Oral Multi-Dose Simulator

What is Accumulation?

When you take repeated doses of a drug, each new dose adds to what remains from previous doses. If the dosing interval (τ) is short relative to the drug’s elimination half-life (t_½), the drug accumulates until it reaches a repeating (periodic) pattern. In linear pharmacokinetics, that repeating pattern corresponds to a practical "steady state" where the average rate of drug entering the body matches the average rate of drug leaving the body [1][3].

The key relationship: the degree of accumulation depends strongly on the relationship between τ and t_½. Shorter intervals and/or longer half-lives generally mean more carryover from prior doses, hence more accumulation [3][1].

Key Metrics Explained

Accumulation Ratio (R)

The accumulation ratio is a compact way to describe how much overall exposure increases under repeated dosing. This simulator uses an AUC-over-a-dosing-interval ratio form (AUC_τ-based), consistent with common reporting of AUC_τ in multiple-dose settings [2].

Where AUC_τ,ss is the area under the concentration-time curve over one full dosing interval at steady state, and AUC_τ,ref is the reference AUC_τ used for comparison (commonly the first-dose interval or an early-dose interval, depending on study/reporting convention) [2].

• R = 1: No meaningful accumulation (interval exposure is similar across doses)
• R = 2: Interval exposure at steady state is 2× the reference interval exposure
• R > 3: Substantial accumulation; often seen when t_½ is long relative to τ

Helpful intuition (exact for multiple IV bolus dosing; often a decent approximation when absorption is rapid vs elimination): the classic accumulation factor is [6]

This is a closed-form result for repeated bolus input with first-order elimination; oral/extravascular cases can deviate when absorption is slow, but the dependence on k_e and τ remains the core driver of accumulation [6][3].

Steady State

Steady state is the practical condition where the concentration-time profile becomes periodic - each dosing interval looks the same (up to small numerical tolerance), reflecting balance between average input and average output [1].

Rule of thumb in linear pharmacokinetics: steady state is approached after approximately 4-5 half-lives of repeated dosing (i.e., most of the eventual accumulation has occurred) [1].

Implementation note: this simulator flags steady state numerically by comparing consecutive interval AUC_τ values and requiring the relative difference to fall below a tolerance (default: 5%). Because this is a tool definition rather than a universal pharmacokinetic law, you should treat the 5% threshold as a configurable engineering criterion.

Recovery Time

When a schedule disruption occurs (missed dose, late dose, etc.), the concentration-time profile deviates from the ideal baseline. Recovery time is defined here as the time required for the disrupted profile to return to within a specified tolerance (default: 5%) of the undisrupted baseline using interval AUC_τ comparison (AUC_τ is a standard per-interval exposure metric) [2].

Specifically, this simulator measures recovery as the time from the disruption until the first complete dosing interval where:

Recovery is typically slower for longer half-life drugs because deviations decay more slowly under first-order elimination [3].

Elimination Timeline (5 and 7 Half-Lives)

For first-order elimination, half-life means the amount declines by 50% each half-life. Therefore the fraction remaining after n half-lives is (½)ⁿ [1].

• 5 × t_½: (½)⁵ = 3.1% remaining → ~97% eliminated [1]
• 7 × t_½: (½)⁷ = 0.78% remaining → ~99% eliminated [1]

These are practical “washout intuition” benchmarks under first-order kinetics, not hard regulatory thresholds.

1-Compartment Model with First-Order Absorption

This simulator uses a standard one-compartment extravascular (oral) model that tracks an absorption-site amount and a central-compartment amount under first-order absorption and first-order elimination [5]. It tracks two state variables:

• A_g(t) - Amount at the absorption site (gut/depot), in mg
• A_c(t) - Amount in the central compartment, in mg

Governing Differential Equations

Under first-order assumptions, absorption is proportional to A_g(t), and elimination is proportional to A_c(t) [5]:

Where:

• k_a = absorption rate constant (h⁻¹)
• k_e = elimination rate constant (h⁻¹)
• F = bioavailability (fraction absorbed, 0-1)

Elimination Rate Constant

Under first-order elimination, the elimination rate constant relates to half-life as [5]:

T_max Relationship (for k_a solving)

For a single extravascular dose with first-order absorption and elimination (and no lag), the time to peak concentration (T_max) is [5]:

If the user provides T_max measured from ingestion and a lag time is enabled, the solver uses:
Tmax,abs = Tmax,ing − tlag (must be > 0)

Absorption Bucket Mapping (Mode A)

The "bucket" approach defines how quickly 90% of the dose is absorbed using a first-order absorption fraction [5]:

Extended Release (ER) Approximation

ER formulations are approximated by splitting a single dose into N micro-doses uniformly spaced over the release duration T_rel. This approximates a more continuous input into the absorption site, but it is still an approximation (discrete micro-dosing, not a true continuous zero-order release process).

Why V_d is Optional

All calculations are performed in amount space (mg). Concentration output (mg/L) requires dividing by the apparent volume of distribution [5]:

If you enter V_d, the plot can show "Model concentration" - but this is derived from user-entered parameters, not measured plasma values.

Saturable Elimination (Michaelis-Menten & Zero-Order)

Most drugs clear first-order: a fixed fraction leaves per hour, giving a constant half-life. A few important drugs saturate their metabolizing enzymes at ordinary doses and instead follow Michaelis-Menten kinetics:

When the amount is well below K_m the elimination term is approximately (V_max/K_m)·A_c, i.e. ordinary first-order. When it climbs well above K_m, the rate flattens toward a constant V_max - that is the zero-order limit (K_m → 0), where a fixed amount is cleared per hour and the curve falls in a straight line. There is no closed form, so this tool integrates these modes numerically (RK4). Two consequences matter clinically:

• Half-life is not constant, so the "5 and 7 half-lives" elimination timeline does not apply (the tool marks it n/a).
• Accumulation is nonlinear. Near saturation a small dose increase can raise the plateau far more than proportionally - the classic reason phenytoin dosing is hazardous. Load the phenytoin preset and step the dose from 300 to 400 mg/day to see it. Ethanol is the pure zero-order case: blood levels fall linearly at ~0.15 g/L/h. [5]

Aspirin Note: Modeling Salicylate

Intact acetylsalicylic acid (ASA) is rapidly converted to salicylate; in repeated-dose contexts, salicylate exposure is commonly the more relevant analyte for persistence/accumulation intuition than intact ASA [7][8].

Assumptions & Limitations (summary)

• First-order kinetics by default, with optional saturable elimination (Michaelis-Menten and its zero-order limit) for drugs like phenytoin and ethanol [5]
• Single compartment - no distribution phase [5]
• No drug interactions or disease effects (model does not change parameters dynamically)
• Parameters represent population-level estimates (not patient-specific)
• ER approximation uses discrete micro-doses, not true zero-order release

References

1. [1] NCBI Bookshelf (StatPearls). Pharmacokinetics. NCBI Bookshelf
2. [2] U.S. FDA. Clinical Pharmacology Review (Example: NDA 200327) - includes explicit definition of AUC_τ and "Accumulation ratio" as an AUC_τ ratio across dosing days. FDA PDF
3. [3] Toutain PL, Bousquet-Mélou A. Plasma terminal half-life. J Vet Pharmacol Ther. 2004;27(6):427-439. PubMed
4. [4] NCBI Bookshelf (StatPearls). Half Life. NCBI Bookshelf
5. [5] DTU (Technical University of Denmark). PK/PD modelling (course notes; 1-compartment oral model, equations, t_max, k_e-half-life relationship). PDF
6. [6] Bourne, D. Multiple IV Bolus Doses (Accumulation Factor). Boomer.org PDF
7. [7] Needs CJ, Brooks PM. Clinical pharmacokinetics of the salicylates. Clin Pharmacokinet. 1985;10(2):164-177. PubMed
8. [8] Levy G. Pharmacokinetics of salicylate in man. Drug Metab Rev. 1979;9(1):3-19. PubMed search

What This Model Does

• Simulates drug accumulation with repeat dosing using AUC-based interval comparisons [A5]
• Models first-order absorption from gut/depot to central compartment [A3][A4]
• Models first-order elimination from central compartment [A2]
• Handles schedule disruptions (missed, late, early, double doses)
• Provides uncertainty bands via Monte Carlo sampling (parameter range sampling)

What This Model Does NOT Do

• No medical recommendations - this is an educational tool only
• No nonlinear kinetics - cannot model saturable metabolism (Michaelis-Menten / capacity-limited clearance). Drugs with saturable metabolism can show disproportionate exposure changes with dose (e.g., phenytoin) [A1].
• No multi-compartment distribution - no peripheral compartments / distribution phase modeling [A4].
• No protein binding - the simulator does not distinguish free (unbound) vs bound drug, even though unbound concentration is often the driver of distribution and pharmacologic effect [A3].
• No drug interactions - cannot model enzyme induction/inhibition or competitive displacement. Many clinically relevant interactions occur via metabolism induction/inhibition and/or binding displacement [A1].
• No individual variability - uses population-like parameters, not patient-specific physiology.

When Outputs May Be Misleading

• Drugs with saturable metabolism at therapeutic doses (e.g., phenytoin) [A1] or cases where kinetics deviate from first-order (ethanol is a classic zero-order example) [A2]
• Drugs with significant distribution phases (often better described by 2+ compartment models) [A4]
• Enteric-coated / delayed-release formulations where absorption timing can be highly variable across people and conditions (example context: enteric-coated aspirin absorption variability) [A6]
• When comparing to measured plasma concentrations without validated/fit PK parameters (inputs here are simplified and user-provided)

References

1. [A1] DailyMed (US label). Phenytoin Sodium Injection - notes susceptibility to interactions due to saturable metabolism and extensive plasma protein binding. PDF
2. [A2] StatPearls. Half Life - discusses first-order half-life relationships and common examples where kinetics may deviate (e.g., ethanol as a zero-order example). NCBI Bookshelf
3. [A3] NCBI Bookshelf. Absorption, Distribution, Metabolism, and Excretion in Pharmacokinetics - covers protein binding concepts and why unbound drug matters for distribution/effect. NCBI Bookshelf
4. [A4] University lecture notes. Compartment Models - overview of 1-compartment vs multi-compartment models and distribution phases. PDF
5. [A5] Open-access PMC article discussing accumulation ratio definitions used in PK/BE contexts (AUC-based). PubMed Central (PMC)
6. [A6] Angiolillo DJ (review context on enteric-coated aspirin absorption variability / delayed absorption issues). SpringerLink