### Abstract

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- References

**Aims ** [1] To quantify the random and predictable components of variability for aminoglycoside clearance and volume of distribution [2] To investigate models for predicting aminoglycoside clearance in patients with low serum creatinine concentrations [3] To evaluate the predictive performance of initial dosing strategies for achieving an aminoglycoside target concentration.

**Methods ** Aminoglycoside demographic, dosing and concentration data were collected from 697 adult patients (>=20 years old) as part of standard clinical care using a target concentration intervention approach for dose individualization. It was assumed that aminoglycoside clearance had a renal and a nonrenal component, with the renal component being linearly related to predicted creatinine clearance.

**Results ** A two compartment pharmacokinetic model best described the aminoglycoside data. The addition of weight, age, sex and serum creatinine as covariates reduced the random component of between subject variability (BSVR) in clearance (CL) from 94% to 36% of population parameter variability (PPV). The final pharmacokinetic parameter estimates for the model with the best predictive performance were: CL, 4.7 l h^{–1} 70 kg^{–1}; intercompartmental clearance (CLic), 1 l h^{–1} 70 kg^{–1}; volume of central compartment (V_{1}), 19.5 l 70 kg^{–1}; volume of peripheral compartment (V_{2}) 11.2 l 70 kg^{–1}.

**Conclusions ** Using a fixed dose of aminoglycoside will achieve 35% of typical patients within 80–125% of a required dose. Covariate guided predictions increase this up to 61%. However, because we have shown that random within subject variability (WSVR) in clearance is less than safe and effective variability (SEV), target concentration intervention can potentially achieve safe and effective doses in 90% of patients.

### Introduction

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- References

The aminoglycosides have been in use for around 50 years and are still widely used in the hospital setting for serious systemic gram-negative bacterial infections [1, 2]. The Target Concentration Intervention (TCI) strategy [3] is commonly employed with the aminoglycosides. The TCI strategy involves firstly selecting the desired target concentration and secondly predicting volume (V) and clearance (CL) for the patient, which can then be used to predict initial doses, required to achieve and maintain the target [4]. TCI then uses information obtained during treatment about doses, concentrations, and clinical effect in an individual and integrates them to estimate pharmacokinetic and pharmacodynamic parameters in that individual [4]. These individually estimated parameters are then used to predict subsequent doses required to achieve the target.

Aminoglycosides are unusual in having at least two target concentrations. For example, using once daily dosing the initial concentration is recommended to be at least 10 mg l^{–1} to achieve infection cure [1, 5, 6] and subsequent concentrations are targeted to achieve an average steady state concentration of 3 mg l^{–1} (equivalent to an area under the concentration time curve (AUC) of 72 mg l^{–1}·h over a 24 h dosing interval) to reduce the risk of toxicity [2]. Trough concentrations greater than 0.5 mg l^{–1} are associated with an increased incidence of nephrotoxicity [7, 8]. The initial concentration is primarily determined by volume of distribution which must rely on patient covariates (typically weight). Subsequent concentration measurements are of no assistance in predicting the dose needed to achieve the initial target concentration but are critical for effective TCI and subsequent dose adjustment. Targets such as average steady state concentration or AUC over a dosing interval depend mainly on clearance, which can be estimated by measuring aminoglycoside concentrations. We examine the quantitative implications of using aminoglycoside pharmacokinetics for initial dose predictions and subsequent dose adjustment aiming for an average steady concentration as an example of a target for TCI.

It is expected that pharmacokinetic parameters can vary widely between individuals, however, they may even differ within an individual during treatment or progression of a disease state. We call the overall parameter variability across all individuals in a defined population the population parameter variability (PPV), which is comprised of between subject variability (BSV) and within subject variability (WSV) [4]. BSV and WSV can be further divided into predictable (BSVP and WSVP) and random (BSVR and WSVR) components. BSVP and WSVP are the variability that can be explained through the use of covariates (e.g. weight, CL_{Cr}), while BSVR and WSVR are unpredictable, given the current state of knowledge, and are modelled as if they arise from a random process. WSVR can be approximated by between occasion variability (BOV) [4]. WSVR includes both BOV and stochastic parameter variation within an occasion. For all practical purposes WSVR and BOV can be considered equivalent. We define all variability values (PPV, BSV, WSV, BSVR, WSVR, SEV) as the standard deviation of a log normally distributed random variable with mean zero (e.g. see Temp. Equation 14). They can be considered approximate coefficients of variation.

Importantly, WSVR must be relatively small in order for the TCI strategy to be of practical relevance [9]. TCI becomes a practical method of forecasting individual doses when WSVR is smaller than Safe and Effective Variability (SEV). SEV reflects a judgement about the acceptable degree of variability of concentration in the target population, e.g. 90% of concentrations should be within an acceptable percentage of the target concentration [4].

The pharmacokinetics of aminoglycoside antibiotics are well known and they are predominantly excreted unchanged via the renal route [10, 11]. Because of the proposed relationship between renal function and aminoglycoside clearance, many methods have been proposed for predicting aminoglycoside clearance using serum creatinine concentration (SCr) or creatinine clearance (CL_{Cr}) [11–13]. These methods rely on empirical models for creatinine production, which are used to predict CL_{Cr} and renal function (RF).

Renal function (RF) is a measure of the ability of the kidneys to eliminate a drug or metabolite. It involves several processes including glomerular filtration, tubular secretion, and active and passive reabsorption. An accurate assessment of RF is important in the dose-individualization of drugs with extensive renal elimination. RF is typically predicted from measures of glomerular filtration, for example using creatinine clearance (CL_{Cr}). CL_{Cr} can be easily predicted from a serum creatinine concentration (SCr) using formulae or nomograms [14]. Measuring CL_{Cr} with urine and plasma samples has logistical and patient compliance factors, which can make accurate assessment problematic [15].

Cockcroft and Gault (C & G) developed a formula to predict CL_{Cr} from SCr using age, weight and sex. They reviewed the records of 534 patients who had CL_{Cr} estimated from two or more 24 h collections of urinary creatinine [14]. Using three previously published formulas [16–18] and a new model for CL_{Cr} (Temp. Equation 1 and Equation 2), they predicted the CL_{Cr} in ml min^{–1} for each patient from their SCr. A 15% reduction in Creatinine Production Rate (CPR) for females was based on previous work from other authors [14].

- (1)

CPR is predicted by Temp. Equation 2, SCr is the measured serum creatinine concentration in mg/dL, WT is the weight in kg of the patient and age is the age in years.

- (2)

Despite being widely studied, there is still no clear consensus as to the best model to predict RF [19, 20]. Most authors use SCr in association with some form of the Cockcroft and Gault (C & G) equation to predict renal clearance [2, 11, 12]. It has been suggested that patients with low SCr concentrations should be dealt with in a different manner. Duffull and colleagues noted that when SCr was low, aminoglycoside clearance was not linearly related to C & G predicted CL_{Cr}[12]. It was suggested that low SCr concentrations are due to a low creatinine production rate (CPR) rather than high renal function [12]. An empirical ‘round up’ method was proposed which rounded up SCr concentrations less than 0.06 mmol l^{–1} to 0.06 mmol l^{–1}. Rosario *et al.* (1998) [11] and Kirkpatrick *et al.* (1999) [2] subsequently used this method. Kirkpatrick *et al.* confirmed that a round up value of 0.06 mmol l^{–1} provided the best correlation using a linear relationship between SCr and predicted aminoglycoside clearance. Both papers confirmed the lack of linearity with CL_{Cr} but did not test the predictive performance, i.e. how well did these models predict individual aminoglycoside clearance based on covariate information alone.

We aim to [1] Quantify the predictable and random components of variability for clearance and volume for the aminoglycosides gentamicin and tobramycin in order to establish quantitative criteria for using TCI with aminoglycosides [2], Examine various methods for dealing with low concentrations because standard approaches appear to overestimate renal function and would be expected to be associated with over-dosing, and [3] Test the performance of models using covariate information alone to predict clearance.

### Discussion

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- References

Several studies on the population pharmacokinetics of aminoglycosides have been undertaken over the last 30 years, however, there is still no consensus as to the best pharmacokinetic model, with many authors using one compartment models [2, 12] while others use two [11, 27]. The use of a two compartment pharmacokinetic model seems reasonable as it provided a better fit to our concentration data as determined by improvement in the Obj. However, sampling was sparse and it is difficult to conclude that a two-compartment model is a better description on the basis of individual profiles.

Rosario *et al.*[11] collected data from patients with cancer who were given gentamicin and also found a two compartment pharmacokinetic model significantly improved the objective function value. Table 6 summarizes the results of the base model without covariates applied to the Christchurch data and the base model applied to the Glasgow data. These data show that the Christchurch data is more variable than that from Glasgow, with the PPV being considerable larger especially for V_{2}. The reason for the greater variability could be due to the wider range of conditions for which patients included in the current analysis were being treated. However, the base model CL and V_{SS} estimates do compare reasonably well.

Table 6. Comparison of population pharmacokinetic models without use of covariates to explain PPV | **Christchurch** | **Glasgow (11)** |
---|

Patients | 697 | 210 |

Concentrations | 2567 | 378 |

CL l h^{−1} 70 kg^{−1} | 4.0 | 4.4 |

PPV CL % | 49 | 34 |

VSS L/70 kg^{−1} | 30.6 | 27.3 |

PPV V % | V1 = 26 V2 = 170 | V1 = 16 V2 = ‘low’ |

Aminoglycosides have a molecular weight of 477.6, are highly polar molecules with negligible plasma protein binding and are expected to cross cell membranes slowly. Renal clearance is therefore expected to be similar to the glomerular filtration rate. Our estimate of renal clearance (CL_{R}, Table 5) is 40% lower than the predicted CL_{Cr} which implies either that aminoglycosides are re-absorbed after filtration or that CL_{Cr} is an over-prediction of glomerular filtration. Wright *et al.*[28] have estimated that the C & G method underestimates filtration estimated by ^{31}Cr-EDTA by 12%, which would make the latter explanation unlikely. Even if the estimate of nonrenal clearance was assumed to be zero the clearance of aminoglycosides is still 22% lower than CL_{Cr}. This provides indirect evidence for renal re-absorption of aminoglycosides and confirms directly measured values. Contrepois *et al.*[29] observed a mean aminoglycoside renal clearance of 5.25 l h^{–1} 70 kg^{–1} in 12 males aged 21–28 y. The CL_{R} was 82% of simultaneous inulin clearance measurement. Regamey *et al.*[30] measured CL_{R} of gentamicin and tobramycin of 6.7 l h^{–1} per 1.73 m^{2} which was 94% of the estimated total clearance (2 healthy subjects). Measured CL_{Cr} and gentamicin CL_{R} in critically ill patients are poorly correlated with CL_{NR} estimates of 1.8 l h^{–1}[13].

Based on the expected poor ability to cross cell membranes the apparent volume of distribution of aminoglycosides is expected to be close to extracellular fluid volume. The estimate of V1 of 19.5 l/70 kg^{–1} is compatible with this assumption. However Vss of 30.8 l/70 kg^{–1} implies tissue uptake or binding. Aminoglycosides are known to be taken-up into certain tissues by active transport mechanisms [31–34], which may account for some of the additional apparent volume beyond extracellular fluid. However, it would seem unlikely that accumulation of aminoglycosides in the ear and kidneys would account totally for this observed difference. This might indicate the uptake into other tissue sites.

#### Quantification of pharmacokinetic variability

To the best of our knowledge this is the first study that has tried to quantify the WSVR in aminoglycosides. Table 3 shows that the WSVR is relatively small for CL, approximately 8% of PPV, and around 19% of PPV for V_{2}. Attempts to quantify the size of the WSVR for CLic and V_{1} were unsuccessful for numerical reasons and probably reflect the sparse sampling design and limitations of the software.

#### Prediction of renal function with low serum creatinine

Our results confirm [2, 11, 12] that having a cut-off value below which all concentrations should be replaced with some other higher value improves the overall fit. None of the low serum creatinine models markedly changed the predictable component of between subject variability for either CL or V. The *F*_{CPR} method incorporates the speculation [12] that the reason for over prediction of CL_{Cr} was due to low CPR rather than high renal function. In addition, unlike the simple round up method [12], it does not discard information about renal function that might be reflected by changes in SCr less than 0.06 mmol l^{–1}. Despite these potential advantages it does not seem to offer any additional benefit to the predictive performance of the model. Nevertheless, we suggest the *F*_{CPR} method [Figure 4] be considered as an alternative to the method [Figure 3] proposed by Duffull *et al.*[12]. We found no evidence that sex is an independent predictor of renal function (Model 1). This observation is quite different from the well-understood association of a lower CPR in females presumably reflecting different body composition compared to males.

#### The importance of within subject variability

The target concentration strategy used by Christchurch hospital [5], assumes that the information in measured concentrations can be used to predict future doses to achieve a target concentration. Implicit in this assumption is that pharmacokinetic parameters estimated from measured concentrations will not vary importantly when the subsequent predicted dose is given. A quantitative description of the extent of this variability is provided by WSVR. We have been able to estimate the WSVR in CL and V_{2}. The WSVR in CL is more important because it will determine the dose-to-dose variability in AUC especially with once a day dosing when the half-life is short relative to the dosing interval.

WSVR alone is not sufficient to decide if TCI is necessary or will succeed. The extent of variability that is safe and effective (Safe and Effective Variability; SEV [4]) around the target concentration defines the need and potential for success. If SEV is large then there is a wide margin for acceptable dosing but if it is small then dose adjustments based only on covariate model predictions (fixed effects) may be inadequate to individualize dose because covariates can only reduce the predictable component of BSV. The random component of BSV can be decreased using TCI to estimate the average individual pharmacokinetic parameter of interest, e.g. clearance. The limiting imprecision of dosing is then determined by random WSV. If we propose that variation of 80–125% around the target concentration, e.g. average steady state concentration, is likely to produce safe and effective aminoglycoside therapy if achieved in 90% of patients then, assuming a lognormal distribution, 90% of concentrations will be > = 80 and < = 125% of the target with SEV = 0.135. This can be compared to the slightly lower bootstrap estimate of WSVR for clearance of 0.13. Clearance is the key parameter determining average steady state concentration. Using covariate based predictions we estimated that 65% of BSV in clearance is predictable. However, because BSVR of 0.29 is substantially more than the proposed SEV of 0.135 we can predict that covariate guided predictions based on weight and renal function will be inadequate for safe and effective aminoglycoside therapy. Because WSVR is smaller than SEV this means that target concentration intervention should be capable of helping to achieve safe and effective therapy by reducing uncertainty in individual CL to a limiting value approaching WSVR. It should be noted that the efficiency of using measured concentrations to reduce BSV will depend on the residual error in the measurement but given enough measurements this source of random noise will average out to zero. Methods that use weight, SCr, and age for initial dosing and nomograms for TCI, e.g [35], have been shown to be better than nothing [36] but are still unable to reliably achieve target concentrations in clinical use [37]. Bayesian TCI approaches are more effective for TCI [12].

#### Predictive performance of covariate models

An improved correlation of predicted CL_{Cr} (using C & G) with a Bayesian estimate of gentamicin clearance along with a closer fit to a linear relationship between these variables [2, 11, 12], a lower bias in the predicted mid-dose concentration [12] and a lower objective function [11] were the basis for the use of the round up method advocated by others. However, none of these studies evaluated how well the round up method helped to predict the dose needed to reach the target concentration.

The performance of group models for predicting clearance from covariates is shown in Table 4. We have evaluated performance in terms of the ability to achieve initially predicted doses within 80%–125% of target. If everyone got the same dose we would expect 35% of patients to receive safe and effective therapy (Table 4; Model 14). Covariate guided dose predictions at best can increase this to only 61% of patients. Weight, SCr and age individually contribute less than half of the benefit (Models 11–13; Table 3). However, with weight, SCr and age in combination, at least half of a population similar to the one we studied would get adequate treatment. There are relatively minor differences between the models, which use weight, age, sex and serum creatinine to predict clearance.

#### Safe and effective variability and rational dose individualization

We are unable to provide anything other than a plausible suggestion for SEV based on a goal of achieving 90% of concentrations within 80–125% of the target. Aminoglycoside target concentrations are only murkily defined and it is even harder to know what degree of imprecision (SEV) in hitting these targets is clinically acceptable. One approach we have tried is to ask experienced clinicians what range of concentrations they think are safe and effective. We have illustrated how SEV might be derived from such a concentration range by assuming the range is a 90% prediction interval and the variability in average steady state concentration is log normally distributed.

Our results support a paradigm for rational choices for dose selection [4]. If SEV is larger than overall variability (PPV) then a standard dose based on an average patient will be adequate. If SEV is less than random BSV, but larger than WSVR, then covariate-guided dose prediction can achieve adequate therapy. Finally, if SEV is less than random BSV but larger than WSVR, then target-concentration intervention may be able to help ensure safe and effective treatment.

Based on these criteria we conclude that covariates alone cannot be used to guide aminoglycoside dosing because, at best, only 61% of patients will get initial doses within 80–125% of the required dose. We predict that individual concentration feedback using TCI can achieve upto 90% of concentrations within 80%–125% of an average steady state target concentration. Our analysis gives quantitative support to using TCI in order to achieve the target concentrations required to maximize the clinical benefit of aminoglycosides.