# Drug Accumulation Index Calculator (R = 1/(1-e^(-kτ)))

Calculate the drug accumulation index R under repeated fixed-interval dosing with linear first-order elimination: R = 1 / (1 - e^(-kτ)), where k = ln2 / t½ and τ is the dosing interval. R is the ratio of the steady-state peak concentration to the peak concentration after the first dose — it quantifies how much the body burden rises at steady state versus a single dose. For example, dosing once per half-life (τ = t½) gives R = 2. The term e^(-kτ) is the fraction of peak concentration remaining at the end of one interval. Valid only for linear first-order elimination with fixed dose and interval — NOT for Michaelis-Menten drugs (phenytoin) or non-linear kinetics. Not medical advice.

> Canonical page: https://elysiatools.com/en/tools/drug-accumulation-index

- **Category:** Health

- **Keywords:** accumulation index, R, Rac, accumulation factor, k, half-life, dosing interval, steady state, linear kinetics, pharmacokinetics, pk, drug, medicine

## Overview

Calculate the drug accumulation index R for repeated fixed-interval dosing with linear first-order elimination. Enter the elimination half-life and dosing interval to calculate R, the ratio of the steady-state peak concentration to the peak after the first dose, along with k and the fraction remaining between doses.

## Inputs

- **Half-life t½ (h)** (number): Elimination half-life. Must be positive.
- **Dosing interval τ (h)** (number): Time between consecutive doses in hours. Must be positive.
- **Decimal Places** (number)

## When to use

- Estimate accumulation when doses are repeated at a fixed interval under linear first-order elimination.
- Compare how dosing intervals relative to the half-life affect the accumulation index.
- Review the calculated elimination rate constant and fraction remaining at the end of each interval.

## How it works

- Enter a positive elimination half-life t½ in hours and a positive dosing interval τ in hours.
- The calculator determines the elimination rate constant using k = ln2/t½.
- It calculates the fraction remaining after one interval as e^(-kτ).
- It returns R = 1/(1-e^(-kτ)), with k, the fraction remaining, and the selected decimal precision in JSON output.

## Use cases

- Pharmacokinetic calculations using a known drug half-life and fixed dosing interval.
- Teaching how half-life and dosing interval influence repeated-dose accumulation.
- Checking whether a schedule spaced several half-lives apart produces relatively limited accumulation.

## Frequently asked questions

### What does the accumulation index R represent?

R is the ratio of the steady-state peak concentration to the peak concentration after the first dose.

### What inputs are required?

A positive half-life in hours and a positive dosing interval in hours are required.

### What does k mean?

k is the first-order elimination rate constant, calculated as ln2 divided by the half-life, and expressed in h⁻¹.

### What happens when the dosing interval equals the half-life?

The accumulation index is R = 2, and the fraction remaining at the end of the interval is 0.5.

### Does this calculator apply to every drug?

No. It is valid only for linear first-order elimination with fixed doses and intervals, not nonlinear kinetics such as Michaelis-Menten elimination.

## Related tools

- [Base Excess Calculator (Metabolic Component)](https://elysiatools.com/en/tools/base-excess-calculator): Calculate base excess (BE) using the Siggaard-Andersen formula. BE < −3 metabolic acidosis, > +3 metabolic alkalosis. HCO₃⁻ auto-calculated from pH + PaCO₂ if omitted. Derived from Siggaard-Andersen 1974, Cornell PICU, Medscape, and Langer 2022. Not medical advice.
- [Blood Gas Acid-Base Interpretation (AG / ΔAG)](https://elysiatools.com/en/tools/blood-gas-anion-gap): Complete ABG analysis: pH, primary disorder, compensation (Winter's formula), anion gap ± albumin correction, delta ratio for mixed disorder detection. Derived from Emmett 2016, Kraut 2007, Rastegar 2007, LITFL, MDCalc, and Adrogué 1998. Not medical advice.
- [Caprini Score (VTE Risk, 2005)](https://elysiatools.com/en/tools/caprini-score): Calculate the 2005 Caprini Risk Assessment Model to stratify venous thromboembolism (VTE) risk in surgical patients. ~40 items weighted 1/2/3/5 points: 1 point (age 41-60, minor surgery, BMI >25, swollen legs, varicose veins, pregnancy/postpartum, recurrent miscarriage, OCP/HRT, sepsis <1mo, lung disease/pneumonia <1mo, COPD, acute MI, CHF <1mo, bedridden, medical bed rest, IBD); 2 points (age 61-74, arthroscopic surgery, major open surgery >45min, laparoscopic >45min, malignancy, bed >72h, plaster cast, central line); 3 points (age ≥75, personal/family VTE history, factor V Leiden, prothrombin 20210A, lupus anticoagulant, anticardiolipin, homocysteine, HIT, other thrombophilia); 5 points (stroke <1mo, elective major lower-extremity arthroplasty, hip/pelvis/leg fracture <1mo, acute spinal cord injury <1mo, multiple trauma <1mo). Management tiers: 0 lowest, 1-4 low-moderate (mechanical), 5-6 high (consider LMWH 7-10d), 7-8 high (LMWH 7-10d), ≥9 highest (LMWH 30d). Derived from Caprini 2005. Not medical advice.
- [CHA₂DS₂-VASc Score (AFib Stroke Risk)](https://elysiatools.com/en/tools/chads2-vasc-score): Calculate the CHA₂DS₂-VASc score to stratify stroke risk in non-valvular atrial fibrillation and guide oral anticoagulation. Components: Congestive heart failure/LV dysfunction (+1), Hypertension (+1), Age ≥75 (+2), Diabetes (+1), prior Stroke/TIA/thromboembolism (+2), Vascular disease — prior MI/PAD/aortic plaque (+1), Age 65–74 (+1), Sex category female (+1). Range 0–9 (women) / 0–8 (men). Thresholds: men ≥2 or women ≥3 → recommend oral anticoagulation (DOAC preferred over warfarin); men 1 or women 2 → consider anticoagulation; men 0 or women ≤1 → omit. Female sex alone is not an independent risk factor (score 1 from sex alone is treated as 0). Includes an approximate annual stroke-risk estimate (Lip 2010 derivation cohort). Derived from Lip 2010 (Chest), 2019 AHA/ACC/HRS update, and 2023 ACC/AHA/ACCP/HRS AF guideline. Combine with bleeding risk (HAS-BLED) and shared decision-making. Not medical advice.
- [Charlson Comorbidity Index (Age-Adjusted)](https://elysiatools.com/en/tools/charlson-comorbidity-index): Calculate the Charlson Comorbidity Index (CCI), including the age-adjusted variant, to predict 1-year mortality from chronic comorbidities. 19 comorbidities are weighted by mortality risk: 1 point (MI, CHF, PVD, cerebrovascular disease, dementia, chronic pulmonary disease, connective tissue disease, peptic ulcer, mild liver disease, diabetes without complications); 2 points (hemiplegia/paraplegia, moderate-to-severe CKD, diabetes with end-organ damage, any non-metastatic tumor, leukemia, lymphoma); 3 points (moderate-to-severe liver disease); 6 points (metastatic solid tumor, AIDS). Age adjustment adds 1 point per decade ≥50 (50-59 +1 … ≥90 +5). Interpretation: 0 ~8%, 1-2 ~25%, 3-4 ~50%, ≥5 ~80-90% 1-year mortality (derivation cohort). Derived from Charlson 1987 (J Chronic Dis) and the 1994 validation. Not medical advice.
- [CURB-65 vs PSI vs SMART-COP (Pneumonia Severity Comparison)](https://elysiatools.com/en/tools/curb-65-vs-pneumonia-severity): Compare three pneumonia severity scores side-by-side from one set of inputs to support concordance reading. CURB-65 (0-5: Confusion, Urea, RR, BP, age ≥65) — quick bedside triage; PSI/PORT (Fine 1997, 5 risk classes) — the most accurate mortality stratification, more complex; SMART-COP (0-10, 2 points each for SBP <90 and hypoxia, 1 point each for multilobar CXR, albumin <3.5, RR ≥25, tachycardia ≥125, confusion, pH <7.35) — predicts need for ICU respiratory/vasopressor support. Each score has a different focus: CURB-65 is fast; PSI is more precise; SMART-COP identifies who needs ICU. The tool reports each score's total/category/interpretation and an agreement analysis (whether all three point to outpatient/admission/ICU). When scores disagree, favor the more conservative disposition. Derived from Lim 2003, Fine 1997, Charles 2008. Not medical advice.
- [Serial Dilution Final Concentration (C_f = C₀ × ∏ Vᵢ/(Vᵢ+V_d))](https://elysiatools.com/en/tools/dilution-factor-final): Compute the final concentration after multi-step serial dilutions, with per-step folds, cumulative folds, and total dilution factor.
- [HAS-BLED Score (Major Bleeding Risk)](https://elysiatools.com/en/tools/has-bled-score): Calculate the HAS-BLED score to estimate major bleeding risk in patients on oral anticoagulation (most commonly combined with CHA₂DS₂-VASc in atrial fibrillation). Each component scores 1 point: Hypertension (uncontrolled, systolic BP >160), Abnormal renal function (dialysis/transplant/Cr >2.26 mg/dL), Abnormal liver function, prior Stroke, prior major Bleeding/predisposition, Labile INR (TTR <60% on warfarin), Elderly (age >65), Drugs (antiplatelet/NSAID), Alcohol (≥8 drinks/week). Range 0–9 (renal/liver and drugs/alcohol can each contribute up to 2). Interpretation: 0 low, 1–2 moderate, ≥3 high risk — regular review and correct reversible factors; a high score alone does NOT justify withholding anticoagulation. Derived from Pisters 2010 (Chest) and the 2023 ACC/AHA/ACCP/HRS AF guideline. The 'labile INR' criterion mainly applies to warfarin users; predictive value is lower for DOACs. Not medical advice.

## Samples

- [Web Image Processing Python Samples](https://elysiatools.com/en/samples/web-image-processing-python): Web Python image processing examples using PIL/Pillow including reading, saving, resizing, and format conversion
- [Android Image Processing Java Samples](https://elysiatools.com/en/samples/android-image-processing-java): Android Java image processing examples including reading/saving images, scaling, and format conversion
- [Android Image Processing Kotlin Samples](https://elysiatools.com/en/samples/android-image-processing-kotlin): Android Kotlin image processing examples including reading/saving images, scaling, and format conversion
- [Web Image Processing Rust Samples](https://elysiatools.com/en/samples/web-image-processing-rust): Web Rust image processing examples including image read/save, scaling, and format conversion

## Related content

- [Pharmacokinetic Teaching and Dose-Model Checks](https://elysiatools.com/en/hubs/pharmacokinetic-dose-planning-calculators): Explore pharmacokinetic formulas, model assumptions, and result consistency for teaching or QA without giving medication dosing or clinical advice.