Vancosource | Empiric Dosing and Bayesian Dose Adjustment of Vancomycin

Overview

VancoSource is an empiric dosing and Bayesian dose optimization calculator for vancomycin.

VancoSource is a personal project developed by a pharmacist, designed exclusively for educational purposes. It is not intended for practical applications beyond learning and exploration.

Features

Choose between 18 population models.
Or reduce selection burden with population model averaging.
Including, during optimization, a published averaging algorithm.
Model a steady-state dose or an entire dosing history.
Incorporate changes in renal function.

Exclusions

Patients < 18 years of age.
Patients on any form of renal replacement therapy (dialysis, CRRT, etc).

About Me

Hello, I'm Nate, a pharmacist. While lacking specific training in statistics or pharmacometrics, my motivation to develop this calculator stemmed from a desire to delve into AUC dosing, Bayesian optimization, and pharmacokinetics.

Contact Information

contact@vancosource.com

Getting started

Input:

Commence by entering vital patient attributes and kinetic parameters:

Patient: Provide details such as height, weight, age, and sex.
Kinetics: Select the calculator type and population model based on your requirements:

Empiric Calculator:

Predict an intial regimen from pharmacokinetic values using only patient attributes and renal function:

Range: Offers a spectrum of potential steady-state dosing regimens.
Specific: Generates predictions for a specific regimen, accommodating irregular doses or significant changes in renal function.

Optimized Calculator:

Conduct Bayesian optimization of pharmacokinetic values using dosing and level history for refined regimen optimization:

Steady-State: Optimizes a single dose at steady-state, considering multiple associated levels.
Full History: Optimizes using the complete dosing history, accommodating irregular doses or significant changes in renal function.

Events:

Enter dose(s), level(s), and serum creatinine(s) as necessary to accurately model patient response.

Settings:

Customize settings to tailor the calculations to your specific needs:

Dose Range: Set limits for displayed steady-state regimens.
Dosing Duration: Determine dose duration when unspecified, aiding in calculating potential regimens.
Recursive Kinetic Equations: Enable or disable when calculating concentration time curves or during optimization for added precision. Read more.

Output:

Predictions:

Review estimated values of Peak, Trough, AUC, and Root-mean-square deviation (RMSD) for both Empiric and Optimized calculations.

Root-mean-square deviation (RMSD): Measure of goodness-of-fit. Lower RMSD indicates closer alignment between model predictions and actual observations.

Levels: Actual vs. Prediction:

Compare actual observations with predicted levels, aiding in assessing the accuracy of the model's fit to the patient's data.

Possible Steady-State Regimen(s):

Explore a table of potential steady-state regimens based on the chosen kinetics model (Empiric or Optimized). Adjust search ranges and parameters in the settings to refine results.

Custom Steady-State Regimen:

Input a dose, duration, and interval of choice to calculate Peak, Trough, and AUC using the model kinetics, allowing for tailored regimen creation.

Weights:

When utilizing the Average Algorithm, view the relative weights of each individual model for a more comprehensive understanding.

Tips, etc.

Entering a Dose — Interval:

If the interval is unknown or the dose is a one-time or loading dose, input the expected interval. VancoSource utilizes this information to calculate AUC and determine the timeframe for concentration-time curve graphs.

Optimize Steady-State — Sawchuk Zaske Method:

When using the Optimized Steady-State calculator and entering more than one level, VancoSource automatically calculates kinetics using the Sawchuk Zaske method. This method requires peak and trough levels, with the tool selecting the first and last levels entered for calculation. Users can toggle between Bayesian optimization-derived kinetics and the Sawchuk Zaske method post-calculation.

Goodness-Of-Fit:

While VancoSource does not explicitly declare model fit, a RMSD ≤ 1.0 typically indicates a good fit. Additionally, the tool provides actual vs. predicted levels for further scrutiny of fit accuracy.

Models

Buelga 2005

Population: Adults (hematological malignancies)

Country: Spain

Kinetics: 1-compartment

Buelga 2005 explores vancomycin (VAN) population pharmacokinetics in adult patients with hematological malignancies using a retrospective analysis of 1,004 serum concentration samples from 215 individuals. Significant factors influencing VAN clearance include total body weight, renal function, and a diagnosis of acute myeloblastic leukemia (AML). Proposed models include a general one and two AML-specific models.

Only the general population model is included in VancoSource.

Model:

\begin{matrix} C l = 1.08 * C r C l * \frac{60}{1000} \end{matrix}

\begin{matrix} V_{1} = 0.98 * W e i g h t \end{matrix}

Buelga re-estimated by Hughes 2021

Population: Adults

Country: USA

Kinetics: 1-compartment

In Hughes 2021, three pharmacokinetic (PK) models—Buelga, Goti, and Thomson—were selected for evaluation. The authors implemented these models in the InsightRX software and used de-identified patient data related to vancomycin treatment. For each model and each vancomycin treatment course, PK parameters were estimated using maximum a posteriori (MAP) Bayesian estimation based on the first n drug levels. The model priors were then re-estimated using new patient data to refine the PK models and improve their predictive performance for precision dosing applications. The re-estimated models were compared against the original models and a hybrid machine learning/pharmacokinetic approach incorporating flattened priors, demonstrating the potential benefits of continuous learning in refining PK models for precision dosing.

VancoSource includes the re-estimated model for Buelga 2005.

Model:

\begin{matrix} C l = 0.713 * C r C l * \frac{60}{1000} \end{matrix}

\begin{matrix} V_{1} = 1.12 * W e i g h t \end{matrix}

Goti 2018

Population: Adults

Country: USA

Kinetics: 2-compartment

Goti 2018 developed a population pharmacokinetic model for vancomycin using real-world data from patients, encompassing those on and not on hemodialysis. Employing the NONMEM software, the model incorporated a two-compartment structure, considering factors such as creatinine clearance (CrCL) and hemodialysis status as significant covariates. The findings revealed distinct pharmacokinetic differences, with hemodialysis patients exhibiting approximately 65% of the clearance observed in non-hemodialysis patients.

VancoSource implementation assumes the patient is not on hemodialysis.

Model:

\begin{matrix} C l = 4.6 * {(\frac{C r C l}{120})}^{0.8} \end{matrix}

\begin{matrix} V_{1} = 58.4 * \frac{W e i g h t}{70} \end{matrix}

\begin{matrix} V_{2} = 48.6 \end{matrix}

\begin{matrix} Q_{2} = 6.5 \end{matrix}

Goti re-estimated by Hughes 2021

Population: Adults

Country: USA

Kinetics: 2-compartment

VancoSource includes the re-estimated model for Goti 2018.

Model:

\begin{matrix} C l = 4.79 * {(\frac{C r C l}{120})}^{0.8} \end{matrix}

\begin{matrix} V_{1} = 36.2 * \frac{W e i g h t}{70} \end{matrix}

\begin{matrix} V_{2} = 38.4 \end{matrix}

\begin{matrix} Q_{2} = 5.32 \end{matrix}

Thompson 2009

Population: Adults

Country: United Kingdom

Kinetics: 2-compartment

Thompson 2009 aimed to develop a population pharmacokinetic model for vancomycin, utilizing routine therapeutic drug monitoring data from adult patients treated with intravenous vancomycin between 1991 and 2007. The model incorporated a two-compartment structure, and the Cockcroft–Gault equation based on total body weight (TBW) was identified as the best fit for estimating creatinine clearance (CLCR).

Model:

\begin{matrix} C l = 2.99 * (1 + 0.0154 * (C r C l - 66)) \end{matrix}

\begin{matrix} V_{1} = 0.675 * W e i g h t \end{matrix}

\begin{matrix} V_{2} = 0.732 * W e i g h t \end{matrix}

\begin{matrix} Q_{2} = 2.28 \end{matrix}

Thompson re-estimated by Hughes 2021

Population: Adults

Country: USA

Kinetics: 2-compartment

VancoSource includes the re-estimated model for Thompson 2009.

Model:

\begin{matrix} C l = 2.42 * (1 + 0.0126 * (C r C l - 66)) \end{matrix}

\begin{matrix} V_{1} = 0.667 * W e i g h t \end{matrix}

\begin{matrix} V_{2} = 0.737 * W e i g h t \end{matrix}

\begin{matrix} Q_{2} = 2.82 \end{matrix}

Colin 2019

Population: General, premature neonates, adults, burn-injured adults, obese, critically ill, trauma patients, excluded CRRT, ECMO, HD

Country: Various (meta-analysis)

Kinetics: 2-compartment

Colin 2019 aimed to address challenges in vancomycin dosing by developing a unified population pharmacokinetic (PK) model based on data from 39 studies. The final two-compartment PK model revealed significant age-related changes in vancomycin clearance, with maturation occurring by 2 years postmenstrual age. Serum creatinine was identified as a key covariate influencing clearance. Simulations showed that current dosing regimens lead to inconsistent efficacy and safety across patient populations, emphasizing the need for age- and kidney function-adjusted dosing to optimize therapeutic outcomes.

Model:

\begin{matrix} P M A_{w k} = \frac{A g e}{0.01923} \end{matrix}

\begin{matrix} P M A_{y r} = A g e \end{matrix}

\begin{matrix} P M A_{50} = 46.4 \end{matrix}

\begin{matrix} A G E_{50} = 61.6 \end{matrix}

\begin{matrix} G a m m a_{1} = 2.89 \end{matrix}

\begin{matrix} G a m m a_{2} = 2.24 \end{matrix}

\begin{matrix} Θ_{S C R} = 0.649 \end{matrix}

\begin{matrix} S C r_{s t d} = e x p^{- 1.228 + l o g 10 (P M A_{y r}) * 0.672 + 6.27 * e x p^{- 3.11 * P M A_{y r}}} \end{matrix}

\begin{matrix} F_{S i z e} = \frac{W e i g h t}{70} \end{matrix}

\begin{matrix} F_{M A T} = \frac{{P M A_{w k}}^{G a m m a_{1}}}{{P M A_{w k}}^{G a m m a_{1}} + {P M A_{50}}^{G a m m a_{1}}} \end{matrix}

\begin{matrix} F_{D E C L I N E} = \frac{{P M A_{y r}}^{G a m m a_{2}}}{{P M A_{y r}}^{G a m m a_{2}} + {P M A_{50}}^{G a m m a_{2}}} \end{matrix}

\begin{matrix} F_{S C R} = e x p^{- Θ_{S C R} * (S C r - S C r_{s t d})} \end{matrix}

\begin{matrix} C l = 5.31 * {(\frac{V_{1}}{42.9})}^{0.75} * F_{M A T} * F_{D E C L I N E} * F_{S C R} \end{matrix}

\begin{matrix} V_{1} = 42.9 * F_{S I Z E} \end{matrix}

\begin{matrix} V_{2} = 41.7 * F_{S I Z E} \end{matrix}

\begin{matrix} Q_{2} = 3.22 * {(\frac{V_{2}}{41.7})}^{0.75} \end{matrix}

Bury 2019

Population: General

Country: Netherlands

Kinetics: 2-compartment

Bury 2019 aimed to quantify the effect of neutropenia on the pharmacokinetics of vancomycin in patients with hematological malignancies. A retrospective, matched cohort design included patients with hematological disease, solid tumors, and those without cancer. Neutropenia, defined as absolute neutrophil count (ANC) < 1.5 cells/nL, was identified as a covariate affecting vancomycin clearance. The study concluded that neutropenic patients with hematological diseases require a 25% higher vancomycin maintenance dose at the start of therapy to achieve therapeutic plasma concentrations promptly, potentially improving treatment effectiveness in this vulnerable population.

Bury 2019 expresses neutropenia as a binary covariate. VancoSource implements this model assuming patients do not have neutropenia.

Model:

\begin{matrix} B M I = \frac{W e i g h t}{{(\frac{H e i g h t}{100})}^{2}} \end{matrix}

if male:

\begin{matrix} F F M = \frac{9270 * W e i g h t}{6680 + (216 * B M I)} \end{matrix}

if female:

\begin{matrix} F F M = \frac{9270 * W e i g h t}{8780 + (244 * B M I)} \end{matrix}

\begin{matrix} C l = 3.22 + (1 + 0.00834 * (C r C l - 104)) \end{matrix}

\begin{matrix} V_{1} = 45.8 * \frac{F F M}{57.2} \end{matrix}

\begin{matrix} V_{2} = 51.7 * \frac{F F M}{57.2} \end{matrix}

\begin{matrix} Q_{2} = 4.03 * {(\frac{F F M}{57.2})}^{0.75} \end{matrix}

Okada 2018

Population: Patients undergoing allo-HSCT

Country: Japan

Kinetics: 2-compartment

This study aimed to develop a population pharmacokinetic (PopPK) model for vancomycin in patients undergoing allogeneic hematopoietic stem-cell transplantation (allo-HSCT) to optimize dosing. The study included 95 patients, and the final PopPK model identified body weight (BW) and creatinine clearance (CLCr) as significant covariates on the distribution volume (V1) and clearance (CL) of vancomycin, respectively. The study found that patients undergoing allo-HSCT had higher V1 and V2 compared to general populations, possibly due to the inflammatory response associated with the transplantation process.

Model:

\begin{matrix} C l = 4.25 * {(\frac{C r C l}{113})}^{0.7} \end{matrix}

\begin{matrix} V_{1} = 39.2 * {(\frac{W e i g h t}{59.4})}^{0.78} \end{matrix}

\begin{matrix} V_{2} = 56.3 \end{matrix}

\begin{matrix} Q_{2} = 1.95 \end{matrix}

Aljutayli 2022

Population: Adults

Country: Canada

Kinetics: 1-compartment

Aljutayli 2022 aimed to develop a local vancomycin population pharmacokinetic (PopPK) model using data from adult patients admitted during 2016 and 2017. A total of 116 patients were included, and their vancomycin pharmacokinetics were modeled using a one-compartment model with linear elimination. Creatinine clearance was identified as a significant covariate.

Model:

\begin{matrix} C l = 4.16 * \frac{C r C l}{84} \end{matrix}

\begin{matrix} V_{1} = 102.46 * \frac{W e i g h t}{70} \end{matrix}

Smit 2020

Population: Morbidly Obese Adults

Country: Netherlands

Kinetics: 3-compartment

Smit 2020 investigated vancomycin pharmacokinetics in obese individuals without renal impairment, aiming to optimize dosing for this population. The researchers found that total body weight (TBW) was a better predictor of vancomycin clearance than renal function estimates.

Model:

\begin{matrix} C l = 5.72 * {(\frac{W e i g h t}{70})}^{0.535} \end{matrix}

\begin{matrix} V_{1} = 16.7 * (1 + 0.0136 * (A g e - 36.5)) \end{matrix}

\begin{matrix} V_{2} = 6.98 * \frac{W e i g h t}{70} + (1 + 0.0136 * (A g e - 36.5)) \end{matrix}

\begin{matrix} Q_{2} = 15.8 \end{matrix}

\begin{matrix} V_{3} = 19.7 \end{matrix}

\begin{matrix} Q_{3} = 5.21 \end{matrix}

Yamamoto 2009

Population: Adults

Country: Japan

Kinetics: 2-compartment

Yamamoto 2009 investigated the population pharmacokinetics (PPK) of vancomycin in adult patients with gram-positive infections, aiming to determine the optimal dosage of the antibiotic. The analysis included 106 subjects, and a two-compartment model was found to better fit the data. The final PPK model incorporated covariates such as age, weight, creatinine clearance, and subject status (healthy volunteers vs. patients with gram-positive infections). The study identified a linear correlation between vancomycin clearance and estimated creatinine clearance for values below 85 mL/min, while clearance remained constant for higher values.

Model:

if CrCl < 85:

\begin{matrix} C l = 0.0322 * C r C l + 0.32 \end{matrix}

else:

\begin{matrix} C l = 3.83 \end{matrix}

\begin{matrix} V_{1} = 0.478 * W e i g h t \end{matrix}

\begin{matrix} V_{2} = 60.6 \end{matrix}

\begin{matrix} Q_{2} = 8.81 \end{matrix}

Medellin Garibay 2016

Population: Trauma Patients

Country: Spain

Kinetics: 2-compartment

Medellin Garibay 2016 aimed to develop a population pharmacokinetic model for vancomycin in trauma patients receiving intravenous infusion. The retrospective analysis included 118 patients for model construction and 40 for external validation. The two-compartment open model considered covariates such as total body weight (TBW), creatinine clearance (CLCR), age, and concomitant use of furosemide.

VancoSource implements this model assuming no exposure to furosemide.

Model:

\begin{matrix} C l = 0.49 * C r C l * \frac{60}{1000} \end{matrix}

if Age ≤ 65:

\begin{matrix} V_{1} = 0.74 * W e i g h t \end{matrix}

else:

\begin{matrix} V_{1} = 1.07 * W e i g h t \end{matrix}

\begin{matrix} V_{2} = 5.86 * W e i g h t \end{matrix}

\begin{matrix} Q_{2} = 0.81 \end{matrix}

Adane 2015

Population: Extremely Obese Adults

Country: USA

Kinetics: 1-compartment

Adane 2015 conducted at a 322-bed acute care community teaching hospital, extremely obese adult patients (BMI ≥ 40 kg/m²) with suspected or confirmed Staphylococcus aureus infection and requiring vancomycin treatment were recruited. The study aimed to determine optimal vancomycin dosing in this population. Various pharmacokinetic parameters were calculated, including creatinine clearance, and a one-compartment intravenous infusion model was applied using NONMEM 7.3. The final model incorporated total body weight (TBW) on volume of distribution and Cockcroft-Gault creatinine clearance (Clcr) on vancomycin clearance.

Model:

\begin{matrix} C l = 6.54 * \frac{\frac{C r C l}{\sqrt{\frac{H e i g h t * W e i g h t}{3600} * 1.73}}}{125} \end{matrix}

\begin{matrix} V_{1} = 0.51 * W e i g h t \end{matrix}

Roberts 2011

Population: Critically Ill Adults

Country: Belgium

Kinetics: 1-compartment

Roberts 2011 examined the pharmacokinetics of vancomycin in 206 critically ill patients diagnosed with sepsis who received continuous infusion (CI) of vancomycin in the Intensive Care Unit (ICU). The key covariates influencing vancomycin volume of distribution were total body weight, and clearance was influenced by creatinine clearance.

Model:

\begin{matrix} C l = 4.58 * \frac{\frac{C r C l}{\sqrt{\frac{H e i g h t * W e i g h t}{3600} * 1.73}}}{100} \end{matrix}

\begin{matrix} V_{1} = 1.53 * W e i g h t \end{matrix}

Bang 2021

Population: Critically Ill Adults

Country: South Korea

Kinetics: 3-compartment

Bang 2021 aimed to construct a new pharmacokinetic model for vancomycin administered through target-controlled infusion (TCI) in critically ill patients. The study involved 22 patients, and a three-compartment model was identified as the most suitable for describing vancomycin pharmacokinetics. Important covariates included ideal body weight (IBW) for the central and slow peripheral volume of distribution, and weight and age (categorized) for clearance.

Model:

if Age ≤ 65:

\begin{matrix} C l = (0.0349 + \frac{W e i g h t}{63} * 0.0271) * 60 \end{matrix}

else:

\begin{matrix} C l = (0.0143 + \frac{W e i g h t}{63} * 0.0271) * 60 \end{matrix}

\begin{matrix} V_{1} = 4.04 + {(\frac{W e i g h t_{I d e a l}}{59})}^{4.21} \end{matrix}

\begin{matrix} V_{2} = 16.6 \end{matrix}

\begin{matrix} Q_{2} = 0.294 * 60 \end{matrix}

\begin{matrix} V_{3} = 42.6 + {(\frac{W e i g h t_{I d e a l}}{59})}^{12.7} \end{matrix}

\begin{matrix} Q_{3} = 0.0816 * 60 \end{matrix}

Llopis-Salvia 2006

Population: Critically Ill Adults

Country: Spain

Kinetics: 2-compartment

Llopis-Salvia 2006 aimed to develop a model for individualizing vancomycin dosage in critically ill patients. The researchers conducted a retrospective cohort study on 50 adult ICU patients receiving vancomycin over a 4-year period. They established a population pharmacokinetic model incorporating creatinine clearance and total body weight.

Model:

\begin{matrix} C l = (0.034 * C r C l) + (0.015 * W e i g h t) \end{matrix}

\begin{matrix} V_{1} = 0.414 * W e i g h t \end{matrix}

\begin{matrix} V_{2} = 1.32 * W e i g h t \end{matrix}

\begin{matrix} Q_{2} = 7.48 \end{matrix}

Revilla 2010

Population: ICU Adults

Country: Spain

Kinetics: 1-compartment

Revilla 2010 conducted a population pharmacokinetic analysis of vancomycin in adult patients (≥18 years old) treated in the medical ICU of a teaching hospital over a 6-year period. The analysis included 191 patients. The study developed a one-compartment model with zero-order input and first-order elimination for vancomycin pharmacokinetics.

Model:

\begin{matrix} C l = 0.67 * (C r C l * \frac{60}{1000}) + A g e^{- 0.24} \end{matrix}

if Scr ≤ 1:

\begin{matrix} V_{1} = 0.82 * W e i g h t \end{matrix}

else:

\begin{matrix} V_{1} = 0.82 * 2.49 * W e i g h t \end{matrix}

Kinetics

1-Compartment Kinetics

Parameterization

\begin{matrix} k_{10} = \frac{C l}{V_{1}} \end{matrix}

Steady-State

if Time ≤ Dur:

\begin{matrix} C t = \frac{D o s e}{V_{1} * D u r} * 1 - e x p^{- k_{10} * T i m e} + \frac{e x p^{- k_{10} * T a u} * 1 - e x p^{- k_{10} * D u r} * e x p^{- k_{10} * (T i m e - D u r)}}{1 - e x p^{- k_{10} * T a u}} \end{matrix}

else:

\begin{matrix} C t = \frac{D o s e}{D u r} * \frac{1}{k_{10} * V_{1}} * \frac{1 - e x p^{- k_{10} * D u r} * e x p^{- k_{10} * (T i m e - D u r)}}{1 - e x p^{- k_{10} * T a u}} \end{matrix}

Non-Steady-State

if Time ≤ Dur:

\begin{matrix} C t = \frac{D o s e}{D u r} * \frac{1}{k_{10} * V_{1}} * 1 - e x p^{- k_{10} * T i m e} \end{matrix}

else:

\begin{matrix} C t = \frac{D o s e}{D u r} * \frac{1}{k_{10} * V_{1}} * 1 - e x p^{- k_{10} * D u r} * e x p^{- k_{10} * (T i m e - D u r)} \end{matrix}

2-Compartment Kinetics

Parameterization

\begin{matrix} k_{10} = \frac{C l}{V_{1}} \end{matrix}

\begin{matrix} k_{12} = \frac{Q_{2}}{V_{1}} \end{matrix}

\begin{matrix} k_{21} = \frac{Q_{2}}{V_{2}} \end{matrix}

\begin{matrix} a_{0} = k_{10} * k_{21} \end{matrix}

\begin{matrix} a_{1} = - (k_{10} + k_{12} + k_{21}) \end{matrix}

\begin{matrix} a l p h a = \frac{- a_{1} + \sqrt{a_{1} * a_{1} - 4 * a_{0}}}{2} \end{matrix}

\begin{matrix} b e t a = \frac{- a_{1} - \sqrt{a_{1} * a_{1} - 4 * a_{0}}}{2} \end{matrix}

\begin{matrix} A = \frac{\frac{k_{21} - a l p h a}{b e t a - a l p h a}}{V_{1}} \end{matrix}

\begin{matrix} B = \frac{\frac{k_{21} - b e t a}{a l p h a - b e t a}}{V_{1}} \end{matrix}

Steady-State

if Time ≤ Dur:

\begin{matrix} A_{1} = 1 - e x p^{- 1 * a l p h a * T i m e} \end{matrix}

\begin{matrix} A_{2} = e x p^{- 1 * a l p h a * T a u} * \frac{1 - e x p^{- 1 * a l p h a * D u r} * e x p^{- 1 * a l p h a * (T i m e - D u r)}}{1 - e x p^{- 1 * a l p h a * T a u}} \end{matrix}

\begin{matrix} B_{1} = 1 - e x p^{- 1 * b e t a * T i m e} \end{matrix}

\begin{matrix} B_{2} = e x p^{- 1 * b e t a * T a u} * \frac{1 - e x p^{- 1 * b e t a * D u r} * e x p^{- 1 * b e t a * (T i m e - D u r)}}{1 - e x p^{- 1 * b e t a * T a u}} \end{matrix}

\begin{matrix} C t = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * (A_{1} + A_{2}) + \frac{B}{b e t a} * (B_{1} + B_{2})) \end{matrix}

else:

\begin{matrix} A_{2} = \frac{1 - e x p^{- 1 * a l p h a * D u r} * e x p^{- 1 * a l p h a * (T i m e - D u r)}}{1 - e x p^{- 1 * a l p h a * T a u}} \end{matrix}

\begin{matrix} B_{2} = \frac{1 - e x p^{- 1 * b e t a * D u r} * e x p^{- 1 * b e t a * (T i m e - D u r)}}{1 - e x p^{- 1 * b e t a * T a u}} \end{matrix}

\begin{matrix} C t = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * A_{2} + \frac{B}{b e t a} * B_{2}) \end{matrix}

Non-Steady-State

if Time ≤ Dur:

\begin{matrix} C t = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * 1 - e x p^{- a l p h a * T i m e} + \frac{B}{b e t a} * 1 - e x p^{- b e t a * T i m e}) \end{matrix}

else:

\begin{matrix} C t = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * 1 - e x p^{- a l p h a * D u r} * e x p^{- a l p h a * (T i m e - D u r)} + \frac{B}{b e t a} * 1 - e x p^{- b e t a * D u r} * e x p^{- b e t a * (T i m e - D u r)}) \end{matrix}

3-Compartment Kinetics

Parameterization

\begin{matrix} k_{10} = \frac{C l}{V_{1}} \end{matrix}

\begin{matrix} k_{12} = \frac{Q_{2}}{V_{1}} \end{matrix}

\begin{matrix} k_{21} = \frac{Q_{2}}{V_{2}} \end{matrix}

\begin{matrix} k_{13} = \frac{Q_{3}}{V_{2}} \end{matrix}

\begin{matrix} k_{31} = \frac{Q_{3}}{V_{3}} \end{matrix}

\begin{matrix} a_{0} = k_{10} * k_{21} * k_{31} \end{matrix}

\begin{matrix} a_{1} = (k_{10} * k_{31}) + (k_{21} * k_{31}) + (k_{21} * k_{13}) + (k_{10} * k_{21}) + (k_{31} * k_{12}) \end{matrix}

\begin{matrix} a_{2} = k_{10} + k_{12} + k_{13} + k_{21} + k_{31} \end{matrix}

\begin{matrix} p = a_{1} - \frac{a_{2} * a_{2}}{3} \end{matrix}

\begin{matrix} q = \frac{2 * a_{2} * a_{2} * a_{2}}{27} - \frac{a_{1} * a_{2}}{3} + a_{0} \end{matrix}

\begin{matrix} r_{1} = \sqrt{- \frac{p * p * p}{27}} \end{matrix}

\begin{matrix} p h i = a c o s (\frac{(- q / 2) / r 1}{3}) \end{matrix}

\begin{matrix} r_{2} = 2 * e x p^{\frac{l o g (r_{1})}{3}} \end{matrix}

\begin{matrix} a l p h a = - c o s (p h i) * r_{2} - \frac{a_{2}}{3} \end{matrix}

\begin{matrix} b e t a = - (c o s (p h i + 2 * \frac{π}{3}) * r_{2} - \frac{a_{2}}{3}) \end{matrix}

\begin{matrix} g a m m a = - (c o s (p h i + 4 * \frac{π}{3}) * r_{2} - \frac{a_{2}}{3}) \end{matrix}

\begin{matrix} A = \frac{1}{V_{1}} * \frac{\frac{Q_{2}}{V_{2} - a l p h a}}{a l p h a - b e t a} * \frac{\frac{Q_{3}}{V_{3} - a l p h a}}{a l p h a - g a m m a} \end{matrix}

\begin{matrix} B = \frac{1}{V_{1}} * \frac{\frac{Q_{2}}{V_{2} - b e t a}}{b e t a - a l p h a} * \frac{\frac{Q_{3}}{V_{3} - b e t a}}{b e t a - g a m m a} \end{matrix}

\begin{matrix} C = \frac{1}{V_{1}} * \frac{\frac{Q_{2}}{V_{2} - g a m m a}}{g a m m a - b e t a} * \frac{\frac{Q_{3}}{V_{3} - g a m m a}}{g a m m a - a l p h a} \end{matrix}

Steady-State

if Time ≤ Dur:

\begin{matrix} A_{1} = 1 - e x p^{- 1 * a l p h a * T i m e} \end{matrix}

\begin{matrix} A_{2} = e x p^{- 1 * a l p h a * T a u} * \frac{1 - e x p^{- 1 * a l p h a * D u r} * e x p^{- 1 * a l p h a * (T i m e - D u r)}}{1 - e x p^{- 1 * a l p h a * T a u}} \end{matrix}

\begin{matrix} B_{1} = 1 - e x p^{- 1 * b e t a * T i m e} \end{matrix}

\begin{matrix} B_{2} = e x p^{- 1 * b e t a * T a u} * \frac{1 - e x p^{- 1 * b e t a * D u r} * e x p^{- 1 * b e t a * (T i m e - D u r)}}{1 - e x p^{- 1 * b e t a * T a u}} \end{matrix}

\begin{matrix} C_{1} = 1 - e x p^{- 1 * g a m m a * T i m e} \end{matrix}

\begin{matrix} C_{2} = e x p^{- 1 * g a m m a * T a u} * \frac{1 - e x p^{- 1 * g a m m a * D u r} * e x p^{- 1 * g a m m a * (T i m e - D u r)}}{1 - e x p^{- 1 * g a m m a * T a u}} \end{matrix}

\begin{matrix} C t = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * (A_{1} + A_{2}) + \frac{B}{b e t a} * (B_{1} + B_{2}) + \frac{C}{g a m m a} * (C_{1} + C_{2})) \end{matrix}

else:

\begin{matrix} A_{2} = \frac{1 - e x p^{- 1 * a l p h a * D u r} * e x p^{- 1 * a l p h a * (T i m e - D u r)}}{1 - e x p^{- 1 * a l p h a * T a u}} \end{matrix}

\begin{matrix} B_{2} = \frac{1 - e x p^{- 1 * b e t a * D u r} * e x p^{- 1 * b e t a * (T i m e - D u r)}}{1 - e x p^{- 1 * b e t a * T a u}} \end{matrix}

\begin{matrix} C_{2} = \frac{1 - e x p^{- 1 * g a m m a * D u r} * e x p^{- 1 * g a m m a * (T i m e - D u r)}}{1 - e x p^{- 1 * g a m m a * T a u}} \end{matrix}

\begin{matrix} C t = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * A_{2} + \frac{B}{b e t a} * B_{2} + \frac{C}{g a m m a} * C_{2}) \end{matrix}

Non-Steady-State

if Time ≤ Dur:

\begin{matrix} C t = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * 1 - e x p^{- a l p h a * T i m e} + \frac{B}{b e t a} * 1 - e x p^{- b e t a * T i m e} + \frac{C}{g a m m a} * 1 - e x p^{- g a m m a * T i m e}) \end{matrix}

else:

\begin{matrix} C t = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * 1 - e x p^{- a l p h a * D u r} * e x p^{- a l p h a * (T i m e - D u r)} + \frac{B}{b e t a} * 1 - e x p^{- b e t a * D u r} * e x p^{- b e t a * (T i m e - D u r)} + \frac{C}{g a m m a} * 1 - e x p^{- g a m m a * D u r} * e x p^{- g a m m a * (T i m e - D u r)}) \end{matrix}

Optimization

All the population kinetics models used by VancoSource were developed using the NONMEN software package. These models (except for Llopis-Salvia 2006) take the general form:

\begin{matrix} T V = Θ * C V * e^{η} \end{matrix}

Where TV is the typcial value, THETA is a fixed effect parameter, CV is the covariate parameter, and ETA is the parameter for between subject variation. ETA is normally distributed with a mean of 0 and a variance given by the covariance matrix OMEGA.

Most population kinetic models are published with the values for between subject variablity reported as percent co-effiecent of variation (CV%). This CV% needs to be converted to a variance before being used to form the covariance matrix OMEGA. To convert CV% to variance, first divide by 100 to remove the percentage and then:

\begin{matrix} S D = \sqrt{\log (C V^{2} + 1)} \end{matrix}

\begin{matrix} V a r i a n c e = S D^{2} \end{matrix}

For example, Goti 2018 reports the CV% of Cl as 39.8% → CV of 0.398 → SD of 0.383 → Variance of 0.147. The full Goti 2018 model is:

\begin{matrix} C l = 4.6 * {(\frac{C r C l}{120})}^{0.8} * e^{η 1} \end{matrix}

\begin{matrix} V_{1} = 58.4 * \frac{W e i g h t}{70} * e^{η 2} \end{matrix}

\begin{matrix} V_{2} = 48.6 * e^{η 3} \end{matrix}

\begin{matrix} Q_{2} = 6.5 * e^{η 4} \end{matrix}

\begin{matrix} [\begin{matrix} η_{1} \\ η_{2} \\ η_{3} \\ η_{4} \end{matrix}] & \sim M V N ([\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0.147 & 0 & 0 & 0 \\ 0 & 0.510 & 0 & 0 \\ 0 & 0 & 0.282 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]) \end{matrix}

The population pharmacokinetic models are then used to individualized kinetic values via penalized maximum likelihood estimation with optimization (Bayesian optimization). These complex calculations are completed by utilizing the Stan platform for statistical modeling and high-performance statistical computation.

Averaging Algorithm

Navigating model selection is a persistent challenge in kinetics calculators. VancoSource, for instance, offers a selection of 18 models, posing the question of how to discern the most suitable model. A plausible resolution to this predicament involves averaging the outcomes from all models. Uster (2021) explores diverse averaging algorithms in their paper. VancoSource adopts a specific algorithm outlined by Uster, which involves weighting the average based on the sum of square error (SSE) of each individual model.

\begin{matrix} W_{S S E_{i}} = \frac{e^{- 0.5 * \sum_{}^{} {(t r u e_{j} - p r e d_{j})}^{2}}}{\sum_{1}^{n} e^{- 0.5 * \sum_{}^{} {(t r u e_{j} - p r e d_{j})}^{2}}} \end{matrix}

Recursive Kinetics

The kinetics equations for concentration taught in pharmacy school assume that everything except time is constant. However, real-world scenarios involve dynamic changes in patients' renal function over time. To address this complexity, VancoSource employs two distinct strategies.

The first strategy involves computing the concentration-time curve at discrete time intervals, utilizing standard kinetic equations while accounting for variations in both time and renal function. This method essentially stitches together segments of individual curves at different time points to create a cohesive curve spanning the entire timeframe. However, a drawback of this approach is that the concentration at each time point is not contingent upon the preceding concentration.

Alternatively, we can modify the conventional kinetics equations to establish a dependency of concentration at any given time point on the preceding concentration. By formulating these recursive equations, VancoSource accommodates 1-, 2-, and 3-compartment kinetics while considering changing renal function over time.

Go to 'Settings' to enable the use of the recursive equations. You can either enable their use when calculating the concentration time curve or during optimization or both. Not used in calculating or optimizing steady-state concentrations. Will increase computation time significantly. Probably best reserved for situations where renal function is significantly unstable. May or may not improve fit.

1-Compartment Recursive Kinetics (Non-steady-state)

for Time_N in Times:

if Time_N ≤ Dur:

if Time_N ≤ 0:

\begin{matrix} C t_{N} = \frac{D o s e}{D u r} * \frac{1}{k_{10} * v_{1}} * 1 - e x p^{- k_{10} * T i m e_{N}} \end{matrix}

else:

\begin{matrix} C t_{N} = (\frac{D o s e}{D u r} * \frac{1}{k_{10} * v_{1}} * 1 - e x p^{- k_{10} * (T i m e_{N} - T i m e_{N - 1})}) + (C t_{N - 1} * e x p^{- k_{10} * (T i m e_{N} - T i m e_{N - 1})}) \end{matrix}

else:

\begin{matrix} C t_{N} = C t_{N - 1} * e x p^{- k_{10} * (T i m e_{N} - T i m e_{N - 1})} \end{matrix}

2-Compartment Recursive Kinetics (Non-steady-state)

for Time_N in Times:

if Time_N ≤ Dur:

if Time_N ≤ 0:

\begin{matrix} C t_{A_{N}} = \frac{D o s e}{D u r} * \frac{A}{a l p h a} * 1 - e x p^{- a l p h a * T i m e_{N}} \end{matrix}

\begin{matrix} C t_{B_{N}} = \frac{D o s e}{D u r} * \frac{B}{b e t a} * 1 - e x p^{- b e t a * T i m e_{N}} \end{matrix}

\begin{matrix} C t_{N} = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * 1 - e x p^{- a l p h a * T i m e_{N}} + \frac{B}{b e t a} * 1 - e x p^{- b e t a * T i m e_{N}}) \end{matrix}

else:

\begin{matrix} C t_{A_{N}} = (\frac{D o s e}{D u r} * \frac{A}{a l p h a} * 1 - e x p^{- a l p h a * (T i m e_{N} - T i m e_{N - 1})}) + (C t_{A_{N - 1}} * e x p^{- a l p h a * (T i m e_{N} - T i m e_{N - 1})}) \end{matrix}

\begin{matrix} C t_{B_{N}} = (\frac{D o s e}{D u r} * \frac{B}{b e t a} * 1 - e x p^{- b e t a * (T i m e_{N} - T i m e_{N - 1})}) + (C t_{B_{N - 1}} * e x p^{- b e t a * (T i m e_{N} - T i m e_{N - 1})}) \end{matrix}

\begin{matrix} C t_{N} = C t_{A_{N}} + C t_{B_{N}} \end{matrix}

else:

\begin{matrix} C t_{A_{N}} = C t_{A_{N - 1}} * e x p^{- a l p h a * (T i m e_{N} - T i m e_{N - 1})} \end{matrix}

\begin{matrix} C t_{B_{N}} = C t_{B_{N - 1}} * e x p^{- b e t a * (T i m e_{N} - T i m e_{N - 1})} \end{matrix}

\begin{matrix} C t_{N} = C t_{A_{N}} + C t_{B_{N}} \end{matrix}

3-Compartment Recursive Kinetics (Non-steady-state)

for Time_N in Times:

if Time_N ≤ Dur:

if Time_N ≤ 0:

\begin{matrix} C t_{A_{N}} = \frac{D o s e}{D u r} * \frac{A}{a l p h a} * 1 - e x p^{- a l p h a * T i m e_{N}} \end{matrix}

\begin{matrix} C t_{B_{N}} = \frac{D o s e}{D u r} * \frac{B}{b e t a} * 1 - e x p^{- b e t a * T i m e_{N}} \end{matrix}

\begin{matrix} C t_{C_{N}} = \frac{D o s e}{D u r} * \frac{C}{g a m m a} * 1 - e x p^{- g a m m a * T i m e_{N}} \end{matrix}

\begin{matrix} C t_{N} = \frac{D o s e}{D u r} * (\frac{A}{a l p h a} * 1 - e x p^{- a l p h a * T i m e_{N}} + \frac{B}{b e t a} * 1 - e x p^{- b e t a * T i m e_{N}} + \frac{C}{g a m m a} * 1 - e x p^{- g a m m a * T i m e_{N}}) \end{matrix}

else:

\begin{matrix} C t_{A_{N}} = (\frac{D o s e}{D u r} * \frac{A}{a l p h a} * 1 - e x p^{- a l p h a * (T i m e_{N} - T i m e_{N - 1})}) + (C t_{A_{N - 1}} * e x p^{- a l p h a * (T i m e_{N} - T i m e_{N - 1})}) \end{matrix}

\begin{matrix} C t_{B_{N}} = (\frac{D o s e}{D u r} * \frac{B}{b e t a} * 1 - e x p^{- b e t a * (T i m e_{N} - T i m e_{N - 1})}) + (C t_{B_{N - 1}} * e x p^{- b e t a * (T i m e_{N} - T i m e_{N - 1})}) \end{matrix}

\begin{matrix} C t_{C_{N}} = (\frac{D o s e}{D u r} * \frac{C}{g a m m a} * 1 - e x p^{- g a m m a * (T i m e_{N} - T i m e_{N - 1})}) + (C t_{C_{N - 1}} * e x p^{- g a m m a * (T i m e_{N} - T i m e_{N - 1})}) \end{matrix}

\begin{matrix} C t_{N} = C t_{A_{N}} + C t_{B_{N}} + C t_{C_{N}} \end{matrix}

else:

\begin{matrix} C t_{A_{N}} = C t_{A_{N - 1}} * e x p^{- a l p h a * (T i m e_{N} - T i m e_{N - 1})} \end{matrix}

\begin{matrix} C t_{B_{N}} = C t_{B_{N - 1}} * e x p^{- b e t a * (T i m e_{N} - T i m e_{N - 1})} \end{matrix}

\begin{matrix} C t_{C_{N}} = C t_{C_{N - 1}} * e x p^{- g a m m a * (T i m e_{N} - T i m e_{N - 1})} \end{matrix}

\begin{matrix} C t_{N} = C t_{A_{N}} + C t_{B_{N}} + C t_{C_{N}} \end{matrix}

References