Equations & derivations

1. Population clearance — Colin 2019

CL 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.

CL (L/h) = θCL × FSize^0.75 × FMat × FDecline × FSCR

  FSize    = WT / 70
  FMat     = PMA(wk)^γ1 / (PMA(wk)^γ1 + PMA50^γ1)
  FDecline = 1 / (1 + (PMA(yr) / AGE50)^γ2)
  FSCR     = exp(-θSCR × (SCr − SCRstd))
  SCRstd   = exp(-1.228 + log10(PMA(yr)) × 0.672 + 6.27 × exp(-3.11 × PMA(yr)))

Constants (Table 3):
  θCL    = 5.31 L/h per 70 kg
  θSCR   = 0.649 (mg/dL scale)
  PMA50  = 46.4 weeks
  γ1     = 2.89
  AGE50  = 61.6 years     ← 50% CL reduction at this age
  γ2     = 2.24

Reference patient (35 yr, 70 kg, SCr 0.83) → CL 4.10 L/h ✓

Colin PJ et al. 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 4–5 terminal half-lives (95–97% 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(6, ceil(4 × 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. Obesity model (BMI ≥ 40)

For patients with BMI ≥ 40, Vancomyzer activates a separate prior derived from morbid-obesity cohorts (Smit 2020 + Zhang 2024). Volumes scale to Fat-Free Mass (vancomycin is hydrophilic; it doesn’t distribute into adipose), while clearance retains a TBW component because renal elimination scales with total body weight. Colin 2019’s age-decline factor is composed on top — a small but defensible bridge of the geriatric-obesity gap that the source cohorts under-represented.

CL = (0.0571 × CrCl + 0.0158 × TBW) × FDecline(age)
V1 = 0.287 × FFM     (central volume — adipose excluded)
V2 = 0.89  × FFM     (peripheral volume — adipose excluded)
Q  = 1.23 L/h         (fixed intercompartmental clearance)

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

Cockcroft-Gault CrCl (TBW, female correction):
  CrCl = ((140 − age) × TBW) / (72 × SCr)   [× 0.85 if female]

IIV (used as prior log-SDs in MAP):
  ωCL = 0.29
  ωV1 = 0.32
  ωV2 = 0.28

Smit C et al. Br J Clin Pharmacol. 2020;86(2):303–317. doi:10.1111/bcp.14144 ↗ ·  Zhang T et al. Clin Pharmacokinet. 2024;63:79–91. doi:10.1007/s40262-023-01324-5 ↗ ·  Janmahasatian S et al. Clin Pharmacokinet. 2005;44(10):1051–1065. doi:10.2165/00003088-200544100-00004 ↗

7. MAP-Bayesian posterior fitting

Given measured levels, Vancomyzer fits the patient’s individual PK by minimizing a Bayesian objective: the sum of negative-log-likelihood of the observations under a normal assay-error model and a log-normal prior penalty on each PK parameter. The minimization runs in log-space with multi-start Nelder-Mead so single-level outlier cases don’t get stuck in local minima.

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 (default):
  ω_CL = 0.35    ω_V1 = 0.25    ω_Q = 0.50    ω_V2 = 0.50
  (overridden by Smit 2020 ωs in obesity branch)

By design, a single observation cannot dominate the prior — that’s the safety property MAP estimation provides. 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. It replaces trough-only monitoring for serious MRSA infections, citing improved nephrotoxicity outcomes at equivalent efficacy. 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 ↗