Remaining Activity Calculator

In nuclear medicine, this formula is used to calculate the activity of a radiopharmaceutical at the time of patient injection, given its activity at the time of calibration (A₀). For example, if a ⁹⁹ᵐTc generator is eluted at 8:00 AM with 10 GBq, and the patient is scheduled for injection at 10:00 AM, the formula (with λ for ⁹⁹ᵐTc = 0.1155 h⁻¹) gives A = 10 × e^{-0.1155×2} ≈ 7.94 GBq. This ensures the prescribed administered activity is accurate, which is critical for diagnostic image quality and patient safety.

The decay constant and half-life are related by λ = ln(2)/T₁/₂. The formula A = A₀e^{-λt} can be rewritten as A = A₀ (1/2)^{t/T₁/₂}. While the exponential form with λ is mathematically direct, using half-life is often more intuitive for estimating remaining activity without a calculator (e.g., after 3 half-lives, it's 1/8). For accurate calculations, either form works, but λ is typically used in software and when dealing with multiple decay branches.

The exponential term e^{-λt} is dimensionless only if λ and t use the same time unit. If λ is given in s⁻¹, t must be in seconds; if λ is in h⁻¹, t must be in hours. Using mismatched units (e.g., λ in s⁻¹ and t in hours) will produce an incorrect exponent. For example, ⁶⁰Co has λ = 4.17×10⁻⁹ s⁻¹; to calculate activity after 1 year (3.15×10⁷ s), you must use seconds. Converting to years: λ_yr = λ × (365.25×24×3600) ≈ 0.131 yr⁻¹, then t in years. Always check unit consistency before plugging numbers.

Rearrange the formula: A₀ = A / e^{-λt} = A × e^{λt}. This is used to back-calculate the original activity at a reference time, for example, when calibrating a source that was measured after some delay. Suppose a source currently has 500 Bq and was known to have decayed for 30 days (with λ = 0.001 day⁻¹). Then A₀ = 500 × e^{0.001×30} = 500 × e^{0.03} ≈ 515.2 Bq. This is essential for source inventory tracking and for correcting decay in environmental samples.

For a mixture of independent radionuclides, the total remaining activity is the sum of the activities of each component: A_total(t) = Σ A₀ᵢ e^{-λᵢ t}. You cannot use a single effective half-life unless the components have similar decay constants. This summation is used in nuclear waste characterization, where a drum may contain multiple fission products with different half-lives. The formula for each isotope is applied separately, and the results are summed to get the total activity at any time. This is also used in decay heat calculations after reactor shutdown.

Remaining activity (A) is the number of decays per second, a source property. Dose rate is the energy deposited per unit mass per time at a point (e.g., mSv/h), which depends on the radiation type, energy, geometry, and shielding. While activity decreases exponentially with time, dose rate also decreases with the same exponential factor for a fixed geometry, but the proportionality constant (dose rate per activity) depends on the radionuclide and distance. You must apply a dose conversion factor (e.g., dose rate constant) to get dose rate from activity; the remaining activity formula alone only gives the decay rate.

Radiopharmaceuticals are prepared and then used at a later time. Quality control assays (e.g., measuring activity with a dose calibrator) are performed at a specific time. To compare results to the expected activity at the time of preparation (or to the label claim), you must apply the remaining activity formula to correct the measured activity back to the preparation time (or forward to the patient injection time). This is known as decay correction. For example, if the label says 10 GBq at 8:00 AM, and you measure 9.2 GBq at 9:00 AM, you use A₀ = A × e^{λt} to see if it matches the expected, ensuring proper dispensing.

The remaining activity uncertainty due to λ is σ_A = A₀ × e^{-λt} × t × σ_λ (for small uncertainties). This uncertainty grows linearly with time t and with the fractional uncertainty in λ. For short-lived isotopes used in medical applications (e.g., ⁹⁹ᵐTc, T₁/₂=6.02 h), the half-life is known very precisely (~0.1%), so the propagation is negligible. For long-lived isotopes like ⁹⁰Sr, the half-life uncertainty may be larger (few %), and over long periods (decades), this can introduce significant uncertainty in the predicted activity. In waste management, this must be accounted for in safety assessments.

The formula A = A₀e^{-λt} only applies after the source is removed from the production environment (i.e., during decay only). Under neutron irradiation, the activity builds up and saturates according to A(t) = (P/λ)(1 - e^{-λt}), where P is the production rate. Once irradiation stops, the remaining activity decays according to the standard exponential law. Therefore, the formula is used for decay after activation, not for the irradiation phase. For example, after a reactor irradiation, the activity at the end of bombardment (EOB) is calculated from the buildup equation, and then the remaining activity at a later time is calculated using the decay formula.

If you have two activity measurements A₁ at time t₁ and A₂ at time t₂, you can solve for λ: A₂ = A₁ e^{-λ (t₂ - t₁)}, so λ = (1/(t₁ - t₂)) ln(A₁/A₂). This is a common method to experimentally determine the half-life of a new isotope or to verify the purity of a sample (e.g., checking for the presence of contaminants with different half-lives by comparing decay over time). For example, if a sample's activity drops from 1000 Bq to 800 Bq in 5 days, λ = (1/5) ln(1000/800) ≈ 0.0446 day⁻¹, giving T₁/₂ ≈ 15.5 days.

Specific activity (Bq/g) is activity per unit mass. As the sample decays, both the total activity A(t) and the mass of the radioactive isotope decrease. The remaining activity formula A(t) = A₀e^{-λt} assumes the mass follows the same exponential decay. Therefore, the specific activity remains constant for a pure radionuclide if the decay product is stable and not counted in the mass. However, if the daughter is stable and accumulates, the total mass of the sample increases (or stays constant), and the specific activity decreases over time because the mass increases while activity decreases. In practice, for a pure isotope, the specific activity at any time is A₀e^{-λt} / (m₀e^{-λt}) = A₀/m₀, so it is constant. But if you include the daughter mass, the specific activity changes.

Remaining activity is the number of decays per second. Decay heat is the thermal power (watts) generated by the absorption of the emitted radiation (beta, gamma, alpha) in the material. Both decrease with the same exponential law: decay heat Q(t) = Q₀e^{-λt}, assuming the energy per decay is constant. However, for a mixture of isotopes, each contributes according to its own decay constant and its own average energy per decay. So the same form applies to heat if it's a single isotope. In spent fuel, the decay heat is the sum over many fission products, each with its own λ and energy, so it follows a more complex curve, not a simple exponential.

The formula A = A₀e^{-λt} is exact for all times, including fractions of a half-life. For short times compared to the half-life (t << T₁/₂), you can use the linear approximation: A ≈ A₀(1 - λt), which is the first-order Taylor expansion. For example, if t = 0.1 T₁/₂, the exact value is A = A₀ × 2^{-0.1} ≈ 0.933 A₀, while the linear approximation gives 1 - 0.693×0.1 = 0.9307, which is close. This approximation is sometimes used for quick estimates, but the exponential formula should be used for accuracy.

When a radionuclide is in the body, it is removed by both physical decay and biological excretion. The effective decay constant is λ_eff = λ_phys + λ_bio, where λ_bio = ln(2)/T_bio. The remaining activity in the body at time t is A(t) = A₀ e^{-λ_eff t}. This is used in the MIRD (Medical Internal Radiation Dosimetry) framework to calculate the cumulated activity (integral of activity over time) for dose estimation. The effective half-life is always shorter than the physical half-life. For example, ¹³¹I in the thyroid has a physical half-life of 8.02 days and a biological half-life of about 100 days, so the effective half-life is ~7.4 days.

After an accident, environmental samples (soil, water, vegetation) are collected and measured for activity. The remaining activity formula allows authorities to project how the activity will decrease over time for each radionuclide. This helps in planning remediation actions, setting exclusion zones, and assessing the long-term radiation exposure to the public. For instance, knowing that ¹³⁷Cs (T₁/₂=30 yr) will take many decades to decay significantly is essential for land-use decisions. The formula A(t) = A₀e^{-λt} is applied to each detected isotope to estimate future contamination levels, which feeds into dose assessment models.