European Review for Medical and Pharmacological Sciences
`
`2002; 6: 33-44
`
`A short introduction to pharmacokinetics
`
`R. URSO, P. BLARDI*, G. GIORGI
`
`Dipartimento di Farmacologia “Giorgio Segre”, University of Siena (Italy)
`*Centre of Clinical Pharmacology, Department of Internal Medicine, University of Siena (Italy)
`
`Abstract. – Phamacokinetics is proposed
`to study the absorption, the distribution, the bio-
`transformations and the elimination of drugs in
`man and animals.
`A single kinetic profile may be well summa-
`rized by Cmax, Tmax, t2 and AUC and, having more
`than one profile, 8 parameters at least, the mean
`and standard deviation of these parameters,
`may well summarize the drug kinetics in the
`whole population.
`A more carefull description of the data can be
`obtained interpolating and extrapolating the
`drug concentrations with some mathematical
`functions. These functions may be used to re-
`duce all the data in a small set of parameters, or
`to verify if the hypotheses incorporated in the
`functions are confirmed by the observations. In
`the first case, we can say that the task is to get a
`simulation of the data, in the second to get a
`model.
`The functions used to interpolate and reduce
`the pharmacokinetic data are the multiexponen-
`tial functions and the reference models are the
`compartmental models whose solutions are just
`the multiexponential functions. Using models,
`new meaningfull pharmacokinetic parameters
`may be defined which can be used to find rela-
`tionships between the drug kinetic profile and
`the physiological process which drive the drug
`absorption, distribution and elimination. For ex-
`ample, compartmental models allow to define
`easily the clearance which is dependent on the
`drug elimination process, or the volume of distri-
`bution which depends on the drug distribution in
`the tissues. Models provide also an easy way to
`get an estimate of drug absorption after ex-
`travasculare drug administration (bioavailability).
`Model building is a complex multistep
`process where, experiment by experiment and
`simulation by simulation, new hypothesis are
`proven and disproven through a continuous in-
`teraction between the experimenter and the
`computer.
`
`Key Words:
`Pharmacokinetic models, Multiexponential func-
`tions, AUC, Half-life, Volume of distribution,
`Clearance, Bioavailability.
`
`Introduction
`
`Pharmacokinetics is proposed to study the
`absorption, the distribution, the biotrasfor-
`mations and the elimination of drugs in man
`and animals1. Absorption and distribution in-
`dicate the passage of the drug molecules from
`the administration site to the blood and the
`passage of drug molecules from blood to tis-
`sues respectively. Drug elimination may oc-
`cur through biotrasformation and by the pas-
`sage of molecules from the blood to the out-
`side of the body through urines, bile or other
`routes. Figure 1 shows two graphic represen-
`tations of these processes.
`Measuring the amounts or the concentra-
`tions of drugs in blood, urines or other fluids
`or tissues at different times after the adminis-
`tration, much information can be obtained on
`drug absorption and on the passage of drug
`molecules between blood and tissues and fi-
`nally on the drug elimination. Figure 2 shows
`the results of a hypothetical pharmacokinetic
`experiment. Notice that the scale of the plot
`is not homogenous, because drug in urines
`and in the absorption site are amounts, while
`the other curves represent drug concentra-
`tions.
`Pharmacokinetics is important because:
`
`a. The studies completed in laboratory ani-
`mals may give useful indications for drug
`research and development. For example
`less powerful molecules in vitro can turn
`out more effective in vivo because of
`their favorable kinetics (greater absorp-
`tion, better distribution, etc.).
`b. Pharmacokinetics supports the studies of
`preclinical toxicology in animals (toxico-
`kinetics) because the drug levels in plas-
`ma or tissues are often more predictive
`than the dose to extrapolate the toxicity
`
`33
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 001
`
`

`

`R. Urso, P. Blardi, G. Giorgi
`
`Drug
`to be
`absorbed
`
`Arterial
`
`blood
`
`Drug in
`blood
`
`Drug in
`tissues
`
`lungs
`
`heart
`
`tissues
`
`kidneys
`
`Venous
`
`blood
`
`liver
`
`intestine
`
`Drug
`excreted
`
`Metabolites
`
`Figure 1. Drug absorption and disposition (ie distribution and elimination). A, Graphic representation of the blood
`circulation: arterial blood is pumped by the heart through all the tissues and after the passage through the organs the
`venous blood reaches the lungs. All the organs except the lungs are in parallel because they are perfused by a fraction
`of the whole blood in each passage, while the lungs are in series with the other organs because all the blood reaches
`the lungs in each passage. Some organs have an arrow to the outside of the body which represents the drug elimina-
`tion. For example the liver may produce drug metabolites wich in turn enter into the systemic circulation. Notice that
`the heart is represented just for the mechanical function associated with it, and not as a perfused tissue. B, Blocks
`which represent the drug absorption, distribution and elimination.
`
`Drug to be
`absorbed
`
`Metabolite in blood
`
`Drug excreted
`
`Drug in blood
`
`Hours
`
`Figure 2. Observations collected during a hypothetical pharmacokinetic experiment: the unit of measure of y-axis
`may be not omogeneous because drug to be absorbed and drug excreted are drug amount, while metabolite and drug
`in blood are concentrations.
`
`34
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 002
`
`

`

`A short introduction to pharmacokinetics
`
`data to man. Toxicokinetics is also im-
`portant to:
`– verify that the animals have measur-
`able levels of drug in plasma and that
`these levels are proportional to the ad-
`ministered dose,
`– estimate the area under the curve and
`the maximum concentration of the
`drug in plasma, because these parame-
`ters can be used to represent the expo-
`sure of the body to the drug,
`– evidence differences in pharmacoki-
`netics between the various groups of
`treatment, the days of treatment and
`other factors,
`– estimate the variability between ani-
`mals and identify cases with abnormal
`levels of the drug.
`c. Knowledge of the kinetics and of the ef-
`fects (pharmacodynamics) of drugs in
`man is necessary for a correct use of
`drugs in therapy (choice of the best
`route of administration, choice of the
`best dose regimen, dose individualiza-
`tion).
`
`Moreover, as the relationship between the
`drug levels and the effects is very often inde-
`pendent on the formulation, formulations
`which produce superimposable drug levels
`can be considered interchangeable and this is
`the basis of the concept of bioequivalence.
`
`Planning and Presenting the
`Results of a Pharmacokinetic Study
`
`The experimental design depends closely
`on the purpose of the investigator. For exam-
`ple some studies may be planned to get accu-
`rate estimates of particular parameters (the
`rate or the extent of drug absorption), or to
`get information on the variability of the phar-
`macokinetic parameters in the population
`(population kinetics), consequently the ex-
`perimental protocols may vary considerably.
`Anyway, in order to plan a pharmacokinetic
`experiment, the following conditions should
`be well defined: route of drug administration,
`dose regimen, tissues to sample, sample
`times, analytical method, the animal species
`or, in clinical settings, the inclusion and exclu-
`sion criteria of the subjects.
`
`All these informations and the purposes of
`the experimenter should always be given
`when presenting the design or discussing a
`pharmacokinetic study.
`Moreover, as in many protocols the sam-
`pling times are equal for all the subjects or
`animals under investigation, it is good prac-
`tice at the beginning of the data analysis, to
`plot not only the observations relative to
`every single subject, but also the mean (and
`standard deviation) concentrations in the
`population at each time.
`Plotting and listing the data may be a big
`job because the size of acquired data during
`a pharmacokinetic study is often huge. As
`an example, in a typical study of bioequiva-
`lence in man, there are generally not less
`than 12 plasma samples in at least 18 sub-
`jects treated with two formulations of the
`same drug. The consequence is that not less
`than 12 × 18 × 2 = 432 set of data (time, plas-
`ma concentration) are produced. The num-
`ber of data increases if also other districts
`have been sampled (urines for example), or
`if the levels of some metabolite have also
`been measured.
`For this reason it is very helpful for evalua-
`tion and communication to sintetize all these
`data without loosing relevant informations,
`and the following few pharmacokinetic para-
`meters can be defined:
`
`– peak concentrations (Cmax)
`- peak time (Tmax)
`- terminal half-life (t2)
`- area under the curve (AUC)
`
`When urines are also sampled, the drug
`amounts excreted unchanged or the percent-
`age of the dose excreted in urines should al-
`so be computed. Notice that the drug con-
`centrations in urines are very rarely of inter-
`est in pharmacokinetics even if this is what
`is measured directly, but the drug amounts
`allow making a mass balance getting the
`fraction of the excreted dose. The drug
`amount can be computed from the concen-
`trations having the volumes, consequently
`when urines are sampled it is important to
`record also the volumes excrete during the
`experiment.
`A single kinetic profile may be well sum-
`marized by Cmax, Tmax, t2 and AUC and, hav-
`ing more than one profile, 8 parameters at
`
`35
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 003
`
`

`

`R. Urso, P. Blardi, G. Giorgi
`
`least, the mean and standard deviation of
`these parameters, may well summarize the
`drug kinetics in the whole population.
`A short description of these parameters is
`presented below.
`
`Tmax and Cmax
`The peak time (Tmax) and the peak concen-
`tration (Cmax) may be directly obtained from
`the experimental observations of each sub-
`jects (see Figure 3).
`After an intravenous bolus these two para-
`meters are closely dependent on the experi-
`mental protocol because the concentrations
`are always decreasing after the dose. On the
`other hand the peak time corresponds to the
`time of infusion if the drug is infused i.v. at
`constant rate.
`After oral administration Cmax and Tmax
`are dependent on the extent, and the rate of
`drug absorption and on the disposition pro-
`file of the drug, consequently they may char-
`acterize the properties of different formula-
`tions in the same subject2.
`
`Half-life of monoexponential functions
`The terminal half-life (t2) is a parameter
`used to describe the decay of the drug con-
`centration in the terminal phase, ie when the
`semilog plot of the observed concentrations
`vs time looks linear. This parameter is de-
`rived from a mathematic property of the mo-
`noexponential functions and its meaning is
`shown in Figure 4.
`
`It can be seen that the monoexponential
`curve halves its value after a fixed time inter-
`val, independently on the starting time.
`Plotting the logarithms of the concentrations
`or using a semilogarithmic scale, a straight
`line can be obtained (see Figure 5). The dia-
`grams in semilogarithmic scale are of fre-
`quent use in pharmacokinetics mainly for two
`reasons. First, because the log trasformation
`widen the scale of the concentrations so as to
`be able to clearly observe the full data plot
`even when the data range over various orders
`of magnitude. Second because a semilog plot
`helps more in the choice of the best pharma-
`cokinetic model to fit the data.
`Tracing with a ruler the straight line which
`interpolate better the data points, it is possi-
`ble to obtain an estimate of the half-life by vi-
`sual inspection of the semilog plot. All what
`is needed is to observe on the diagram the
`time at which the line halves its starting val-
`ue, anyway, for a more rigorous estimation,
`the best line can be obtained applying the lin-
`ear regression technique on log-transformed
`data.
`
`Terminal half-life of
`multiexponential functions
`Very often in pharmacokinetics the drug
`profile is not monoexponential, however it
`has been observed that the log-concentra-
`tions of many drugs in plasma and tissues de-
`cay linearly in the terminal phase, ie after a
`sufficently long time from the administration
`
`Cmax
`
`ug/ml
`
`Tmax
`
`Hours
`
`Figure 3. Getting the estimate of Cmax (peak concentration) and Tmax (peak time) from the observed data. (Cmax =
`35 µg/ml and Tmax = 4 h).
`
`36
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 004
`
`

`

`A short introduction to pharmacokinetics
`
`Conc
`
`Time (hours)
`
`Figure 4. Plot of the observed drug concentrations vs time data (points) interpolated by a monoexponential function
`(continuous line). It can be seen that at any time the curve halves its values after 2 hours and this happens because
`the half-life of the curve is just 2 hours.
`
`time. This means that the kinetic profile of
`many drugs is well approximated by a mono-
`exponential function in the terminal phase
`and consequently it make sense to define the
`half-life, or terminal half-life, in order to
`characterize the slope of the curve in this
`phase. Two examples of multiesponential
`curves are shown in Figure 6.
`
`In these cases the estimation of the termi-
`nal half-life may be highly subjective because
`the experimenter must choose the number of
`points to use in the computation by visual in-
`spection of the plot. Adding or discarding
`one point may have big influence on the esti-
`mate when few data are available and the ex-
`perimental error is high. To avoid confusion,
`
`Time
`
`Conc
`
`B
`
`Time
`
`og (Conc)
`
`A
`
`Figure 5. Semilog plot of the monoexponential function. A, Plot of the log-concentration vs time data and the inter-
`polating monoexponential function. B, Plot of concentration vs time data in a semilog scale. Plot A and B are super-
`imposable, but in plot A the log-concentration levels are reported on the y-axis while in plot B the drug levels can be
`read without the need of the antilog transformation.
`
`37
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 005
`
`

`

`R. Urso, P. Blardi, G. Giorgi
`
`Observed values
`Interpolated values
`
`H
`
`ng/ml
`
`B
`
`Observed values
`Interpolated values
`
`H
`
`ng/ml
`
`A
`
`Figure 6.Two examples of biexponential curves. A, Plot of the drug levels after intravenous administration. B, Plot
`of the drug levels after oral administration. In both plots the continuous line represents the data interpolated by a bi-
`exponential function. Plot A and plot B show that the terminal phase of the washout curves is log-linear and conse-
`quently it can be approximated by a monoexponential function.
`
`the points used in the estimation should al-
`ways be declared when the estimate of the
`terminal half-life is presented.
`Notice, also, that the accuracy of the esti-
`mates are very dependent on the time sam-
`pling range. Planning many sampling times
`over a time interval of three or more half-
`lives, allows to get a good estimate of the ter-
`minal half-life, while the same number of
`samples in shorter time intervals makes the
`estimate less accurate. For example, the esti-
`mate of a 24 hours half-life cannot be very ac-
`
`curate having points up to only 12-24 hours,
`even when many data points have been col-
`lected.
`
`The area under the curve (AUC)
`The under the curve (AUC) is a parameter
`that may be used in different ways depending
`on the experimental context. This parameter
`may be used as an index of the drug exposure
`of the body, when referred to the plasma drug
`levels, or as an index of the drug exposure of
`particular tissues if referred to the drug levels
`
`ng/ml
`
`Time (hours)
`
`Figure 7. The area under the curve (AUC). The area of the trapezoid A is given by’: (Cn-1 + Cn) × (tn – tn-1)/2. The ex-
`trapolation to infinity (B) is computed by the terminal half-life: Clast / (0.693 / t2). The AUC is given by the sum of all
`the trapezoids and of the terminal extrapolation B. The dimension of the AUC are always given by time × concentra-
`tion and, in this example, we have: AUC = ng × h × ml-1.
`
`38
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 006
`
`

`

`A short introduction to pharmacokinetics
`
`in tissues. Under very general assumptions,
`the area under the plasma or blood drug con-
`centrations is a parameter that is closely de-
`pendent on the drug amount that enter into
`the systemic circulation and on the ability
`that the system has to eliminate the drug
`(clearance). Therefore it can be used to mea-
`sure the drug amount absorbed or the effi-
`ciency of physiological processes that charac-
`terize the drug elimination.
`In most cases a sufficiently accurate esti-
`mate of the AUC can be obtained applying
`the trapezoidal rule as illustrated in Figure 7.
`It is good practice to calculate the AUC
`from time 0 (administration time) to infinity af-
`ter single drug administration, and within the
`dose interval after multiple dose treatment.
`In the first case the extrapolation from the
`last measurable concentration to infinity is
`computed assuming that the wash-out in the
`terminal phase follows a monoexponential
`profile, and for the calculation the terminal
`half-life is needed. In the second case extrap-
`olations are not necessary provided that the
`beginning and the end of the dose intervals
`are also sampled.
`
`Data Interpolation and
`Multiexponential Functions
`
`The estimation of Cmax, Tmax, t2 and AUC is
`the first step in the analysis of the pharmaco-
`kinetic data because these parameters can
`well represent the data without the need of
`any complex mathematical model. This is the
`reason why Tmax, Cmax, AUC and the terminal
`half-life are often called model-independent
`parameters even though their definition may
`be well dependent on some very general
`pharmacokinetic assumption.
`A more carefull description of the data can
`be obtained interpolating and extrapolating
`the drug concentrations with some mathe-
`matical functions. These functions should be
`choosen properly according to the task of the
`experimenter, who may want to reduce all the
`data in a small set of parameters, or to verify
`if the hypotheses incorporated in the func-
`tions are confirmed by the observations. In
`the first case we can say that the task is to get
`a simulation of the data, in the second to get
`a model3.
`
`The functions used to interpolate and re-
`duce the pharmacokinetic data are the multi-
`exponential functions and the reference mod-
`els are the compartmental models whose so-
`lutions are just the multiexponential func-
`tions. Below, some property of the multiex-
`ponential functions will be introduced and it
`will be shown how these functions may help
`in getting some relevant pharmacokinetic pa-
`rameters.
`
`Monoexponential function
`A monoexponential function can be writ-
`ten as:
`
`c(t) = C0 · e–λ·t
`
`(1)
`
`where c(t) is the drug concentration at time t
`and Co and λ are the parameters to be esti-
`mated.
`This function has been used to interpolate
`the plasma concentration profiles of different
`drugs after intravenous administration, that is
`estimating the parameters Co and λ by fitting
`the curve to the experimental data.
`Co is the drug concentration at time 0 and
`λ is dependent on the half-life of the curve
`because the following relationship holds:
`t2 = 0.693 / λ
`
`(2)
`
`Integrating eq. 1 between 0 and infinity, we
`get:
`
`Co
`AUC = ––––λ
`
`(3)
`
`which says that the area under the curve can
`also be computed easily by the ratio between
`Co e λ.
`Co and λ can be estimated by e non-linear
`regression technique or by linearizing the da-
`ta using the log-trasformation, and the esti-
`mates may well be used to synthetize all the
`observed drug concentration of one subject in
`a particular pharmacokinetic study (usually
`no less than 10-12 set of time-concentration
`data). When many subject are involved, then
`the interest is in the population kinetics of
`the drug and consequently the population
`pharmacokinetic parameters have to be esti-
`mated. These are not only the mean vales of
`Co and λ, but also some measure of their
`
`39
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 007
`
`

`

`R. Urso, P. Blardi, G. Giorgi
`
`variability in the population, for example
`their standard deviation. Having the esti-
`mates of the parameters in each subjects, the
`population estimates may be obtained
`straightfore-ward using standard statistical
`formulas and this is called the two stage
`method. Sometimes very few data points per
`subject are collected, consequently the first
`step of this procedure is problematic, if not
`impossible, even when many subjects have
`been investigated and a lot of data are avail-
`able. In this case, sophisticated statistical tec-
`niques can be applied to get the population
`parameters (bayesian methods, non-linear
`mixed effect models). Anyway, at the end of
`the procedure, the mean and the standard de-
`viations of the pharmacokinetic parameters
`are computed and can be used to summarize
`a very large number of experimental data and
`to represent the pharmacokinetic properties
`of any drug.
`
`Biexponential functions
`In many cases a monoexponential function
`like equation 1 is unable to fit accurately a
`particular drug profile. For example, it has
`been shown that after oral administration the
`plasma drug concentrations are increasing
`just after the dose and decreasing after the
`peak time, or, even after an intravenous bo-
`lus, a clear bias may be present between the
`observed and the predicted data when fitted
`by a monoexponential function. In this case,
`a good interpolation of an oral or iv profile
`may be obtained by adding a new exponen-
`tial term to this function, ie by fitting the data
`with the following equation:
`
`c(t) = A1 · e–λ1·t + A2 · e–λ2·t
`
`(4)
`
`which is called a biexponential function. This
`equation is frequently parametrized as:
`
`c(t) = A · e–α·t + B · e–β·t
`(5)
`where it is assumed that α > β, c(t) and t are
`the dependent and the independent variable
`respectively, and A, B, α and β (or A1, A2, λ1,
`λ2) are the parameters to be estimated.
`Notice that when α is much greater than β
`and when t is sufficiently high, then the term
`A·e–α·t is much smaller than the second term
`in eq. 5, and c(t) is well approximated by the
`monoexponential function c(t) = B·e–β·t.
`
`40
`
`Consequently it may be stated that after a
`sufficiently long time interval the biexponen-
`tial function declines to 0 with a half-life
`characterized by the lower exponent β and
`that it make sense to extend the idea of ter-
`minal half-life even to a biexponential func-
`tion.
`C(t) at time 0 is given by A+B and when
`A+B = 0 then C(0) = 0. This means that when
`A is negative and equal to –B, the function
`c(t) is increasing at the beginning, reaches a
`peak level and then decreases to 0. When
`both A and B are positive, the curve is de-
`creasing in all the range t > 0. In both cases
`the terminal half-life is given by 0.693 / β.
`The parameters A, B, α and β can also be
`used to compute the AUC, because integrat-
`ing the eq. 5 from 0 to infinity, it holds:
`
`A B
`AUC = ––– + –––
`α
`β
`
`or AUC = B · (–– – ––)
`
`1
`α
`
`1
`β
`
`Having a set of experimental data, the esti-
`mates of the parameters can be obtained by a
`non-linear fitting procedure or by the resid-
`ual (or peeling) method which is based on a
`sequential log-linearization of the curve. This
`method gives only approximate estimates of
`the parameters and usually is used to get the
`initial estimates of the parameters needed to
`run the non-linear fitting procedure.
`
`Multiexponential functions
`The majority of the drugs have approxima-
`tively a biexponential profile in plasma after
`an intravenous bolus, but there are excep-
`tions expecially after oral administration, be-
`cause it is common that, if n exponential
`terms are needed to fit the plasma concentra-
`tions after an iv bolus, then, to get a good in-
`terpolation of the data after oral or extravas-
`cular administration, one exponential term
`should at least be added to the equation.
`In general it can be stated that the kinetics
`of all drugs in plasma is well described by
`multiexponential functions which may be
`written as:
`
`c(t) = ∑ Ai · e–λi·t
`
`i
`
`(6)
`
`The number of the exponential terms in
`this equation should be choosen time to time
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 008
`
`

`

`A short introduction to pharmacokinetics
`
`in order to get the best fit of the experimental
`data and the best accuracy in the estimated
`parameters. Then the initial drug level and
`the AUC can be computed by the parameters
`of the curve as:
`Co = ∑ Ai
`
`i
`
`e AUC = ∑ ––––
`Ai
`λi
`
`i
`
`where Co is equal to 0 after oral or extravas-
`cular administration, which means that in
`these cases some coefficients should be nega-
`tive, and where the terminal half-life is al-
`ways given by 0.693/λn where λn is the lowest
`exponent.
`The parameters of eq. 6 can still be esti-
`mated by a non-linear fitting procedure, but
`remember that in tipical pharmacokinetic set-
`tings, the exponents λi should be different at
`least of one order of magnitude to be the esti-
`mates sufficiently accurate.
`Multiexponential function are a very general
`tool in pharmacokinetics because they may be
`adapted to describe not only the profile of the
`drugs in plasma, but also in all other tissues or
`fluids after almost any kind and route of drug
`administration. Exceptions are very rare.
`
`Clearance, Volume of Distribution
`and Compartmental Models
`
`We have seen in the previous paragraphs
`how it is possible to compute some parame-
`ters that allow to describe synthetically the re-
`sults of a pharmacokinetic experiment. It has
`been said also that the kinetic profile of a
`drug depends on various biological processes
`which modulate the drug absorption, elimina-
`tion and distribution. For practical purposes it
`should be usefull to correlate in some way the
`observed concentrations, and therefore the ki-
`netic parameters, with these processes. The
`mathematical models, and in particular the
`compartmental models, can help to this scope
`and for this reason they have been extensively
`used in pharmacokinetics. Moreover the com-
`partmental models allow to make predictions4
`and to describe more complex experiments
`where, for example, having more fluids and
`tissues sampled in the body at the same time,
`it is possible to find relevant relationships be-
`tween the drug profiles.
`
`We can start working with models by intro-
`ducing two new parameters, the volume of
`distribution (V) and the clearance (CL)
`whose definitions are largely dependent on a
`pharmacokinetic model5.
`A tipical problem in pharmacokinetics is
`that on one hand the measured variables
`are usually the drug concentrations in tis-
`sues or, more frequently, in plasma and, on
`the other hand, the experimenter may be
`interested in knowing the amount of the
`drug present in the body, or the amount
`eliminated at time t after administration of
`a known dose.
`V and CL have been thought to cope
`with this problem when the reference vari-
`able is the drug concentration in plasma,
`and the following very general definitions
`can be given:
`
`CL = drug amount eliminated per unit of time/
`drug concentration in plasma
`
`(7)
`
`V = drug amount in the body/
`drug concentration in plasma
`
`(8)
`
`Assuming that CL is not time dependent, it
`may be shown that:
`
`D
`CL = –––––––––
`AUC
`
`(9)
`
`V cannot be time independent because of
`drug distribution in tissues outside the plas-
`ma, but under some general assumptions it
`may be shown that the ratio of eq. 8 ap-
`proaches a limit value given by:
`
`D
`Varea = –––––––––––
`λ · AUC
`
`(10)
`
`D is the drug amount that enters into the
`systemic circulation (after an intravenous ad-
`ministration, D is the dose) and λ is the low-
`est exponent used to get the terminal half-
`life. Eq. 9 and 10 may be derived appling very
`general compartmental models to the drug
`kinetics.
`Varea has been used to characterize the drug
`distribution in the total body, while CL as a
`measure of the efficiency of the drug elimina-
`tion process. The first may be shown getting
`
`41
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 009
`
`

`

`R. Urso, P. Blardi, G. Giorgi
`
`eq. 9 from a monoexponential function. In
`this particular case V is time independent and
`the following equation holds:
`
`D
`D
`V = –––––––––– = –––––
`λ · AUC
`C0
`
`(11)
`
`After an iv bolus, D is just the drug
`amount present in the body at time 0 and it is
`given by the product between V and the con-
`centration at time 0. This means that V may
`be thought as that ideal volume where the
`dose should dilute istantaneously at time 0 in
`order to get a drug concentration equal to
`Co. As V is not time dependent, then eq. 11
`is equivalent to eq. 8, consequently the mo-
`noexponential function can be rewritten as:
`
`D
`c(t) = C0 · e–λ·t = –––––·e–λ·t
`V
`
`The new parameters V and λ (the dose D
`is known) are said to be invariant, which
`means that they do not change changing the
`dose and time, and consequently they charac-
`terize the drug kinetics in a particular sub-
`jects.
`It can be noticed that the new parametriza-
`tion adds new meanings to the monoexpo-
`nential equation, for example now it is explic-
`itly stated that the drug concentrations are
`proportional to the dose, and these meanings
`are a consequence of a model assumption.
`Equation 11 is no more true for a multiex-
`ponential function, consequently it may be
`convenient to introduce two time and dose
`independent volumes:
`
`D
`D
`Vc = –––– and Varea = –––––––––– (λ1 = lowest λ)
`λ1 · AUC
`C0
`
`Vc is usually called the volume of the
`central compartment or the initial volume
`of distribution, while the second is said Varea
`(or Vβ when derived from a biexponential
`function).
`Vc is always equal to Varea in a monoexpo-
`nential function and always lower than Varea
`in a multiexponential function, while at any
`time after the dose the ratio of eq. 8 is always
`higher or equal to Vc and lower than Varea.
`
`42
`
`Varea can be obtained measuring the drug
`concentrations in plasma and as it may be
`highly correlated with the tissues to plasma
`ratio (ie the ratio between the drug levels in
`tissues and blood), it may be used as a mea-
`sure of drug distribution: higher estimates of
`Varea mean higher levels of the drug in tissues
`compared to plasma and vice versa.
`Vc may be used to predict Co for a given iv
`dose or, conversely, to estimate the non toxic
`loading doses when the toxic levels are
`known.
`The second aspect of our pharmacokinetic
`model refers to CL. Eq. 7 can be written us-
`ing the following symbols6:
`
`- da(t) / dt = CL · c(t)
`
`(12)
`
`where a(t) is the drug amount present in the
`body at time t, c(t) is the drug concentration
`in plasma at time t and da(t)/dt is the deriva-
`tive of a(t) respect to time. Eq. 12 and eq. 7
`are equivalent, because the drug amount
`eliminated per unit time must be equal to the
`absolute value of the rate of change of the
`drug amount in the body.
`Intergrating eq. 12 from time 0 to infinity
`on the assumption that CL is constant, we
`get:
`
`Amount eliminated =
`(Dose entered into the systemic circulation) =
`CL × AUC
`
`which is equivalent to eq. 9.
`CL is commonly used to characterize the
`efficiency of drug elimination from the body
`being higher values of CL associated with
`higher efficiency of drug elimination.
`Models have also been developed to find
`relationships between the volumes and the
`reversible drug-protein binding in plasma and
`between CL and the blood flow through the
`eliminating organs (liver and kidneys) and
`many experimental observations support the
`utility of these models.
`Without going into the details, it is interest-
`ing to notice that the following relationship
`can be derived from the previous equations:
`. λ1 = CL
`Varea
`and remembering that λl depends on the half-
`life, we get:
`
`Mylan v. Janssen (IPR2020-00440) Ex. 1015 p. 010
`
`

`

`A short introduction to pharmacokinetics
`
`Varea
`t2
`––––––– = –––––––
`0.693
`CL
`
`which means that in our models the terminal
`half-life is dependent on both the drug distri-
`bution and the drug elimination. For exam-
`ple, it may happen that the half-life increases
`not because the drug elimination is impaired
`but just because the drug distribution im-
`proves.
`
`Bioavailability
`
`The extent and the rate of drug absorption
`play an important role in pharmacokinetics,
`and this parameters are usually referred as
`the drug bioavailability7,8. For example, a
`fraction of the dose may be metabolized dur-
`ing the early passage through the gastroin-
`testinal tract or through the liver after an oral
`dose, or part of the dose may not reach the
`blood due to drug malabsorption. The conse-
`quence is an incomplete absorption of the
`drug into the systemic circulation and an in-
`complete drug availability may produce inef-
`fectiveness of the treatment.
`Absorption is a complex process which
`cannot be monitored experimentally in a sim-
`ple way, and consequently it is not easy to get
`the extent of drug absorption by direct obser-
`vation. Anyway pharmacokinetic models al-
`low to estimate this parameter with a simple
`experimental design. Look