i.e. concentrations). Validated assays are expected to have an accuracy less than or equal to ±15% at all concentrations in the assay range. The accuracy can be determined for each sample, then the mean accuracy can be reported. Accuracy is calculated using the following equation

$\textrm{Accuracy (\%)}={\frac{\textrm{Measurement}-\textrm{Theoretical}}{\textrm{Theoretical}}}*100$

If three measurements of a 50 ng/mL sample are 41.5, 51.2, and 47.6 ng/mL, then the mean accuracy would be -6.47% (average of -17%, 2.4%, and -4.8%), which is within the acceptable range.

## Calculating Urine PK Parameters

Pharmacokinetic analysis normally focuses on systemic exposure to a drug; however, much can be learned from urinary pharmacokinetic parameters. Urinary PK parameters tell you about how much drug was absorbed (at a minimum), and how much drug is eliminated through the kidney. Often it provides easy access to metabolites that are also eliminated in the urine. To understand how to collect urine data and calculate urine PK parameters, it is important to understand what you are measuring.

Urine PK parameters are calculated by relating the amount of drug eliminated in the urine relative to the concentration of drug in the plasma. The bladder is a collection vessel, therefore drug that is eliminated from the body through the kidney collects in the kidney until it is voided from the body with the urine. Thus the concentration of drug in the urine changes with time, and is largely irrelevant. However, the amount of drug in the urine is constantly increasing until all drug is eliminated. Because of this we measure the amount of drug in the urine over a time period rather than the concentration in a single sample.

To properly calculate the amount of drug in a urine sample, you will need to collect all of the urine in an interval following dose administration. Collection intervals normally range from 4 hours to 24 hours. Shorter intervals are used immediately after dosing, and longer intervals are used after a majority of drug has been eliminated. A common sample collection scheme for urine is 0-4 hr, 4-8 hr, 8-12 hr, 12-18 hr, 18-24 hr, 24-48 hr, 48-72 hr. For each collection interval, the following information is required:

1. The volume of urine collected. This can be directly measured, or calculated based on the mass of urine, and an average density of urine (1.01 g/mL) using the equation $\textrm{Volume}=\frac{\textrm{Mass}}{\textrm{1.01 g/mL}}$
2. The concentration of urine in each collection interval sample. Once the volume is determined, only a small aliquot is needed for the drug concentration assay. Normally 1 – 10 mL of urine is retained for analysis.
3. The start and stop time of each interval is required. Timing to the minute is preferred, and patients should try to void at the end of each interval.

After the data is collected, the amount of drug in each collection interval can be calculated using the following equation:

$\textrm{Amount ng} = \textrm{Concentration ng/mL} \cdot \textrm{Volume mL}$

The cumulative amount of drug excreted after each collection interval is then plotted against the median of the collection interval. Nominal or actual times can be used as appropriate. An example dataset is shown below:

Collection Interval (h)   Midpoint for Plots (h) Volume (mL) Concentration (ng/mL) Amount excreted (mg) Cumulative amount excreted (mg)
0 – 4 2 400 111 44.4 44.4
4 – 8 6 290 91.3 26.5 70.9
8 – 12 10 390 40.3 15.7 86.6
12 – 18 15 460 27.2 12.5 99.1
18 – 24 21 415 13.8 5.7 104.8
24 – 48 36 1720 2.69 4.6 109.4
48 – 72 60 2020 0.1 0.2 109.6

Cumulative amount of drug excreted in the urine

As you can see the drug reaches an asymptote as the elimination of drug slows at around 24-hours after the dose. The asymptote is called the total amount excreted (Ae). You can also calculate the renal clearance (CLR) using the following equation:

$CL_R=\frac{Excretion;rate}{C}$

You can also estimate the elimination half-life of the drug by linear regression of the semi-logarithmic plot of the rate of excretion versus the midpoint of the urine collection time. This is based on the following equations:

$Excretion\;rate=\frac{CL_R}{V}\cdot Dose \cdot e^{-k \cdot t}$

$\ln{(Excretion\;rate)}=\ln{(\frac{CL_R}{V}\cdot Dose)} - {k \cdot t}$

There are additional ways to calculate a variety of urinary PK parameters, but these are the basics. I hope that this is helpful!

## What can we learn from a human mass balance study?

Mass balance studies are also called “C-14 studies” or “Absorption, Metabolism, and Excretion (AME) studies”. It is important to understand what you are trying to learn from the experiment. The primary objectives of a mass balance study are generally:

1. To determine the mass balance of drug-related material following dose administration
2. To determine the ratio of parent drug to metabolite(s) in circulation
3. To determine the primary route of excretion of drug-related material

Let’s discuss each of these in order … First, mass balance is a term that refers to balancing the amount of drug administered as a dose to the amount of drug-related material collected in human excreta (normally feces and urine, but also could include expired air and sweat). One would expect if we administered 100 drug molecules to a human subject, we should collect 100 drug-related molecules in the excreta to achieve mass balance. Because there are always errors in any measurement technique, we normally use recovery of >90% as reference for nearly complete recovery. The difference between the theoretical 100% and the actual 90% could be due to measurement errors, sample processing errors, or missing samples. Since the experiment requires accurate collection and measurement of all feces and urine for up to 14 days, there are many opportunities for errors.

To calculate mass balance, we need an accurate method for measuring drug-related material in the various human excreta. While LC/MS-MS methods are sensitive, they can only be used to quantify the amount of an analyte with a known structure. In this situation, we need to measure parent drug AND all metabolites (even ones that are previously unknown). Thus, a different tool is used. The most common tool is “labeling” the parent drug molecule with a Carbon-14 atom. Carbon-14 (C-14) only represents 0.1% of carbon in the world, so it is not commonly found in any molecule. But, we can make a drug product with extra C-14 to “label” it in a way that we can follow it with sensitive radiometric detection methods (liquid scintillation or accelerated mass spectroscopy). Thus we can compare the amount of “radioactivity” in the original dose to the amount of radioactivity in the excreta to calculate the mass balance. Radioactivity measurements are independent of chemical structure, thus total radioactivity measurements can be thought of as “parent + all metabolites”.

Second, we want to learn how much of the circulating drug is parent drug versus metabolites. This is important to evaluate the safety of each metabolite, and identify unique human metabolites. The blood or plasma can be analzyed for parent drug concentrations using standard techniques (e.g. LC/MS-MS) to allow for estimates of total exposure (AUCparent). Then the blood or plasma can be analyzed for total drug product (parent + metabolites) using radiometric detection methods to allow for estimates of total exposure (AUCparent+metabolites). The ratio of the two AUC measurements gives the proportion of total exposure represented by parent drug. Similarly, if specific assays are available for some metabolites, the proportion of each metabolite relative to total drug exposure can be calculated. These ratios are important for addressing development questions around safety metabolite testing and drug-drug interaction studies. Further, the presence of the “label” allows for identification of metabolites using LC/MS-MS methods combined with radiometric detection.

Third, the primary route of excretion (feces or urine) can be determined in a mass balance study. Normally only feces and urine are collected as human excreta, but in certain situations expired air and sweat might be obtained if excretion by those routes is expected. Depending on the specific excretion profile of the drug, the majority of radioactivity will normally end up in the urine or the feces. Radioactivity can only appear in the urine after systemic absorption, suggesting that the bioavailability is at least equal to the fraction of drug appearing in the urine. The amount of drug in the feces is a mixture of unabsorbed drug (assuming oral administration), drug excreted in the GI tract, and drug excreted in the bile.

A properly designed human mass balance study will allow you to address these three main objectives with a small number of healthy volunteers.

## Accumulation: What it means and how to calculate it

A reader, Michael, asked me to discuss the concept of accumulation. This term is used frequently in both the nonclinical and clinical setting. Some people use the word with fear, while others explain it in complicated terms. Accumulation represents the relationship between the dosing interval and the rate of elimination for the drug. When the dosing interval is long relative to the time needed to eliminate the drug, accumulation is low. When the dosing interval is short relative to the time needed to eliminate the drug, accumulation is high. Thus, changing the dosing interval can change accumulation. The value for accumulation is not “good” or “bad”, despite what people may say. It simply “is”. The important piece to remember regarding accumulation is what are the actual drug levels at steady-state (maximum accumulation), and are those levels associated with efficacy and/or toxicity? If those levels are associated with toxicity, you can increase the dosing interval to lower the accumulation and hopefully avoid the toxicity.

Imagine a funnel under a water faucet. If you turn on the water faucet slowly, and run the water through the funnel, the water level in the funnel will not rise as long as the faucet output (input rate) is slower than the funnel output (output rate). If you increase the amount of water flowing through the faucet, the level of water in the funnel will rise until it reaches “steady-state” where the input and output rate are equal. Further increases in the faucet flow rate will cause the funnel to overflow. The level of water in the funnel is considered the accumulation, and it rises with increases in the input rate.

#### How to Calculate Accumulation Ratio (AR)

The accumulation ratio can be calculated using PK parameters, or from observed data. All of these methods give reasonable estimates, but have slightly different drawbacks. The first method is to use the dosing interval and elimination rate constant and the following equation to calculate the accumulation ratio (AR):

$AR = \frac{1}{1-e^{-k*tau}}$

This method requires knowledge of the terminal elimination rate constant (k) following a single dose of the compound. This elimination rate constant can be calculated from the clearance and volume, terminal half-life, or the terminal slope of the concentration-time profile. The dosing interval (τ) is the time between successive doses. For once-daily (qd), this would be 24 hours. Make sure that the units for k and τ are the same before you complete the calculation. This equation assumes first-order elimination of the drug. The denominator estimates the proportion of drug eliminated after one dosing interval. The advantage of this method is that you can predict the accumulation ratio after many different dosing regimens by inserting different dosing intervals. For example, you could estimate the accumulation ratio after once-, twice-, and thrice-daily dosing very quickly if you know the elimination rate constant and use this equation. The disadvantage is that the calculation is highly dependent upon the estimate for the elimination rate constant. If that parameter is poorly estimated, then values for the accumulation ratio will be biased.

The second method is to use observed data from a study where you have measurements after a single dose and at steady-state, using one of the following equations:

$AR = \frac{C_{max-multiple dose}}{C_{max-single dose}}$

$AR = \frac{AUC_{multiple dose}}{AUC_{single dose}}$

$AR = \frac{C_{trough-multiple dose}}{C_{trough-single dose}}$

All of these equations are similar in that they take the ratio of an exposure parameter at steady-state and divide it by that same parameter after a single dose. The assumption is that once steady-state has been achieved, no further accumulation will occur. At that point, the ratio of any measure of exposure at steady-state will be proportional to the same measure after a single dose in the amount of the accumulation ratio. The advantage of this method is that it can be easily calculated directly from the data. If you measure AUC on Day 1 and Day 28 of a toxicokinetic study, you can calculate the accumulation ratio. In addition, multiple measures can be used to verify the calculations. The disadvantage is that you may generate multiple values for accumulation ratio if the PK parameters vary widely. For example, Cmax could be poorly estimated at steady-state because tmax is delayed and the sampling scheme is too sparse at the new tmax. Another difficulty with this method is that one often has to assume steady-state was achieved without independent confirmation from multiple measurements at steady-state. Even with these disadvantages, this method offers a quick way to calculate accumulation ratio from observed data.

#### Using the Accumulation Ratio

Combining these two types of equations for accumulation ratio provides the pharmacokineticist an opportunity to use observational information to make predictions. For example, the accuracy of the elimination rate constant calculation can be evaluated by comparing the accumulation ratios calculated from the first and second methods. If the AR values are similar, then it is likely that the elimination rate constant value used is accurate. If the elimination rate constant cannot be calculated (e.g. t1/2 is > 6 hours, and τ is 24 hrs), one could calculate the accumulation ratio using method 2 (e.g. AUC or Cmax), and then input that AR value into the equation for method 1, and solve for k. As mentioned previously, using method 1, the AR can be calculated for a variety of dosing intervals. The resulting AR values can then be used to predict exposure parameters (i.e. AUC, Cmax, Ctrough) at steady-state for those dosing intervals. These exposure parameters can be predicted using the equations for method 2 along with the AR and the associated parameter following a single dose.

In conclusion, the accumulation ratio is a simple, but useful calculation of the relationship between the dosing interval and the elimination rate constant. The manifestation of this relationship is a rise in steady-state drug exposure parameters as the dosing interval shrinks relative to the elimination rate constant. Accumulation is often wrongly associated with toxicity, thus it is often spoken of in the context of toxicokinetic analysis. Accumulation is simply a reflection of how much drug is being added to the body relative to how much is being eliminated from the body during a defined period of time. And that ratio can be controlled by changing the dosing frequency.

## Bioavailability

The term bioavailability is used very frequently in pharmacokinetic discussions. Often it is misused and complicated by those who don’t understand its meaning. Bioavailability simply means the fraction of administered drug that reached the systemic circulation (blood). It can range from 0% (no drug) to 100% (all of the administered drug).

#### Absolute vs Relative

The adjectives “absolute” and “relative” are commonly added to the bioavailability term. “Absolute” bioavailability is the amount of drug from a formulation that reaches the systemic circulation relative to an intravenous (IV) dose. The IV dose is assumed to be 100% bioavailable … since you are injecting the drug directly into the systemic circulation. “Relative” bioavailability is the amount of drug from a formulation that reaches the systemic circulation relative to a different formulation (non-IV) such as oral solution, reference formulatione, etc. Relative bioavailability is commonly used when an IV formulation does not exist, or cannot be made.

#### First-pass effect

When working with oral formulations, you may hear something about the “first-pass” effect. This refers to the drug lost between oral administration and first appearance in the systemic circulation. The drug must survive the milieu in the gastrointestinal (GI) tract, cross the gut wall, and then pass through the portal vein to the liver. If a drug molecule survives that guantlet, it will reach the systemic circulation.

Let’s take an example with 100 drug molecules that are ingested in pill form. Only 90 of those molecules survive the GI tract. Then 81 make it past the gut wall and into the portal vein. Of the 81 that enter the liver, only 41 make it to the systemic circulation. Thus the bioavailability is 41/100 = 41%. We also know the fraction that passed the gut (90/100 = 90%), the gut wall (81/90 = 90%), and the liver (41/81 = 50%). If you multiply each of those fractions together (90% * 90% * 50% = 41%) you arrive at the total bioavailability for the drug.

This information is very helpful because we can see that the liver metabolism is the biggest challenge with increasing bioavailability. Increases in solubility and gut wall permeability will not significantly improve the bioavailability. If, however, we could modify the drug molecule to avoid some liver metabolism, we might be able to increase bioavailability significantly.

### Calculating bioavailability

Bioavailability is calculated as the ratio of area under the curve (AUC) for the test and reference formulation/route of administration. If you are calculating the absolute bioavailability, it would be calculated as:

$F = \frac{AUC_{oral}}{AUC_{IV}}$

If you were calculating the relative bioavailability, it would be:

$F_{rel} = \frac{AUC_{formulation_/}}{AUC_{formulation_2}}$

So keep bioavailability simple in your mind, and you will be successful in your future PK discussions.

## What is hysteresis in PK/PD analysis?

I apologize to all my readers for such a long lapse between posts. After a very busy summer and fall, I am back to posting regularly to my blog about PK/PD topics.

When analyzing PK/PD data, one of the most important plots used to visualize the data is to plot time-matched PK/PD data on a scatter plot. The X-axis has the PK concentration and the Y-axis has the PD data. Two examples of these scatter plots are shown below.

No Hysteresis

Hysteresis

The first plot shows a relationship with no hysteresis, and the second shows hysteresis. The easiest way to identify hysteresis is by drawing a vertical line on the concentration-effect plot. If that line crosses the curve in 2 places, indicating 2 different response levels for a single drug concentration, then you have hysteresis. In the first plot (no hysteresis) a vertical line at 40 ng/mL corresponds to a single effect level (20%). However, in the second plot (hysteresis) a vertical line at 40 ng/mL corresponds to both effect levels of 40% and 100%.

A hysteresis is neither good nor bad when reviewing PK/PD data. A hysteresis loop simply means that there is a time delay between the measured concentration and the effect response. Normally this means that the measured effect is indirectly affected by the measured concentration. To properly model this relationship, you would want to use an effect compartment or an indirect PK/PD model.

So a hysteresis loop simply provides information on how to model your PK/PD data.