Equations & derivations

1. Population model — Colin 2019

Every adult is calculated with the Colin 2019 two-compartment population PK model. Clearance is built from four covariate functions composed multiplicatively: allometric size scaling, sigmoidal maturation (effectively 1.0 for adults), an age-decline sigmoid, and a serum-creatinine exponential effect.

Structure:

Two-compartment model, intermittent IV infusion, first-order elimination

Source data:

Pooled population analysis of data from 14 studies (2,554 individuals), from neonates to elderly adults.

Scope in Vancomyzer:

Adults (18 years and older) receiving intermittent IV vancomycin who are not on renal replacement therapy. Used for every body size, including BMI 40 kg/m2 and above.

Renal covariate:

Serum creatinine is used directly (mg/dL) as the Colin 2019 renal covariate: FSCR = exp(−0.649 × (SCr − SCRstd)), where SCRstd is the age-standardised reference creatinine. Cockcroft-Gault does not enter this population prior; a separate post-fit safety policy can cap fitted clearance at twice estimated CrCl.

CL (L/h) = 5.31 × (WT/70)^0.75 × FMat × FDecline × FSCR
V1 (L) = 42.9 × (WT/70)
V2 (L) = 41.7 × (WT/70)
Q (L/h) = 3.22 × (WT/70)^0.75
PMA (years) = age (years) + 40/52
FMat = PMAwk^2.89 / (PMAwk^2.89 + 46.4^2.89) (≈1 for adults)
FDecline = 1 / (1 + (PMA/61.6)^2.24)
FSCR = exp(−0.649 × (SCr − SCRstd)), SCr in mg/dL
SCRstd = exp(−1.228 + 0.672 × log10(PMA) + 6.27 × exp(−3.11 × PMA))

Typical values (Colin 2019, Table 3):
  θCL   = 5.31 L/h per 70 kg
  θV1   = 42.9 L per 70 kg
  θV2   = 41.7 L per 70 kg
  θQ    = 3.22 L/h per 70 kg
  PMA50 = 46.4 weeks (maturation half-point)
  γ1    = 2.89 (maturation Hill coefficient)
  AGE50 = 61.6 years (PMA at which ageing halves CL)
  γ2    = 2.24 (age-decline Hill coefficient)
  θSCR  = 0.649 per mg/dL

Typical-adult reference check (Age 35 y, weight 70 kg, SCr 0.83 mg/dL):
  CL ≈ 4.10 L/h   V1 = 42.9 L   V2 = 41.7 L   Q = 3.22 L/h

Colin PJ, et al. Vancomycin pharmacokinetics throughout life: results from a pooled population analysis and evaluation of current dosing recommendations. Clin Pharmacokinet. 2019;58(6):767-780. doi:10.1007/s40262-018-0727-5 ↗

2. Two-compartment rate constants

Vancomycin behaves as a two-compartment drug: a central compartment (V₁) that contains the measured concentration, and a peripheral compartment (V₂) the drug distributes into and slowly returns from. The hybrid rate constants α (fast, distribution) and β (slow, terminal elimination) are the eigenvalues of the system.

k10 = CL / V1
k12 = Q / V1
k21 = Q / V2

α + β = k10 + k12 + k21
α × β = k10 × k21

α = ½ [(k10 + k12 + k21) + √((k10 + k12 + k21)² − 4·k10·k21)]
β = ½ [(k10 + k12 + k21) − √((k10 + k12 + k21)² − 4·k10·k21)]

A = (α − k21) / [V1 × (α − β)]
B = (k21 − β) / [V1 × (α − β)]

Half-lives:
  t½α = ln(2) / α    ← distribution half-life (~0.5–4h)
  t½β = ln(2) / β    ← terminal elimination half-life (~6–80h)

3. Single-dose concentration (IV infusion)

Closed-form solution for a constant-rate IV infusion of duration T_inf. During infusion the concentration builds; after the pump stops, it falls as a sum of two exponentials.

R0 = dose_mg / T_inf            (infusion rate, mg/h)

During infusion (0 ≤ t ≤ T_inf):
  C(t) = R0 × [ A/α × (1 − e^(−α·t))
              + B/β × (1 − e^(−β·t)) ]

After infusion (t > T_inf):
  C(t) = R0 × [ A/α × (1 − e^(−α·T_inf)) × e^(−α·(t − T_inf))
              + B/β × (1 − e^(−β·T_inf)) × e^(−β·(t − T_inf)) ]

4. Multi-dose superposition (accumulation to steady state)

For repeated dosing the total concentration at any time is the linear sum of single-dose contributions from every prior dose. The number of doses simulated is chosen so the curve spans at least 5 terminal half-lives (about 97% of steady state).

C_total(t) = Σ C_single(t − k·τ)   for k = 0, 1, 2, …, N−1
             where t ≥ k·τ

Number of doses simulated:
  t½β  = ln(2) / β
  N    = max(10, ceil(5 × t½β / τ) + 2)

This ensures the graph spans enough time for concentrations
to approach steady state.

5. Steady-state AUC₂₄

By linear pharmacokinetics, the steady-state daily exposure depends only on the daily dose and the patient’s clearance — independent of how the dose is split across the day.

AUC₂₄ = (dose_mg / CL) × (24 / τ)

Equivalently:
  AUC₂₄ = TDD / CL    (TDD = total daily dose)

This is exact under linear PK; peak and trough use the
two-compartment steady-state superposition formula
(not the multi-dose simulation) for maximum numerical accuracy.

6. Body size: no model switch

Colin 2019 is used for every adult at every body size, including BMI 40 kg/m² and above. Clearance and volumes scale with total body weight exactly as in section 1: CL and Q with (WT/70)^0.75, V1 and V2 with WT/70. There is no BMI threshold at which a different model or different equations are used.

At BMI 40 kg/m² or more, the calculator adds an advisory because published evaluation of this model at that body size is limited. The text comes from the same registry function the calculator uses; for a patient weighing 130 kg with a height of 175 cm it reads:

BMI 42.4 kg/m² (40 or more): estimates use the Colin 2019 model with total-body-weight scaling, the same model used for all adults. Published evaluation of this model at BMI 40 or more is limited (Colin 2021: 15 of 49 obese adults), so obtain vancomycin levels early to individualize. Fat-free mass and alternative creatinine-clearance estimates are shown for information only and do not change the calculation.

If height is not entered for a heavier patient, it reads:

Height was not entered, so BMI was not assessed. If BMI is 40 or more, published evaluation of the Colin 2019 model at that body size is limited; obtain vancomycin levels early.

Shown for information only

At BMI 40 kg/m² or more, the calculator also displays fat-free mass and alternative creatinine-clearance estimates so they can be compared with your own assessment. They use the formulas below and do not change the Colin 2019 calculation.

Fat-free mass (Janmahasatian 2005):
  Male:    FFM = (9270 × TBW) / (6680 + 216 × BMI)
  Female:  FFM = (9270 × TBW) / (8780 + 244 × BMI)

Cockcroft-Gault CrCl, on TBW, adjusted body weight or FFM:
  CrCl  = ((140 − age) × weight) / (72 × SCr)   [× 0.85 if female]
  IBW   = 50 kg (male) or 45.5 kg (female) + 2.3 kg per inch over 60 in
  AdjBW = IBW + 0.4 × (TBW − IBW)

Janmahasatian S, et al. Clin Pharmacokinet. 2005;44(10):1051–1065. doi:10.2165/00003088-200544100-00004 ↗ ·  Cockcroft DW, Gault MH. Nephron. 1976;16(1):31–41. PubMed ↗

7. MAP-Bayesian posterior fitting

Given measured levels, Vancomyzer fits the patient’s individual PK using maximum a posteriori (MAP) Bayesian estimation. It balances two things: how well the estimate fits the patient’s measured concentrations (under a normal assay-error model), against how far it strays from the population priors (a log-normal prior on each PK parameter). The fit is repeated from several starting points and the lowest-objective result is kept, which makes it less likely that the optimizer stops at a poor local solution.

minimize  Σᵢ ½·((Cᵢ_obs − Cᵢ_pred) / σᵢ)² + ln(σᵢ)
        + ½·(ln(CL/CL_prior) / ω_CL)²
        + ½·(ln(V1/V1_prior) / ω_V1)²
        + ½·(ln(Q /Q_prior ) / ω_Q )²
        + ½·(ln(V2/V2_prior) / ω_V2)²

Assay error model:
  σᵢ = max(1.0 mcg/mL, 0.15 × max(Cᵢ_obs, Cᵢ_pred))

Bounds: each posterior parameter clamped to [0.1×, 10×] of prior.

Prior log-SDs (Vancomyzer settings, all adults):
  ω_CL = 0.35    ω_V1 = 0.25    ω_Q = 0.50    ω_V2 = 0.50

With MAP estimation, a single observation is weighed against the prior rather than replacing it. The prior widths above are Vancomyzer settings, not the published Colin 2019 variability shown in section 1. When the residual exceeds 25% relative error, the calculator surfaces a Fit Quality Advisory and recommends a confirmatory level rather than overriding the prior.

8. AUC₂₄ target — ASHP/IDSA/PIDS/SIDP 2020

The therapeutic target of AUC₂₄ 400–600 mg·h/L (assuming MIC = 1 mg/L) follows the 2020 revised consensus guideline for serious MRSA infections. The guideline no longer recommends trough-only monitoring and recommends AUC-guided dosing, preferably with Bayesian estimation, citing data associating AUC-guided dosing with less acute kidney injury than trough-guided dosing. The engine’s recommendation search picks the dose × interval combination whose predicted steady-state AUC sits closest to the midpoint of this range.

Rybak MJ et al. Am J Health Syst Pharm. 2020;77(11):835–864. doi:10.1093/ajhp/zxaa036 ↗