Symposium on Pediatric Pharmacology
`
`Pharmacokinetic Principles in
`Pediatric Pharmacology
`
`William J. Jusko, Ph.D.*
`
`Pharmacokinetics is gaining increasing importance for the practical
`interpretation and prediction of pharmacologic data in human beings.
`The cornerstone of pharmacologic principles is the concept of a drug
`molecule interacting with a receptor to produce a response. Because the
`drug levels which elicit this response are transient, the effect is likewise
`transient. Increasing the dose can enhance the intensity and duration of
`the effect but may produce undesired toxicity, while too low a dose may
`yield inadequate therapy. Many pharmacologic effects are difficult to
`measure directly and therapy must await delayed judgment. Thus, dos-
`age and time are the factors which make the pharmacologic use of drugs
`both useful and hazardous. Because of this, pharmacokinetic techniques
`have been developed to seek and utilize mathematical methods to relate
`drug dosage, pharmacologic effects, and time as a means of evaluating
`and controlling drug therapy.
`The primary variables which determine the time course of phar-
`macologic effects are the absorption, distribution, metabolism, and ex-
`cretion of the drug and the intrinsic affinity of the drug for the receptor.
`Therapeutic difficulties can arise when one or more of these factors are
`complicated by patient variables such as age, pathology, genetics, and
`environment. The purpose of this report is to consider several aspects of
`pharmacokinetics which are applicable to pediatric pharmacology.
`
`GENERAL DYNAMIC PROCESSES
`
`The major factors which determine the time -course of phar-
`macologic effects of a drug are depicted in Figure 1. In general, drug
`levels and pharmacologic effects are dependent on the physicochemical
`properties of the particular drug and the physiology of the patient.
`During the process of absorption, the rate at which the drug dis-
`
`Assistant Professor of Pharmacology, Boston University Medical Center, and Section of
`Clinical Pharmacology, Veterans Administration Hospital, Boston, Massachusetts
`Pediatric Clinics of North America -Vol. 19, No. 1, February 1972
`
`81
`
`Page 1
`
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`82
`
`IGastric Emptying
`and GI Motility
`
`PHARMACOLOGIC
`RECEPTOR SITE
`
`il
`
`Distribution
`Factors
`
`WILLIAM J. JUSKO
`
`1
`
`OTHER DISTRIBUTION
`SPACE
`
`Active
`Metabolites
`
`BIO-
`TRANS -
`FORMA -
`TION
`SITE
`
`DRUG
`DOSAGE
`FORM
`
`Drug
`Release
`
`Drug 9
`Absorption'
`
`XA
`
`I
`
`CENTRAL DISTRIBUTION
`UNBOUND DRUG
`T
`BOUND
`PROTEIN
`DRUG
`
`GI Degradation
`or
`Excretion
`
`1
`
`X
`
`.MM .-1---1- .MM -1-
`
`Biliary
`Renal
`Inactive
`Excretion Excretion T Meta-
`
`bolites J
`
`Cycling
`
`ELIMINATED
`
`DRUG
`
`XE
`
`Figure 1.Schematic representation of the major factors which control body levels of
`drugs and affect the intensity and duration of pharmacologic effects.
`
`solves will often determine the bioavailability of the agent. The release
`properties of the drug are related to the acid -base -salt form, its water sol-
`ubility and degree of ionization in gastrointestinal fluids, and the for-
`mulation of the dosage form. Since infants and children usually receive
`drugs in solution, suspension, or chewable form, formulation palatability
`and drug stability are of additional concern. Physiologic factors such as
`gastric and intestinal pH, composition of the gastrointestinal contents,
`gastric emptying, intestinal motility, the integrity of the gastrointestinal
`membranes, and the mesenteric blood flood affect drug absorption and
`are subject to variability with age.3' 'L 25
`The distribution of drugs in the body is partially dependent on the re-
`tention of drug in the blood owing to plasma protein- binding. The low
`concentration of plasma proteins found in the neonate can thus alter
`drug response. Drug availability to receptor, tissue, and elimination sites
`is also related to the lipid solubility and degree of ionization of the drug
`as well as physiologic factors which can vary with age, including local
`blood flow, the type and integrity of membrane barriers, and the water
`and protein content of tissues.3' 4' 22
`The predominant mechanisms of drug elimination are renal excre-
`tion, metabolism, and biliary excretion. The net effect of glomerular fil-
`tration, renal tubular secretion, and tubular reabsorption results in renal
`clearance of both natural and foreign compounds. In addition to the
`excretory mechanism, the renal clearance of drugs is also dependent on
`blood flow to the kidneys, plasma protein- binding and the distribution of
`the drug while factors such as urine pH, urine volume, and glomerular
`
`Page 2
`
`

`

`83
`
`PHARMACOKINETIC PRINCIPLES IN PEDIATRIC PHARMACOLOGY
`integrity are sometimes of importance. Developmental changes in renal
`function are well recognized in pediatrics.2, 3
`Drug metabolism involves a large number of reaction mechanisms
`which result in alteration of foreign substrates .6 Detoxification usually
`involves drug conjugation with carbohydrates, amino acids, acetate, sul-
`fate, and methyl groups, to form products which are more water soluble
`than the original drug and are thus more readily excreted in urine and
`bile. A large number of oxidation processes result in removal from or at-
`tachment to drugs of moieties such as hydroxyl, amino, hydrogen, alkyl,
`or halogen groups. Reduction reactions can produce amino groups from
`nitro or azo compounds, while hydrolysis involves splitting of ester or
`amide linkages. All mechanisms except conjugation reactions can pro-
`duce either an active or an inactive pharmacologic compound. Done3 has
`noted that hydrolytic processes are least likely to be affected by imma-
`turity.
`Of major importance is the relationship between age and intrinsic
`sensitivity to drugs. Goldenthal5 has compiled LD50 values for over 200
`drugs in newborn and adult animals. Almost every drug listed is more
`
`Table 1. Characteristics of the Principal Rate Processes
`Observed in Pharmacokinetics
`
`FIRST -ORDER
`
`ZERO -ORDER
`
`CAPACITY -LIMITED
`
`Rate constant
`
`k,
`
`k
`
`Vmax, Km
`
`Dimensions
`
`time -'
`
`amount /time
`
`dose -dependent
`
`Rate of drug
`loss
`
`Proportional to the
`amount present (X)
`
`Constant amount per
`unit time
`
`Varies with amount:
`First -order at low
`levels; 0 -order at
`high levels
`
`dX - Vma,X
`dt
`Km + X
`
`t'Vmax-D-X+K¡ln
`
`dX
`dt
`
`-k
`
`X=D-kt
`
`Differential
`dX
`equation (DE) dt
`for drug loss:
`
`-k; X
`
`X= De';
`
`Solution to DE:
`(D = dose)
`
`Graphical
`behavior: (at
`low and high
`doses)
`
`log- linear
`
`linear
`
`log\X
`
`x
`
`log
`x
`
`Half -life
`
`Constant
`
`Increases with dose
`
`time
`
`time
`
`Common
`Examples
`
`Renal clearance,
`Drug diffusion,
`Blood transport,
`Drug metabolism
`
`Drug dissolution,
`Drug infusion,
`Multiple- dosing
`
`Generally increases
`with dose
`
`Saturable metabolic,
`transport, or reac-
`tion processes.
`
`Page 3
`
`

`

`84
`WILLIAM J. JUSKO
`toxic in the newborn than in the adult animal. This can be attributed to
`altered drug distribution and elimination, or to greater reactivity of the
`newborn to drugs, or to both.
`In order to consider the role of pharmacokinetics in pediatrics, it is
`necessary to understand both mechanisms of drug movement in the
`body and their expression as mathematical rate processes. The principal
`types of rate processes encountered in pharmacokinetics are first -order,
`zero -order, and capacity -limited. The definition and the mathematical
`and graphical behavior of a single mechanism involving drug removal
`by each of these rate processes is shown in Table 1. Rate processes which
`are first -order are easiest to quantitate since they are dose -independent.
`For example, if drug absorption and elimination are both first -order, then
`body levels of drug will always be proportional to the dose at a specific
`time. In contrast, both the amount of drug involved and the duration of
`the process determine the time course of body levels for zero -order
`processes. Capacity -limited or Michaelis -Menten type processes exist
`when an enzyme or transport mechanism has a limited number of recep-
`tors available for drug substrate. Because of this, capacity -limited pro-
`cesses behave in a zero -order manner at high drug levels but are first -
`order at very low drug levels. Fortunately, most of the rate -processes
`which are represented as arrows in Figure 1 are either first -order or drug
`dosages are low enough so that Michaelis -Menten processes can be
`regarded as first -order.
`The basic information sought in a pharmacokinetic analysis of phar-
`macologic data is listed in Table 2. These parameters are available either
`from direct measurement of drug levels in the body or from the elicited
`pharmacologic response. Therapeutic control based on the patient re-
`sponse is obviously desirable and feasible for drug effects such as anes-
`thesia, cardiac function, or antipyresis, but it is difficult to control ther-
`apy for effects such as analgesia or chemotherapy. In either case, it is
`useful for the physician to be guided by drug level -effect information
`gained from previous experience, particularly when age and develop-
`ment are critical factors in choosing a dosage regimen. Thus, the clinical
`pharmacologic investigation of a drug has come to involve the phar-
`macokinetic analysis of blood level data (usually plasma or serum con-
`centrations), urinary excretion data, and occasionally feces, cerebro-
`spinal fluid, bile, milk, lymph, and tissue biopsy samples in addition to
`the type and time -course of pharmacologic and toxicologic effects.
`
`KINETICS OF DRUG DISPOSITION
`
`In many instances, the behavior of the drug is such that the body can
`be assumed to be a single homogeneous compartment. The dose (Do) can
`thus be accounted for as being unabsorbed (XA), in the body (XB), or
`eliminated (XE) in the manner:
`
`Absorption
`
`XA
`
`Elimination XE
`KE
`
`(Scheme I)
`
`Page 4
`
`

`

`PHARMACOKINETIC PRINCIPLES IN PEDIATRIC PHARMACOLOGY
`Table 2. General Parameters of Concern in the Elaboration
`of Pharmacodynamic Data
`
`85
`
`Control of body levels of drug
`Elimination rate constants and mechanisms
`Rate constant for renal excretion.
`Rate constants for metabolism.
`Absorption parameters
`Fraction of dose absorbed by various routes.
`Rate constants for absorption.
`Distribution parameters
`Volume(s) of distribution.
`Distribution rate constants.
`Protein -binding constants.
`Single and multiple dose -dependence of absorption, distribution, and elimination.
`
`Control of the pharmacologic effect
`Relationships between the amount in the body and time and the pharmacologic effect.
`Relationship between amount in the body and occurrence of side or toxic effects.
`
`Evaluation of patient variables such as: physiologic, pathologic, genetic, developmental,
`and environmental factors affecting body levels of drug and pharmacological effects.
`
`where KE is the first -order rate constant for overall drug elimination. The
`parameters which are most often associated with the evaluation of data
`using this model are listed in Table 3. Except for the fraction absorbed
`(F) and the absorption constant (ka), these parameters can be obtained by
`following the time course of blood levels and /or urinary excretion rates of
`drug following administration by a route involving instantaneous or very
`rapid absorption. In this case, the amount of drug in the body will be
`proportional to the dose, according to the exponential (e) relationship:
`X-D e
`Table 3. Pharmacokinetic Parameters of the One -Compartment
`Model For Drug Absorption, Distribution, Metabolism, and Excretion
`
`(Eq. 1)
`
`KH;
`
`t
`
`SYMBOL
`
`DIMENSIONS
`
`PARAMETER'"
`
`SPECIMENS
`USUAL
`SOURCE** REQUIRED ***
`
`B or U or O
`B or U or O
`B
`B or U
`B and U
`U
`
`UB
`
`B
`B
`
` or U
`
`D
`C
`C
`C
`D
`C
`
`C C
`
`D
`C
`
`Elimination half -life
`Elimination rate constant
`Apparent volume of distribution
`Fraction of dose absorbed
`Renal clearance
`Rate constant for renal excretion
`Rate constant for metabolism
`Rate constant for absorption
`Plasma level area
`Body clearance
`
`t1/2
`
`KE;
`V,,
`F
`Cl
`k.
`k,,,
`ka
`AUC
`Cl,;
`
`time
`time -'
`volume
`-
`volume /time
`time -'
`time -'
`time -'
`conc. time
`volume /time
`
`All rate constants are assumed first -order.
`'Source: D = Experimental data, C = Calculated from D.
`***Specimens: B = Serum or Plasma: U = urine; 0 = other
`
`Page 5
`
`

`

`86
`WILLIAM J. JUSKO
`The value of the elimination constant can be obtained directly from the
`half -life (ti12) of the drug according to:
`
`KE _
`
`0.693
`Li/2
`
`(Eq. 2)
`
`where the t112 is the time required for body levels of drug to decrease by
`one -half of a previous level. The elimination constant has dimensions of
`1 /time and is also defined as the fraction of the dose removed from the
`body during that time interval.
`Since direct measurements are usually made of serum, plasma, or
`blood concentrations rather than amounts of drug in the body, the appar-
`ent volume of distribution (Vd) has been used to correlate the amount of
`drug in the body with the concentration in the plasma (Cr):
`
`Cr = XB/Vd
`
`(Eq. 3)
`
`An exponential expression can be linearized graphically by plotting
`the data on a logarithmic versus linear scale. In this way, Equations 1
`and 3 become:
`
`log C =1og C; -23 t
`
`(Eq. 4)
`
`where KE /2.3 is the slope of the line and C/ is the zero -time intercept on
`the logarithmic plasma concentration axis. This provides a method of
`calculating the volume of distribution from the intercept value:
`
`Vd = Do/C1,
`
`(Eq. 5)
`
`The apparent volume of distribution is primarily a mathematical
`constant which is useful for relating amounts and concentrations of
`drug. The magnitude of the value obtained is related partly to the char-
`acteristics of the drug and partly to patient variables such as body size,
`body water composition, hemodynamics, and plasma protein concentra-
`tion. The major properties of the drug which are of importance are the
`lipid solubility and the degree of plasma and tissue protein binding. In
`general, drugs which are highly plasma protein bound exhibit small Vd
`values while highly lipid- soluble compounds which are able to penetrate
`into fatty tissue exhibit a large Vd.6
`Figure 2 shows an example where several useful parameters of the
`one -compartment model were obtained from pediatric data. Intravenous
`doses of bromsulphalein (BSP) were administered to patients of various
`ages and plasma samples were collected and assayed.28 The dye, BSP,
`serves as a measure of general hepatic function since it is eliminated al-
`most entirely by liver metabolism and biliary excretion. The semi -
`logarithmic plot of plasma concentrations versus time represents Equa-
`tion 4. Pharmacokinetic parameters for subjects of various age ranges
`were calculated and are listed in Table 4. The apparent volume of dis-
`
`Page 6
`
`

`

`PHARMACOKINETIC PRINCIPLES IN PEDIATRIC PHARMACOLOGY
`
`87
`
`o D
`R/LCp. V
`
`D
`
`d 20
`zo
`
`P. 10
`
`ccI-ZWVZ 5oU
`
`SLOPE =-21.E
`
`-i-
`
`---T-
`
`t
`5
`
`:
`
`10
`15
`TIME, MIN.
`
`20
`
`2
`
`o
`
`a.
`
`Na
`
`o 2c
`
`nJa
`
`Figure 2. Plasma concentrations of brom-
`sulphalein (BSP) as a function of time after
`intravenous administration. The data (from
`Wichman et a1.2 ") show monoexponential decline
`which can be described using Equation 4.
`
`tribution increases slightly while the elimination constant increases
`markedly with age. The interaction of the two factors can be evaluated
`by calculation of the body clearance (C1B) where:
`
`C1B = KE.Vd
`
`(Eq. 6)
`
`Body clearance is defined as the portion of the volume of distribution
`which is cleared of drug per unit time. It is equal to the sum of all of the
`clearance or elimination mechanisms, and just as the upper limit of
`renal clearance is the renal plasma flow, the upper limit of body clear-
`ance is the plasma flow through all of the elimination sites. Thus, for
`BSP, the maximum possible C1B is the hepatic plasma flow. It can be
`noted that BSP, because of its high degree of plasma protein binding,
`exhibits a distribution volume slightly larger than the plasma volume
`(approximately 0.04 ml. per gm.).
`The half -life and elimination constant have the numerical advan-
`
`Table 4. Elimination and Distribution Parameters of BSP
`in Normal Infants and Children28
`
`AGE
`RANGE
`
`NUMBER
`STUDIED
`
`t112,
`MIN
`
`1 -10 days
`11 -20 days
`21 -30 days
`31 -60 days
`61 -90 days
`3 mo. -14 yr.
`
`7
`10
`5
`5
`5
`16
`
`9.6 ± 0.8
`8.9 ± 0.6
`7.7 ± 0.6
`7.0 ± 0.6
`6.1 ± 0.4
`5.5 ± 0.2
`
`KE,
`MIN
`
`0.072
`0.078
`0.090
`0.099
`0.113
`0.126
`
`ML /GM
`
`0.056
`0.057
`0.061
`0.070
`0.068
`0.068
`
`CLH,
`ML /KG /HR
`
`246 ± 11
`268 ± 14
`341 ± 36
`420 ± 42
`469 ± 44
`508 ± 28
`
`Page 7
`
`

`

`WILLIAM J. JUSKO
`
`88
`tage that they do not require correction for body size in comparing sub-
`jects of various sizes or ages. Although the half -life is a term which is
`easily understood, the elimination constant is more useful for compari-
`sons of data and in making predictions. The elimination constant, easily
`calculated from the half -life (Equation 2), is an overall rate constant
`which is equal to the sum of the individual or specific constants for elimi-
`nation processes such as renal excretion (ke), metabolism (km), biliary
`excretion (kb), and others:
`
`KE = ke + km + kb +
`
`(Eq. 7)
`
`The numerical value of each constant can be determined from the
`total fraction (fr) of the dose removed from the body by the particular
`rate process (ks) as calculated with the equation:
`
`kX = fX KE
`
`(Eq. 8)
`
`The fraction, fX, must be obtained by collecting urine and feces samples
`until all drug and metabolites have been removed from the body. These
`rate constants, in comparison with KE, quantitate the contribution of a
`given rate process to overall drug removal. For example, if a half -life was
`calculated from:
`
`"t1/ . , _
`
`0.693
`KE-kX
`
`(Eq. 9)
`
`the resultant time value would provide a reasonable estimate of the half -
`life of a drug if the particular kX mechanism was completely immature or
`severely impaired.
`The elimination constant provides a factor for adjusting drug dosage
`according to age and pathology as can be shown with the data for am-
`picillin in Table 5. This antibiotic, as seen from the KE values in normal
`and anephric adults, is almost entirely excreted in the urine. Thus the
`age- dependence of ampicillin elimination closely reflects the degree of
`maturation of renal function. The ratio of half -lives provides the mul-
`
`Table 5. Biological Disposition Parameters of Ampicillin in
`Infants' and Adults]°
`
`AGE
`RANGE
`
`NUMBER
`STUDIED
`
`HALF -LIFE,
`(HRS ± SD)
`
`KE,
`(HRS. -1)
`
`1 -10 days
`11 -20 days
`21 -30 days
`31 -90 days
`
`Normal adults
`Anephric adults
`
`9
`8
`6
`9
`
`8
`4
`
`3.5 ± 1.1
`2.0 ± 0.6
`1.7 ± 0.8
`1.6 ± 0.3
`
`0.1
`1.3 -±-
`20.0 ± 12.0
`
`0.198
`0.346
`0.408
`0.433
`
`0.533
`0.035
`
`RATIO OF
`KE:
`0.533
`
`0.37
`0.65
`0.77
`0.81
`
`1.00
`0.07
`
`Page 8
`
`

`

`89
`PHARMACOKINETIC PRINCIPLES IN PEDIATRIC PHARMACOLOGY
`tiple of the adult dosing interval which should be used in infants and
`children to compensate for their deficient renal function and thus pro-
`duce equivalent overall body levels of drug.
`The urinary excretion rate (dXu /dt) of a drug is related to the plasma
`concentration (C ) by the relationship:
`
`dXu
`dt
`
`(Eq. 10)
`
`This equation predicts that the rate of urinary excretion of drug will par-
`allel the plasma drug level, since renal clearance (Cl) is a direct propor-
`tionality factor. In turn, renal clearance is related to the renal excretion
`rate constant, ke, and the volume of distribution:
`
`C1= keVd
`
`(Eq. 11)
`
`Renal clearance can be compared to body clearance as k,. is com-
`pared to KE to ascertain the contribution of the other processes to re-
`moval of drug from the body.
`As defined by Equations 10 and 11, renal clearance is a phar-
`macokinetic constant, but it is also identical to renal clearance as clas-
`sically measured by the physiologist:
`
`C1=UVC
`where the product of U (urine drug concentration) and V (urine volume)
`is the urinary excretion rate over the specific time interval. It is usually
`worthwhile to obtain several clearance values by analyzing plasma and
`urine samples over an appreciable plasma concentration range of drug.
`This permits determination of possible changes in renal clearance due to
`saturation of renal tubular secretory or reabsorption processes.
`
`(Eq. 12)
`
`DRUG ABSORPTION
`
`Bioavailability
`When a drug is administered orally, rectally, or by another route
`where absorption is not likely to be instantaneous, then F, the fraction of
`the dose absorbed or the "bioavailability of the drug, and ka, the first -
`order (or any other) rate constant for absorption are important factors.
`The area under the plasma level curve (AUC) is a useful parameter since
`it correlates several variables, including F:
`AUC = F D
`KEVd
`
`(Eq. 13)
`
`It is usually reasonable to assume that KE and Vd do not vary with the
`route of drug administration, and thus F can be obtained by measure-
`
`Page 9
`
`

`

`WILLIAM J. JUSKO
`
`Figure 3.
`Linear representa-
`tion of plasma concentrations of
`ampicillin as a function of time in
`0 to 7 day old infants after intra-
`muscular (0) and oral suspension
`(0) doses of 10 mg. per kg. of the
`antibiotic. The data represent
`mean values from 35 infants for
`the
`intramuscular doses'
`and
`median values from 16 infants for
`the oral doses.'
`
`90
`
`j 20
`
`Va
`z 16
`
`oá
`
`cc
`
`r
`
`z
`ó
`
`12
`
`8
`
`2 4
`CCw
`
`8
`TIME, HOURS
`
`12
`
`16
`
`ment of the AUC of the test dose and comparison with the AUC obtained
`after a parenteral dose where all of the drug is, of course, absorbed:
`AUC test
`AUC Iv
`
`(Eq. 14)
`
`Alternatively, the total recovery of unchanged drug in the urine (XII -)
`after administration by the two routes, also yields F:
`
`XU 'test
`Xu`°Iv
`
`(Eq. 15)
`
`A graphical comparison of differences in drug availability from two
`routes is shown in Figure 3. Ampicillin was administered in doses of 10
`mg. per kg. to 0 to 7 day old newborn infants, by intramuscular injection
`by Axline and co- workers,' and by oral suspension by Grossman and
`Ticknor,' and serum was obtained until 12 hours. For the plotted data,
`the plasma level areas were found to be 97 and 64 microgram hours per
`ml, with the intramuscular and oral doses, respectively. According to
`Equation 14, this indicates that about 67per cent of the oral dose was ab-
`sorbed which can be favorably compared to about 30 per cent availability
`of ampicillin from capsules in adults.10 The areas can be determined by
`dividing the curve into measurable trapezoids or by the simple method of
`weighing the respective portions of the graph paper. However, the areas
`
`Page 10
`
`

`

`91
`PHARMACOKINETIC PRINCIPLES IN PEDIATRIC PHARMACOLOGY
`must be obtained from a linear rather than from a semilogarithmic
`graph of the data.
`Data obtained after oral dosage such as shown in Figure 3 for am-
`picillin depicts the rising and falling time -course of serum level of drug
`with the characteristic absorption phase. This requires an equation with
`an additional exponential term for mathematical description of the data:
`FDaka
`ka - KE
`
`(Eq. 16)
`
`C
`
`(e-KEt-e-kat)
`
`A parallel decline in serum levels between 6 and 12 hours can be noted
`for the two routes of administration. By this time, absorption was com-
`plete and the downcurve portion of the data, if plotted on a semi -
`logarithmic graph, would yield a half -life value of about 3 hours from
`both curves.
`Constant -Rate Absorption
`Zero -order input or absorption rates may be encountered if a drug is
`administered by constant -rate infusion, if a tablet is slow in dissolving
`(or is sustained- release), or if an easily saturated specialized absorption
`process is involved. In such instances, the increase in plasma level of
`drug during the course of drug absorption at a constant rate of k
`(amount /time) is described by:
`
`k_
`
`vd KE
`
`(1-e-KEt)
`
`(Eq. 17)
`
`If the absorption process continues for at least 4 to 6 elimination half -
`lives (t *), then the maximum plasma concentration will reach a steady -
`state level (Cr) of:
`
`C;
`
`ko
`
`(Eq. 18)
`
`However, when absorption ceases, plasma levels fall mono- exponen-
`tially from this maximum with the usual half -life.
`The total amount absorbed (XA) or the dose is:
`
`X , = kt'``
`
`(Eq. 19)
`
`where t* is the duration of the absorption process. The time course of
`urinary excretion rate parallels the plasma level and the maximum
`urinary excretion rate is:
`
`dXu
`dtss
`
`ko
`
`ke
`KE
`
`(Eq. 20)
`
`Similar expressions have been used to interpret the capacity -limited
`absorption of riboflavin from the small intestine as a function of age.`'
`
`Page 11
`
`

`

`WILLIAM J. JUSKO
`
`Figure 4. Urinary excretion
`rate (corrected for dose) of ribo-
`flavin as a function of time after
`oral administration of 150 mg. per
`M2. body surface area doses of the
`vitamin to a 5 day old infant ()
`and to a 10 month old infant (0).
`From Jusko, W. J., et al.: Pediat-
`Tics, 45:945, 1970, reproduced
`with permission.
`
`92
`
`0.25
`
`w'
`
`oo
`
` 0.20
`
``.
`
`20
`10
`TIME, HOURS
`
`w
`
`o:
`z o.15
`ó
`~
`cc
`U
`
`0.10
`
`0.05 .
`
`,
`
`}c
`
`c4
`
`Figure 4 shows the time -course of urinary excretion of the vitamin after
`administration of 150 mg. per sq. meter oral saturation doses to a 5 day
`old and a 10 month old infant.8 As shown above, the maximum excretion
`rate reflects the absorption capacity while the duration from time zero to
`the end of the peak is t'`. The urinary recovery was similar at both ages (6
`to 8 per cent of the dose) which indicates that the neonate, in spite of a
`poor absorption capacity, absorbs an adequate amount of the vitamin
`because of the prolonged duration of absorption.
`Multiple- Dosing
`The administration of a dosage regimen consisting of constant doses
`of drug at routine intervals produces a "picket- fence" type of plasma
`level curve and urinary excretion rate profile. This rise and fall sequence
`of drug levels can be quantitated exactly with complex pharmacokinetic
`equations.18' t0 However, a reasonable estimate of the average drug level
`in the body ()L) after any route of administration can be obtained from
`the approximation:
`
`Table 6. Ratio of the Average Body Level of Drug to Dose for Drugs
`with Various Half -lives and Given at Regular Time Intervals (T)
`
`HALF -LIFE
`(hr.)
`
`r= 3 hr.
`
`ACCUMULATION RATIO: X5 /D
`r= 6 hr.
`
`T=12hr.
`
`3
`6
`12
`24
`
`1.44
`2.89
`5.77
`11.5
`
`0.72
`1.44
`2.89
`5.77
`
`0.36
`0.72
`1.44
`2.89
`
`Page 12
`
`

`

`PHARMAC OKINETIC PRINCIPLES IN PEDIATRIC PHARMACOLOGY
`
`FD
`
`K E
`
`X13
`
`93
`
`(Eq. 21)
`
`where T is the time interval between doses. This expression reflects the
`interaction of the principal factors which control the plateau body level
`of drug: the bioavailability, the dose, the elimination constant, and the
`dosing interval. An approximate rule is that drug will accumulate in the
`body if it is given at intervals less than 1.4 times the half -life. Assuming
`complete absorption, this can be expressed as an accumulation ratio (RA)
`which is:
`
`RA_XH_1.4t12
`
`T
`
`D
`
`(Eq. 22)
`
`Accumulation ratios for drugs with various half -lives and given at regu-
`lar intervals are shown in Table 6. This type of information can be used
`as a basis for adjusting the timing or dose of a drug for which estimates
`of half -life are available in age- dependent or pathological situations. For
`example, if the half -life of a drug is increased two -fold, then either the
`dosing interval can be doubled, or, more appropriately, the maintenance
`dose can be halved in order to obtain the desired plateau drug level.
`
`KINETICS OF DRUG LOCALIZATION
`
`Drug concentrations in various tissues and pharmacologic sites can
`differ manyfold and are not always directly reflected by blood level data.
`This is due to the drug distribution factors presented previously. It is
`feasible to measure actual drug localization in animals,30 but human
`studies often require an indirect approach. One method involves mul-
`tiple- compartment drug distribution models which are employed to
`quantitate the kinetics of drug localization using plasma drug concen-
`tration data.
`The simplest and most utilized multiple compartment model is the
`two -compartment open modell" which can be represented as:
`
`(Scheme II)
`
`where X1 and X2 are amounts of drug in the central (plasma and viscera)
`and peripheral (tissue) compartments, respectively, KE is the first -order
`rate constant for drug elimination, and k12 and k21 are rate constants for
`drug distribution between compartments. This model is used to describe
`pharmacokinetic data where logarithmic plasma concentrations (C )
`decline in a curvilinear manner with time after intravenous dosage ac-
`cording to the equation:
`
`Page 13
`
`

`

`WILLIAM J. JUSKO
`
`Figure 5. Plasma concentrations (CPM/
`ml. x 10-2) of radioactive sodium iothalamate
`I -125 (0) as a function of time after intra-
`venous injection of about 0.1 mg. I (4.55 x
`10' CPM) to an 11 year old child with
`nephrotic syndrome. The data21 and the first -
`phase residuals (0) can be quantitated with
`Equation 23 and the two -compartment model
`where: A = 105, B = 51, a = 0.937 min. -',
`ß = 0.0103 min. -', and Cl= 90 ml. per min.
`
`94
`
`200
`
`50
`
`20
`
`zO r
`
`rZ
`
`wUzo
`
`U
`
`40
`TIME, MIN.
`
`80
`
`120
`
`H 24Ja
`
`C = A. e-at + B. e-at
`
`(Eq. 23)
`
`where A and B are zero -time plasma concentration intercepts, and a and
`ß are slopes of drug disappearance from the plasma. A graphical ex-
`ample of this pattern is shown in Figure 5. Such drug behavior is consis-
`tent with theory that the drug goes through an initial phase where it is
`eliminated and distributed simultaneously from the central compart-
`ment followed by the second phase where drug levels in all compart-
`ments decline in parallel due to drug elimination from the central corn -
`partment. This is the expression of differences in blood perfusion of
`organs and tissues located in each compartment as well as differences in
`the degree of drug accessibility to intravascular, extravascular, and in-
`tracellular body water.19
`Age- dependent changes in drug distribution are likely to be reflected
`by data fitted to multi- compartment models, but this application to pedi-
`atric pharmacology has been limited. The data in Figure 5 show plasma
`levels of radioactive sodium iothalamate after intravenous injection in an
`11 year old gir1.21 The data were fitted to Equation 23 and, in turn, the pa-
`rameters of the two -compartment model as listed in Table 7 were calcu-
`lated. Sakai and co- workers used this approach for determination of
`renal clearance from plasma data alone. This was possible since all of
`the compound is excreted in the urine and thus renal clearance equals
`body clearance. Excellent correlation was obtained with creatinine
`clearances measured by the conventional methods involving blood and
`urine collections.
`A variety of other parameters are listed in Table 7 for the two-corn-
`partment model. This model, and those of succeeding complexity, yield
`increasing information concerning drug distribution as it affects the cen-
`tral or plasma compartment.
`
`Page 14
`
`

`

`PHARMACOKINETIC PRINCIPLES IN PEDIATRIC PHARMACOLOGY
`Table 7. Summary of Principal Calculated Parameters* of the
`Two -Compartment Open Model: Pharmacokinetics of
`Sodium Iothalamate I -125
`
`95
`
`PARAMETER (UNITS)
`
`METHOD OF CALCULATION
`
`CALCULATED VALUE
`
`Central compartment volume (Vol.)
`
`Ve - A1?---
`
`Biological half -life (time)
`
`Distribution constant (time -')
`
`Elimination constant (time -')
`Distribution constant (time -')
`Total distribution volume (Vol.)
`
`Plasma level area (Conc. time)
`
`Renal clearance (volume /time)
`
`k21 = A. ßA
`
`t12 = 0.6931ß
`B.«
`+ B
`KE = a.ß /k2,
`k12 = a + ß - k2, - KE
`V0 = Ve.KE /ß
`
`AUC = a + ß = D /V,.. KE
`Cl = ke.Vc = d C° /dt
`
`P
`
`Body clearance (volume /time)
`
`Cl,, = KEN,
`
`L.
`
`67 min.
`
`0.313 min.
`
`0.0308 min.
`0.604 min.
`8.74 L.
`-
`
`90 ml. /min.
`
`90 ml. /min.
`
`"Hybrid constants A, a, B, and ß were obtained from graphical data in Figure 5 according
`to Equation 23.
`
`DOSE -DEPENDENT KINETICS
`
`The pharmacokinetics of a drug are dose -dependent when the rate of
`absorption, distribution, or elimination changes with the dose adminis-
`tered.12 These phenomena usually occur when a process involves Mi-
`chaelis -Menten kinetics and an increased drug level exceeds the first -
`order region of such a mechanism. A notable example of this concerns
`the biotransformation of salicylate. The capacity of the process of salicy-
`late conjugation with glycine begins to be exceeded at doses of approxi-
`mately 5 mg. per kg. in the adult. Because the overall rate of elimination
`is decreased with increasing dose, the half -life increases with the body
`level. This is demonstrated by the serum level data shown in Figure 6 for
`a 2 year 9 month old child who ingested about 100 mg. per kg. of as-
`pirin.13 As serum levels decline, the apparent half -life decreases from
`about 25 hours to about 3 hours. This situation creates a therapeutic
`problem when multiple doses are administered. Instead of a plateau
`body level of drug being reached after several doses of aspirin, body levels
`of salicylate continue to increase until toxic effects often occur. In gener-
`al, dose -dependent kinetics should be anticipated more often in infants
`than in adults since the immaturity of various drug transport and reac-
`tion mechanisms makes them prone to saturation.
`
`KINETI