Multi-exponential Model Research Articles

On the glow of cremated remains: long-lived green photo-luminescence of heat-treated human bones.

The long-lived green luminescence of human bone (that has been heated to 600°C for a short duration) is attributed to a carbon quantum dot material (derived from collagen) encapsulated and protected by an inorganic matrix (derived from bone apatite) and is more intense in dense rigid and crystalline parts of (healthy) human bones. The strong collagen-apatite interaction results (upon decomposition) in a protective inorganic environment of the luminescent centers allowing long-lived triplet-based emission of a carbon (quantum) dot-like material at room temperature, as well as resilience against oxidation between 550 and 650°C. The graphitic black phase (obtained upon heating around 400°C) is a precursor to the luminescent carbon-based material, that is strongly interacting with the crystalline inorganic matrix. Human bone samples that have been heated to 600°C were subjected to steady-state and time-resolved spectroscopy. Excitation-emission matrix (EEM) luminescence spectroscopy revealed a broad range of excitation and emission wavelengths, indicating a heterogeneous system with a broad density of emissive states. The effect of low temperature on the heat-treated bone was studied with Cryogenic Steady State Luminescence Spectroscopy. Cooling the bone to 80K leads to a slight increase in total emission intensity as well as an intensity increase towards to red part of the spectrum, incompatible with a defect state model displaying luminescent charge recombination in the inorganic matrix. Time-resolved spectroscopy with an Optical Multichannel Analyzer (OMA) and Time Correlated Single Photon Counting (TCSPC) of these samples showed that the decay could be fitted with a multi-exponential decay model as well as with second-order decay kinetics. Confocal Microscopy revealed distinct (plywood type) structures in the bone and high intensity-fast decay areas as well as a spatially heterogeneous distribution of green and (fewer) red emissive species. The use of the ATTO 565 dye aided in bone-structure visualization by chemical adsorption. Conceptually our data interpretation corresponds to previous reports from the material science field on luminescent powders.

Integrating plant morphological traits with remote-sensed multispectral imageries for accurate corn grain yield prediction.

Sustainable crop production requires adequate and efficient management practices to reduce the negative environmental impacts of excessive nitrogen (N) fertilization. Remote sensing has gained traction as a low-cost and time-efficient tool for monitoring and managing cropping systems. In this study, vegetation indices (VIs) obtained from an unmanned aerial vehicle (UAV) were used to detect corn (Zea mays L.) response to varying N rates (ranging from 0 to 208 kg N ha-1) and fertilizer application methods (liquid urea ammonium nitrate (UAN), urea side-dressing and slow-release fertilizer). Four VIs were evaluated at three different growth stages of corn (V6, R3, and physiological maturity) along with morphological traits including plant height and leaf chlorophyll content (SPAD) to determine their predictive capability for corn yield. Our results show no differences in grain yield (average 13.2 Mg ha-1) between furrow-applied slow-release fertilizer at ≥156 kg N ha-1 and 208 kg N ha-1 side-dressed urea. Early season remote-sensed VIs and morphological data collected at V6 were least effective for grain yield prediction. Moreover, multivariate grain yield prediction was more accurate than univariate. Late-season measurements at the R3 and mature growth stages using a combination of normalized difference vegetation index (NDVI) and green normalized difference vegetation index (GNDVI) in a multilinear regression model showed effective prediction for corn yield. Additionally, a combination of NDVI and normalized difference red edge index (NDRE) in a multi-exponential regression model also demonstrated good prediction capabilities.

Myelin water imaging at 0.55 T using a multigradient-echo sequence.

To investigate the prospects of a multigradient-echo (mGRE) acquisition for in vivo myelin water imaging at 0.55 T. Scans were performed on the brain of four healthy volunteers at 0.55 and 3 T, using a 3D mGRE sequence. The myelin water fraction (MWF) was calculated for both field strengths using a nonnegative least squares (NNLS) algorithm, implemented in the qMRLab suite. The quality of these maps as well as single-voxel fits were compared visually for 0.55 and 3 T. The obtained MWF values at 0.55 T are consistent with previously reported ones at higher field strengths. The MWF maps are a considerable improvement over the ones at 3 T. Example fits show that 0.55 T data is better described by an exponential model than 3 T data, making the assumed multi-exponential model of the NNLS algorithm more accurate. This first assessment shows that mGRE myelin water imaging at 0.55 T is feasible and has the potential to yield better results than at higher fields.

Predicting TERT promoter mutation status using 1H-MR spectroscopy and stretched-exponential model of diffusion-weighted imaging in IDH-wildtype diffuse astrocytic glioma without intense enhancement.

Isocitrate dehydrogenase (IDH)-wildtype diffuse astrocytic glioma with telomerase reverse transcriptase (TERT) promoter mutation is defined as glioblastoma by the WHO 2021 criteria, revealing that TERT promotor mutation is highly associated with tumor aggressiveness. The aim of this study was to identify features from MR spectroscopy (MRS) and multi-exponential models of DWI distinguishing wild-type TERT (TERTw) from TERT promoter mutation (TERTm) in IDH-wildtype diffuse astrocytic glioma. Participants comprised 25 adult patients with IDH-wildtype diffuse astrocytic glioma. Participants were classified into TERTw and TERTm groups. Point-resolved spectroscopy sequences were used for MRS data acquisition. DWI was performed with 13 different b-factors. Peak height ratios of NAA/Cr and Cho/Cr were calculated from MRS data. Mean apparent diffusion coefficient (ADC), perfusion fraction (f), diffusion coefficient (D), pseudo-diffusion coefficient (D*), distributed diffusion coefficient (DDC), and heterogeneity index (α) were obtained using multi-exponential models from DWI data. Each parameter was compared between TERTw and TERTm using the Mann-Whitney U test. Correlations between parameters derived from MRS and DWI were also evaluated. NAA/Cr and Cho/Cr were both higher for TERTw than for TERTm. The α of TERTw was smaller than that of TERTm, while the f of TERTw was higher than that of TERTm. NAA/Cr correlated negatively with α, but not with other DWI parameters. Cho/Cr did not show significant correlations with any DWI parameters. The combination of NAA/Cr and α may have merit in clinical situation to predict the TERT mutation status of IDH-wildtype diffuse astrocytic glioma without intense enhancement.

The influence of flip-flop in population pharmacokinetic analyses.

Flip-flop is a term used to describe the scenario where the rate constants for multiexponential models appear to be switched. However, in reality, it is a permutation of the rank order of the parameter values. In this perspective, we revisit flip-flop pharmacokinetics and discuss its influence in population pharmacokinetic analyses.

Multi-exponential model to describe pressure-dependent P- and S-wave velocities and its use to estimate the crack aspect ratio

We present new quantitative model describing the pressure dependence of acoustic P- and S-wave velocities. Assuming that a variety of individual mechanisms or defects (such as cracks, pore collapse and grain crushing) can contribute to the pressure-dependent change of the wave velocity, we order a characteristic pressure to all of them and allow a series of exponential terms in the description of the (P- and S-waves) velocity-pressure function. We estimate the parameters of the multi-exponential rock physical model in inversion procedures using laboratory measured P- and S-wave velocity data. As is known, the conventional damped least squares method gives acceptable results only when one or two individual mechanisms are assumed. Increasing the number of exponential terms leads to highly nonlinear ill-posed inverse problem. Due to this reason, we develop the spectral inversion method (SIM) in which the velocity amplitudes (the spectral lines in the characteristic pressure spectrum) are only considered as unknowns. The characteristic pressures (belonging to the velocity amplitudes) are excluded from the set of inversion unknowns, instead, they are defined in a set of fixed positions equidistantly distributed in the actual interval of the independent variable (pressure). Through this novel linear inversion method, we estimate the parameters of the multi-exponential rock physical model using laboratory measured P- and S-wave velocity data. The characteristic pressures are related to the closing pressures of cracks which are described by well-known rock mechanical relationships depending on the aspect ratio of elliptical cracks. This gives the possibility to estimate the aspect ratios in terms of the characteristic pressures.

Go’s and no-go’s towards more correct reverberation room measurements of absorption

Measurements of random-incidence absorption coefficients have been hotly contested since the ASA was founded in 1929. Yes, you read correctly, since 1929! Theories, explanations, myths, and strange interpretations appear, disappear, and reappear over the decades. What is wrong with this method of measurement? This paper discusses some of the methodological difficulties and reasons for discrepancies when comparing results between different reverberation chambers. It also presents possibilities arising from recent developments, such as the isotropy and angle dependence of the sound field incident on the absorber, the multi-exponential model for calculating decay times, and reference absorbers.

Vascular ageing and peripheral pulse: an improved model for assessing their relationship

Objective. Vascular ageing is associated with several alterations, including arterial stiffness and endothelial dysfunction. Such alterations represent an independent factor in the development of cardiovascular disease (CVD). In our previous works we demonstrated the alterations occurring in the vascular system are themselves reflected in the shape of the peripheral waveform; thus, a model that describes the waveform as a sum of Gaussian curves provides a set of parameters that successfully discriminate between under (≤35 years old) and over subjects (>35 years old). In the present work, we explored the feasibility of a new decomposition model, based on a sum of exponential pulses, applied to the same problem. Approach. The first processing step extracts each pulsation from the input signal and removes the long-term trend using a cubic spline with nodes between consecutive pulsations. After that, a Least Squares fitting algorithm determines the set of optimal model parameters that best approximates each single pulse. The vector of model parameters gives a compact representation of the pulse waveform that constitutes the basis for the classification step. Each subject is associated to his/her ‘representative’ pulse waveform, obtained by averaging the vector parameters corresponding to all pulses. Finally, a Bayesan classifier has been designed to discriminate the waveforms of under and over subjects, using the leave-one-subject-out validation method. Main results. Results indicate that the fitting procedure reaches a rate of 96% in under subjects and 95% in over subjects and that the Bayesan classifier is able to correctly classify 91% of the subjects with a specificity of 94% and a sensibility of 84%. Significance. This study shows a sensible vascular age estimation accuracy with a multi-exponential model, which may help to predict CVD.

Error performance optimization utilizing the dynamic detection cycle in a SPAD-based free space optical communication system.

Afterpulsing is a critical non-ideal factor of the single photon avalanche diode (SPAD) at telecommunication wavelength, which limits the performance of a SPAD-based free space optical communication (FSO) system. Afterpulsing probability (AP) is highly dependent on the SPAD detection cycle. In a conventional SPAD-based system, the detection cycle is set to a relatively large constant time length to mitigate afterpulsing. However, it will limit the SPAD counting rate and degrade system performance. In order to improve system performance, a new scheme of the dynamic detection cycle is proposed to adapt to different operation conditions. Then, a multi-exponential model of AP is built and fitted with the test data of the single photon detector QCD-300. Furthermore, based on the joint model of the multi-exponential model of AP and bit error rate (BER) model of the SPAD-based FSO system, a simple and effective optimization algorithm is developed to optimize the detection cycle. And the optimization of the detection cycle under different operation conditions is also investigated. The results indicate that optimal detection cycle is dominated by signal light, background radiation, and SPAD gate length. Compared with a conventional scheme with a constant detection cycle, the proposed scheme with the dynamic detection cycle can improve system error performance effectively.

Two-Photon Excited Fluorescence Dynamics in Enzyme-Bound NADH: the Heterogeneity of Fluorescence Decay Times and Anisotropic Relaxation.

The dynamics of polarized fluorescence in NADH in alcohol dehydrogenase (ADH) in buffer solution has been studied using the TCSPC spectroscopy. A global fit procedure was used for determination of the fluorescence parameters from experiment. The interpretation of the results obtained was supported by ab initio calculations of the NADH structure. A theoretical model was developed describing the polarized fluorescence decay in ADH-NADH complexes that considered several interaction scenarios. A comparative analysis of the polarization-insensitive fluorescence decay using multiexponential fitting models has been carried out. As shown, the origin of a significant enhancement of the decay time in the ADH-NADH complex can be attributed to the decrease of nonradiative relaxation rates in the nicotinamide ring in the conditions of the apolar binding site environment. The existence of a single decay time in the ADH-NADH complex in comparison with two decay times observed in free NADH was attributed to a single NADH unfolded conformation in the ADH binding site. Comparison of the experimental data with the theoretical model suggested the existence of an anisotropic relaxation time of about 1 ns that is related with the rotation of fluorescence transition dipole moment due to the rearrangement of the excited state NADH nuclear configuration.

Continuous diffusion spectrum computation for diffusion-weighted magnetic resonance imaging of the kidney tubule system

The use of rigid multi-exponential models (with a priori predefined numbers of components) is common practice for diffusion-weighted MRI (DWI) analysis of the kidney. This approach may not accurately reflect renal microstructure, as the data are forced to conform to the a priori assumptions of simplified models. This work examines the feasibility of less constrained, data-driven non-negative least squares (NNLS) continuum modelling for DWI of the kidney tubule system in simulations that include emulations of pathophysiological conditions. Non-linear least squares (LS) fitting was used as reference for the simulations. For performance assessment, a threshold of 5% or 10% for the mean absolute percentage error (MAPE) of NNLS and LS results was used. As ground truth, a tri-exponential model using defined volume fractions and diffusion coefficients for each renal compartment (tubule system: Dtubules , ftubules ; renal tissue: Dtissue , ftissue ; renal blood: Dblood , fblood ;) was applied. The impact of: (I) signal-to-noise ratio (SNR) =40-1,000, (II) number of b-values (n=10-50), (III) diffusion weighting (b-rangesmall =0-800 up to b-rangelarge =0-2,180 s/mm2), and (IV) fixation of the diffusion coefficients Dtissue and Dblood was examined. NNLS was evaluated for baseline and pathophysiological conditions, namely increased tubular volume fraction (ITV) and renal fibrosis (10%: grade I, mild) and 30% (grade II, moderate). NNLS showed the same high degree of reliability as the non-linear LS. MAPE of the tubular volume fraction (ftubules ) decreased with increasing SNR. Increasing the number of b-values was beneficial for ftubules precision. Using the b-rangelarge led to a decrease in MAPE ftubules compared to b-rangesmall. The use of a medium b-value range of b=0-1,380 s/mm2 improved ftubules precision, and further bmax increases beyond this range yielded diminishing improvements. Fixing Dblood and Dtissue significantly reduced MAPE ftubules and provided near perfect distinction between baseline and ITV conditions. Without constraining the number of renal compartments in advance, NNLS was able to detect the (fourth) fibrotic compartment, to differentiate it from the other three diffusion components, and to distinguish between 10% vs. 30% fibrosis. This work demonstrates the feasibility of NNLS modelling for DWI of the kidney tubule system and shows its potential for examining diffusion compartments associated with renal pathophysiology including ITV fraction and different degrees of fibrosis.

Blind deconvolution estimation by multi-exponential models and alternated least squares approximations: Free-form and sparse approach.

The deconvolution process is a key step for quantitative evaluation of fluorescence lifetime imaging microscopy (FLIM) samples. By this process, the fluorescence impulse responses (FluoIRs) of the sample are decoupled from the instrument response (InstR). In blind deconvolution estimation (BDE), the FluoIRs and InstR are jointly extracted from a dataset with minimal a priori information. In this work, two BDE algorithms are introduced based on linear combinations of multi-exponential functions to model each FluoIR in the sample. For both schemes, the InstR is assumed with a free-form and a sparse structure. The local perspective of the BDE methodology assumes that the characteristic parameters of the exponential functions (time constants and scaling coefficients) are estimated based on a single spatial point of the dataset. On the other hand, the same exponential functions are used in the whole dataset in the global perspective, and just the scaling coefficients are updated for each spatial point. A least squares formulation is considered for both BDE algorithms. To overcome the nonlinear interaction in the decision variables, an alternating least squares (ALS) methodology iteratively solves both estimation problems based on non-negative and constrained optimizations. The validation stage considered first synthetic datasets at different noise types and levels, and a comparison with the standard deconvolution techniques with a multi-exponential model for FLIM measurements, as well as, with two BDE methodologies in the state of the art: Laguerre basis, and exponentials library. For the experimental evaluation, fluorescent dyes and oral tissue samples were considered. Our results show that local and global perspectives are consistent with the standard deconvolution techniques, and they reached the fastest convergence responses among the BDE algorithms with the best compromise in FluoIRs and InstR estimation errors.

Transcapillary escape rate of 125I-albumin in relation to timing of blood sampling: the need for standardization

BackgroundIncreased vascular permeability is an early sign of vascular damage and can be measured with the transcapillary escape rate of albumin (TERalb). Although TERalb has a multi-exponential kinetic model, most published TERalb data are based on mono-exponential kinetic models with variation in blood sampling schemes. Aim of this posthoc study was to evaluate the influence of variation in blood sampling schemes and the impact of mono- or bi-exponential analyses on the calculation of TERalb. Study participants were part of a cross-over intervention study protocol, investigating effects of sodium loading on blood pressure, endothelial surface layer and microcirculation. Multiple blood samples were drawn between 3 and 60 min after injection of radioactive iodide labeled human serum albumin (rHSA).ResultsIn total 27 male participants with 54 measurements were included. For all participants the maximum serum radioactivity was reached within 20 min, while 85% of the participants had their maximum serum activity within 10 min. The TERalb calculated with the subsequently chosen T20–60 min reference scheme (6.19 ± 0.49%/h) was significantly lower compared to the TERalb of the T3–60 min, T5–60 min, and Tmax – 60 min schemes. There was no significant difference between the T20–60 min reference scheme and the T10–60 min and T15–60 min schemes. Bi-exponential kinetic modeling did not result in significant different observations compared to the mono-exponential kinetic analysis.ConclusionsAs there is variation in the timing of the maximum serum radioactivity of rHSA, blood sampling schemes starting before 10 min after administration of rHSA will result in a significant overestimation of TERalb. In addition, variation in kinetic modeling did not result in significant changes in TERalb. Therefore, we emphasize the need to standardize TERalb and for practical and logistical reasons advocate the use of a mono-exponential model with blood sampling starting 20 min after rHSA administration.

One-Dimensional Deep Learning Architecture for Fast Fluorescence Lifetime Imaging

We present a hardware-friendly deep learning architecture with one-dimensional convolutional neural networks (1D CNN) for fast analyzing fluorescence lifetime imaging (FLIM) data. A 1D CNN shows unparalleled advantages; they are more straightforward, quicker to train, and faster than high dimensional CNNs. 1D CNNs can be easily applied to multi-exponential fluorescence decay models. Compared with traditional least-square methods, superior performances of 1D CNNs on fluorescence lifetime image reconstruction have been validated using simulated data. We also employ the proposed 1D CNN to analyze two-photon FLIM images of functionalized gold nanoprobes in Hek293 and human prostate cancer cells. The results further demonstrate that 1D CNNs are fast and can accurately extract lifetime parameters from fluorescence signals. Our study shows that 1D CNNs have great potential in various real-time FLIM applications.

Distinguishing Electron and Hole Dynamics in Functionalized CdSe/CdS Core/Shell Quantum Dots Using Complementary Ultrafast Spectroscopies and Kinetic Modeling

The evolution of excitation energy and photogenerated charges in semiconductor quantum dots (QDs) functionalized with molecular acceptors can be probed on ultrafast time scales using techniques such as transient absorption (TA) spectroscopy. However, historical interpretations that the 1S(e)-1S3/2(h) transition in Cd-chalcogenide QDs is fully attributable to electrons may be misleading, and multiexponential models used to fit TA kinetics do not correspond directly to specific photophysical processes. Here, we present visible-wavelength and mid-IR TA and time-resolved photoluminescence measurements to inform a comprehensive kinetic model of the photoexcited CdSe/CdS core/shell QDs functionalized with passivating oleic acid (OA), hole-accepting ferrocene, or electron-accepting naphthalene bisimide (NBI). We show that ∼30% of the 1S signal and 72% of the IR signal can originate from holes in well-passivated core/shell QDs. We also demonstrate evidence of electron trapping in OA-capped core/shell QDs, with additional electron transfer and hole trapping in the QDs functionalized with NBI. Electron (hole) trapping and detrapping occur in 450 ± 100 ps (430 ± 70 ps) and 340 ± 100 ps (1.1 ± 0.4 ns) respectively, while the time constant for electron transfer to NBI is ∼1.8 ns. The comprehensive picture of photophysical processes provided by the complementary ultrafast techniques and kinetic modeling can accelerate both the fundamental science and application development of nanostructured and molecular systems.

Modeling of Thin-Film Lateral SOI PIN Diodes with an Alternative Multi-Branch Explicit Current Model

We propose the use of an alternative multi-exponential model to describe multiple conduction mechanisms in thin-film SOI PIN diodes with parasitic series resistance over a wide operating temperature range, from 90 to 390 K. This alternative multi-exponential model can be used for semiconductor junctions which exhibit multiple conduction mechanisms with series and shunt resistances. Using Thevenin’s theorem and the Lambert W function, the terminal current is expressed explicitly as a function of the terminal voltage. Its explicit nature allows higher computational efficiency and makes this model better suited for repetitive simulation applications than conventional implicit models. Additionally, direct analytic differentiation and integration are possible. This alternative model is used to describe the I-V characteristics of real SOI PIN diodes.

Investigations on Average Fluorescence Lifetimes for Visualizing Multi-Exponential Decays

Intensity- and amplitude-weighted average lifetimes, denoted as τI and τA hereafter, are useful indicators for revealing Förster resonance energy transfer (FRET) or fluorescence quenching behaviors. In this work, we discussed the differences between τI and τA and presented several model-free lifetime determination algorithms (LDA), including the center-of-mass, phasor, and integral equation methods for fast τI and τA estimations. For model-based LDAs, we discussed the model-mismatch problems, and the results suggest that a bi-exponential model can well approximate a signal following a multi-exponential model. Depending on the application requirements, suggestions about the LDAs to be used are given. The instrument responses of the imaging systems were included in the analysis. We explained why only using the τI model for FRET analysis can be misleading; both τI and τA models should be considered. We also proposed using τA/τI as a new indicator on two-photon fluorescence lifetime images, and the results show that τA/τI is an intuitive tool for visualizing multi-exponential decays.

Research on dynamic thermal performance of high-power ThinGaN vertical light-emitting diodes with different submounts

Investigation of dynamic thermal performance is a key to improve the heat management of high-power (HP) vertical light-emitting diodes (VLEDs). Specifically, the thermal time constant is a crucial parameter for optimizing the design and reliability of HP LEDs. Herein, the dynamic thermal behavior of seven HP ThinGaN VLEDs with different constructions was demonstrated. The LEDs’ thermal parameters were measured through the thermal transient tester system by a forward voltage technique. A three-stage of multiexponential function model was applied to divide the transient response curve into three regions with different thermal properties. This study focused on analyzing the first region that involved the chip region (epitaxial layer, wafer bonding layer, and submount) and chip bonding layer. The submounts of the LEDs under consideration include silicon carbide (SiC), silicon (Si), sapphire (Al2O3), and germanium (Ge). The results revealed that with a qualified wafer bonding layer, the LED packages with SiC, Si, and Al2O3 submount presented the optimum thermal time constant, which was 85, 69, 75, and 81 ms, respectively.

Second-order simultaneous components model for the overshoot and “slow component” in V̇O2 kinetics

The human oxygen uptake responses to exercise step on-transients present different shapes depending on the overshoot and/or the “slow component” manifestations. The conventional First-Order Multi-Exponential (FOME) model incorporates delayed add-on terms to comprise these phenomena, increasing parameter quantity, requiring a delayed recruitment of type II fibers to explain the “slow component,” and not offering a unified structure for different individuals and intensity domains. We hypothesized that a model composed of two Second-Order Simultaneous Components (SOSC) would present a better overall fitting performance than the FOME. Fourteen well-trained male cyclists performed repeated step on-transitions to moderate, heavy, and severe cycling intensities, whose responses were fitted with FOME and SOSC models. The SOSC presented significantly smaller (p < 0.05) root mean squared errors for moderate, supra-moderate, and all intensities combined. Along with conceptual analyses, these findings suggest the SOSC as a comprehensive alternative to the FOME model, explaining all oxygen uptake step responses with as many parameters and without delayed add-on components.

A general framework for non-exponential delayed fluorescence and phosphorescence decay analysis, illustrated on Protoporphyrin IX

Delayed fluorescence (DF) is a long-lived luminescence process used in a variety of applications ranging from oxygen sensing in biological tissues to organic Light Emitting Diodes. In common cases, DF results from the de-excitation of the first excited triplet state via the first excited singlet state of the chromophore, which produces a mono-exponential light signal whose amplitude and lifetime give an insight into the probed environment. However, non-linear de-excitation reactions such as triplet-triplet annihilation, which can cause decays to lose their mono-exponential nature, are often neglected. In this work, we derive a global framework to properly interpret decays resulting from a combination of linear and non-linear de-excitation processes. We show why the standard method of using multi-exponential models when decays are not mono-exponential is not always relevant, nor accurate. First, we explain why the triplet de-excitation and light production processes should be analyzed individually: we introduce novel concepts to precisely describe these two processes, namely the deactivation pathway – the reaction which mainly contributes to the triplet state de-excitation – and the measurement pathway – the reaction which is responsible for light production. We derive explicit fitting functions which allow the experimenter to estimate the reaction rates and excited state concentrations in the system. To validate our formalism, we analyze the in vitro Transient Triplet Absorption and DF of Protoporphyrin IX, a well-known biological aromatic molecule used in photodynamic therapy, cancer photodetection and oxygen sensing, which produces DF through various mechanisms depending on concentration and excitation intensity. We also identify the precise assumptions necessary to conclude that triplet-triplet annihilation DF should follow a mono-exponential decay with a lifetime of half the triplet state lifetime. Finally, we describe why the commonly used definitions of triplet / DF lifetime are ill-defined in the case where second-order reactions contribute to the deactivation process, and why the fitting of precise mixed-orders DF kinetics should be preferred in this case. This work could allow the correct interpretation of various long-lived luminescence processes and facilitate their understanding.