Comparison of three a-priori models in the prediction of serum lithium concentration.

CONTEXT: Mathematical models are valuable for optimizing drug dose and dosing regimens. AIMS: To compare the precision and bias of three a-priori methods in the prediction of serum level of lithium in patients with bipolar disorder, and to determine their sensitivity and specificity in detecting serum lithium levels outside the therapeutic range. SETTINGS AND
DESIGN: Hospital-based, retrospective study.
MATERIALS AND METHODS: In a retrospective study of 31 in-patients, the serum level of lithium was calculated using three different a-priori methods. Mean Prediction Error was used as a measure of bias while Mean Absolute Error and Root Mean Squared Error were used as a measure of precision. The sensitivity and specificity of the methods was calculated.
RESULTS: All three models underestimated serum lithium level. Precision was best with the model described by Pepin et al., while bias of prediction was the least with the method of Abou Auda et al. The formula by Pepin et al. was able to predict serum lithium level with a mean error of 36.57%. The sensitivity and specificity of the models in identifying serum lithium levels outside the therapeutic range was 80% and 76.19% for Pepin et al., 90% and 74.19% for Zetin et al., and 90% and 66.67% for Abou-Auda et al., respectively.
CONCLUSION: The study demonstrates the difference in precision and bias of three a-priori methods, with no one method being superior to the other in the prediction of serum concentration.

Introduction

Mathematical models can be useful in guiding pharmacotherapy. The mathematical description of the relationship between the pharmacokinetics of a drug and its pharmacodynamics, is gaining relevance in pharmacology, especially with regard to drugs with a narrow therapeutic window or those with a dose-dependent side effect. Such methods have become increasingly useful in medicine, eg, in predicting renal side effects of vancomycin[1] or in predicting response with chemotherapeutic agents.[2] Prediction of therapeutically effective plasma levels is also valuable in designing rational dosage regimes in clinical psychopharmacology.[34] Models have also been devised for optimizing the dose of capecitabine for breast cancer chemotherapy,[5] prediction of therapeutically effective plasma levels of antipsychotics,[6] and prediction of the onset and duration of the pharmacological effect of diazepam.[7]

Lithium is an important drug in the treatment of bipolar disorders, being effective for both, the treatment of acute mania and for prophylaxis.[89] However, it has a narrow therapeutic index and its toxicity can be fatal.[10] The recommended serum lithium level at steady state (Css) is from 0.5–1.2 mEq/L for acute treatment of bipolar disorder,[11] and 0.6 and 0.8 mEq/L for prophylaxis.[12] A serum level below 0.4 mEq/L is found to be no better than placebo and greater than 1.5 mEq/L is associated with potential toxicity.[10] The risk factors that contribute to lithium toxicity are varied and include dehydration, high grade fever, concurrent treatment with diuretics, Nonsteroidal Anti-inflammatory Drugs (NSAIDs), and Angiotensin-Converting-Enzyme (ACE) inhibitors, and intentional overdose.[1314] The contribution of lithium-induced nephrotoxicity, though controversial, underlines the need for drug monitoring even in long-term use.[1516] Hence, the use of mathematical models to estimate the serum level of lithium is relevant.

A-priori methods use formulae derived from patient's demographic, laboratory, and treatment-related data to predict serum lithium level and dosage of lithium. Among the models used to predict serum lithium concentration,[17-21] those described by Pepin et al.,[19] Zetin et al.,[20] and Abou-Auda et al.,[21] have been shown to predict the dose and serum level of lithium with reasonable accuracy. Abou-Auda et al.,[21] have shown their model to be more precise than other formulae. However, the sensitivity and specificity of the models in detecting values that lie outside the therapeutic range has not been examined. This may be more relevant than accuracy and bias of prediction for a drug like lithium, as there is no linear relationship between serum lithium level and outcome as long as the serum level lies within the therapeutic range. Furthermore, the ‘harm’ resulting from serum lithium levels outside the therapeutic range, both from toxicity and the risk of relapse due to sub-therapeutic levels, far outweighs the ‘benefits’ of a small milliequivalent accuracy in serum levels, which lie within the therapeutic range.

The aim of this study was to compare the precision and bias of three a-priori methods, namely, those of Pepin et al.,[19] Zetin et al.,[20] and Abou-Auda et al.,[21] in predicting the serum level of lithium in patients with bipolar disorder. The study also evaluated the sensitivity and specificity of these models in detecting serum lithium values outside the therapeutic range.

Materials and Methods

This retrospective study was conducted at the psychiatric unit of a general hospital in Bangalore. Data regarding age, sex, dose of lithium, serum lithium level, blood urea, serum creatinine, and concomitant medication was obtained from the chart review of inpatients with International Classification of Diseases (ICD)-10 diagnosis of bipolar affective disorder treated with lithium. Blood chemistries were analyzed by Dimension® RxL-Max® Integrated Chemistry System (Siemens Healthcare Diagnostics). Serum lithium was determined as a trough at steady state, ie, 5 days after initiation of the drug.

Predictive models of Pepin et al.,[19] Zetin et al.,[20] and Abou Auda et al.,[21] were used to generate calculated values of serum lithium and dose. The formulae for the three methods are as follows:

Pepin et al.:[19]

Daily Dose (mg/day of lithium carbonate) = (300/8.12)(Css × Vd)/(F × e–kdt)(1 – e–kdt)

where D=daily dosage of lithium (mg), min Css=the desired steady state trough concentration (mmol/L), Vd=the apparent volume of distribution (L) (calculated as CLLi/kd), F=oral bioavailability of lithium (1.0), τ=dosing interval (24 h), and kd=elimination rate constant (h-1) [calculated as ln(2)/lithium t½].

Zetin et al.:[20]

Daily Dose (mg/day of lithium carbonate)= 486.8 + (746.83×serum lithium level) – (10.08 × age) + (5.95 × weight) + (92.01 × status) + (147.8 × sex) – (74.73 × TCA)

Abou-Auda et al.:[21]

Daily dose (mg/day of lithium carbonate)=382.54 + 348.29 (desired serum lithium level)+67.19 (CL Cr).

Creatinine clearance was estimated by the formula:

Creatinine clearance=[[140 – age (yr)]*weight (kg)]/[72*serum Cr(mg/dL)] *(0.85 for women).

Precision and bias are better parameters than correlation analysis in comparing predictive models.[22] Root Mean Squared Error (RMSE) and Mean Prediction Error (MPE) were used to measure precision and bias, respectively. The percentage error of prediction was calculated. Data were analyzed using SPSS 15.0. One-way Analysis Of Variance (ANOVA) was used to compare the three predictive models. The sensitivity and specificity of the models in detecting serum levels outside the therapeutic range were calculated. For a type 1 error rate (α=0.05), power (1-β)=80%, and conservative estimates of sensitivity and specificity of 80%, we calculated the sample size required for paired comparison to be (n)=31.

Results

The sample comprised 31 in-patients (male=20, female=11). Mean age of the sample was 33.09 ± 11.13 years and a mean weight was 57.9 ± 9.1 kg. The patients received a mean lithium dose of 840.3 ± 236.4 mg. The mean steady state serum lithium level was 0.77 ± 0.24 mEq/L [Table 1].

Table 1

Demographics of the sample (n=31)

In the estimation of serum concentration of lithium (in mEq/L), RMSE was 0.37, 0.59, and 0.57 for the predictive models of Pepin et al.,[19] Zetin et al.,[20] and Abou Auda et al.,[21] respectively. MPE for serum lithium level was 0.27, 0.46, and 0.24 for Pepin et al.,[19] Zetin et al.,[20] and Abou Auda et al.,[21] models, respectively. The error in prediction of serum lithium level was 36.57%, 61.33%, and 61.52% for the Pepin et al.,[19] Zetin et al.,[20] and Abou Auda et al.,[21] models, respectively. One-way ANOVA for MPE showed no significant differences among methods [F(2,92)=2.8, P=0.065]. A significant difference was found in one-way ANOVA comparing the three methods on RMSE [F(2,92)=3.3, P=0.04]. Post hoc tests revealed a significant difference between Pepin et al.,[24] and Zetin et al.,[25] (P=0.05) for RMSE [Table 2].

Table 2

Comparison of three a-priori methods to predict serum lithium levels

The data were analyzed to evaluate the sensitivity and specificity of the models in correctly classifying serum lithium as being ‘outside the therapeutic range’ (ie, serum lithium <0.6 mEq/L or >1.2 mEq/L). There were 10 patients [serum lithium >1.2 mEq/L (n=1), <0.6 mEq/L (n=9)] whose serum lithium level was ‘outside the therapeutic range’. The Pepin et al.,[19] method was able to correctly identify 8 of the 10 values and incorrectly classified 5 ‘within therapeutic range’ values as being outside the therapeutic range (sensitivity=80%, specificity=76.19%). The Zetin et al.,[20] and Abou-Auda et al.,[21] formulae correctly identified 9 of the 10 values (sensitivity=90%), but misclassified 6 (specificity=71.42%) and 7 (specificity=66.67%) values, which were ‘within the therapeutic range’, respectively [Figure 1].

Figure 1

Sensitivity and specificity of three a-priori models

Discussion

The study compared three a-priori predictive models for estimation of serum level of lithium in the Indian setting. The sample consisted of in-patients, and hence compliance with medication was ensured. Estimation of serum lithium levels was conducted at a common laboratory using flame photometry.

All three models underestimated serum lithium level. Precision was best with the model described by Pepin et al.,[19] while bias of prediction was least with the Abou Auda et al.,[21] method. The Pepin et al.,[19] formula was able to predict serum lithium level with a mean error of 36.57%.

These findings are at variance with those of Abou-Auda et al.[21] In a retrospective study of 60 adult psychiatric patients, Abou Auda et al.,[21] developed an equation using stepwise multiple linear regression. The equation was found to be more accurate than the empirical method and a-priori methods developed by Zetin et al.,[20] Pepin et al.,[19] Jermain et al.,[18] and Terao et al.[23] The loss of precision in estimating serum lithium levels with the Abou Auda et al.,[21] method in our sample is a limitation of its clinical utility as a sole method of drug monitoring.

It has been observed[24] that the predictive models by Zetin et al.,[20] Jermain et al.,[18] and Pepin et al.,[19] are biased. The Jermain et al.,[18] and the empirical method[17] significantly overestimate serum lithium concentration, while Pepin et al.,[19] method underestimate it.

All three methods tested by us were able to correctly identify serum lithium levels that were outside the therapeutic range with a sensitivity of 80%-90%, though the methods had lower levels of specificity (60%-76%). The paired design is the strength of the analysis, as it accounts for inter-individual differences and other confounding factors. Despite the high sensitivity, the harm that would result from a single false-negative outweighs the potential benefit of picking up true-positives, for a drug like lithium. This is especially true when resources to monitor serum lithium are available and affordable. It appears that the models do not take into account ethnic factors that could influence serum lithium levels, such as variation in environmental temperature, seasonal variation,[25] and genetic composition. Incorporating these factors into an ideal model may improve the sensitivity and specificity of prediction.

In conclusion, a-priori models of lithium estimation can be useful as a screening tool, although they may need to be adapted to the regional settings.