- Research
- Open Access
Establishment of a new initial dose plan for vancomycin using the generalized linear mixed model
- Yasuyuki Kourogi^{1, 2},
- Kenji Ogata^{2},
- Norito Takamura^{2, 3}Email author,
- Jin Tokunaga^{2},
- Nao Setoguchi^{2},
- Mitsuhiro Kai^{1},
- Emi Tanaka^{1} and
- Susumu Chiyotanda^{1}
https://doi.org/10.1186/s12976-017-0054-9
© The Author(s). 2017
Received: 8 September 2016
Accepted: 20 March 2017
Published: 8 April 2017
Abstract
Background
When administering vancomycin hydrochloride (VCM), the initial dose is adjusted to ensure that the steady-state trough value (Css-trough) remains within the effective concentration range. However, the Css-trough (population mean method predicted value [PMMPV]) calculated using the population mean method (PMM) often deviate from the effective concentration range. In this study, we used the generalized linear mixed model (GLMM) for initial dose planning to create a model that accurately predicts Css-trough, and subsequently assessed its prediction accuracy.
Methods
The study included 46 subjects whose trough values were measured after receiving VCM. We calculated the Css-trough (Bayesian estimate predicted value [BEPV]) from the Bayesian estimates of trough values. Using the patients’ medical data, we created models that predict the BEPV and selected the model with minimum information criterion (GLMM best model). We then calculated the Css-trough (GLMMPV) from the GLMM best model and compared the BEPV correlation with GLMMPV and with PMMPV.
Results
The GLMM best model was {[0.977 + (males: 0.029 or females: -0.081)] × PMMPV + 0.101 × BUN/adjusted SCr – 12.899 × SCr adjusted amount}. The coefficients of determination for BEPV/GLMMPV and BEPV/PMMPV were 0.623 and 0.513, respectively.
Conclusion
We demonstrated that the GLMM best model was more accurate in predicting the Css-trough than the PMM.
Keywords
Background
Methods
Subject extraction
Summary of patient characteristics
Characteristic | |
---|---|
No. of patients (female/male) | 46 (14/32) |
Age (years) | 77.37 ± 8.79 |
Height (cm) | 157.66 ± 8.59 |
Weight (kg) | 46.66 ± 9.91 |
BMI (kg/m^{2}) | 18.70 ± 3.34 |
SCr (mg/dL) | 0.82 ± 0.35 |
CLcr (mL/min) | 45.37 ± 18.31 |
BUN (mg/dL) | 19.15 ± 11.76 |
AST (IU/L) | 34.70 ± 24.64 |
ALT (IU/L) | 30.46 ± 36.94 |
CRP (mg/dL) | 8.98 ± 7.32 |
Calculation of CLcr
The PMM requires the CLcr for calculating the VCM Css-trough values and, therefore, we first calculated the CLcr for each patient from their sex, age, weight, and serum creatinine (SCr) at initial dose planning using the Cockcroft-Gault formula (CG formula, Eq. 1) [9]. SCr was affected by the patient’s muscle mass. Therefore, because patients with low muscle mass have low SCr levels, we estimate that the CLcr calculated using the CG formula would high, which overestimates the renal function. In Japan, to estimate CLcr calculated using the CG formula accurately, if the patient’s SCr is < 0.6 mg/dL, it is commonly adjusted to 0.6 mg/dL (adjusted SCr) [10]. Therefore, we used the same method here.
To calculate BEPV, the CLcr is calculated using Eq. 1 even when trough values are measured. If the SCr level has not reached 0.6 mg/dL then, it is adjusted accordingly.
Calculation of PMMPV and BEPV
The PMMPV was calculated using CLcr at initial dose planning, weight, VCM dose conditions (dose, infusion time, and administration interval), and mean values for population pharmacokinetic parameters [11]. Furthermore, the calculations were based on the two-compartment model because the distribution of VCM is divided into the central compartment (blood and tissues which equilibrate rapidly with blood) and the peripheral compartment (tissues which equilibrate slowly with blood) [11].
The BEPV was calculated by CLcr at measuring trough value, weight, VCM administration conditions (dose, infusion time, and administration interval), and the estimating the patients’ pharmacokinetic parameters based on the two-compartment model using the Bayesian estimate.
The PMMPV and BEPV were calculated using the TDM analytical software, Vancomycin MEEK Ver. 3.0 (Meiji Seika Pharma).
Definition of difference (PMM prediction deviation quantity, PMMPDQ) between BEPV and PMMPV
Establishing the basic model that the aimed model is based on
β_{1}: PMMPV coefficient
Creating the predictor model (GLMM best model) for VCM Css-trough based on GLMM
Collection of subject medical data (Fig. 3, procedure 1)
To obtain medical data that can potentially be added to the basic model as explanatory variables, we collected the following subject data: Clinical findings (age, age range [10-year intervals], aged ≥ 75 or not, sex, height, weight, hospital days since drug administration commenced), blood test findings (total protein, serum albumin [Alb], aspartate transaminase [AST], alanine transaminase [ALT], lactate dehydrogenase [LDH], total bilirubin, blood urea nitrogen [BUN], SCr, adjusted SCr, BUN/SCr, BUN/adjusted SCr, SCr adjustment amount, SCr adjusted or not, serum Na, serum K, serum Cl, blood glucose level, c-reactive protein [CRP], white blood cell [WBC], red blood cell [RBC], hemoglobin [Hb], hematocrit [Ht], platelet [PLT], mean corpuscular volume [MCV], mean corpuscular hemoglobin [MCH], mean corpuscular hemoglobin concentration [MCHC]), and VCM administration schedule (initial dose, initial daily dose, single dose, daily dose, infusion time, and number of doses; whether doses were irregularly spaced; and number of days until blood concentration trough values were measured since drug administration commenced). Then, to extract effective explanatory variables from the medical data, we conducted the following investigation.
Extraction of medical data (fixed effect candidate 1) correlated to the difference between BEPV and PMMPV (PMMPDQ, Fig. 3, procedure 2)
When using GLMM, multiple explanatory variables can be included in the model. However, the creation of a model including all the medical data we obtained would have produced an inordinate number of model types. Therefore, we first extracted the patient medical data (explanatory variables) that would be effective when added to the basic model. Thus, the two GLMM explanatory variables types were the fixed effect (equivalent to single and multiple regression analyses explanatory variables), which were elements that predict BEPV (response variable), and random effect, which were elements that changed the fixed effect coefficient and the intercept values of the model. In accordance with the software specifications (Stan) used for the GLMM analysis, we used continuous variables (continuous, such as height and weight) for the fixed effect and discrete variables (qualitatively non-continuous, such as sex and all conditions) for random variables.
Since the medical data that correlated highly with the PMMPDQ had an appropriate fixed effect for use in the model, we calculated the Spearman’s rank correlation coefficient for all medical data and the PMMPDQ.
Medical data (continuous variable) that correlated highly to the PMMPDQ (absolute value of the correlation coefficient of ≥ 0.2) were identified as fixed effect candidate 1.
Extraction of appropriate medical data (fixed effect candidate 2) for use as fixed effect in model (Fig. 3, procedure 3)
β_{1}: PMMPV (fixed effect) coefficient, β_{FE1}: each fixed effect candidate 1 (fixed effect) coefficients, and FE1: each fixed effect candidate 1 (fixed effect).
WAIC is used to select the model with a high degree of predictive accuracy from multiple models and is an index for generalization errors (predictive error when making predictions using the model on unknown patients other than the subjects of this study). The smaller WAIC is, the higher predictive accuracy of a model is and it is determined to apply to unknown patients [13]. Therefore, fixed candidate 1 items used in a model made smaller WAIC than the basic model were considered an appropriate fixed effect in the GLMM best model and were designated as fixed effect candidate 2.
Determination of fixed effect model (Fig. 3, procedure 4)
β_{1}: PMMPV (fixed effect) coefficient, β_{FE2i }: ith fixed effect candidate 2 coefficient, and FE2i: ith fixed effect candidate 2 (fixed effect).
However, n is the upper limit of the number of medical data items corresponding to fixed effect candidate 2.
Of all the models created, that with the smallest WAIC was selected as the fixed effect model.
Extracting applicable medical data (random effect candidate) as random effect in the model (Fig. 3, procedure 5)
Since medical data that is highly correlated to the PMMPDQ has a major effect on predictive accuracy, we calculated the intra-class correlation coefficient (ICC). Since medical data with a large ICC related to the PMMPDQ is a likely discrete variable that can be applied to the model [14], we identified the medical data (discrete variables) with the largest ICC as random effect candidates.
Determination of GLMM best model (Fig. 3, procedure 6)
To determine the most appropriate predictive model (GLMM best model) with the smallest predictive error, we created multiple models including random effect candidate items in the fixed effect model (the model created using Procedure 4 in Fig. 3) and calculated WAIC for each model. Of the created models, that with the smallest WAIC was selected as the GLMM best model.
Assessing predictive accuracy of GLMM best model
Data processing method
We used the statistical analysis software R (ver. 3.2.3) and Microsoft Excel for Mac (ver. 15.22) to statistically analyze the data. We used functions included in R for our Spearman’s rank correlation coefficient calculations and Shapiro-Wilk test and the R package ICC (ver. 2.3.0) for ICC calculations. We used Excel for Mac for simple linear regression and R^{2} calculations. A P <0.05 was considered significant for all tests.
We used R, Stan, the R packages rstan, and brms (ver. 2.9, 2.9.0-3, and 0.8.0, respectively) for GLMM analysis and WAIC calculations. We used the Bayesian estimation with Hamilton Monte Carlo to estimate the model coefficient. We used Rhat for the convergence test of the Bayesian estimation and determined that its convergence with Rhat was ≤ 1.1 [15]. The settings of brm function in brms package were as follows: Chains = 3, Iter = 30000 (100000 when random variables were included in the model), Warmup = 15000 (50000 when random variables were included in the model), Thin = 2, and Family = “normal.” When using the Shapiro-Wilk test on the BEPV, the null hypothesis that followed the normal distribution was not rejected (P = 0.19). Thus, the probability distribution for the response variable was a normal distribution.
Results
This study aimed to create a model (GLMM best model) that highly accurately predicts the Css-trough of the initial dose plan for VCM using patient medical data in the GLMM. Additionally, we assessed whether the VCM Css-trough values (GLMMPV) calculated using the GLMM best model were closer to the BEPV than the PMMPV. First, because we thought the medical data correlating to the difference (PMMPDQ) between BEPV and PMMPV would decrease the predictive error, we extracted the medical data (fixed effect candidate 1) that was highly correlated with the PMMPDQ. Next, to extract the medical data that could be applied to the GLMM best model, we created a model (Eq. 4) including each the fixed effect candidate 1 item in the basic model (Eq. 3). Then, we selected the medical data (fixed effect candidate 2) that made the WAIC of the model smaller (the smaller the WAIC, the higher the predictive accuracy of the model and the smaller the generalization error). Then, we created a model (Eq. 5) that included multiple fixed effect candidate 2 items in the basic model and selected the model with the smallest WAIC as the fixed effect model. We designated the medical data with the largest PMMPDQ-related ICC as the random effect candidate items. We created multiple models that included the random effect candidate items in the fixed effect model and selected the model with the smallest WAIC as the GLMM best model. Finally, in to assess the GLMMPV accuracy, we investigated the simple linear regression and R^{2} when the response variable was BEPV and the explanatory variable was either the PMMPV or the GLMMPV. Details of the results are below.
GLMM best model construction
Extraction of medical data (fixed effect candidate 1) that correlated with the difference (PMMPDQ) between BEPV and PMMPV (Fig. 3, procedure 2)
Correlation coefficient for fixed effect candidate 1 and PMMPDQ
Fixed effect candidate 1 (medical data) | Correlation coefficient | p-value |
---|---|---|
BUN/adjusted SCr | 0.398 | 0.006^{*} |
BUN | 0.372 | 0.011^{*} |
BUN/SCr | 0.332 | 0.024^{*} |
AST | 0.253 | 0.090 |
Age | 0.248 | 0.096 |
SCr | 0.215 | 0.152 |
CLcr | -0.233 | 0.119 |
SCr adjusted amount | -0.239 | 0.110 |
Single dose | -0.263 | 0.078 |
Daily dose | -0.279 | 0.060 |
Extracting medical data (fixed effect candidate 2) that was applicable as fixed effect (Fig. 3, procedure 3)
WAIC and the Coefficients of the variables when all fixed effect candidate 1 items are included in basic model
Fixed effect candidate 1 (medical data) | Coefficient (l-95% CI, u-95% CI) | WAIC |
---|---|---|
None (Basic model) | - | 258.42 |
BUN/adjusted SCr | 0.1 (0.02, 0.17) | 254.52^{a} |
BUN | 0.09 (0.01, 0.17) | 256.12^{a} |
BUN/SCr | 0.08 (0.00, 0.16) | 256.15^{a} |
AST | 0.01 (-0.03, 0.05) | 260.46 |
Age | 0.04 (-0.01, 0.10) | 257.51^{a} |
SCr | 1.13 (-1.48, 3.78) | 259.73 |
CLcr | -0.01 (-0.07, 0.04) | 260.6 |
SCr adjusted amount | -16.09 (-32.54, 0.26) | 256.18^{a} |
Single dose | 0.00 (-0.01, 0.00) | 260.6 |
Daily dose | 0.00 (0.00, 0.00) | 259.63 |
Fixed effect model determination (Fig. 3, procedure 4)
WAIC when multiple fixed effect candidate 2 are included in the basic model
Fixed effect candidate 2 (medical data) | WAIC |
---|---|
None (Basic model) | 258.42 |
BUN/adjusted SCr and BUN | 254.94 |
BUN/adjusted SCr and SCr adjusted amount | 253.45^{a} |
BUN/adjusted SCr and Age | 256.26 |
BUN and SCr adjusted amount | 254.33 |
BUN and Age | 256.05 |
SCr adjusted amount and Age | 256.03 |
BUN/adjusted SCr and BUN and SCr adjusted amount | 254.58 |
BUN/adjusted SCr and BUN and Age | 256.83 |
BUN/adjusted SCr and SCr adjusted amount and Age | 255.25 |
BUN and SCr adjusted amount and Age | 256.14 |
BUN/adjusted SCr and BUN and SCr adjusted amount and Age | 256.56 |
Extracting medical data (random effect) that was applicable as random effect (Fig. 3, procedure 5)
ICC for medical data (discrete variables) related to PMMPDQ
Medical data (discrete variables) | ICC | l-95% CI | u-95% CI |
---|---|---|---|
Sex | 0.057 | -0.032 | 0.991 |
Adjusted SCr | 0.036 | -0.041 | 0.989 |
Aged 75 or above | 0.023 | -0.039 | 0.987 |
No. of days from start of administration to blood test for blood concentration trough | -0.043 | -0.065 | 0.484 |
Age group (10-year intervals) | -0.044 | -0.117 | 0.392 |
Irregular interval administration | -0.047 | -0.047 | -0.037 |
No. of doses | -0.071 | -0.140 | 0.358 |
GLMM best model determination (Fig. 3, procedure 6)
WAIC when random effects are included in the fixed effect model
Fixed effect including random effect (Sex) | WAIC |
---|---|
None (fixed effect model) | 253.45 |
PMMPV | 252.01^{a} |
BUN/adjusted SCr | 252.29 |
SCr adjusted amount | 252.34 |
PMMPV and BUN/adjusted SCr | 253.65 |
PMMPV and SCr adjusted amount | 253.39 |
BUN/adjusted SCr and SCr adjusted amount | 252.69 |
PMMPV and BUN/adjusted SCr and SCr adjusted amount | 253.11 |
Based on the above results, we determined the GLMM best model (Eq. 7) would predict the Css-trough with high accuracy when establishing the VCM initial dose plan.
All explanatory variables for the GLMM best model and their coefficient
Explanatory variables | Coefficient | l-95% CI | u-95% CI |
---|---|---|---|
PMMPV (fixed effect) | 0.977 | 0.314 | 1.960 |
BUN/adjusted SCr (fixed effect) | 0.101 | 0.020 | 0.180 |
SCr adjusted amount (fixed effect) | -12.899 | -28.700 | 2.652 |
Sex: Female (random effect) | -0.081 | -1.201 | 0.592 |
Sex: Male (random effect) | 0.029 | -1.123 | 0.711 |
Assessing predictive accuracy of GLMM best model
Discussion
Figure 4b shows that the simple linear regression slope of BEPV and GLMMPV was closer to 1 than that of BEPV and PMMPV was (Fig. 4a, GLMMPV, 1.060 and PMMPV, 0.902). Additionally, the simple linear regression intercept of BEPV and GLMMPV was closer to 0 than that of the BEPV and PMMPV was (GLMMPV, -1.511 and PMMPV, 2.522). Additionally, because the R^{2} of BEPV and GLMMPV was higher than that of BEPV and PMMPV (GLMMPV, 0.623 and PMMPV, 0.513), we were able to determine that the GLMM best model created in this study predicted the VCM Css-trough with better accuracy than the PMM did for the study subjects. Table 6 shows that the WAIC of the GLMM best model (252.01) was smaller than that of the basic model (258.42, equivalent to the model that predicted Css-trough from the PMM). This indicates that generalization error is decreased in the GLMM best model. Therefore, we believe that the GLMM best model can predict the VCM Css-trough of unknown patients with greater accuracy than the PMM can.
Figure 4 shows that 4.35% (2/46) of patients had PMMPDQ of ≥10 μg/mL, but none had GLMMPDQ of ≥10 μg/mL. Considering the effective blood concentration range of the VCM Css-trough, a difference of ≥ 10 μg/mL in Css-trough predictions would raise concerns that the drug may be less effective and cause adverse effects. However, we believe that the GLMM best model controls large prediction deviations like this.
Nevertheless, there were also cases where the GLMM best model created in this study did not improve the predictive accuracy of the Css-trough values. These patients had large deviations (PMMPDQ) between PMMPV and BEPV, and the GLMM best model did not improve the Css-trough prediction accuracy. For example, PMMPDQ and GLMMPDQ of patient d showed large positive deviations (Fig. 4a and b, 11.2 and 9.4 μg/mL, respectively), and those of patient e also showed positive deviations (Fig. 4a and b, 6.9 and 5.5 μg/mL, respectively). PMMPDQ and GLMMPDQ of patient f showed negative deviations (Fig. 4a and b, -7.0 and -8.8 μg/mL, respectively). We believe that these were likely attributable to the effect of changes in SCr after VCM administration commenced. Our results showed that SCr of patient d was 0.60 mg/dL before VCM administration, but rose to 0.85 mg/dL after VCM administration, and SCr of patient e was risen from 1.05 mg/dL to 1.52 mg/dL. We considered that whose renal functions were declined. Our results also showed that SCr of patient f was 1.20 mg/dL before VCM administration, but decreased to 0.82 mg/dL after VCM administration, which we considered that whose renal function was improved. Therefore, since the renal function of these patients changed after VCM administration started (change in CLcr), the Css-trough prediction accuracy worsened, and the absolute PMMPDQ and GLMMPDQ values increased. Furthermore, PMMPDQ and GLMMPDQ of patient g showed large positive deviations (Fig. 4a and b, 8.8 and 6.4 μg/mL, respectively), and those of patient h also showed large positive deviations (Fig. 4a and b, 10.3 and 8.5 μg/mL, respectively). We thought these were due mainly to involvement of hypoalbuminemia. It has been reported that kidney function is overestimated because of proximal tubule secretion of creatinine increases in patients with hypoalbuminemia [17]. The serum albumin levels of patients g and h were 2.4 and 2.0 g/dL, respectively. We considered that overestimation of kidney function in patients g and h led to excessive VCM doses, and rose Css-trough unexpectedly, resulting the absolute PMMPDQ and GLMMPDQ values increased. To solve these problems, new medical data must be extracted and included in the GLMM best model.
Conclusions
This study demonstrated that the GLMM best model we created for use with the GLMM method in initial VCM dose planning allowed a more accurate Css-trough prediction than PMM did. The GLMM best model increased the rate of achieving the effective VCM blood concentration range. This may lead to reduce the revised dose planning requirement and increase the therapeutic effect of VCM safely.
Declarations
Acknowledgements
Not applicable.
Funding
Self-funded.
Availability of data and materials
Please contact author for data requests.
Authors’ contributions
YK conceptualized and developed the models for GLMM. YK, KO and NT wrote the manuscript. JT and NS contributed to the composition of the manuscript. MK and ET collected medical data and performed TDM. SC planed the clinical protocol. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
This study was conducted after receiving approval from the Institutional Review Boards of Chiyoda Hospital and Kyushu University of Health and Welfare.
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