Document purpose

This document explains the Bayesian mathematics, population pharmacokinetic models, concentration-time equations, and clinical interpretation used by the GlobalRPh Vancomycin Dosing and Bayesian AUC Calculator Advanced. It is written for physicians, clinical pharmacists, pharmacometric reviewers, software validators, and medically sophisticated users who need to understand what the Bayesian option actually computes.

The Advanced calculator contains two distinct calculation families. The Bayesian pathway combines one of five adult population priors with zero, one, or two measured concentrations. The conventional pathway uses first-order one-compartment equations, timed levels, and optional trapezoidal cross-checks. This document focuses primarily on the Bayesian pathway while explaining how the conventional pathway is used as an alternative when Bayesian model requirements are not met.

The exact software implementation is treated as the authority for equations, constants, parameter bounds, model-selection rules, and optimization behavior. Primary publications and the 2020 multidisciplinary vancomycin monitoring guideline are used to explain source populations, clinical context, and limitations.[1,4-8]

The guideline recommends AUC-guided monitoring for serious invasive MRSA infection, generally targeting an AUC over 24 hours of 400–600 mg·h/L when a broth-microdilution MIC of 1 mg/L is assumed. Bayesian estimation is preferred because it can incorporate prior pharmacokinetic knowledge and can estimate exposure before steady state, although accurate dosing and sampling information remain essential.[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. Population-model selection, Bayesian fit, conventional fallback, and every generated regimen require clinical confirmation and local validation before unrestricted deployment.

1. Scope and Advanced method architecture

1.1 Supported calculation methods

The Advanced calculator permits the user to choose among three method settings:

1. Bayesian AUC dosing. The software requires a supported population prior. If no suitable prior can be established, Bayesian calculation stops and a conventional option may be offered when conventional assumptions are independently satisfied.
2. Bayesian dosing with automatic conventional fallback. The software first attempts the Bayesian pathway. If model selection or Bayesian estimation fails, it automatically attempts the conventional pathway only when the patient and data meet conventional requirements.
3. Conventional first-order PK/equation-based AUC dosing. The software bypasses Bayesian population priors and uses one-compartment first-order equations with an assumed or level-derived volume of distribution.

The legacy GlobalRPh Advanced AUC form supplied four clinical workflows: empiric calculation, two levels at steady state, two levels after the first dose, and one level at steady state. The new Advanced calculator retains these concepts while making the Bayesian-versus-conventional method choice explicit.[17]

1.2 Bayesian input states

The Bayesian pathway can operate with:

• No measured level: a population-prior estimate. This is model-informed dosing, but it is not individualized by a measured vancomycin concentration.
• One measured level: the software estimates the individual clearance deviation while retaining central volume at the population prior.
• Two measured levels: the software estimates clearance and central volume when the selected population model contains an estimable central-volume random effect. The Tsai CRRT model updates clearance only.

The dose history is represented by exact infusion start times, dose amounts, and infusion durations. Each concentration is linked to its exact collection time rather than being treated simply as a generic peak or trough.

1.3 Supported Bayesian 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 adults with obesity
• Critically ill adults receiving CRRT without ECMO
• Adults with hematologic malignancy when the Belabbas model is selected manually

1.4 Unsupported or blocked Bayesian populations

Bayesian calculation is blocked for pregnancy, pediatric age, intermittent hemodialysis, SLED, peritoneal dialysis, ECMO, unstable native renal function outside CRRT, or a manually selected population model that does not match the entered patient. A mathematical optimizer cannot correct an invalid population prior.

When a suitable Bayesian prior cannot be identified, the interface displays “No suitable population model located.” The Advanced calculator may then offer conventional first-order PK only if those separate assumptions are satisfied. A Bayesian failure does not by itself establish that conventional dosing is valid.

2. Bayesian mathematics: conceptual foundation

2.1 Bayes’ theorem

Bayesian inference begins with Bayes’ theorem:

$$P(\theta\mid y)=\frac{P(y\mid\theta)P(\theta)}{P(y)}$$

Equation BF1.

where:

• $\theta$ represents unknown individual PK parameters, such as clearance and central volume.
• $y$ represents the observed vancomycin concentrations.
• $P(\theta)$ is the prior distribution supplied by a population PK model.
• $P(y\mid\theta)$ is the likelihood, the probability of observing the measured concentrations for a proposed parameter set.
• $P(\theta\mid y)$ is the posterior, the updated distribution after considering the patient’s concentrations.
• $P(y)$ is a normalizing constant and does not affect the location of the posterior maximum.

Bayesian dosing does not simply average a population estimate and a measured concentration. It evaluates how well many possible parameter values agree with both the source population and the patient’s observed concentration-time data.

2.2 Prior, likelihood, and posterior

The relationship can be summarized as:

$$Posterior\propto Prior\times Likelihood$$

Equation BF2.

A narrow prior exerts stronger pull toward the population mean. A broad prior permits larger patient-specific departures. A precise and correctly timed concentration has more influence than a concentration associated with a large residual-error variance. With sparse or poorly informative sampling, the prior may dominate.

2.3 Maximum a posteriori estimation

The Advanced calculator uses maximum a posteriori estimation, or MAP. It identifies:

$$\hat\theta_{MAP}=\underset{\theta}{\operatorname{argmax}}\;P(\theta\mid y)$$

Equation BF3.

Equivalently, because logarithms preserve the maximum:

$$\hat\theta_{MAP}=\underset{\theta}{\operatorname{argmin}}\left[-\log P(y\mid\theta)-\log P(\theta)\right]$$

Equation BF4.

MAP provides one best-supported parameter vector. It is not the same as the posterior mean, median, or a full Bayesian simulation. The Advanced calculator does not run Markov-chain Monte Carlo.

2.4 Why a prior is clinically useful

One or two concentrations generally cannot identify every parameter of a multicompartment vancomycin model without additional information. The population prior supplies plausible values for clearance, distribution volumes, and intercompartmental clearances. The measured levels then individualize the parameters that the implementation permits to vary.

The prior is particularly influential when:

• Only one level is entered
• Both levels occur at similar times in the same kinetic phase
• Sampling times are uncertain
• The residual-error model is broad
• The patient differs substantially from the source population
• Distribution parameters are fixed by the selected model

2.5 Bayesian shrinkage

The prior penalty pulls extreme individual estimates toward the population mean. This behavior is often called shrinkage. In MAP estimation, shrinkage occurs naturally because large random effects reduce the prior density.

For an individual random effect $\eta$ with variance $\omega$:

$$Penalty_{prior}=\frac{1}{2}\frac{\eta^2}{\omega}$$

Equation BF5.

A smaller $\omega$ imposes a larger penalty for the same deviation. Therefore, two models with different between-patient variability can respond differently to the same concentration data even when their population means are similar.

2.6 Identifiability and information content

A concentration mainly informs the combinations of parameters that control the concentration at its sampling time. A later post-distribution level is often strongly informative about clearance. An early post-infusion level can contain more information about central volume and distribution. Two well-separated levels generally provide more parameter information than two closely spaced levels.

However, additional levels do not guarantee a correct result. A wrong dose time, sample time, missed dose, contaminated line draw, rapidly changing renal function, or structurally inappropriate model can produce a numerically converged but clinically misleading posterior.

3. Application of Bayesian forecasting to vancomycin

3.1 Population PK model as the patient-specific starting point

Each population model converts patient descriptors into typical PK parameters. These typical values become the prior centers $CL_{pop}$ and $V_{c,pop}$. Model-specific variances determine how far individualized values may reasonably move from those centers.

The Advanced calculator includes five priors:

3.2 From dose history to predicted concentrations

For every proposed parameter set, the software reconstructs the concentration contribution from each entered infusion. Multicompartment predictions use the exact matrix-derived disposition rates. Contributions from previous doses are added by superposition.

The predicted concentration at sample time $t_i$ is therefore:

$$C_{pred,i}=f(\theta,\;DoseHistory,\;t_i)$$

Equation BV1.

The likelihood compares $C_{pred,i}$ with the measured concentration $C_{obs,i}$ using the selected model’s residual-error equation.

3.3 Individualized exposure

After MAP estimation, the Advanced calculator recomputes the current regimen and every candidate regimen using individualized clearance and, when applicable, individualized central volume. For linear vancomycin PK:

$$AUC_{24}=\frac{DailyDose}{CL_i}$$

Equation BV2.

A measured concentration that shifts clearance downward increases the predicted AUC for a fixed daily dose. A shift in central volume can materially alter predicted peak, trough, and early post-infusion concentrations without directly changing AUC when clearance is unchanged.

3.4 AUC target selection

The Advanced interface allows a target AUC24 of 400, 450, 500, 550, or 600 mg·h/L. Candidate regimens are ranked by proximity to the selected point target, while the broader target classification remains 400–600 mg·h/L. The selected point is a software ranking preference, not a claim that one exact AUC is universally optimal.

3.5 Bayesian model failure and conventional fallback

A Bayesian calculation can fail because no supported prior matches the patient, the optimizer cannot find a plausible fit, the entered levels are incompatible with the dose history, or the resulting parameters violate hard bounds. In the Advanced calculator:

• Bayesian-only mode stops and may offer conventional calculation.
• Automatic-fallback mode attempts conventional first-order PK if the patient is eligible and the required inputs are available.
• Conventional mode bypasses the population priors entirely.

Conventional fallback is blocked for pregnancy, CRRT, unstable renal physiology, intermittent hemodialysis, SLED, peritoneal dialysis, ECMO, or invalid level/dose inputs. These restrictions prevent the calculator from treating a Bayesian model failure as permission to use an equally unsuitable conventional assumption.

4. 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.

5. Input normalization and patient-derived equations

5.1 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.

5.2 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.

5.3 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.

5.4 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.

5.5 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.

5.6 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 Advanced calculator substitutes BSA-indexed Cockcroft-Gault.

5.7 Adjusted body weight

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

Equation P7.

Use: In the Advanced 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.

6. 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.

6.1 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.

6.2 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.

6.3 BSA-indexed Cockcroft-Gault

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

Equation R2.

Use: This is the Advanced 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.

6.4 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]

7. Population-model selection

7.1 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. In the Advanced calculator, this model-selection step occurs only when a Bayesian method is selected; conventional first-order PK uses a separate one-compartment pathway.

7.2 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.

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

8.1 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 Advanced calculator threshold, and are not marked critically ill. It is a two-compartment model with first-order elimination.[4]

8.2 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.

8.3 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.

8.4 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

8.5 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.

9. Model 2: Roberts 2011 ICU prior

9.1 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]

9.2 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.

9.3 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.

9.4 Bayesian behavior

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

9.5 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.

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

10.1 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]

10.2 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 Advanced calculator centers the linear term at 90 kg. Advanced Version 1.0 follows the formal equation used in the source parameterization and documents the discrepancy.

10.3 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.

10.4 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.

10.5 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

10.6 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 Advanced calculator issues an overlap warning and recommends early levels and model comparison when results are discordant.[10]

10.7 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.

11. Model 4: Tsai 2024 CRRT prior

11.1 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]

11.2 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.

11.3 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, Advanced Version 1.0 does not estimate Vc from two levels in CRRT.

11.4 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]

11.5 Bayesian behavior

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

11.6 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.

12. Model 5: Belabbas hematologic-malignancy prior

12.1 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]

12.2 Actual-weight Cockcroft-Gault

This model uses Cockcroft-Gault calculated with actual body weight, rather than the Advanced 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.

12.3 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.

12.4 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.

12.5 Bayesian behavior

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

12.6 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.

13. Drug-related and compartmental pharmacokinetic calculations

13.1 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.

13.2 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.

13.3 Exact disposition matrix used by the Advanced 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.

13.4 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).

13.5 Infusion rate

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

Equation D12.

Unit: mg/h.

13.6 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.

13.7 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.

13.8 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.

13.9 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.

13.10 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.

13.11 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.

13.12 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]

13.13 Daily dose

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

Equation D20.

13.14 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 Advanced calculator does not require steady state for Bayesian estimation.

14. Bayesian MAP implementation in the Advanced calculator

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 Advanced 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 Advanced 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. Loading-dose and infusion-duration logic

15.1 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.

15.2 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.

16. Advanced level workflows and Bayesian data handling

16.1 Empiric or pre-level Bayesian calculation

When no concentration is entered, the Bayesian pathway sets all individual random effects to zero:

$$\eta_{CL}=0,\qquad \eta_{Vc}=0$$

Equation W1.

The output is therefore a population-model estimate. It can support initial regimen selection and prior-to-dosing AUC targeting, but it does not demonstrate that the patient follows the model.

16.2 Existing regimen without measured levels

The exact current regimen is simulated using population parameters. This can estimate expected AUC24, peak, trough, and terminal half-life before the first measured concentration becomes available. The result remains prior-dependent and should be reassessed after a suitably timed level.

16.3 One measured concentration

With one level, the Advanced calculator estimates only $\eta_{CL}$. Central volume remains fixed:

$$V_{c,i}=V_{c,pop}$$

Equation W2.

This restriction avoids attempting to identify two major parameters from one observation. The single level may be obtained before steady state because the full entered dose history is simulated. Its usefulness depends heavily on accurate timing and whether the level is informative about elimination.

16.4 Two measured concentrations

With two levels, the software generally estimates both $\eta_{CL}$ and $\eta_{Vc}$. The two concentrations should be temporally separated enough to contain distinct information. Two concentrations that are almost coincident may add little identifiability.

The Tsai CRRT prior is an exception:

$$\eta_{Vc}=0\quad\text{for Tsai CRRT}$$

Equation W3.

because the final source model did not retain an estimable random effect on central volume.

16.5 Levels after the first dose

Bayesian estimation can use first-dose levels because it does not require steady state. The complete finite-infusion model predicts concentrations from the first dose and subsequent doses. Nevertheless, very early samples may be dominated by distribution and may strongly depend on the structural model and sampling accuracy.

The Advanced calculator also contains a separate conventional two-level first-dose workflow. That workflow is not Bayesian and uses one-compartment first-order equations. The result pages identify which method was used.

16.6 Timed level entry

The Bayesian engine uses the actual concentration collection timestamp and the exact dose/infusion history. It does not require the user to label a level as a peak or trough. Correct time-zone handling, daylight-saving transitions, delayed doses, interrupted infusions, and missed doses remain critical data-quality concerns.

16.7 What the number of levels does not prove

• Zero levels do not individualize the prior.
• One level does not independently identify volume.
• Two levels do not guarantee reliable volume if timing is poor.
• A small residual does not prove the source population is appropriate.
• More levels cannot rescue an incorrect dose history or rapidly changing physiology without appropriate modeling.

17. Candidate-regimen generation and ranking

17.1 Dose and interval grid

The Advanced engine examines standard intermittent regimens with intervals of:

$$\tau\in\{8,12,24,36,48\}\;hours$$

Equation D1.

Doses are generated in 250-mg increments. The maximum candidate dose per administration is:

$$Dose_{max}=\min\left\{3000,\;\max\left[500,\;250\,\lfloor(30\times TBW)/250\rfloor\right]\right\}$$

Equation D2.

The total daily dose must not exceed 6000 mg/day:

$$DailyDose= Dose\times\frac{24}{\tau}\le6000\;mg/day$$

Equation D3.

These limits are software search constraints, not universal dosing recommendations.

17.2 Candidate infusion duration

The generated infusion duration is:

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

Equation D4.

This produces 0.5-hour increments with a minimum of one hour. Local infusion-rate policies supersede the generated duration.

17.3 Preferred intervals by renal estimate

The interval preference used for ranking is:

This is a ranking heuristic. It does not replace institutional dosing protocols or clinical assessment of renal trajectory.

17.4 Point-target ranking score

For the user-selected target AUC, the Advanced calculator uses:

$$Score=\frac{|AUC_{24}-AUC_{target}|}{100}+P_{interval}+P_{dose}$$

Equation D5.

where:

$$P_{interval}=\begin{cases}0,&preferred\ interval\\0.25,&other\ interval\end{cases}$$

Equation D6.

and:

$$P_{dose}=\begin{cases} 0.45,&DailyDose>4500\;mg\\ 0.18,&4000<DailyDose\le4500\;mg\\ 0,&DailyDose\le4000\;mg \end{cases}$$

Equation D7.

The regimen with the lowest score is preferred. All regimens with AUC24 from 400 through 600 mg·h/L are marked as within the target range, even when another regimen is closer to the selected point target.

17.5 Why the ranking is not a treatment recommendation

The score does not include infection source, organism susceptibility, nephrotoxin burden, renal trajectory, fluid balance, source control, expected treatment duration, or institutional administration constraints. It ranks modeled exposure options; it does not replace a clinician’s regimen decision.

18. Output interpretation

18.1 Population-prior estimate versus Bayesian MAP 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.

18.2 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.

18.3 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.

18.4 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.

18.5 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.

18.6 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.

18.7 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.

19. Model fit, safety warnings, and fallback behavior

The Advanced calculator displays warnings when any of the following are present:

• Population/model overlap, such as obesity plus critical illness
• A model-development covariate is approximated rather than directly measured
• Entered patient characteristics are near or beyond a source population range
• CRRT prescription characteristics are collected but not encoded numerically
• A standardized level residual exceeds the warning threshold
• Optimization fails to converge
• The MAP solution reaches or approaches a parameter bound
• Predicted AUC is outside 400–600 mg·h/L
• Total daily dose is unusually high
• A conventional fallback depends on an assumed Vd or single-level identifiability

19.1 Standardized residual warning

The software flags a measured concentration when:

$$|StdResidual_i|>3$$

Equation F1.

This is a fit diagnostic, not a definitive outlier test. The correct response is to verify dose history, infusion duration, collection time, assay validity, line contamination, changing renal function, and model appropriateness.

19.2 Hard stop: no suitable Bayesian model

Bayesian results are suppressed when a supported prior cannot be established or when parameter estimates are invalid. The warning states that no suitable population model was located and instructs the user not to use a calculated Bayesian regimen until model appropriateness is confirmed.

19.3 Conventional fallback decision

A conventional alternative is attempted only when:

• The patient is an adult and not pregnant
• CRRT and unsupported extracorporeal/dialysis modalities are absent
• Native renal physiology is sufficiently stable
• The required dose and timed-level inputs are valid
• An assumed Vd is clinically acceptable when the workflow requires it

When these conditions are not met, the calculator blocks both Bayesian and conventional output rather than forcing a result.

19.4 Method labeling

Every result should be interpreted with its displayed method label:

• Population-prior Bayesian estimate
• One-level Bayesian MAP estimate
• Two-level Bayesian MAP estimate
• Conventional empiric first-order estimate
• Conventional single-level estimate with assumed Vd
• Conventional two-level steady-state estimate
• Conventional two-level first-dose estimate
• Trapezoidal cross-check

These methods are not interchangeable. Similar numerical AUC values do not imply identical evidentiary support.

20. Worked Bayesian examples

The examples below are hypothetical software-verification cases. They demonstrate the implemented mathematics and are not patient-specific dosing recommendations.

20.1 Example 1: Goti population-prior estimate without levels

Patient: 65-year-old male, 80 kg, 175 cm, SCr 1.0 mg/dL, not critically ill, no CRRT.

BMI:

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

Cockcroft-Gault CrCl using actual weight:

$$CrCl=\frac{(140-65)\times80}{72\times1.0}=83.3\;mL/min$$

Goti population clearance:

$$CL=4.5\left(\frac{83.3}{120}\right)^{0.8}=3.36\;L/h$$

Central volume:

$$V_c=58.4\left(\frac{80}{70}\right)=66.74\;L$$

A current regimen of 1000 mg every 12 hours provides 2000 mg/day:

$$AUC_{24}=\frac{2000}{3.361}=595.0\;mg\cdot h/L$$

The engine’s preferred population-prior candidate is 750 mg every 12 hours over one hour:

$$AUC_{24}=\frac{1500}{3.361}=446.2\;mg\cdot h/L$$

Because no concentration was entered, the result remains a Goti population-prior estimate.

20.2 Example 2: two-level Zhang Bayesian update in obesity plus ICU illness

Patient: 57-year-old female, 125 kg, 165 cm, SCr 1.1 mg/dL, critically ill, no CRRT. Current regimen: 1250 mg every 12 hours over 1.5 hours. Measured concentrations: 28 mg/L at 50.5 hours and 14 mg/L at 59.5 hours from the modeled origin.

The Zhang prior gives:

$$CL_{pop}=3.200\;L/h,\qquad V_{c,pop}=16.7\;L$$

The two-level MAP solution is:

$$CL_i=3.884\;L/h,\qquad V_{c,i}=31.77\;L$$

For 2500 mg/day:

$$AUC_{24,current}=\frac{2500}{3.884}=643.7\;mg\cdot h/L$$

The preferred candidate is 1000 mg every 12 hours:

$$AUC_{24,new}=\frac{2000}{3.884}=515.0\;mg\cdot h/L$$

The standardized residuals are approximately -0.49 and -0.02, indicating that both observations are compatible with the fitted model. The local posterior approximation gives an $\eta_{CL}$ SD of approximately 0.129 and an $\eta_{Vc}$ SD of approximately 0.653. The larger volume uncertainty is consistent with weaker information about Vc and substantial Zhang Vc variability.

20.3 Example 3: one-level Belabbas update

Patient: 45-year-old male, 75 kg, 178 cm, SCr 0.65 mg/dL, hematologic malignancy with AML and neutropenia. Actual-weight Cockcroft-Gault CrCl is approximately 152.2 mL/min.

Belabbas population clearance:

$$CL_{pop}=3.09\left(\frac{152.2}{90}\right)^{0.973}(1.15)(1.13)=6.697\;L/h$$

After a single level of 20 mg/L under a 1000-mg-every-8-hour regimen, the MAP estimate is:

$$CL_i=6.740\;L/h$$

Volume remains at the population prior because only one level is entered. The current 3000 mg/day regimen gives:

$$AUC_{24}=\frac{3000}{6.740}=445.1\;mg\cdot h/L$$

This example illustrates that a one-level result can refine clearance while leaving volume entirely prior-driven.

20.4 Example 4: Tsai CRRT prior

Patient: 68-year-old male, 80 kg, receiving CRRT without ECMO.

$$CL=2.30\left(\frac{80}{70}\right)^{0.73}=2.535\;L/h$$

$$V_c=13.49e^{0.54}=23.15\;L$$

A candidate 1250 mg every 24 hours gives:

$$AUC_{24}=\frac{1250}{2.535}=493.0\;mg\cdot h/L$$

Effluent rate, residual urine output, modality, and downtime are displayed for clinical review but do not alter this equation. The estimate must be reassessed after meaningful CRRT changes or interruptions.

21. 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

22. Validation status and recommended deployment controls – Beta Testing

GlobalRPh goals:

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, plus PHP-to-JavaScript parity tests
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

23. Bayesian versus conventional AUC calculation

23.1 Fundamental distinction

Bayesian dosing begins with a population distribution and updates individual random effects using a likelihood. Conventional first-order PK generally derives $k_{el}$ and Vd directly from timed concentrations or assumes Vd when the data cannot identify it.

23.2 Conventional two-level elimination slope

For two post-infusion concentrations $C_1$ and $C_2$:

$$k_{el}=\frac{\ln(C_1/C_2)}{t_2-t_1}$$

Equation COV1.

The conventional pathway then extrapolates end-of-infusion peak and end-of-interval trough and derives Vd and CL. The Advanced calculator may also report a linear/log trapezoidal AUC cross-check.

23.3 Why conventional fallback is not always safer

Conventional equations can be attractive because they do not require a named population prior. Nevertheless, they still assume stable linear PK, correct dose and level timing, an appropriate one-compartment approximation, and adequate sampling. A single-level conventional estimate is especially dependent on the selected Vd. Therefore, the Advanced calculator does not automatically force a conventional result for CRRT, unstable renal function, pregnancy, or unsupported extracorporeal therapies.

23.4 When agreement is reassuring and when it is not

Agreement between Bayesian and conventional AUC estimates can support confidence when both methods’ assumptions are met. Disagreement should prompt review of model selection, dose history, level timing, distribution-phase sampling, renal trajectory, and Vd assumptions. The calculator does not resolve disagreement by automatically averaging the two estimates.

24. 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.

• The conventional fallback is not a substitute for an absent Bayesian model in CRRT, unstable renal function, pregnancy, or unsupported extracorporeal therapy.

• A selectable point target from 400 through 600 mg·h/L is a regimen-ranking preference within a clinically interpreted target range, not proof that the exact selected number is optimal for every infection.

• MAP plus a Hessian approximation is not a full posterior simulation and may understate uncertainty when the posterior is skewed, multimodal, or constrained by parameter bounds.

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17. GlobalRPh Inc. GlobalRPh Vancomycin Dosing and Bayesian AUC Calculator Advanced, source code, Advanced Version 1.0, August 2026.

Documentation status: This document describes the Bayesian implementation in the GlobalRPh Vancomycin Dosing and Bayesian AUC Calculator Advanced, Version 1.0, as implemented in August 2026. Equations, model behavior, and screenshots should be reviewed whenever calculator source code changes.