Document purpose

This document explains the clinical and mathematical methods used by Version 7 of the GlobalRPh Vancomycin Initial Dosing and Bayesian AUC Calculator. It is written for physicians, clinical pharmacists, advanced practice clinicians, pharmacometric reviewers, software validators, and other medically sophisticated users who need to understand how an entered patient profile becomes a population-model estimate, an individualized Bayesian estimate, and a ranked list of candidate vancomycin regimens.

The document describes the exact implemented calculations, not merely the general concepts behind vancomycin dosing. Where the software deliberately approximates or extrapolates a published population model, the approximation is stated explicitly. The calculator uses five adult population priors: Goti for general hospitalized adults, Roberts for non-obese critically ill adults, Zhang for adults with obesity including critically ill patients with obesity, Tsai for adults receiving continuous renal replacement therapy (CRRT), and Belabbas as an optional prior for hematologic malignancy, acute myeloid leukemia (AML), neutropenia, and augmented renal clearance.

The current multidisciplinary consensus guideline recommends AUC-guided monitoring for serious invasive MRSA infection, with an AUC over 24 hours of 400–600 mg·h/L when a broth-microdilution MIC of 1 mg/L is assumed. The target should generally be assessed early, preferably within 24–48 hours. Bayesian estimation is useful because it can estimate exposure before steady state and with limited sampling.[1]

Clinical-use limitation: This calculator is an independently implemented clinical decision-support tool. It has not undergone prospective clinical validation, regulatory review, or software-as-a-medical-device verification. Results require clinical confirmation, accurate dosing and sampling times, and local validation before unrestricted deployment.

1. Scope and intended use

1.1 Supported use

The calculator supports intermittent intravenous vancomycin dosing in adults. It can perform two related tasks:

1. Initial-regimen estimation without prior doses or measured concentrations. The calculator selects or accepts a population PK prior, estimates patient-specific population parameters from demographics and renal function, and generates regimens predicted to produce an AUC24 of 400–600 mg·h/L.
2. Evaluation of an existing regimen. The calculator reconstructs the exact dose history and can use zero, one, or two measured vancomycin concentrations. Zero levels retain the population prior. One level usually estimates individual clearance. Two levels usually estimate individual clearance and central volume, except when the selected source model does not support central-volume variability.

1.2 Supported adult populations

• General hospitalized adults with clinically stable native renal function
• Critically ill or septic ICU adults with BMI below 30 kg/m²
• Adults with BMI at least 30 kg/m², including critically ill patients with obesity
• Critically ill adults receiving CRRT
• Adults with hematologic malignancy when the Belabbas model is selected manually

1.3 Unsupported or blocked populations

Calculation is blocked when the entered patient is outside the implemented scope, including:

• Age below 18 years or above the software limit
• Pregnancy
• Unstable native renal function when the patient is not receiving CRRT
• Intermittent hemodialysis
• Sustained low-efficiency dialysis (SLED)
• Peritoneal dialysis
• Extracorporeal membrane oxygenation (ECMO)
• A manually selected model that does not match the patient
• A selected model that generates invalid PK parameters

When a supported model cannot be established, the calculator suppresses regimen results and displays “No suitable population model located.”

2. End-to-end calculation workflow

The calculator follows the sequence below.

1. Normalize entered units. Weight is converted to kilograms, height to centimeters, and serum creatinine to mg/dL.
2. Validate clinical scope. The software confirms adult age, renal stability or CRRT status, absence of unsupported renal replacement methods or ECMO, and model-specific eligibility.
3. Derive patient descriptors. BMI, body surface area, ideal body weight, adjusted body weight, and several renal-function estimates are calculated.
4. Select a population model. CRRT has first priority; obesity has second priority; non-obese critical illness has third priority; otherwise the general hospitalized-adult model is used. The hematologic-malignancy model is manual-only.
5. Calculate population PK parameters. Clearance, central volume, and any peripheral volumes or intercompartmental clearances are generated from the chosen model.
6. Reconstruct dose and sample history when applicable. Exact dates and times are converted to hours from the first relevant dose.
7. Apply Bayesian MAP estimation when measured levels are available. The model combines population priors with the likelihood of the observed concentrations.
8. Calculate exposure and concentration metrics. The program predicts AUC24, AUC/MIC, steady-state peak, steady-state trough, and terminal half-life.
9. Generate candidate regimens. Standard doses and intervals are simulated and ranked.
10. Display warnings and fit diagnostics. Model limitations, level residuals, CRRT caveats, and overlap-population concerns are surfaced before clinical use.

3. Symbols, terminology, and units

Unit equivalence: For vancomycin, 1 mg/L equals 1 mcg/mL. Therefore, numerical serum-concentration values are identical in mg/L and mcg/mL.

4. Input normalization and patient-derived equations

4.1 Weight conversion

When weight is entered in pounds, the software converts it to kilograms:

$$TBW_{kg}=\frac{TBW_{lb}}{2.2046226218}$$

Equation P1.

Use: All population-model size covariates, loading-dose calculations, BMI, BSA, and dose-per-kilogram checks require kilograms.

4.2 Height conversion

When height is entered in inches:

$$Height_{cm}=Height_{in}\times 2.54$$

Equation P2.

Use: Height is needed for IBW, BMI, and BSA.

4.3 Serum-creatinine conversion

When serum creatinine is entered in micromoles per liter:

$$SCr_{mg/dL}=\frac{SCr_{\mu mol/L}}{88.4}$$

Equation P3.

Use: SCr in mg/dL is required by Cockcroft-Gault and CKD-EPI 2009.

4.4 Ideal body weight: Devine method

The software first converts height to inches. For males:

$$IBW_{male}=50+2.3\times(Height_{in}-60)$$

Equation P4a.

For females:

$$IBW_{female}=45.5+2.3\times(Height_{in}-60)$$

Equation P4b.

The implementation prevents the result from falling below the sex-specific base value.

Use: IBW helps determine the weight used in the general Cockcroft-Gault estimate and is also displayed for clinical review.

4.5 Body mass index

$$BMI=\frac{TBW_{kg}}{(Height_{m})^2}$$

Equation P5.

Use: BMI drives the obesity-model selection rule. In automatic mode, BMI at least 30 kg/m² selects the Zhang obesity model unless CRRT has higher priority.

4.6 Body surface area: Mosteller equation

$$BSA=\sqrt{\frac{Height_{cm}\times TBW_{kg}}{3600}}$$

Equation P6.

Use: BSA is used to index or de-index renal estimates. The Roberts model was published with BSA-normalized measured urinary CrCl; the calculator substitutes BSA-indexed Cockcroft-Gault.

4.7 Adjusted body weight

$$AdjBW=IBW+0.4\times(TBW-IBW)$$

Equation P7.

Use: In the calculator’s general Cockcroft-Gault pathway, adjusted body weight is selected when BMI is at least 30 kg/m². This renal estimate is calculated even though the Zhang obesity model itself uses CKD-EPI rather than Cockcroft-Gault.

5. Renal-function equations

Renal function is not handled by one universal equation. The calculator computes several estimates because different population models were developed with different renal covariates.

5.1 Weight selection for the general Cockcroft-Gault estimate

The selected Cockcroft-Gault weight is:

• Actual body weight when TBW is below IBW
• Actual body weight when BMI is below 30 kg/m²
• Adjusted body weight when BMI is at least 30 kg/m²

The calculator also computes an actual-weight Cockcroft-Gault estimate for the Belabbas hematologic-malignancy model.

5.2 Cockcroft-Gault creatinine clearance

For males:

$$CrCl=\frac{(140-Age)\times Weight_{kg}}{72\times SCr}$$

Equation R1a.

For females:

$$CrCl_{female}=0.85\times\frac{(140-Age)\times Weight_{kg}}{72\times SCr}$$

Equation R1b.

Variables and units: Age is in years, weight in kg, and SCr in mg/dL. The result is mL/min.[2]

Implementation: The primary displayed Cockcroft-Gault estimate is constrained to 1–150 mL/min. Separate actual-weight and adjusted-weight estimates are retained with broader internal limits for model-specific use.

5.3 BSA-indexed Cockcroft-Gault

$$CrCl_{indexed}=CrCl\times\frac{1.73}{BSA}$$

Equation R2.

Use: This is the calculator’s approximation for the BSA-normalized 24-hour urinary CrCl covariate used by the Roberts ICU model. It is not equivalent to a measured 24-hour urinary clearance and is explicitly disclosed as an extrapolation.

5.4 CKD-EPI 2009 creatinine equation

The race coefficient is not used. Define:

$$k=\begin{cases}0.7,&female\\0.9,&male\end{cases} \qquad \alpha=\begin{cases}-0.329,&female\\-0.411,&male\end{cases}$$

Then:

$$eGFR=141\times\min\left(\frac{SCr}{k},1\right)^{\alpha} \times\max\left(\frac{SCr}{k},1\right)^{-1.209} \times0.993^{Age}\times SexFactor$$

Equation R3.

where:

$$SexFactor=\begin{cases}1.018,&female\\1,&male\end{cases}$$

Unit: mL/min/1.73 m².[3]

Use: The Zhang obesity model uses indexed CKD-EPI because this renal descriptor was the strongest clearance covariate in the model-development analysis.[6]

6. Population-model selection

6.1 Automatic hierarchy

The calculator applies the following priority:

1. CRRT documented: Tsai CRRT model
2. No CRRT and BMI at least 30 kg/m²: Zhang obesity model
   • If critically ill, the Zhang ICU clearance factor is applied
3. BMI below 30 kg/m² and critically ill: Roberts ICU model
4. All other supported adults: Goti general hospitalized-adult model

The Belabbas hematologic-malignancy prior is manual-only because hematologic malignancy can overlap with obesity, ICU status, and CRRT. The clinician must decide whether the disease-specific prior is preferable.

6.2 Why model selection matters

A population model supplies the prior expected values and variability of CL, Vc, peripheral volumes, and Q. Different models can produce materially different AUC estimates from the same dose and concentration. Bayesian updating reduces but does not eliminate model dependence, especially with sparse levels. Model selection therefore remains a clinical and pharmacometric decision rather than a purely administrative dropdown.

7. Model 1: Goti 2018 general hospitalized-adult prior

7.1 Intended population and structure

The Goti model is the automatic default for supported hospitalized adults who are not receiving CRRT, are not obese by the calculator threshold, and are not marked critically ill. It is a two-compartment model with first-order elimination.[4]

7.2 Implemented equations

Vancomycin clearance:

$$CL=4.5\times\left(\frac{CrCl}{120}\right)^{0.8}$$

Equation G1.

Central volume:

$$V_c=58.4\times\left(\frac{TBW}{70}\right)$$

Equation G2.

Fixed peripheral parameters:

$$V_p=38.4\;L$$

Equation G3.

$$Q=6.5\;L/h$$

Equation G4.

Use: CL determines AUC for a given daily dose. Vc, Vp, and Q determine distribution, predicted peak, trough, and terminal half-life.

7.3 Variability and residual error

The source-reported coefficients of variation are converted to log-scale variances for Bayesian estimation:

$$\omega_{CL}=\ln(1+0.398^2)=0.14705$$

Equation G5.

$$\omega_{Vc}=\ln(1+0.816^2)=0.51019$$

Equation G6.

The residual-error variance for a predicted concentration is:

$$Var(\epsilon)=3.4^2+(0.227\times C_{pred})^2$$

Equation G7.

Interpretation: The additive component is important at lower concentrations; the proportional component increases with the predicted concentration.

7.4 Bayesian behavior

• No level: population CL and Vc are retained
• One level: CL is updated; Vc remains at the population prior
• Two levels: CL and Vc are updated
• Vp and Q remain fixed at source-model values

7.5 Limitations

The Goti model was developed from routine hospitalized-patient TDM data. Although it has performed well in several external comparisons, its population average may not fully capture abrupt ICU physiology, major fluid shifts, unsupported dialysis modalities, or special disease states. The calculator does not use the model’s hemodialysis covariate because intermittent hemodialysis is outside scope.

8. Model 2: Roberts 2011 ICU prior

8.1 Intended population and structure

The Roberts model is automatically selected for a critically ill or septic ICU adult with BMI below 30 kg/m² when CRRT is not present. The source analysis used continuous infusion and a one-compartment model in 206 septic critically ill adults.[5]

8.2 Implemented equations

Clearance:

$$CL=4.58\times\frac{CrCl_{indexed}}{100}$$

Equation ICU1.

Volume:

$$V=1.53\times TBW$$

Equation ICU2.

Important implementation substitution: The published covariate was BSA-normalized measured 24-hour urinary CrCl. The calculator uses Equation R2, BSA-indexed Cockcroft-Gault, because measured urinary clearance is not entered. This approximation must be independently reviewed in unstable or augmented renal clearance.

8.3 Variability and residual error

$$\omega_{CL}=\ln(1+0.389^2)$$

Equation ICU3.

$$\omega_{V}=\ln(1+0.374^2)$$

Equation ICU4.

$$Var(\epsilon)=2.4^2+(0.199\times C_{pred})^2$$

Equation ICU5.

8.4 Bayesian behavior

• No level: population CL and V
• One level: individual CL; V remains at the population prior
• Two levels: individual CL and V

8.5 Limitations

The source model was developed for continuous infusion. The GlobalRPh implementation uses its population parameters within an intermittent-infusion concentration engine. This is a clinically meaningful extrapolation. In addition, serum creatinine may lag rapidly changing renal function in critical illness. The calculator blocks non-CRRT use when native renal function is declared clinically unstable.

9. Model 3: Zhang obesity and morbid-obesity prior

9.1 Intended population and structure

The Zhang model is automatically selected at BMI at least 30 kg/m², including when the patient is critically ill. It is a three-compartment model developed from overweight and obese hospitalized patients, including ward and ICU patients, plus rich data from morbidly obese bariatric-surgery subjects.[6]

9.2 Implemented clearance equation

Define the ICU factor:

$$F_{ICU}=\begin{cases}0.845,&critically\ ill/ICU\\1,&not\ ICU\end{cases}$$

Equation Z1.

Then:

$$CL=3.36\times\left(\frac{eGFR_{CKD-EPI}}{75}\right)^{0.658} \times\left[1+0.0106\times(TBW-90)\right]\times F_{ICU}$$

Equation Z2.

The software prevents the linear weight factor from falling below 0.15, although the model is not intended for low-BMI patients and manual selection requires BMI at least 30 kg/m².

Published-text discrepancy: The article narrative describes 3.36 L/h in a typical 100-kg patient, while the formal equation implemented in the calculator centers the linear term at 90 kg. Version 7 follows the formal equation used in the source parameterization and documents the discrepancy.

9.3 Fixed distribution parameters

$$V_c=16.7\;L$$

Equation Z3.

$$Q_2=16.1\;L/h,\qquad V_2=23.1\;L$$

Equation Z4.

$$Q_3=2.44\;L/h,\qquad V_3=83.0\;L$$

Equation Z5.

9.4 Variability and residual error

$$\omega_{CL}=0.0730$$

Equation Z6.

$$\omega_{Vc}=0.978$$

Equation Z7.

The implemented patient-data residual model is additive:

$$Var(\epsilon)=9.32 \qquad\Rightarrow\qquad \sigma_{add}=\sqrt{9.32}\approx3.05\;mg/L$$

Equation Z8.

9.5 Bayesian behavior

• No level: population CL and Vc
• One level: individual CL
• Two levels: individual CL and Vc
• V2, V3, Q2, and Q3 remain fixed

9.6 Obesity plus critical illness

The same Zhang equation is used, but the ICU factor reduces the population clearance estimate by 15.5%. This is why an obese ICU patient does not automatically use the non-obese Roberts prior. Because external validation has sometimes favored Goti or Roberts in critically ill adults with obesity, the calculator issues an overlap warning and recommends early levels and model comparison when results are discordant.[10]

9.7 Limitations

The source excluded renal replacement therapy. Evidence was limited at very low eGFR and at the highest body weights with renal dysfunction. The large Vc variability indicates uncertainty in peak prediction. The source authors emphasized AUC rather than trough-based adjustment.

10. Model 4: Tsai 2024 CRRT prior

10.1 Intended population and structure

The Tsai model is automatically selected whenever CRRT is documented. It is a two-compartment model developed in critically ill adults with time-varying CRRT or temporary mechanical circulatory support status.[7]

10.2 Implemented CRRT equations

Clearance:

$$CL=2.30\times\left(\frac{TBW}{70}\right)^{0.73}$$

Equation C1.

Central volume under CRRT:

$$V_c=13.49\times e^{0.54}\approx23.15\;L$$

Equation C2.

Peripheral volume:

$$V_p=32.68\;L$$

Equation C3.

Intercompartmental clearance when temporary mechanical circulatory support is absent:

$$Q=18.81\;L/h$$

Equation C4.

10.3 Variability and residual error

$$\omega_{CL}=0.100$$

Equation C5.

$$Var(\epsilon)=1.39^2$$

Equation C6.

The final source model did not retain a random effect on central volume. Therefore, Version 7 does not estimate Vc from two levels in CRRT.

10.4 Collected CRRT settings

The calculator records modality, total effluent rate, 24-hour urine output, and downtime. These values support warnings and clinical review but do not numerically alter Equations C1–C4. Modern CRRT literature shows that effluent intensity, residual diuresis, interruptions, and modality may influence vancomycin clearance; the selected prior cannot represent all combinations.[11,12]

10.5 Bayesian behavior

• No level: population CL and Vc
• One level: individual CL
• Two levels: individual CL only; Vc remains fixed

10.6 Limitations

The source development cohort was small and concentrated in a cardiac ICU. ECMO and other temporary mechanical-circulatory-support states remain excluded from this calculator. CRRT prescriptions can change over hours, so early and repeated levels are essential and recalculation is needed after meaningful changes or interruptions.

11. Model 5: Belabbas hematologic-malignancy prior

11.1 Intended population and structure

The Belabbas model is a manual optional prior for adults with hematologic malignancy, especially AML, neutropenia, or augmented renal clearance. It is a one-compartment model developed from 148 adults and 681 concentrations.[8]

11.2 Actual-weight Cockcroft-Gault

This model uses Cockcroft-Gault calculated with actual body weight, rather than the calculator’s adjusted-weight rule for obesity:

$$CrCl_{actual}=CG(Age,Sex,TBW,SCr)$$

Equation H1.

Augmented renal clearance is defined in this implementation as:

$$ARC=\begin{cases}1,&CrCl_{actual}\ge130\;mL/min\\0,&CrCl_{actual}<130\;mL/min\end{cases}$$

Equation H2.

11.3 Clearance modifiers

AML factor:

$$F_{AML}=\begin{cases}1.15,&AML\ documented\\1,&otherwise\end{cases}$$

Equation H3.

Neutropenia factor:

$$F_{NEUT}=\begin{cases} 1,&no\ neutropenia\\ 1.07,&neutropenia\ and\ CrCl_{actual}<130\\ 1.13,&neutropenia\ and\ CrCl_{actual}\ge130 \end{cases}$$

Equation H4.

Clearance:

$$CL=3.09\times\left(\frac{CrCl_{actual}}{90}\right)^{0.973} \times F_{AML}\times F_{NEUT}$$

Equation H5.

Volume:

$$V=122\times\left(\frac{TBW}{70}\right)$$

Equation H6.

11.4 Variability and residual error

The reported coefficients of variation are converted to log variances:

$$\omega_{CL}=\ln(1+0.254^2)=0.06252$$

Equation H7.

$$\omega_{V}=\ln(1+0.474^2)=0.20268$$

Equation H8.

$$Var(\epsilon)=4.92^2$$

Equation H9.

11.5 Bayesian behavior

• No level: population CL and V
• One level: individual CL
• Two levels: individual CL and V

11.6 Limitations

The source dataset was trough-heavy, which limits distribution characterization. Hematologic patients can have rapidly changing renal function during sepsis, chemotherapy, fluid shifts, and marrow recovery. The model should not automatically override a better-matching CRRT, ICU, or obesity prior.

12. Drug-related and compartmental pharmacokinetic calculations

12.1 One-compartment elimination rate constant

For a one-compartment model:

$$k_{el}=\frac{CL}{V}$$

Equation D1.

and:

$$t_{1/2}=\frac{\ln(2)}{k_{el}}\approx\frac{0.693}{k_{el}}$$

Equation D2.

Use: These equations apply directly to the Roberts and Belabbas structural models. They are also useful conceptually, but they must not be used as the terminal half-life formula for a multicompartment model unless the terminal slope equals CL/V.

12.2 Multicompartment microconstants

For a two-compartment model:

$$k_{10}=\frac{CL}{V_c},\qquad k_{12}=\frac{Q}{V_c},\qquad k_{21}=\frac{Q}{V_p}$$

Equation D3.

• k10 is the first-order elimination microconstant from the central compartment.
• k12 and k21 describe distribution between central and peripheral compartments.
• k10 is not generally the terminal elimination rate. The terminal rate is the smallest positive eigenvalue of the full disposition system.

For a three-compartment model, additional microconstants are:

$$k_{13}=\frac{Q_3}{V_c},\qquad k_{31}=\frac{Q_3}{V_3}$$

Equation D4.

with k12 and k21 calculated from Q2 and V2.

12.3 Exact disposition matrix used by the calculator

The engine transforms the compartment system into a symmetric matrix so that stable real eigenvalues and coefficients can be obtained.

One compartment:

$$A_1=\begin{bmatrix}-CL/V_c\end{bmatrix}$$

Equation D5.

Two compartments:

$$A_2=\begin{bmatrix} -(CL+Q)/V_c & Q/\sqrt{V_cV_p}\\ Q/\sqrt{V_cV_p} & -Q/V_p \end{bmatrix}$$

Equation D6.

Three compartments:

$$A_3=\begin{bmatrix} -(CL+Q_2+Q_3)/V_c & Q_2/\sqrt{V_cV_2} & Q_3/\sqrt{V_cV_3}\\ Q_2/\sqrt{V_cV_2} & -Q_2/V_2 & 0\\ Q_3/\sqrt{V_cV_3} & 0 & -Q_3/V_3 \end{bmatrix}$$

Equation D7.

A Jacobi eigensolver calculates eigenvalues μj and eigenvectors vj. The positive disposition rates are:

$$\lambda_j=-\mu_j$$

Equation D8.

The serum-concentration coefficient for each term is:

$$a_j=\frac{v_{1j}^2}{V_c}$$

Equation D9.

Use: The set of pairs (λj, aj) forms the multi-exponential impulse response used for every dose, peak, trough, chart point, and level prediction.

12.4 Terminal elimination rate and half-life

$$\lambda_z=\min_j(\lambda_j)$$

Equation D10.

$$t_{1/2,z}=\frac{\ln(2)}{\lambda_z}$$

Equation D11.

Interpretation: λz is the slowest disposition process and governs the terminal slope. In a two- or three-compartment model, it may be much smaller than k10, yielding a longer terminal half-life than 0.693/(CL/Vc).

12.5 Infusion rate

$$R_0=\frac{Dose}{T_{inf}}$$

Equation D12.

Unit: mg/h.

12.6 Concentration contribution during an infusion

Let u be the elapsed time after the dose started. For 0<u≤Tinf, the contribution of one dose is:

$$C_{dose}(u)=\sum_j\frac{R_0a_j}{\lambda_j} \left(1-e^{-\lambda_j u}\right)$$

Equation D13.

12.7 Concentration contribution after an infusion

For u>Tinf:

$$C_{dose}(u)=\sum_j\frac{R_0a_j}{\lambda_j} \left(1-e^{-\lambda_j T_{inf}}\right) e^{-\lambda_j(u-T_{inf})}$$

Equation D14.

Use: Equations D13 and D14 model finite intravenous infusion rather than treating a dose as an instantaneous bolus.

12.8 Superposition of multiple doses

Because the encoded PK models are linear, total concentration is the sum of the contribution from every dose that has started:

$$C(t)=\sum_{i=1}^{N}C_{dose,i}(t-t_i)$$

Equation D15.

Use: Exact maintenance and loading-dose timestamps are converted into dose events. Superposition permits irregular histories and measured samples before steady state.

12.9 Steady-state peak

At the end of the infusion for repeated dosing every τ hours:

$$C_{max,ss}=\sum_j\frac{R_0a_j}{\lambda_j} \frac{1-e^{-\lambda_j T_{inf}}}{1-e^{-\lambda_j\tau}}$$

Equation D16.

12.10 Steady-state trough

Immediately before the next dose:

$$C_{min,ss}=\sum_j\left[ \frac{R_0a_j}{\lambda_j} \frac{1-e^{-\lambda_j T_{inf}}}{1-e^{-\lambda_j\tau}} \right]e^{-\lambda_j(\tau-T_{inf})}$$

Equation D17.

Interpretation: Peak and trough are consequences of dose, infusion duration, interval, and the full compartment model. They are displayed to aid interpretation but are not used as the primary therapeutic target.

12.11 AUC over 24 hours

For linear PK at steady state:

$$AUC_{24}=\frac{Dose\times(24/\tau)}{CL}$$

Equation D18.

The numerator is the total daily dose. This identity is independent of the number of distribution compartments.

12.12 AUC/MIC

$$\frac{AUC_{24}}{MIC}=\frac{AUC_{24}}{MIC}$$

Equation D19.

The calculator targets AUC24 400–600 mg·h/L assuming MIC 1 mg/L. An entered MIC is displayed in the AUC/MIC result, but the therapeutic AUC target is not reduced when MIC is below 1 and should not be escalated indiscriminately when MIC is above 1. The consensus guideline advises caution because MIC methods vary and conventional dosing may not safely achieve AUC/MIC at MIC above 1.[1]

12.13 Daily dose

$$DailyDose= Dose\times\frac{24}{\tau}$$

Equation D20.

12.14 Accumulation and time to steady state

For a single first-order rate λ, the fraction of steady state after time t is:

$$Fraction_{ss}=1-e^{-\lambda t}$$

Equation D21.

After n half-lives:

$$Fraction_{ss}=1-\left(\frac{1}{2}\right)^n$$

Equation D22.

Approximately 94% and 97% of steady state are reached after four and five half-lives, respectively. In multicompartment PK, the clinically relevant approach to steady state may reflect more than one disposition phase; the calculator does not require steady state for Bayesian estimation.

13. Loading-dose and infusion-duration logic

13.1 Loading-dose range

A weight-based range is calculated:

$$LoadingDose_{min}=TBW\times mg/kg_{min}$$

Equation L1.

$$LoadingDose_{max}=TBW\times mg/kg_{max}$$

Equation L2.

Both values are rounded to the nearest 250 mg and capped at 3000 mg.

Implemented ranges:

• General adult or obesity reference: 20–25 mg/kg
• CRRT: 20–25 mg/kg
• Critically ill: 20–35 mg/kg

The calculator presents this as guidance, not as an automatically administered dose. Loading-dose use depends on infection severity, urgency, prior therapy, hemodynamics, and local protocol.

13.2 Candidate infusion duration

The candidate-regimen engine applies:

$$T_{inf}=\max\left(1,\frac{\lceil2\times Dose/1000\rceil}{2}\right)$$

Equation L3.

This rounds the dose in grams upward to the next 0.5-hour infusion increment with a minimum of 1 hour. Examples:

• 500–1000 mg → 1 hour
• 1250–1500 mg → 1.5 hours
• 1750–2000 mg → 2 hours
• 2250–2500 mg → 2.5 hours

This is a software regimen-generation rule and does not replace institution-specific maximum infusion-rate policies.

14. Bayesian principles and implementation

14.1 What Bayesian updating means here

A population model provides a prior distribution for PK parameters. Measured concentrations provide a likelihood: parameter values that predict the observations closely receive greater support than values that predict them poorly. Bayes’ theorem combines both sources:

$$Posterior(\theta\mid y)\propto Likelihood(y\mid\theta)\times Prior(\theta)$$

Equation B1.

The calculator does not perform random Markov-chain Monte Carlo sampling. It finds the maximum a posteriori (MAP) parameter values: the values that minimize the negative log posterior.

14.2 Log-normal individual parameters

Individual clearance is:

$$CL_i=CL_{pop}\times e^{\eta_{CL}}$$

Equation B2.

When central volume is estimated:

$$V_{c,i}=V_{c,pop}\times e^{\eta_{Vc}}$$

Equation B3.

The random effects are normally distributed:

$$\eta_{CL}\sim N(0,\omega_{CL})$$

Equation B4.

$$\eta_{Vc}\sim N(0,\omega_{Vc})$$

Equation B5.

Why log scale is used: Exponentiation ensures that individualized CL and Vc remain positive and makes multiplicative variation symmetric on the log scale.

14.3 Residual-error variance

For each observed concentration:

$$Var_i=\sigma_{add}^2+(\sigma_{prop}\times C_{pred,i})^2$$

Equation B6.

Models with no proportional component set σprop to zero.

14.4 Negative log-posterior objective

Ignoring constants that do not affect the optimum, the calculator minimizes:

$$NLP=\frac{1}{2}\frac{\eta_{CL}^2}{\omega_{CL}} +I_V\left(\frac{1}{2}\frac{\eta_{Vc}^2}{\omega_{Vc}}\right) +\frac{1}{2}\sum_i\left[ \ln(Var_i)+\frac{(C_{obs,i}-C_{pred,i})^2}{Var_i} \right]$$

Equation B7.

where IV equals 1 when Vc is being estimated and 0 otherwise.

The first terms penalize implausibly large deviations from the population prior. The summation rewards parameter values that predict measured concentrations within the model-specific residual error.

14.5 One-level and two-level behavior

• No level: ηCL=0 and ηVc=0; the result is a population-prior estimate, not individualized Bayesian monitoring.
• One level: ηCL is estimated; ηVc is fixed at zero.
• Two levels: ηCL and ηVc are estimated when the source model contains usable Vc variability.
• Tsai CRRT: Vc is retained at the population prior even with two levels because the source final model did not retain a random effect on Vc.

A second level does not guarantee reliable volume estimation if the sample times are poorly separated or both observations contain the same limited information.

14.6 Optimization method and bounds

The calculator uses the derivative-free Nelder-Mead simplex algorithm.[13] The search bounds are:

$$-2.3\le\eta_{CL}\le2.3$$

Equation B8.

and, when Vc is estimated:

$$-1.8\le\eta_{Vc}\le1.8$$

Equation B9.

These correspond approximately to multiplicative CL limits of e⁻²·³ to e²·³, or 0.10 to 9.97 times the population value, and Vc limits of 0.17 to 6.05 times the population value. Additional absolute plausibility checks require CL between 0.03 and 40 L/h and Vc between 2 and 800 L during optimization.

The optimizer allows up to 600 iterations and uses a strict objective and simplex-size convergence tolerance. A nonconvergence warning is displayed.

14.7 Posterior uncertainty approximation

After finding the MAP estimate, the calculator approximates the local curvature of the negative log posterior with finite differences. For one parameter:

$$H_{11}\approx\frac{f(\eta+h)-2f(\eta)+f(\eta-h)}{h^2}$$

Equation B10.

and:

$$SD(\eta_{CL})\approx\sqrt{\frac{1}{H_{11}}}$$

Equation B11.

For two parameters, the Hessian matrix is inverted. If:

$$H=\begin{bmatrix}H_{11}&H_{12}\\H_{12}&H_{22}\end{bmatrix}$$

then:

$$H^{-1}=\frac{1}{H_{11}H_{22}-H_{12}^2} \begin{bmatrix}H_{22}&-H_{12}\\-H_{12}&H_{11}\end{bmatrix}$$

Equation B12.

The square roots of the diagonal elements approximate posterior SDs for ηCL and ηVc. This is a local normal approximation, not a full posterior distribution.

14.8 Level-fit diagnostics

Residual:

$$Residual_i=C_{obs,i}-C_{pred,i}$$

Equation B13.

Standardized residual:

$$StdResidual_i=\frac{C_{obs,i}-C_{pred,i}}{\sqrt{Var_i}}$$

Equation B14.

Percent difference:

$$PercentDifference_i=100\times\frac{C_{obs,i}-C_{pred,i}}{C_{pred,i}}$$

Equation B15.

Large standardized residuals may indicate incorrect dose or sample times, an inapplicable model, a contaminated sample, rapidly changing physiology, an unrecorded dose, assay error, or true patient behavior that the prior cannot represent.

15. Candidate-regimen generation and ranking

15.1 Dose and interval grid

The engine examines:

• Intervals: 8, 12, 24, 36, and 48 hours
• Dose increments: 250 mg
• Maximum dose per administration:

The number of 250-mg dose steps permitted by the 30-mg/kg ceiling is:

$$DoseSteps=\operatorname{floor}\left(\frac{30\times TBW}{250}\right)$$

Equation RG1a.

The per-administration maximum is then:

$$MaxDose=\min\left(3000,\max\left(500,250\times DoseSteps\right)\right)$$

Equation RG1b.

• Maximum generated daily dose: 6000 mg/day
• Infusion duration from Equation L3
• Infusion duration must be shorter than the interval

15.2 Preferred intervals by renal estimate

For most non-CRRT models:

CRRT uses q12h and q24h as preferred candidate intervals. The renal descriptor is model-specific: CKD-EPI for Zhang, indexed Cockcroft-Gault for Roberts, actual-weight Cockcroft-Gault for Belabbas, and the selected Cockcroft-Gault estimate for Goti.

15.3 Target classification

$$InTarget=\begin{cases}true,&400\le AUC_{24}\le600\\false,&otherwise\end{cases}$$

Equation RG2.

15.4 Ranking score

The base exposure score is:

$$ExposurePenalty=\frac{|AUC_{24}-500|}{100}$$

Equation RG3.

For a nonpreferred interval:

$$IntervalPenalty=0.2\times\min_p\left|\ln\left(\frac{\tau}{p}\right)\right|$$

Equation RG4.

where p ranges over the preferred intervals.

Daily-dose penalty:

$$DailyPenalty=\begin{cases} 0.35,&DailyDose>4500\\ 0.15,&4000<DailyDose\le4500\\ 0,&DailyDose\le4000 \end{cases}$$

Equation RG5.

Total score:

$$Score=ExposurePenalty+IntervalPenalty+DailyPenalty$$

Equation RG6.

Regimens within target are ranked first. The software displays up to five options, prioritizing target attainment, proximity to AUC24 500, and practical intervals. This is a software ranking method, not a guideline-mandated hierarchy.

16. Output interpretation

16.1 Population-model estimate versus Bayesian estimate

• Population-model estimate: No measured level was entered. Patient covariates modify the population typical values, but no concentration has individualized the parameters.
• Bayesian MAP estimate: One or two measured concentrations were entered and the software optimized individual random effects.

16.2 Estimated vancomycin clearance

CL is the primary determinant of AUC for a given total daily dose. A lower CL increases AUC. A higher CL decreases AUC. Clearance should be interpreted in the context of renal trajectory, urine output, CRRT prescription, fluid status, and model applicability.

16.3 Volume of distribution

Vc governs the immediate central concentration response to an infusion and contributes to peak prediction and distribution kinetics. Total distribution volume in a multicompartment model is not represented by Vc alone. Vp, V2, and V3 describe peripheral spaces.

16.4 Terminal half-life

The displayed terminal half-life is based on λz, not simply CL/Vc for two- and three-compartment models. A long terminal half-life can coexist with a larger initial distribution rate. It informs accumulation and washout but does not by itself select the dose.

16.5 Predicted peak and trough

These are model-predicted steady-state values. They are not direct measured concentrations and are sensitive to Vc and distribution parameters. AUC is the principal target in this calculator.

16.6 AUC24 and AUC/MIC

AUC24 is the total systemic exposure predicted over 24 hours. AUC/MIC divides exposure by the entered MIC. The primary target display remains AUC24 400–600 mg·h/L under the MIC=1 assumption used in current guidance for serious MRSA.

16.7 Standardized residual

A value near zero means the observation is close to the model prediction relative to the residual-error model. Values around ±2 or larger deserve review. A small residual does not prove that the model is clinically correct, particularly with only one level.

17. Model-fit and safety warnings

The calculator may calculate a regimen but still issue warnings. Examples include:

• Auto-selected model requires clinical confirmation
• Cockcroft-Gault estimate was capped
• Roberts continuous-infusion source is being extrapolated to intermittent infusion
• Obesity and critical illness overlap
• Low eGFR subgroup caution for Zhang
• CRRT effluent intensity or downtime is not directly represented
• AML or neutropenia modifiers were applied
• Actual CrCl outside the Belabbas development range
• No measured level was entered
• Only one measured concentration was entered
• Optimizer did not converge
• Measured levels poorly fit the selected model
• Daily dose exceeds a monitoring threshold

17.1 Hard stop: no suitable model

The calculation is blocked when the patient falls outside supported scope or when a manually selected model is incompatible. The message is intentionally stronger than a routine warning:

No suitable population model located. This tool could not locate a supported population PK model that adequately fits the entered patient. Please recheck the patient characteristics and model criteria. Do not use a calculated regimen until an appropriate model is confirmed.

18. Worked example 1: initial regimen without levels

Consider a hypothetical 65-year-old male, 80 kg, 175 cm, SCr 1.2 mg/dL, not critically ill, not receiving CRRT, and with stable renal function.

18.1 Patient descriptors

BMI:

$$BMI=\frac{80}{1.75^2}=26.1\;kg/m^2$$

IBW:

$$IBW=50+2.3\times(68.9-60)=70.5\;kg$$

BSA:

$$BSA=\sqrt{\frac{175\times80}{3600}}\approx1.97\;m^2$$

Cockcroft-Gault using actual weight:

$$CrCl=\frac{(140-65)\times80}{72\times1.2}=69.4\;mL/min$$

18.2 Model selection and population parameters

BMI is below 30 and the patient is not critically ill, so Goti is selected.

$$CL=4.5\times(69.4/120)^{0.8}=2.91\;L/h$$

$$V_c=58.4\times(80/70)=66.7\;L$$

The fixed values are Vp 38.4 L and Q 6.5 L/h.

18.3 Preferred regimen

The candidate engine selects 750 mg q12h over 1 hour:

$$DailyDose=750\times24/12=1500\;mg/day$$

$$AUC_{24}=1500/2.905=516\;mg\cdot h/L$$

The calculator predicts a steady-state peak of approximately 27.5 mg/L, trough of 17.5 mg/L, and terminal half-life of approximately 26.7 hours. These are population-model predictions because no level was entered.

19. Worked example 2: one-level Bayesian update

Using the same patient, suppose the entered regimen is 1000 mg q12h over 1 hour, four exact doses are recorded, and a concentration of 18.5 mg/L is collected 35.5 hours after the first maintenance dose.

The population Goti clearance is 2.91 L/h. The one-level MAP process estimates ηCL while retaining Vc at 66.7 L:

$$CL_i=2.905\times e^{\eta_{CL}}=2.56\;L/h$$

The model predicts 15.63 mg/L at the sample time. Therefore:

$$Residual=18.5-15.63=2.87\;mg/L$$

The model-specific residual SD at that concentration yields a standardized residual of approximately 0.58, indicating that the observation is reasonably compatible with the model.

For the entered 2000 mg/day regimen:

$$AUC_{24}=2000/2.56=781\;mg\cdot h/L$$

A candidate 750 mg q12h regimen provides 1500 mg/day:

$$AUC_{24}=1500/2.56=586\;mg\cdot h/L$$

This example demonstrates the Bayesian mechanism: a measured level higher than the population prediction shifts clearance downward and increases the estimated AUC for the current dose.

20. Data-quality requirements

Bayesian software cannot correct inaccurate source data. Before using a result, verify:

• Every dose amount
• Infusion start time and duration
• Missed, delayed, interrupted, or partially infused doses
• Loading doses
• Sample collection time, not laboratory result time
• Whether the sample was drawn from a line contaminated by vancomycin
• Serum-creatinine timing and trajectory
• Body weight and fluid status
• CRRT start/stop times, downtime, and prescription changes
• Whether the selected population model remains clinically applicable

21. Validation status and recommended deployment controls

Before unrestricted clinical use, GlobalRPh should complete:

1. Independent line-by-line pharmacometric code review
2. Unit tests for every equation and boundary condition
3. Comparison against trusted NONMEM simulations or validated Bayesian software across representative virtual patients
4. Retrospective validation using local, de-identified concentration-time data
5. Separate subgroup analyses for ICU, obesity, CRRT, and hematologic malignancy
6. Verification of dose and sample timestamp handling across daylight-saving changes and time zones
7. Human-factors and usability testing with pharmacists and physicians
8. Prospective silent-mode evaluation before clinical recommendations are acted upon
9. Medical-legal and institutional review
10. Version-controlled release notes and regression testing after every model or interface revision

22. Key limitations

• Population models are approximations developed in specific cohorts.
• A good numerical fit to one or two levels does not prove structural correctness.
• Serum creatinine may not reflect rapidly changing renal function.
• The Roberts implementation substitutes indexed Cockcroft-Gault for measured urinary CrCl and applies a continuous-infusion prior to intermittent infusion.
• The Zhang model has limited evidence at very low renal function and at extreme body weight combined with renal dysfunction.
• The Tsai equation does not directly include effluent rate, residual diuresis, or downtime.
• The Belabbas model is based largely on trough data and is manual-only.
• Candidate ranking is a software heuristic, not a replacement for clinical assessment.
• The calculator does not support pregnancy, pediatrics, intermittent HD, SLED, peritoneal dialysis, or ECMO.
• The tool does not calculate microbiologic appropriateness, source control, duration, de-escalation, or the need for an alternative antimicrobial.

References

1. Rybak MJ, Le J, Lodise TP, et al. Therapeutic monitoring of vancomycin for serious methicillin-resistant Staphylococcus aureus infections: a revised consensus guideline and review by ASHP, IDSA, PIDS, and SIDP. Am J Health-Syst Pharm. 2020;77(11):835-864. doi:10.1093/ajhp/zxaa036.
2. Cockcroft DW, Gault MH. Prediction of creatinine clearance from serum creatinine. Nephron. 1976;16(1):31-41. doi:10.1159/000180580.
3. Levey AS, Stevens LA, Schmid CH, et al. A new equation to estimate glomerular filtration rate. Ann Intern Med. 2009;150(9):604-612. doi:10.7326/0003-4819-150-9-200905050-00006.
4. Goti V, Chaturvedula A, Fossler MJ, Mok S, Jacob JT. Hospitalized patients with and without hemodialysis have markedly different vancomycin pharmacokinetics: a population pharmacokinetic model-based analysis. Ther Drug Monit. 2018;40(2):212-221. doi:10.1097/FTD.0000000000000490.
5. Roberts JA, Taccone FS, Udy AA, Vincent JL, Jacobs F, Lipman J. Vancomycin dosing in critically ill patients: robust methods for improved continuous-infusion regimens. Antimicrob Agents Chemother. 2011;55(6):2704-2709. doi:10.1128/AAC.01708-10.
6. Zhang T, Krekels EHJ, Smit C, et al. How to dose vancomycin in overweight and obese patients with varying renal (dys)function in the novel era of AUC 400-600 mg·h/L-targeted dosing. Clin Pharmacokinet. 2024;63:79-91. doi:10.1007/s40262-023-01324-5.
7. Tsai MT, Wang WC, Roan JN, Luo CY, Chou CH. Population pharmacokinetics of vancomycin in intensive care patients with the time-varying status of temporary mechanical circulatory support or continuous renal replacement therapy. Infect Dis Ther. 2024;13(12):2617-2635. doi:10.1007/s40121-024-01071-5.
8. Belabbas T, Yamada T, Egashira N, et al. Population pharmacokinetic model and dosing optimization of vancomycin in hematologic malignancies with neutropenia and augmented renal clearance. J Infect Chemother. 2023;29(4):391-400. doi:10.1016/j.jiac.2023.01.010.
9. Mosteller RD. Simplified calculation of body-surface area. N Engl J Med. 1987;317(17):1098. doi:10.1056/NEJM198710223171717.
10. Alsultan A, et al. External validation of obese/critically ill vancomycin population pharmacokinetic models in critically ill patients who are obese. J Clin Pharmacol. 2024;64:353-361. doi:10.1002/jcph.2375.
11. Yu Z, Liu J, Yu H, et al. Population pharmacokinetics and individualized dosing of vancomycin for critically ill patients receiving continuous renal replacement therapy: the role of residual diuresis. Front Pharmacol. 2023;14:1298397. doi:10.3389/fphar.2023.1298397.
12. Optimizing vancomycin dosing in continuous renal replacement therapy: a systematic review of population pharmacokinetic studies in adult critically ill patients. 2026. PMID:41900808; PMCID:PMC13029633.
13. Nelder JA, Mead R. A simplex method for function minimization. Comput J. 1965;7(4):308-313. doi:10.1093/comjnl/7.4.308.
14. Broeker A, Nardecchia M, Klinker KP, et al. Towards precision dosing of vancomycin: a systematic evaluation of pharmacometric models for Bayesian forecasting. Clin Microbiol Infect. 2019;25(10):1286.e1-1286.e7. doi:10.1016/j.cmi.2019.02.029.
15. Turner RB, Kojiro K, Shephard EA, et al. Review and validation of Bayesian dose-optimizing software and equations for calculation of the vancomycin area under the curve in critically ill patients. Pharmacotherapy. 2018;38(12):1174-1183. doi:10.1002/phar.2191.

Documentation status: This help document describes GlobalRPh calculator Version 7 as implemented in August 2026. Equations and model behavior should be reviewed whenever the calculator source code changes.

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