KINETICSVerified Reference Parity
Bi-Exponential Decay
Two-phase exponential decay model resolving fast and slow relaxation lifetimes.
Fluorescence SpectroscopyPhotophysicsPharmacokineticsMaterial Science
Mathematical Formulation
$$y(t) = y_0 + A_1 \exp\left(-\frac{t}{\tau_1}\right) + A_2 \exp\left(-\frac{t}{\tau_2}\right)$$
Plaintext:
y = y0 + A1 * exp(-t / tau1) + A2 * exp(-t / tau2)Parameters & Physical Meanings
Detailed parameter breakdown, standard units, valid mathematical constraints, and initial guess heuristic algorithms.
| Symbol | Name | Unit | Constraints | Physical Interpretation | Initial Guess Heuristic |
|---|---|---|---|---|---|
| $y_0$ | Asymptotic Baseline Offset | Signal units | unconstrained | Residual background signal at infinite time | Estimated from the average of the last 5% of time points |
| $A_1$ | Fast Component Amplitude | Signal units | A1 >= 0 | Initial amplitude contribution from the fast decay pathway | 0.6 * (max(y) - y0) |
| $\tau_1$ | Fast Lifetime / Time Constant | Time (ns, μs, s) | tau1 > 0 | Characteristic lifetime of the rapid decay process | Estimated from initial tangent slope: 0.1 * total_time |
| $A_2$ | Slow Component Amplitude | Signal units | A2 >= 0 | Initial amplitude contribution from the slow decay pathway | 0.4 * (max(y) - y0) |
| $\tau_2$ | Slow Lifetime / Time Constant | Time (ns, μs, s) | tau2 > tau1 | Characteristic lifetime of the slow decay process | 0.5 * total_time |
When to Choose This Model
- Time-Correlated Single Photon Counting (TCSPC) fluorescence lifetime analysis
- Two-compartment pharmacokinetic clearance modeling (distribution vs elimination phases)
- Transient photoluminescence decay in semiconductor quantum wells and perovskites
- Multi-exponential NMR relaxation rate deconvolution
Typical Scientific Applications
- Fluorescence Lifetime Imaging (FLIM)
- Perovskite solar cell carrier recombination
- Two-compartment drug clearance modeling
- Phosphorescence decay lifetime fitting
Expected Fit Profile & Curve Morphology
Shape: exponential_decayExample Dataset & Expected Fit Output
Synthetic 20-point TCSPC fluorescence decay curve resolving fast lifetime tau1 = 1.45 ns and slow lifetime tau2 = 6.82 ns
R²:0.9997RMSE:4.1200
Expected Converged Parameters
- y03.650
- A1652.400
- tau11.450
- A2348.900
- tau26.820
Sample Experimental Vectors (18 points)
| # | x (Independent) | y (Observed) |
|---|---|---|
| 1 | 0.000 | 1005.000 |
| 2 | 0.500 | 684.000 |
| 3 | 1.000 | 482.000 |
| 4 | 1.500 | 351.000 |
| 5 | 2.000 | 264.000 |
| 6 | 2.500 | 204.000 |
| 7 | 3.000 | 161.000 |
| 8 | 4.000 | 108.000 |
| + 10 more points available in AltaiPlot preset | ||
Python / SciPy Reference Implementation
Reference library: scipy.optimize (Levenberg-Marquardt).Exact parameterization parity verified.
import numpy as np
from scipy.optimize import curve_fit
def bi_exponential_decay(t, y0, A1, tau1, A2, tau2):
return y0 + A1 * np.exp(-t / tau1) + A2 * np.exp(-t / tau2)
y0_init = np.mean(y_data[-max(int(len(y_data)*0.05), 3):])
amp_tot = np.max(y_data) - y0_init
t_span = np.max(t_data) - np.min(t_data)
p0 = [y0_init, amp_tot * 0.6, t_span * 0.1, amp_tot * 0.4, t_span * 0.5]
bounds = ([-np.inf, 0, 1e-12, 0, 1e-12], [np.inf, np.inf, np.inf, np.inf, np.inf])
popt, pcov = curve_fit(bi_exponential_decay, t_data, y_data, p0=p0, bounds=bounds)Comparative & Alternative Models
Key decision trade-offs between this model and related functional alternatives:
Frequently Asked Questions (Bi-Exponential Decay)
Why do bi-exponential fits sometimes fail to distinguish two lifetimes?
When tau1 and tau2 are separated by less than a factor of 2 to 3, the parameters become mathematically collinear. In AltaiPlot, multi-start solver initialization and residual map diagnostics help identify true global minimum lifetimes.
Scientific References & Citations
- Lakowicz, J. R. (2006). Principles of Fluorescence Spectroscopy (3rd ed.). Springer US.[Source / DOI]