Introduction

Growth and development are two major aspects of children not seen in adults. These aspects can be investigated by using size and age as covariates. One advantage of population modelling programs such as NONMEM [12, 13] is the ability to investigate covariates. Problems exist because covariates can exhibit co-linearity. Clearance, for example, may increase with weight, height, age, body surface area and creatinine clearance. All of these covariates may show a high degree of correlation and they are not mutually exclusive [15]. Any one covariate may or may not predict another. This current paper explores the use of covariates and priors in paediatric pharmacokinetic modelling and introduces some examples of the sigmoidal Emax model used to describe pharmacodynamic effects in children.

Covariate pharmacokinetic considerations in paediatrics

Size adjustment

The range of body weights in children is far greater than that seen in adults and can vary 200-fold (e.g. 0.5–100 kg). Size may be closely linked to age and both covariates may have an impact on the prediction of clearance (Fig. 1a,b). Size, however, is the primary covariate used in the investigation of co-linearity. Debate exists concerning the methodology used to adjust pharmacokinetic (PK) parameters to body size. Empirical approaches that focus on body weight (W) or body surface area (BSA) have traditionally been used. However, the linear per kilogram and surface area models are inappropriate for scaling small children to adults [32]. The linear per kilogram model under-predicts clearance, while the surface area model over-predicts clearance in children. These errors increase with decreasing weight [32, 44]. Linear size models for young children have led to the idea that there is an enhanced capacity of children to metabolise drugs due to proportionally larger livers and kidneys than their adult counterparts [58]. This idea arises because clearance, expressed per kg of body weight, is larger in children than in adults.

Fig. 1a, b
figure 1

Data from studies investigating paracetamol clearance. a Clearance increases with weight. Individual predicted paracetamol clearances (CL), from the NONMEM post hoc step, are plotted against weight. b Clearance also increases with age. Individual predicted paracetamol clearances (CL), from the NONMEM post hoc step, are plotted against postnatal age

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It now widely recognised that there is a nonlinear relationship between weight and drug elimination capacity [25]. Body surface area (BSA) was proposed in 1950 to be a more satisfactory index of drug requirements than body weight or age, particularly during infancy and childhood [24]. The surface area is calculated from height and weight [26], but it can also be estimated from weight using a power exponent of 0.67 [18, 32]. However, infants are not morphologically similar to adults—infants have short legs, relatively big heads and large body trunks. The surface area formula is inaccurate in children with a predicted surface area of less than 1.3 m2 (an average 12-year-old) by direct photometric measurement [44]. The mass of empirical evidence suggests that the appropriate scaling factor is significantly different from 0.67 and is compatible with the theoretically expected value of 0.75 [37, 52, 65, 66].

It is possible to show that, in almost all species, including humans, the log of the basal metabolic rate (BMR) plotted against the log of body weight produces a straight line with a slope of 0.75 (Fig. 2). West et al. [65, 66] have used fractal geometry to mathematically explain this phenomenon. The 3/4 power law for metabolic rates was derived from a general model that describes how essential materials are transported through space-filled fractal networks of branching tubes. A great many physiological-, structural- and time-related variables scale predictably within and between species with weight exponents (PWR) of 0.75, 1 and 0.25, respectively. These allometric “1/4 power” models can be applied to pharmacokinetic parameter estimates in children, e.g. clearance (0.75), half-life (0.25), volume of distribution (1):

$$Fsize = {\left( {\frac{W}{{W_{{{\text{std}}}} }}} \right)}^{{{\text{PWR}}}} $$

where Fsize is the factor for size, W is the weight of an individual and W std is the weight standard (e.g. a 70-kg person).

Fig. 2
figure 2

A comparison of the temperature-standardised relation for whole-organism metabolic rate as a function of body mass. The “allometric 3/4 power model” fits for unicells, poikilotherms and homeotherms, uncorrected for temperature, are also shown. From Gillooly et al. [28], with permission

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Allometric size modelling is used with increasing frequency in paediatric PK population analyses. The model building procedure can be accomplished by either fixing the allometric exponent (PWR) [9, 20, 50] or by estimating it as a parameter of a nested model [53, 67]. Estimation of an allometric exponent from typical human data is fraught with problems because of the relatively narrow weight ranges and the estimation error involved in such a nonlinear parameter. It is safer to rely on theory and observation from much broader data and use the “1/4 power” allometric models. Allometric scaling is a mechanistic approach that has a strong theoretical and empirical basis [5, 49, 50, 65, 66]. Fixing the power exponent allows investigators to delineate secondary covariate effects from the effect of size. Allometric scaling also allows the direct comparison of paediatric estimates with adults when a weight standard of 70 kg is used. For example, a clearance of 0.2 L/h/kg, when standardised to a 70-kg person (14 L/h/70 kg), is reduced in a 1.5-kg intrauterine growth retarded term neonate (5.4 L/h/70 kg) when compared to a 4-kg term neonate (6.9 L/h/70 kg). Clearance is not the same in the three individuals, although the linear per kilogram model would suggest that this is so. The emergence of a relationship between clearance and age can be seen in Fig. 3a, where clearance is scaled allometrically, compared to Fig. 3b, where the linear per kilogram model was used.

Fig. 3a, b
figure 3

Data from a study investigating paracetamol clearance. a Individual predicted paracetamol clearances (CL) from the NONMEM post hoc step, expressed using the allometric 3/4 power model, are plotted against postnatal age. The trend line shows an increase in clearance with age. b Individual predicted paracetamol clearances (CL) from the NONMEM post hoc step, expressed using the linear per kilogram model, are plotted against postnatal age. There is no obvious relationship between age and clearance

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Allometric scaling is commonly used to predict paediatric dosing based on adult dose (Table 1). Estimates are commonly taken from the surface area model but fail in infancy because age, which is commonly used to describe clearance maturation, is unaccounted for.

Table 1 Paediatric maintenance doses of drugs expressed as a percentage of adult dose using an allometric 34 power model. The neonatal estimate based on size has been reduced further by 50% to account for age-related maturational changes of clearance. (Modified from Holford and Anderson (1997) Paediatric dosages. In: New ethicals catalogue. Adis International, Auckland, New Zealand, C9)
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Effects of age

Clearance

Once size is standardised, the effects of other covariates, such as age [9], temperature [3] and renal function [17], can be investigated. Age is used to describe the maturation of clearance. The quantitative models used to describe this maturation process vary depending on the span of the ages under investigation. A linear model is commonly used for a population sample limited to a small defined age band. An exponential model may describe the gradual increase of clearance in premature neonates [1], while a first-order process has been used to describe clearance maturation from birth to adolescence [9].

The maturation of clearance begins before birth, suggesting that postmenstrual age (PMA) would be a better predictor of drug elimination than postnatal age (PNA). The foetus is capable of metabolising morphine (hepatic enzyme uridine 5′-diphosphate glucuronosyl transferase-2B7, UGT2B7) from 15 weeks gestation [46, 47]. The neonate can use sulphate conjugation as an alternative route for substrates, such as morphine or acetaminophen, before glucuronidation matures. There are distinct patterns associated with isoform-specific developmental expression of the cytochrome P450 (CYP) enzymes. CYP2D6 has been detected in premature neonates as young as 25 weeks PMA [2]. Although some CYPs appear to be switched on by birth, while in others birth is necessary but not sufficient for the onset of expression [31, 35, 39], there are no direct demonstrations that clearance changes as a consequence of being born. Maturation of clearance in neonates may be described by both PMA and PNA, but PMA is a more physiologically appropriate covariate to explain the time course of changes in clearance.

An example of a linear model investigating age-related changes for clearance is:

$$F_{{{\text{PMA}}}} = 1 + SLPCL \times {\left( {PMA - 40} \right)}\;{\text{L}} \mathord{\left/ {\vphantom {{\text{L}} {\text{h}}}} \right. \kern-\nulldelimiterspace} {\text{h}}$$

where F PMA is the factor for PMA and is centred on 40 weeks PMA (full term gestation), PMA is the postmenstrual age in weeks and SLPCL is a slope parameter describing changes of clearance with PMA.

Alternative models (exponential, first-order, variable slope sigmoidal) can approximate the linear model over a narrow age range (e.g. premature neonates) (Fig. 4a). The maturation of clearance must develop before birth and extrapolation of a linear function does not allow this. An exponential function allows for a gradual increase in clearance at earlier postmenstrual ages (Fig. 4b). This exponential function was used by Kimura et al. [36] in their study of vancomycin pharmacokinetics in neonates:

$$F_{{{\text{PMA}}}} = \exp {\left( {SLPCL \times {\left( {PMA - 40} \right)}} \right)}$$
Fig. 4
figure 4

a The four models used to investigate the relationship between age and clearance maturation are similar in the age range of the population studied. The boxed area is shown in more detail in Fig. 5b. b The first-order and linear models intercept the x axis at fixed times during the second trimester. This seems unlikely, as organogenesis is occurring during the second trimester of pregnancy. The exponential and variable slope sigmoidal model (Hill equation) allow for gradual clearance maturation that starts early in foetal life. c, d Individual predicted paracetamol clearances (CL) from the NONMEM post hoc step, expressed using the allometric 34 power model, are plotted against postmenstrual age: (c) a first-order model is used to describe CL maturation with age; (d) a variable slope sigmoid model is used to describe CL maturation with age

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An exponential model may be a more robust empirical model than the linear model because the prediction is always positive. This exponential function, in common with the linear model, predicts increasing clearance with age. This model is empirical and does not extrapolate to adult values, although, physiologically, clearance must rise and plateau at some age. A first-order model, common in biological systems, has been previously used to investigate the clearance maturation of other drugs during infancy [9, 17]. A “mature” clearance can be estimated (Fig. 4c) at a plateau by:

$$F_{{{\text{PMA}}}} = {\left( {1 - {\left( {1 - \beta cl} \right)}l \times \exp {\left( { - {\left( {PMA - 40} \right)} \times \frac{{\ln {\left( 2 \right)}}}{{Tcl}}} \right)}} \right)}$$

where βcl is a parameter estimating the fractional CLstd at 40 weeks PMA and Tcl describes the maturation half-life of the age-related changes of CL. This model has been used to investigate paracetamol age-related clearance and consequent dosing predictions [8]. Clearance was low (0.74 L/h/70 kg) at 28 weeks PMA, but reached 86% of adult values (i.e. plateau 12.5 L/h/70 kg) by 60 weeks. A mean steady state target concentration above 10 mg/L at trough can be achieved by a maintenance oral dose of 25 mg/kg/day in 30-week-gestation premature neonates, rising to 90 mg/kg/day at 6 months. The dose required in infancy to achieve a set target concentration is greater than in older children because clearance is nonlinearly related to weight [8].

The first-order model has been successfully used to compare clearance maturation rates for different clearance enzyme processes. The maturation half-time for morphine (uridine 5′-diphosphate UGT2B7) of 12 weeks [17] is quite different from the rapid clearance maturation of 5 days described for the opioid, alfentanil (CYP3A4) [5]. This model has also been used to investigate the pharmacogenetics of tramadol in premature neonates [2]. Maturational clearance of tramadol was almost completed by 44 weeks PMA. CYP2D6 activity, measured by metabolite formation, was observed as early as 25 weeks PMA, but the impact of CYP2D6 polymorphism on the variability in the pharmacokinetics, metabolism and pharmacodynamics of tramadol could not be established.

This first-order model suffers because it assumes zero clearance at the x intercept (Fig. 4b,c). A variable slope sigmoidal model (Hill equation) allows the gradual maturation of clearance in early life and a “mature” clearance to be achieved at a later age (Fig. 4b,d):

$$F_{{{\text{PMA}}}} = \frac{{PMA^{{{\text{Hill}}CL}} }}{{{\left( {PMA^{{{\text{Hill}}CL}} + EMATCL^{{{\text{Hill}}CL}}_{{50}} } \right)}}}$$

where EMATCL 50 is the PMA at which clearance is 50% that of the mature value and HillCL is the Hill coefficient for clearance. A variable slope model has been used to investigate vancomycin clearance and covariate effects in premature neonates [4]. Size explained 49%, postmenstrual age 18% and renal function 14% of the clearance variability. The use of a variable slope sigmoidal model to describe the relationship between clearance and postmenstrual age predicted a reasonable adult clearance of 3.79 (95% CI: 2.76-4.98) L/h/70 kg from premature neonatal data [4]. Scaling adult data to predict infant clearance is unreliable in infancy because maturation is unaccountable; scaling neonatal data may predict adult clearance when size and age are included as covariates.

Plasma esterases are responsible for the hydrolysis of propacetamol to paracetamol. This rate of hydrolysis has been investigated [6]. The rate was the same in all age groups when standardised for size (physiological time) [5]. The rate in real time (chronological time) is fastest in the youngest patients. Remifentanil is also cleared by plasma esterases and its clearance is similarly fastest in the newborn period when expressed as per kilogram [34, 51].

These models for the effect of age on clearance may prove useful in teasing out information concerning possible temporal switches that may “turn up” clearance at birth. It is postulated that birth itself may have an effect on the ontogeny of clearance. The disappearance of the placenta as “external” clearance may also play a role. The placenta has enzyme capacity to clear drugs (e.g. cytochrome P450 and conjugation) and the interpretation of hepatic maturation remains unravelled in the foetus.

Renal function

The clearance maturation of drugs almost exclusively cleared by renal elimination using PMA as a predictor predicts the time course of development of the glomerular filtration rate (GFR). Estimates for amikacin clearance [1] mirror GFR estimates in premature neonates [57, 59]. GFR matures during infancy and approaches an adult rate (6 L/h/70 kg) by 6 months postnatal age [11, 14]. Difficulties arise in determining the renal function in children, although a number of formulae have been published that allow the estimation of GFR from clinical characteristics [45] (e.g. Schwartz et al. [54, 55]).

Creatinine concentration decreases with age in the newborn. Consequently, vancomycin clearance estimates have been made using simply an inverse relationship to creatinine concentration in premature neonates [29]. Creatinine concentration in the first few days of life reflects maternal concentrations more than neonatal renal function and subsequent concentrations are influenced by tubular reabsorption [30]. Attempts to use the Cockcroft and Gault models [22] to predict the creatinine production rate (CPR) fail. Creatinine clearance (CLcr) can be predicted using postmenstrual age (PMA) as a covariate to predict the creatinine production rate (CPR) with a scaling constant (Kage) for age. This is based on assuming a CPR of 516 μmol/h in a 70-kg, 40-year-old male [22]:

$$CPR = 516 \times \exp {\left( {Kage \times {\left( {\frac{{{\left( {PMA - 40} \right)}}}{{52 - 40}}} \right)}} \right)}\;{{\text{ $ \mu $ mol}}} \mathord{\left/ {\vphantom {{{\text{ $ \mu $ mol}}} {\text{h}}}} \right. \kern-\nulldelimiterspace} {\text{h}}$$
$$CLcr = \frac{{CPR}}{{Serum\;Creatinine}}\;{\text{L}} \mathord{\left/ {\vphantom {{\text{L}} {\text{h}}}} \right. \kern-\nulldelimiterspace} {\text{h}}$$

The renal function (RF) can then be standardised to an adult creatinine clearance of 6 L/h/70 kg to enable the comparison of neonatal with adult data [17, 43]. The increase of CPR with age in neonates is assumed to be a consequence of increasing muscle bulk with age, as opposed to the decrease in muscle bulk that occurs with age in adults.

The population approach has been used to directly estimate GFR in patients with renal disease in children aged 0.8–18 years. The use of the squared length compensates for the nonlinear relationship between muscle mass and age-related changes of serum creatinine [40]:

$$CPR = 56.7 \times W\;{\left( {{\text{kg}}} \right)} + 0.142 \times Length\;{\left( {{\text{cm}}} \right)}^{2} \;{{\text{ $ \mu $ mol}}} \mathord{\left/ {\vphantom {{{\text{ $ \mu $ mol}}} {{\text{min}}}}} \right. \kern-\nulldelimiterspace} {{\text{min}}}$$

Cole et al. [23] used a population approach to investigate CLcr in 50 paediatric cancer patients of similar age range (0.8–19.8 years) and reported a linear model with good predictive properties:

$$CLcr = 36.76 + 1.91 \times W - 0.47 \times Serum\;Creatinine\;{{\text{ml}}} \mathord{\left/ {\vphantom {{{\text{ml}}} {{\text{min}}}}} \right. \kern-\nulldelimiterspace} {{\text{min}}}$$

Volume of distribution

The total body water content constitutes 85% of the body weight in the preterm neonate and 75% in term neonates. This decreases to approximately 60% at 5 months and remains relatively constant from this age onwards [27]. The major component contributing to this reduction in body water is the decrease in extracellular fluid (ECF). ECF constitutes 45% of the body weight at birth and 26% at 1 year. There is a further ECF reduction during childhood until adulthood, where it contributes 18%. The percentage of body weight contributed by fat is 3% in a 1.5-kg premature neonate and 12% in a term neonate; this proportion doubles by 4–5 months of age. “Baby fat” is lost when the infant starts walking and protein mass increases (20% term neonate, 50% adult). Albumin, globulin, lipoprotein and glycoprotein concentrations change over the first year, affecting drug binding [16]. Relative body proportions change dramatically over the first few years of life and may affect the volumes of the distribution of drugs.

Volume of distribution (V) changes and their relation to age that occur as a consequence of body composition changes can be described using quantitative functions similar to those used for clearance maturation. Clarification of these changes has debunked myths such as respiratory drive sensitivity in neonates given morphine that is attributable to the leaky blood–brain barrier [64]. Morphine volume of distribution is reduced in neonates and increases exponentially with a maturation half-life of 26 days from 83 l per 70 kg at birth [17]. Neonates given a similar dose as adults (mg/kg) achieve higher serum concentrations because of the reduced V. Respiratory depression is the same in children from 2 to 570 days of age at the same morphine concentration [41]. Pethidine causes less respiratory depression in neonates and it was thought that pethidine, unlike morphine, is lipid soluble and, therefore, crossed the immature or mature BBB equally. However the lesser respiratory depression in neonates given an equivalent mg/kg adult dose is probably consequent upon an increased pethidine V and reduced serum concentration.

Integrating priors

Estimates from prior analyses of data (e.g. adult, paediatric or animal work) can be incorporated into paediatric studies when these values cannot be estimated from the available current data. The use of 0.75 as the power parameter to scale clearance is based on prior knowledge from animal [38, 49] and fractal geometry [65, 66]. Paediatric studies investigating metabolite data can determine inter-compartment rate constants, but knowledge of inter-compartment CL estimates may be of greater practical use. The amount of drug that is metabolised down different clearance pathways is unknown. This may be calculated from a complete urine collection, but observed data such as parent and metabolite concentrations only do not allow the estimation of the metabolite volume of distribution. Adult priors for metabolite V have been used in paediatric studies [17, 60].

Pharmacodynamics

Pharmacokinetics (PK) is what the body does to the drug, while pharmacodynamics (PD) is what the drug does to the body. Variability in PD response may be even greater than that in PK. The precise boundary between PK and PD is ill defined and often requires a link describing the movement of drug from the plasma to the biophase and its target. It is postulated that PD factors such as the number, affinity and type of receptors or the availability of natural ligands alter the drug–receptor interaction and are altered in childhood, but there are few data concerning PD differences between adults and children. Bronchodilators are less effective in infants because of the immaturity of bronchial smooth muscle fibres. Cyclosporin [42], warfarin [35] and benzodiazepine [21] are all thought to have altered PD in children, but further examples are rare. Pharmacodynamics may be further complicated by the development of functional tolerance and the additive, synergistic or antagonistic effect of drug metabolites or isomers.

Observed effects may not be directly related to serum concentration. There may be a delay due to the transfer of the drug to the effect site (tubocurarine), a lag time (frusemide), physiological response (antipyresis), active metabolite (propacetamol) or elimination of physiological substances (warfarin). A plasma concentration effect plot can form a hysteresis loop because of this delay in effect.

Hull et al. [33] and Sheiner et al. [56] introduced the effect compartment concept for neuromuscular blocking drugs. The effect compartment concentration is not the same as the blood or serum concentration and is not a real measurable concentration. A single first-order parameter (Teq, T1/2ke0) describes the equilibration half-time. It is assumed that the concentration in the central compartment is the same as that in the effect compartment at equilibration, but that a time delay exists before the drug reaches the effect compartment. The concentration in the effect compartment is used to describe the concentration–effect relationship.

Measures of response are not always as straightforward as neuromuscular blockade monitoring in adults and can be more difficult to quantify in children. Analgesic effect measures, for example, vary between ages and even within one age [61–63]. Covariate effects may be subtle and unaccounted for in many studies. The placebo effect contributed 50% of analgesia attributed to paracetamol a few hours after tonsillectomy [10]. Similarly, Brown et al. [19], in a study of paracetamol antipyresis, noted the confounding effect of the initial temperature and normal temperature fluctuations on antipyretic efficacy assessment. A circadian night rhythm effect was noted in an investigation of propofol sedation after major craniofacial surgery [48].

The relation between drug concentration and the magnitude of effect may be described by a variable slope sigmoid curve (Fig. 5) according to the following equation:

$$Effect = \frac{{E\max \times Ce^{N} }}{{EC^{N}_{{50}} + Ce^{N} }}$$

where Emax is the maximum effect change, Ce is the concentration in the effect compartment, EC 50 is the concentration producing 50% effect and N is the Hill coefficient defining the steepness of the concentration–response curve. At low concentrations, the nonlinear relationship may approach linearity. A variable slope sigmoid model is commonly used to describe the concentration–response relationship. Ordinal data may be described using logistic regression [7].

Fig. 5
figure 5

The sigmoid Emax model for three drugs, each with a different Hill coefficient. The EC 50 and efficacy are the same, but the EC 80 is different for each drug. A higher concentration of the drug with the lower Hill coefficient will be required if an effect of 80% of the maximal effect is desired

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An understanding of the paracetamol analgesic effect has come from population modelling. PD estimates for an Emax model, in which the greatest possible pain relief (VAS 0–10) equates to an Emax of 10, were Emax 5.17 (64%) and EC 50 9.98 (107%) mg/l. The equilibration half-time (Teq) of the analgesic effect compartment was 53 (217%) min [10]. These estimates confirm paracetamol as a mild analgesic and that there is a delay after the peak serum concentration before maximal pain relief becomes apparent. An effect site concentration of 10 mg/L resulted in a pain reduction of 2.6 pain units. An increased effect site concentration of 40 mg/L reduced pain by 4.2 mg/L. Higher concentrations are associated with a decrease in the incremental pain relief and the maximum pain reduction can only be 5.17 pain units. Increasing the dose will not maximise benefit but will increase propensity to side effects such as hepatotoxicity.

Conclusions

Growth and development are two major aspects of children not seen in adults. Problems attributable to covariate co-linearity can be approached using size as the first covariate. Size standardisation is achieved using allometric scaling, a mechanistic approach that has a strong theoretical and empirical basis. Age is used to describe the maturation of clearance. The quantitative models (linear, exponential, first-order, variable slope sigmoidal) used to describe this maturation process vary depending on the span of the ages under investigation. Postmenstrual age (PMA) is a more physiologically appropriate covariate to explain the time course of changes in clearance and studies using PMA have improved the understanding of clearance maturation. It is anticipated that this exploration of covariates will assist in improving the understanding cellular clearance pathways, the role of the placenta and differences attributable to pharmacogenetic polymorphisms. Covariate investigation in children is improving the understanding of developmental aspects of drug disposition and effects in the paediatric population, ultimately leading to more effective use of medications.