Aminoglycoside Nephrotoxicity: Modeling, Simulation, and Control

The main constraints on the administration of aminoglycosidesare the risks of nephrotoxicity and ototoxicity, which can leadto acute, renal, vestibular, and auditory toxicities. In thepresent study we focused on nephrotoxicity. No reliable predictorof nephrotoxicity has been found to date. We have developeda deterministic model which describes the pharmacokinetic behaviorof aminoglycosides (with a two-compartment model), the kineticsof aminoglycoside accumulation in the renal cortex, the effectsof aminoglycosides on renal cells, the resulting effects onrenal function by tubuloglomerular feedback, and the resultingeffects on serum creatinine concentrations. The pharmacokineticparameter values were estimated by use of the NPEM program.The estimated pharmacodynamic parameter values were obtainedafter minimization of the least-squares objective function betweenthe measured and the calculated serum creatinine concentrations.A simulation program assessed the influences of the dosage regimenson the occurrence of nephrotoxicity. We have also demonstratedthe relevancy of modeling of the circadian rhythm of the renalfunction. We have shown the ability of the model to fit with49 observed serum creatinine concentrations for a group of eightpatients treated for endocarditis by comparison with 49 calculatedserum creatinine concentrations (r² = 0.988; P < 0.001).We have found that for the same daily dose, the nephrotoxicityobserved with a thrice-daily administration schedule appearsmore rapidly, induces a greater decrease in renal function,and is more prolonged than those that occur with less frequentadministration schedules (for example, once-daily administration).Moreover, for once-daily administration, we have demonstratedthat the time of day of administration can influence the incidenceof aminoglycoside nephrotoxicity. The lowest level of nephrotoxicitywas observed when aminoglycosides were administered at 1:30p.m. Clinical application of this model might make it possibleto adjust aminoglycoside dosage regimens by taking into accountboth the efficacies and toxicities of the drugs.

INTRODUCTION

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Introduction

Aminoglycoside (AG) nephrotoxicity is a well-known occurrence.However, 50 years after the discovery of the first AG (streptomycin),nephrotoxicity is still very difficult to predict and avoid(16).

To date, no model has completely described the pharmacodynamicbehavior of AG nephrotoxicity, even though the mechanism ofthis toxicity has been widely studied (4, 7, 15, 27). AGs areretained in the epithelial cells lining the proximal tubuleafter glomerular filtration. AGs become attached to the brush-bordermembrane in their cationic form. The initial points of attachmentare probably the acidic phospholipids, especially phosphatidylserine.In this way, aminoglycosides accumulate and cause leakage ofintracellular ions (K⁺, Mg²⁺, Ca²⁺), proteins (beta-2-microglobulin,alpha-2-macroglobulin, lysozyme), and enzymes (alanylaminopeptidase,N-acetylglucosaminidase). Thus, the resulting decline in glomerularfiltration has a multifactorial origin and involves a combinationof tubular and nontubular mechanisms. The most important factorseems to be a tubuloglomerular feedback (19, 24). The kidneyshave a strong capacity to compensate for tubular injuries. Theimportance of regeneration for protection against renal injuryis clearly demonstrated by the survival of laboratory rats exposedto repeated administrations of high daily doses of AG (40 mgof gentamicin per kg of body weight per day for at least 42days). After an initial episode of acute tubular necrosis thatoccurs within 8 to 10 days and that is associated with markedazotemia, the renal function returns almost to normal, as ifthe kidney had become refractive (6). Regenerating cells areless differentiated and apparently less susceptible to AG (thelevel of accumulation of AG is actually reduced in the corticesof animals treated for long periods).

Even though AG pharmacokinetic (PK) behavior in humans has beenaccurately described (22), the relation between PKs and theconditions of occurrence of AG nephrotoxicity is not clearlyestablished. Many PK studies show that at any given cumulativearea under the curve (AUC) for the serum AG concentrations,the risk of nephrotoxicity is lower for once-daily dosing ofan AG than for traditional AG dosing (AG administration every8 or 12 h) (1, 20, 21). This phenomenon is due to the renaluptake of AGs, which is nonlinear and saturable, as shown instudies with animals (10) and humans (1, 19). Studies with megalinindicate that uptake of AGs by tubule cells is saturable (23).

Unlike investigators who have recently reported their findings(20, 21), we believe that AG nephrotoxicity can be representedby a single toxicity effect model that does not depend eitheron the AUC for the serum AG concentrations or on the administrationschedule. We believe that a PK model alone probably cannot explainthe clinical or toxicity outcome. We believe that the PK modelmust include the drug distribution in a potential second compartmentand/or a pharmacodynamic (PD) (effect and/or toxicity) model.Also, modeling of AG nephrotoxicity must take into account boththe amounts of an AG in the renal cortex, the resulting PD effectrepresented by leakage from cells into the renal tubule, andthe stimulation of the tubuloglomerular feedback that causesthe decrease in glomerular filtration.

The objectives of the present study were (i) to develop a deterministicmodel of AG nephrotoxicity which takes into account both PKand PD variabilities, (ii) to estimate the PK parameter valuesand, for the first time, the PD parameter values from the deterministicmodel for a group of patients treated with amikacin over a longperiod, and (iii) to simulate AG nephrotoxicity and observethe influence of the dosage regimen and the time of administration.

MATERIALS AND METHODS

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Materials and Methods

PK model. We chose to use a bicompartmental PK model (8) with intravenousadministration. The amount of AG in the serum or central compartment[Q_S_(t); in milligrams] depends on the amount of AG from intravenousadministration, the peripheral compartments, and renal tubularreabsorption. Q_S₍_t₎ also depends on the amount of AG in serumwhich is transferred out of the central compartment to the peripheralcompartment or which is eliminated. The AG elimination rateconstant can be divided into a principal renal elimination route,which depends linearly on creatinine clearance, and a nonrenalelimination constant (22). Q_S₍_t₎ varies according to the following:

(1)
where C(t) is equal to Q_S₍_t₎/(V · B_W)

(2)
where t is time (in hours), i_(t) (in milligramsper hour) is the infusion rate, K_i (per hour) is the AG nonrenalelimination rate constant, K_s (in milliliters per minute perhour) is the renal elimination constant, k_reabs is the tubularreabsorption constant, CL_CR(t) (milliliters per minute) is thecreatinine clearance, C_(t) (in milligrams per liter) is theserum AG concentration, V (in liters per kilogram) is the volumeof distribution of AG in the central compartment, B_W (in kilograms)is the body weight, K_cp (per hour) is the transfer constantfrom the central compartment to the peripheral compartment,K_pc (per hour) is the transfer constant from the peripheralcompartment to the central compartment, and Q_P₍_t₎ (in milligrams)is the amount in the peripheral compartment.

PD model. The amount of AG left in the renal cortex [Q_C₍_t₎] is the differencebetween the amount of AG that accumulates according to a nonlinearand saturable mechanism (1, 10, 19) such as Michaelis-Mentenkinetics (18), as demonstrated in rats (10), and the amountof AG which is eliminated from the renal cortex (10). Q_C₍_t₎varies as shown below:

(3)
where V_(t)(in milligrams per hour) is the maximum accumulation rate inthe renal cortex, Q_S₍_t₎ (in milligrams) is the amount of AGin serum, k_M (in milligrams) is the amount of AG in serum forwhich V_(t)/2 is obtained, and k₁ (per hour) is the eliminationrate constant for AG from the renal cortex.

The AG renal accumulation effect [E_(t); in millimolar] is representedby a classic Hill equation (11, 17), which links the amountof AG in the renal cortex to the intracellular components leakinginto the renal tubule. In addition, we added a threshold ofthe amount of AG in the renal cortex under which no effect isobserved. E_(t) varies as shown below:

(4)

(5)
where Q_min (in milligrams) is the amount of AGin the renal cortex under which no effect is observed, E_max(in millimolar) is the maximum accumulation effect observed,Q₅₀ (in milligrams) is the amount of AG in the renal cortexfor which E_(t) is equal to E_max/2, and ${gamma}$ (which is dimensionless)is the Hill sigmoidicity parameter.

When the AG treatment lasts more than 1 month, it has been shownin the rat that, despite the occurrence of nephrotoxicity duringthe first few weeks, renal function returns almost to normalfrom 6 or 7 weeks after the beginning of treatment (6). Theregeneration of tubular cells during AG nephrotoxicity inducesa decrease in the level of AG accumulation in the renal cortex(26), i.e., a decrease in V_(t), as shown below:

(6)
where V₀ (in milligrams per hour) is the maximumaccumulation rate at the beginning of the treatment and ${alpha}$ (inmillimolar^-1) is the maximum accumulation rate decrease constant.

The glomerulotubular feedback action can also be representedby a Hill equation (24). This effect is observed on creatinineclearance [CL_CR(t); in milliliters per minute] and is associatedwith a circadian variation, represented here in a sinusoidalform (28), which is an independent factor. CL_CR(t) varies duringa typical 24-h day as shown below:

(7)
whereCL_CR0 (in milliliters per minute) is the creatinine clearancevalue at the beginning of the treatment, CL_CRMAX (in millilitersper minute) is the maximum decrease in creatinine clearance,E₅₀ (in millimolar) is the accumulation effect value for whichCL_CR(t) is equal to CL_CRMAX/2, ${delta}$ (which is dimensionless) isthe Hill sigmoidicity parameter, ${eta}$ is the amplitude of the circadianvariation in renal function, ${omega}$ (which is dimensionless) is theperiod of the circadian rhythm of the renal function, and ${varphi}$ (whichis dimensionless) is the value describing the phase relationshipof the circadian rhythm.

At this stage, we have introduced a second model without thecircadian variation in renal function (model 2), i.e., withoutthe sinusoidal function in equation 7, in order to compare itwith model 1, which includes the circadian variation in renalfunction (equation 7), and to investigate the relevancy of thisperiodic variation.

The decrease in renal function involves a rise in the serumcreatinine level, which depends on the amount produced by muscledaily and the daily renal elimination (13). The amount of creatininein serum [SCr_(t); in milligrams] varies as shown below:

(8)
where CL_CR(t) (in milliliters per minute) isthe creatinine clearance, Vol (in liters) is the volume of distributionof creatinine, and k₂ (in milligrams per hour) is the mean amountof creatinine produced by muscle daily.

Model development. We used the Matlab-Simulink program (Math Works Inc.) for allsimulations and estimations. The program makes it possible toselect each PK and PD parameter value. The system allows theuser to choose as inputs the administration schedule, includingthe first loading dose, the number of doses, the infusion time,the time of day of the infusion, and the time delay betweentwo infusions. Every simulation began at 12:00 a.m. The glomerularfiltration rate shows a repeating pattern of variation over24 h, i.e., a circadian rhythm (14). This corresponds to an ${omega}$ value of 2 ${pi}$ /24 (equation 7). We chose a sinusoidal adjustmentso that the creatinine clearance value was highest at 2:00 p.m.and lowest at 2:00 a.m. (14). This corresponds to a ${varphi}$ value of-2 ${pi}$ /3 (equation 7). The amplitude of the circadian rhythm wasfixed at 10% ( ${eta}$ = 0.1) (equation 7) of the creatinine clearancevalue, which is the most widely used value (14).

Experimental framework. For the PK and PD parameter estimations, we used data for patientsfor whom renal toxicity had been observed, as shown by increasesin the serum creatinine concentrations (i.e., a decrease increatinine clearance) with no clinical explanation other thanAG administration. The large number of parameters to be estimatedled us to select patients for whom a large number of measurementsof serum creatinine concentrations were available, i.e., patientstreated over a long period. Eight patients who had receivedamikacin and vancomycin for over 1 month for the treatment ofendocarditis were selected. The patients' general characteristicsand the treatment conditions are described in Table 1. Amikacinwas administered once daily via 30-min infusions. Vancomycinwas administered twice daily during the first 2 to 3 days oftreatment and then once-daily afterwards via 2-h infusions.Amikacin doses were calculated as a function of the CL_CR0, i.e.,15 mg/kg/day when CL_CR0 was above 80 ml/min and 7.5 mg/kg/daywhen CL_CR0 was between 30 and 80 ml/min. The doses of amikacinand vancomycin were constant and not adapted during the treatment.Except for vancomycin, no other nephrotoxic drug was administeredduring the amikacin treatments. The serum creatinine concentrationswere measured at different times during treatment and also aftertreatment had ended in order to observe the overall evolutionof the kinetics of the serum creatinine concentrations.

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TABLE 1. General characteristics of eight patients treated for endocarditis with amikacin and vancomycin over a long period and used for estimations

PK and PD parameter values for estimations and simulations. For the estimations, the K_i parameter value (equation 1) wasfixed at 0.006932/h, which corresponds to the K_i value observedfor anuric patients (half-life, 100 h) (12). We chose to fixthe K_i parameter value in order to put the whole variabilityof elimination on K_s because nonrenal elimination is very smallin comparison with K_s. The k_reabs parameter value was fixedat 0.05 (dimensionless) (12); otherwise, the system is not identifiable.Other individual PK parameter values (K_s, V, K_cp, and K_pc) wereestimated by use of the NPEM program (25). For the PD parametervalues, E₅₀ was fixed at 33.5 mM (24); otherwise, ${alpha}$ and E_maxare not identifiable. Moreover, V₀ was fixed at 1 mg/h in orderto obtain Q_c_(t) values in the range of the data in the literature(10); otherwise, Q₅₀ is not identifiable. Other PD parameters(k_M, k₁, Q_min, E_max, Q₅₀, ${gamma}$ , CL_CRMAX, ${delta}$ , ${alpha}$ , k₂, and Vol) were estimatedfor the eight-patient group by use of the Matlab program withthe estimated PK parameter values and the fixed PD parametervalues described above (E₅₀ and V₀) and by minimization of theleast-squares objective function (OF), as shown below:

(9)
where OF is the objective function to be minimized,SCr_obs (in milligrams per liter) is the measured serum creatinineconcentration, and SCr_calc (in milligrams per liter) is thecalculated serum creatinine concentration.

To investigate the relevancy of the circadian rhythm of therenal function of model 1 in comparison with model 2 (withouta circadian rhythm), we calculated the following errors: err₁= SCr_obs - SCr_calc1 for model 1 (where SCr_obs is the serum creatinineconcentration measured for the eight-patient group treated withamikacin for endocarditis and SCr_calc1 [in milligrams per liter]is the corresponding serum creatinine concentration calculatedby use of model 1) and err₂ = SCr_obs - SCr_calc2 for model 2(where SCr_calc2 [in milligrams per liter] is the correspondingserum creatinine concentration calculated by use of model 2).Their absolute values are designated |err₁| and |err₂|, respectively.In order to test the hypothesis that |err₂| is greater than|err₁|, we applied the Wilcoxon signed-rank test.

The parameter values used for the simulations are presentedin Table 2. PK parameter values (K_i, K_s, V, K_cp, K_pc, and k_reabs)correspond to data from the literature (12). ${alpha}$ was defined sothat renal function returns almost to normal after 42 days oftreatment with amikacin at 1,000 mg per day, which is the delayobserved in the rat (6) and which corresponds to an ${alpha}$ value of0.02 mM^-1. Other PD parameter values were mainly drawn frompast studies: V₀ and k_M from reference 10; CL_CRMAX, E₅₀, and ${delta}$ from reference 24; and k₂ and Vol from reference 13. In theabsence of data from the literature, other PD parameter values(k₁, Q_min, E_max, Q₅₀, and ${gamma}$ ) were those estimated with the Matlabprogram for the eight-patient group by use of model 1. The bodyweight was fixed at 70 kg. CL_CR0 and the initial serum creatinineconcentration (SCr₀) were fixed at normal values, i.e., 100ml/min and 8 mg/liter, respectively.

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TABLE 2. PK and PD parameter values used for simulations and estimated values (median, maximum, minimum) from model 1 (which includes the circadian variation in renal function) for eight patients treated for endocarditis over a long period

RESULTS

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Results

Relevance of model 1 in comparison with that of model 2. Forty-nine datum points were obtained for the eight patientsfor use in each of the two models (i.e., 49 paired |err₁| and|err₂| values), and the Wilcoxon signed-rank test was appliedto the differences between |err₁| and |err₂| and yielded a Pvalue <10^-5. As a consequence, we consider model 1, whichincludes the circadian variation in renal function, to be themore relevant.

Fit of the data with model 1 (which includes the circadian variation in renal function). Table 2 shows the medians, minima, and maxima of the PK parametervalues (V, K_s, K_cp, and K_pc) estimated with the NPEM programby use of a two-compartment model. Table 2 also provides themedians, minima, and maxima of the PD parameter values (k_M,k₁, Q_min, E_max, Q₅₀, ${gamma}$ , CL_CRMAX, ${delta}$ , ${alpha}$ , k₂, and Vol) estimated forthe eight patients by minimizing the objective function (equation9) with the Matlab program. Figure 1 shows the relation betweenthe 49 observed serum creatinine concentrations and the 49 serumcreatinine concentrations calculated by use of model 1, as representedby a linear regression with a high and significant correlationcoefficient (r² = 0.9884 with P < 0.001 if the hypothesisthat r² is equal to 0 is put to the test).

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FIG. 1. Comparison of SCr_obs and SCr_calc (n = 49) by use of model 1 (which includes the circadian variation in renal function) for an eight-patient group treated with amikacin and vancomycin for endocarditis for a long period. Different symbols are used for each patient.

Evolution of nephrotoxicity and health conditions of the patients after treatment. After the treatments had ended, the renal functions of the eightpatients almost returned to their initial states. The eightpatients recovered from the clinical and bacterial perspectives,indicating the efficacy of the treatment.

Simulations with model 1 (which includes the circadian variation in renal function). Figure 2 shows the simulated differences between the amountsof amikacin in the renal cortex for three different infusionschedules (which had identical cumulative AUCs for the serumamikacin concentrations) administered over a 30-day period atidentical daily doses (800 mg). The doses were 1,600 mg every48 h, 800 mg once daily, and 267 mg three times daily. The infusiontime was fixed at 30 min for all doses administered. Figure2 shows that the lowest level of accumulation in the renal cortexand, therefore, the lowest decrease in creatinine clearanceoccurred when amikacin was administered less frequently. Figure2 shows that with a more frequent dosage regimen, nephrotoxicity(i) appears more rapidly, (ii) induces a greater decrease inrenal function, and (iii) is more prolonged. Figure 3 showsthe influences of different k_M values on the accumulation ofAG in the renal cortex with the same dosage regimen, i.e., 1,000mg of amikacin per day over a 30-day period. Higher levels ofaccumulation were observed with lower k_M values (Q_c was equalto 106.26 mg for k_M equal to 1 mg, Q_c was equal to 99.52 mgfor k_M equal to 5 mg, Q_c was equal to 86.08 mg for k_M equalto 15 mg, and Q_c was equal to 68.08 mg for k_M equal to 30).Figure 4 describes the influence of only the circadian variationin renal function by use of a 30-day amikacin treatment (1,000mg once daily) and different times of day of administration(1:30 p.m. versus 1:30 a.m.). The lowest level of nephrotoxicitywas obtained when amikacin was administrated at 1:30 p.m. ThePK and PD parameter values for all simulations are describedin Table 2.

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FIG. 2. Evolution of amounts of amikacin in the renal cortex for three different dosage regimens with identical total doses (A) and the resulting effects on mean renal function evolution (B) by use of model 1 (which includes the circadian variation in renal function).

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FIG. 3. Comparison of evolution of the amount of AG in the renal cortex for different possible values of k_M for treatment with amikacin at 1,000 mg once daily over a 30 day-period by use of model 1 (which includes a circadian variation in renal function).

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FIG. 4. Difference in creatinine clearance evolution for two dosage regimens by use of model 1 (which includes a circadian variation in renal function), with 1,000 mg of amikacin administered once daily at 1:30 p.m. or 1:30 a.m. over a 30 day-period.

DISCUSSION

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Discussion

Many methods for reducing AG nephrotoxicity have been proposed.Molecular modeling can bring about an intrinsically less toxicAG (19), but these methods are very expensive. Protective approachessuch as the administration of polyaspartic acid or deferroxaminecan reduce renal injuries caused by an AG. Unfortunately, theseapproaches could not be translated into clinical applicationsbecause of a lack of efficacy and/or intrinsic toxicity (19).In addition, population PK computer programs, used to controlserum AG concentrations, are able to provide good predictionsof efficacy, as they calculate PD indices such as the AUC/MICor the maximum concentration in serum/MIC ratio (3). By contrast,the estimated concentrations in the second compartment are notgood predictors of nephrotoxicity because they do not take intoaccount nonlinear processes such as the amount of AG taken upin the renal cortex or the tubuloglomerular feedback. Then,a new methodology more suitable for clinical practice was developed.This methodology is based on a PK-PD model. A deterministicmodel seems to be sufficient to represent AG nephrotoxicity.Our model is complex because it takes into account the renalphysiology and describes most precisely the mechanisms involvedin AG nephrotoxicity. This approach is very interesting becauseit gives much information about the PK and PD behaviors of AGsin individual patients.

The relevance of taking into consideration the circadian variationin renal function has been demonstrated by fitting the modelto the observed data and comparison of that model with a modelthat does not include the circadian variation (P < 10^-5).This shows the importance of taking into account these periodicphenomena in biological models.

The high and significant correlation coefficient (r² = 0.9884;P < 0.001) (Fig. 1) shows that model 1 (which includes thecircadian variation in renal function) is a powerful tool forthe representation of renal injuries caused by AG. It mightbe useful as a predictor of the occurrence of AG nephrotoxicity.

Goldbeter and Claude (9) have shown that the modeling approachcan explain the influences of time-patterned drug administrationon toxicity in general. AG nephrotoxicity can be influencedby the administration schedule (5), the times of day of administration(2), the parameter k_M (equation 3) (10), and inter- and intraindividualvariabilities (16). All these aspects are explored below.

For an identical total dose of amikacin, the cumulative AUCfor the serum amikacin concentrations remained unchanged forall administration schedules, but the amounts of amikacin inthe renal cortex differed (Fig. 2A). As is well known (21),less toxicity is observed when AGs are administered less frequently(Fig. 2B), while the efficacy remains the same (the AUC forthe serum AG concentrations and the AUC/MIC ratio remain unchanged).In addition, our model shows that nephrotoxicity (i) appearsmore rapidly, (ii) induces a greater decrease in renal function,and (iii) is more prolonged when administration schedules aremore frequent (Fig. 2B). Accordingly, the differences in thereductions of creatinine clearance between the three dosageregimens for an identical daily dose are maximum on day 30.On that day, the creatinine clearance still represents 99% ofthe initial creatinine clearance for the dosage regimen of 1,600mg every 48 h, whereas it represents only 82% of the initialcreatinine clearance for the dosage regimen of 267 mg threetimes daily. Although less frequent administration results ina smaller amount of AG in the renal cortex, the resulting effectson renal function can be very close for the different regimensbecause of nonlinear relationships. Hence, very similar effectscan be observed, and the ability to select a less frequent dosingregimen in order to obtain a lower level of toxicity is notpossible when the treatments are prolonged.

Because AG uptake by renal cells is saturable (10, 27), theamount of AG in the renal cortex is also saturable. The morequickly saturated it is (i.e., the lower the k_M) (equation 3),the less the amount of AG in the renal cortex is affected bypeak concentrations in serum (Fig. 3). However, less renal protectionis observed with a lower saturability level (i.e., a lower thek_M) (Fig. 3). When the k_M is lower, the amount of amikacin inthe renal cortex increases. When the k_M is higher, the amountsof amikacin in the renal cortex decrease. In fact, 1/k_M canbe considered an index of the sensibility of using a particulardrug for a patient. With a higher k_M, the sensibility, the levelof accumulation in the renal cortex, and the nephrotoxicityare lower. Patients with high k_M values are less sensitive tothe nephrotoxic effects of AG.

In accordance with Beauchamp and Labrecque (2), our model showsthat AG administration at 1:30 p.m., i.e., during the activityperiod or food intake, is preferable to administration at 1:30a.m., i.e., during the rest period, as demonstrated in humansand animals. The mechanisms associated with temporal variationsin AG nephrotoxicity are not completely understood, but twohypotheses can be given. First, temporal variations in PK parametersmay be of crucial importance, and second, food intake or theactivity period may be of crucial importance (2). The relationbetween food intake and the interactions of AGs with renal tubularcells seems to be linked to the urine pH (2). The interactionis stronger at low pH than at high pH, since these moleculesare fully protonated at low pH. It has been shown that the urinepH is higher during the activity period or during the periodof food intake and lower during the rest period (2). This couldpartly explain the lower level of nephrotoxicity during theactivity period. Our model confirms that a simple change inthe time of administration (at 1:30 p.m. instead of 1:30 a.m.)of an identical dosage causes a decrease in toxicity from day8. This is quantified by the difference between the two evolutionsof the creatinine clearance values (Fig. 4). Administrationat 1:30 a.m. induces a higher level of accumulation in the renalcortex and therefore a higher level of nephrotoxicity. The differencebetween the two creatinine clearance evolutions is positiveand becomes maximum at day 16 (5.7 ml/min). After that, thedifference in toxicity decreases until the treatment is stoppedat day 30. This phenomenon may be explained by the fact thatthe diminution of creatinine clearance follows a nonlinear process.Two very different amounts of AG in the renal cortex have twovery similar effects on creatinine clearance when CL_CRMAX, i.e.,the maximum diminution of the creatinine clearance (equation7), is attained. After the end of treatment at day 30, a lowerlevel of reduction of creatinine clearance is still observedwhen AG is administered at 1:30 p.m. In fact, over time, renalfunction returns to its initial value for both administrationschedules but does so more rapidly with daily administrationat 1:30 p.m. After the treatment is stopped, because the amountof AG in the renal cortex is lower when AG is administered at1:30 p.m. than when it is administered at 1:30 a.m., there isa lower level of decrease in creatinine clearance. The differencebetween the two creatinine clearance evolutions is maximum atday 32 (6.2 ml/min), i.e., 2 days after the end of the treatment;then, it decreases to zero at day 35, i.e., 5 days after theend of the treatment. It corresponds to the time required forthe complete elimination of the AG from the renal cortex. Atthat time, the creatinine clearance values are equal to thoseobserved before the beginning of treatment because AG nephrotoxicityis reversible, as is well known (19).

The deterministic aspect of this nonlinear model is shown inparticular in Fig. 4, which shows the periodic oscillations(period of 24 h) corresponding to the administration schedules.Hence, the nonlinear deterministic model (but not a stochasticmodel) is forced by periodic oscillations.

Some of the PD parameters estimated for the eight patients treatedwith amikacin for endocarditis for a long period show higherdegrees of variability than others (Table 2). The great variabilityobserved during the time of AG nephrotoxicity (16) is partlydue to interindividual variability in the level of accumulationin the renal cortex (equation 3), especially for k_M (53%) andk₁ (50%) (Table 2). Another source of interindividual variabilityis the resulting effect of the accumulation of the AG in therenal cortex, i.e., the leaking of AG from the intracellularcomponents into the renal tubule, as shown by the parametersE_max (110%) and Q₅₀ (81%). Some other PD parameters showed slightinterindividual variabilities, such as CL_CRMAX (37%) and ${delta}$ (18%),which are tubuloglomerular feedback parameters. These resultsmay explain why some patients are more sensitive than othersto AG nephrotoxicity. In fact, a high level of interindividualvariability is observed for parameters linked to the AG itself.By contrast, small interindividual variability is observed forthe physiological process that induces a decrease in renal function.We point out the fact that the PD parameters described in Table2 are those estimated for the concurrent administration of amikacinand vancomycin, the latter of which is considered an additionalrisk factor for nephrotoxicity. As a consequence, the valuesof the PD parameters estimated here apply only to the groupof patients evaluated in the present study. The model also takesinto account intraindividual variability, i.e., changes in parametervalues during a treatment. For example, intraindividual variabilitycan be observed in some clinical situations when the infectionitself induces a decrease in renal function. When the infectionis treated, the renal function is restored. In addition, experimentalstudies show that the level of accumulation of gentamicin isactually reduced in the cortices of animals treated for longperiods by a mechanism that involves a decrease in the rateof accumulation of AG in the renal cortex, i.e., a decreasein V_(t) (equation 6) (26).

One of the most interesting aspects of this deterministic modelis that it makes it possible, for the first time, estimationof some PD parameter values with the help of serum creatinineconcentrations. However, in the clinical routine, the use ofserum creatinine concentrations as a marker of renal functionis not 100% reliable. In fact, serum creatinine concentrationsincrease only when renal mechanisms fail to compensate for allthe toxicity. In short, the serum creatinine concentration isa belated marker of renal function. The use of an early markerof renal injuries like enzymuria (alanylaminopeptidase) as acomplement to the serum creatinine concentrations is necessaryto better estimate the PD parameter values, chiefly, those withgreat interindividual variability.

The deterministic model presented in this report is a powerfultool that can be used to simulate and control AG nephrotoxicity.Its use brings rationality to the empirical observations madeover the past 50 years. However, we must point out the extremelimitation of this study: it only applies to eight humans. Moreover,the model takes into account only nephrotoxicity, not efficacyor other toxicities. As a consequence, these results shouldbe confirmed by the joint use of an efficacy model (such asthe model of Zhi [29] or other models) and the toxicity modeldescribed in this report. The population PK computer programscannot be used to achieve an overall optimum therapy. The aimof any therapy is to obtain the greatest efficacy without exceedingtoxicity limits. In future, it might not be idle to considerincorporation of this model into the population PK programs,which would enable them to achieve this type of optimization.

FOOTNOTES

* Corresponding author. Mailing address: ADCAPT, Service Pharmaceutique, Hôpital Antoine Charial, 40 Avenue de la Table de Pierre, 69340 Francheville, France. Phone: 33 472 32 34 87. Fax: 33 472 32 39 08. E-mail: adcapt{at}cismsun.univ-lyon1.fr.

REFERENCES

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