1. Introduction

Flourescence anisotropy (FA) is a powerful tool for medical testing and biomedical research, probing changes that arise straightforwardly in cell biology. These include variations in local micro-viscosity or other constraints to diffusional motion of biomolecules, complex formation, and molecular proximity expressed by homo- or hetero-transfer of electronic energy [1,2].

Measurements of FA decay provide additional and complementary information to traditional steady state FA ($\bar{r}$) measurements. The form of the FA decay reflects the variety of rotational correlation times (τ_r), and thus reveals heterogeneity in the molecular population and changes in the size, shape, and segmental flexibility of the labeled biomolecule [3,4]. Therefore, measurements of time dependent FA are extremely popular in a range of applications including studying the dynamic properties of proteins [5–7], estimating the internal viscosities of membranes [8], performing fluorescence-polarization immunoassays [9–11] hetero- or homo-FRET assays [7,12,13], and monitoring cellular activities [14,15].

The FA is dependent upon the rate and extent of rotational diffusion during the fluorescence lifetime (FLT) of the excited state. FLT measurements, by themselves, provide unique information about the photo-physical properties of the fluorescing molecule, and the chemical and physical nature of its microenvironment due to its sensitivity to intra- and intermolecular processes [16]. Furthermore, FLT measurements are not subject to quantification difficulties as is common in the more traditional fluorescence intensity (FI) and FA measurements. Consequently, FLT measurements have been implemented for a large variety of applications in our lab [17–21] and others [16,22–26].

In recent decades, the implementation of time-resolved fluorescence (TRF) measurements in the frequency domain (FD) has become increasingly popular [24,27,28]. The basic principles of the FD analysis were previously discussed [29]. Compared to the more common time domain (TD) method, the measurement procedures and the data analysis in the FD method are significantly faster, and hence more suitable for clinical applications. Furthermore, there are a variety of works which signify the use of FD measurements for the resolution of FI and FA decays thus probing cell behavior and monitoring changes in its microenvironment [30–39].

However, applying a FD method of FI and FA decays for clinical research is more complex than the more conventional TD analysis as it requires the translation of the frequency response data (FRD) into TD data (elaboration on this matter is discussed in section S.1 in Supplement 1). This drawback is exacerbated as biological systems are inherently complex. The cellular milieu is naturally characterized by molecular heterogeneity. In addition, since biological molecules regularly exhibit molecular asymmetry and anisotropic rotational modes [27,40], the τ_r values in the FA decay are determined by the rates of rotation about the various molecular axes [41]. Consequently, the FI and the FA are usually characterized by complex decays (typically, multi-exponential as in Eq. (S1) and Eq. (S2) in Supplement 1, respectively) [24,27]. Complex FA decay is expected from non-spherical or long flexible molecules (like DNA) and segmental motions besides the overall rotational diffusion of fluorophore-biomolecule conjugates. Complex FI and FA decays are predictable for a single fluorophore in two or more environments or a mixture of fluorophores (known as associated FA decay) [42]. In addition, FA decay analysis requires the resolution of the FLT data and the determination of the fundamental anisotropy (r₀), each of which may be unknown [29].

All the above pose a significant challenge to resolving the underlying parameters describing the FI decay (FLT values, τ_i, and their species fraction, α_i) and the FA decay (τ_r values, θ_j, and their amplitudes, r_0j) [27]. Consequently, the reliability of TRF techniques can occasionally be inadequate for medical diagnostic procedures.

To address the FD-TD transformation challenge, in the last few decades a large variety of analytical tools have been developed and utilized [43]. The most popular is the least-squares fitting [44,45], which estimates the parameters describing the FA and FI decays based on the chi-square model ($\mathrm{\chi }$²), as presented in Eq. (S3) and Eq. (S4) in Supplement 1, respectively. Other dominant techniques include maximum likelihood estimation [44,46], global analysis [47,48], Bayesian analysis [49], phasor analysis [50], crossing point [51] and deconvolution methods. The latter includes stretched exponential analysis [52], moment analysis [53], and Laguerre deconvolution [54]. While these are commonly used methods, each involves several challenges. Common examples include poor accuracy for low signal-to-noise ratio (SNR) data, susceptibility to error from initial assumptions of the decay parameters or measurement conditions and the assumption of Gaussian- or Poisson-distributed noise. Furthermore, they require long computation times and do not utilize most of the frequency response information [43].

It is well-known in recent years that machine learning (ML) algorithms can be used to perform samples classifications and hence perform medical diagnosis with high accuracy [55]. Numerous ML methods have been used for medical diagnosis such as deep artificial neural network [56], Bayesian classifier [57], classification and regression tree [58] and linear discriminant analysis [59]. However, this paper deals with a different direction, therefore these methods will not be elaborated in the present work.

The FRD of the FI decay, by themselves, can achieve quantitative, valuable, diagnostic classification in a rapid and efficient manner with the D² approach (as previously demonstrated in [60]). The novelty of this approach stems from the circumvention of the estimation methods, since the D² characterizes and classifies samples without requiring the determination of the TD data through analytical tools such as nonlinear fittings. As such, it avoids the possible blurring of the data in the analysis process and the effect of analysis and model assumptions on the obtained results. These advantages were previously demonstrated by applying the D² method for bacterial and viral infection detection and classification [60].

This paper demonstrates the potential of using the D² approach on the FRD of the FA decay with the same principles as the FI decay. The following section discusses the theory for the D² method adjusted to analyze the FRD of the FA decay.

2. Theory: squared Euclidean distance (D²)

Today, the accepted method of characterizing FA decay in the FD requires the transformation of the FRD into FA decay data by nonlinear fitting algorithms. However, this method has a major drawback, namely the potential fail to extract the full correct TD data (the complete FLT or τ_r components as well as their amplitudes). In other words, the temporal resolution of the TD data is inherently limited using the estimation techniques as closely spaced FLT or τ_r components practically cannot be extracted [24,27]. Therefore, this paper discusses an alternative method of characterization based on squared Euclidean distance measured with respect to the raw FRD.

The standard Euclidean distance (D) is normalized (according to the variance) to proper and uniform scale based on the number of the modulation frequencies (n), where each frequency is defined by the angular modulation frequency, ω_i, used to achieve the FRD. The representative FRD values of each sample are dependent on the method of measurement and the type of sample. If an imaging system is used to acquire the FRD, then each sample is classified by first taking the spatial average FRD in a chosen region of interest. If the sample contains several subgroups (e.g., a sample with several cells), then taking the average between the different subgroups is necessary. Finally, the representative FRD values are compared between different samples/groups to determine similarity.

D² is calculated by:

The terms $\overline {\Delta \phi } _{{\omega _i}}^A$ and $\bar{\Lambda }_{{\omega _i}}^A$ refer to the arithmetical means of the measured phase shift difference between the two polarized components of the fluorescence emission and their AC ratio, respectively for sample A. For sample B, $\overline {\Delta \phi } _{{\omega _i}}^B$ and $\bar{\Lambda }_{{\omega _i}}^B$ are similarly described (for more information, see section S.1 in Supplement 1).

The means are not weighted and are calculated (at each ω_i) as follows:

The terms $\sigma _{sample\; A}^F$ and $\sigma _{sample\; B}^F$ refer to the SDs from calculating the means of the FRD (Δϕ and $\Lambda $) for each sample A and B.

Unlike the $\mathrm{\chi }$² model [Eq. (S3) in Supplement 1], the D² model is not based on a least-squares fit nor does it require any estimation techniques, which require a long computation time as well as may fail to resolve the TD data. Any two samples can be compared based on their raw FRD. In addition, the indices A or B can represent the mean FRD of a requested group where a series of samples is evaluated based on this group. In this case, the SD of this group is adjusted accordingly.

3. System design and methods

This section will describe the analyzed samples as well as the optical setup used to extract the FRD and the method by which D² is calculated and implemented.

3.1 Sample preparation

As a fluorescein-glycerol system is a popular model to study rotational dynamics [1,14], 4 fluorescein solutions (50 µM) were analyzed in this research. They were prepared by dissolving fluorescein (Sigma, St. Louis, MO) in Phosphate Buffered Saline (Biological industries, Kibbutz Beit Haemek 25115, Israel) solutions having different viscosity (different glycerol in Phosphate Buffered Saline concentrations: 0%, 30%, 60%, 80%). All samples had a pH value of 7.4.

3.2 Optical setup and analysis

The FRD were acquired using a time-resolved FA imaging (FD TR-FAIM) system, as is presented in Fig. 1. An existing FD-FLT imaging microscopy (FLIM) technology of Lambert instruments (Groningen, The Netherlands) [61] was adapted by mainly adding a linear polarizer and a polarized beam splitter (PBS) (Thorlabs Inc., New Jersey, United States) as will be explained below.

 

Fig. 1. An image of the FD TR-FAIM. (a) FA measurements are implemented by adding entrance polarizers to our FD-FLIM system. A vertical polarizer is deposited at the output of the LED source and a PBS at the input of the image intensifier. A pinhole at the input of the PBS adjusts the width of the FI to avoid overlap of the two polarization components of the fluorescence emission. (b) Using a mirror, the two polarized beams, which are extracted by the PBS, arrive in parallel to the CCD camera whose FOV is divided between the two.

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In this system, a light-emitting diode (LED) source with three different wavelengths of 403 nm, 468 nm (used in this work) or 537 nm, is modulated in a sinusoidal fashion by a signal generator (Prior OptiScan I, Rockland, MA). This sine wave (AC modulation superimposed upon DC) is characterized by a high frequency (in the range of 1-120 MHz) [61]. A vertical polarizer is placed at the output of the LED to excite the sample with vertically polarized modulated light.

An Olympus IX-81 inverted microscope with a 10X, NA = 0.4 objective (OLYMPUS, Tokyo, Japan) is used to focus the sample. Then, a dichroic mirror within the microscope filters the excitation so that only the FI arrives at the PBS that divides the polarized fluorescence emission to the vertical (${I_\parallel }$) and horizontal (${I_ \bot }$) orientations at the input of the image intensifier.

An added pinhole in the input of the PBS [Fig. 1(a)], is used to control the width of the fluorescence emission to avoid an overlapping between the two polarized beams. The output of the image intensifier is coupled to a charge-coupled device (CCD) camera (LI²CAM MD). The field of view (FOV) is divided between the two polarization components of the emission [Fig. 1(b)].

Each component of the emission responds with the same modulation frequency as the excitation but presents a different phase shift (ϕ) and a different decrease in modulation (m) with respect to the excitation. These FRD can be acquired by using a gain-modulated detection device (image intensifier) operating at the same frequency as the fluorescence emission but with numerous different phase angles between them (known as a homodyne phase-sensitive detection) [61]. In addition, these FRD depend on the decay constants of the fluorescent material and the modulation frequency. They are calculated in each individual pixel of the image in the FLIM software package computer program (LI-FLIM). This program fully controls the settings of the signal generator, light source, intensifier and camera, and presents the data in 2D presentations (with a resolution of 1392 × 1040 pixels).

Finally, the FRD of the FA decay is extracted by the phase shift difference between the two polarized emission components (Δϕ) and their AC ratio (Λ), which is calculated by multiplying their modulation and DC ratios:

3.3 Measurement procedure

The FRD of the FI and FA decay data were analyzed with MATLAB 2020b software (MathWorks, MA, United States). The mean FLT for each FI image and the mean τ_r, r₀ and $\bar{r}$ for each FA image were calculated after background subtraction and pixel registration of the two polarized emission images by finding the center of gravity of each polarized component and calculating the mean pixel values in a region of interest of 21X21 pixels. The correction function that compensates for any electrical or optical distortion of the polarization components (G factor) was measured with horizontal polarizer excitation and was ${\approx} $1. The FLTs were extracted through a separate experiment by replacing the PBS with a linear polarizer oriented 54.7° from the vertical (magic angle setting) [62]. Measurements were performed at ambient temperature (25°C).

3.4 Classification procedure

The following two subsections describe the steps for implementing the D² approach on a given FI and FA decay data (whether artificially created by simulation or received by experiment) or given the FRD directly (through FD instrumentation).

3.4.1 Simulated or experimental FI and FA decay data

The procedure by which the simulation analyses (section 4.1) were performed is presented hereinafter:

Table 1. Simulated data of single and multi-anisotropy decay.

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3.4.2 Experimental FRD

A reference of a fluorescein dye (the first sample in section 3.1) was used for calibration of the TR-FAIM system, described in section 3.2. This system was used to acquire the FRD parameters and their SDs. They were obtained in six frequencies between 10-120 MHz linearly spaced, and the specific steps are:

The apparent FI and FA decay parameters as well as the steady state FA of the four groups were extracted using Eqs. (S3)-(S11) in section S.2 in Supplement 1. This step is excluded from the D² classification but is used to validate the FRD measures.

3.5 Confusion matrix

The performance of D² was visualized via a confusion matrix plot, a technique for summarizing the performance of a classification algorithm in statistical classification [63].

On this plot, used in Fig. 2(e-f) and Fig. 3(c), the rows and columns correspond to the class yielded from the D² classification (output class) and the actual true class (target class), respectively [63]. The diagonal cells (green) present the number of correct predictions and their percentages relate to the total number of experiments. The off-diagonal cells (red) present the incorrect classifications. The far-right column presents the accuracy per predicted class (light gray), whereas the bottom row presents the accuracy per true class (light gray). The bottom right of the plot (dark gray) shows the overall accuracy. The confusion matrix easily reveals misclassifications between the tested classes/samples (i.e., mislabeling one class as another or one sample from one group as belongs to another group).

4. Results

In this chapter, the D² model was validated based on two simulated datasets of multi-exponential FA decay (Table 1 and Fig. 2), distinguished by the proximity between the τ_r values (θ_is). Furthermore, the robustness of the D² was demonstrated by the two simulations through suboptimal experimental conditions (increasing the noise level and changing the number of modulation frequencies, Table 2). Finally, the D² was used to classify experimental FD TR-FAIM measurements of 4 fluorescein- glycerol samples with increasing viscosity (Fig. 3 and Table 3). The TD data of these 4 samples were also extracted (Table 4).

4.1 Simulated data

Two simulations (I and II) were created to examine the D² model (the procedure by which the TD data were used to extract the FRD is elaborated on in section 3.4.1). Simulation I represents typical FA decay data of a fluorophore bound to protein and displaying segmental motion besides the overall rotational diffusion [64]. Simulation II represents closely spaced τ_r values, which typically cannot be distinguished (less than 20% difference as discussed in section S.1 in Supplement 1). The two simulations investigate classifications between six classes presenting similar FI and FA decay data (Table 1, Simulations I and II), according to observations of their FRD [Fig. 2(a-d)] and performing the D² classification [Fig. 2(e-f)].

 

Fig. 2. FRD (Δϕ_ω and r_ω) of simulation I (a and b) and simulation II (c and d) that result from the FA decay of one or two widely (simulation I) and closely (simulation II) spaced τ_rs. The FRD emphasize the close proximity of the data (some classes are not shown since they are almost identical to the others). In simulation I and simulation II, the FRD were extracted from the TD data using 100 modulation frequencies between 1MHz-5000 MHz only for the FRD presentation (a-d). However, the D² analysis used the FRD of only 10 modulation frequencies between 10MHz-120 MHz, which corresponds with our FD TR-FAIM system limitation. The D² method classified the different six classes with a total accuracy of 95% (e) and 96% (f), respectively.

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A single FLT was set to 1.00 ns in all classes in the two simulations. This FLT is a relatively difficult condition for performing classifications of the 2 simulated datasets as well as using our FD system. First, the FRD distribution shifts to higher modulation frequencies with shorter FLTs, and hence gets further away from our system limit. Second, the FRD are typically less sensitive to modulation frequency variations since the effect of the longer τ_r (about 5 ns) becomes less dominant. The fundamental anisotropy (${r_0} = \mathop \sum \nolimits_{j = 1}^2 {r_{0j}}$) was set to 0.4.

The presence of two or more τ_r values has a dramatic effect on the appearance of the FRD of the FA decay with respect to one τ_r. This is illustrated in the appearance of the modulated anisotropy (r_ω) and the differential phase angle (Δϕ) of simulation I, as presented in Fig. 2(a-b), respectively (in classes 3-6 compared to classes 1 and 2) for θ₁= 5 ns and θ₂= 50ps. Such τ_r values are expected for independent segmental motions of an aromatic ring within the protein or on the surface of the protein (such as tryptophan residue), which also undergoes overall slower rotational diffusion [64].

In addition, in simulation I the two τ_r values were widely spaced and therefore classes characterized by single and double τ_r values are clearly distinct by their FRD [Fig. 2(a-b)]. Yet, the 2 classes with a single exponential decay (classes 1-2) and the 4 classes with double exponential decays (classes 3-6) are indistinguishable. The presence of an unresolved motion (θ₂) can be detected by a failure of r_ω to approach the expected limiting value of r₀ [classes 3-6, Fig. 2(a)]. In addition, in a case of two τ_r values, instead of a single Lorentzian distribution for Δϕ, the differential phase angles show two such distributions, one for each τ_r [classes 3-6, Fig. 2(b)]. The peak of the first Lorentzian distribution is relatively low due to the short FLT (increasing the FLT to 5 ns gave 15° whereas increasing the FLT to 10 ns gave 20°, data is not shown). These phenomena are in agreement with the literature [27].

In simulation II, the proximity between the two τ_r values resulted in nearly identical r_ω and Δϕ in all six classes [Fig. 2(c-d), respectively]. One may easily miss the presence of the second component, especially when translating this FRD into TD data with the fitting algorithms.

Next, in each simulation, the 300 tests were classified by the D² approach. The accuracy and error of the classifications are presented through a confusion matrix [Fig. 2(e-f)]. The architecture of a confusion matrix was previously discussed in section 3.5. For example, in Fig. 2(e), the top left green diagonal cell denotes that 50/50 tests were correctly classified as class 1 (100% accuracy, bottom left corner). This corresponds to 16.7% of all the 300 tests. However, only 45/50 tests were correctly classified as class 3. This corresponds to 15% of all the 300 tests. One test was misclassified as class 4 and four tests were misclassified as class 5. This corresponds to 0.3% and 1.3% of the 300 experiments, respectively. In addition, the total accuracy in class 3 was 90% (column 3 in bottom line). Classes 2, 4-6 as well as the six classes in Fig. 2(f) are similarly described.

The D² classification categorized the 300 tests in each simulation into their six rightful classes with an overall accuracy of 95% for simulation I and 96% for simulation II [Fig. 2(e-f), respectively]. This is an acceptable result considering the extreme proximity between the FI and FA decay data. Less proximity would dramatically increase the accuracy as, for example, appeared in classes 1-2 in simulation I. The D² classification was repeated 10 times for the same classes in simulation I and II (with AWGN) and the total accuracy was 96.60$\pm$1.06% and 94.80$\pm$1.11%, respectively.

Table 2. Overall D² classification accuracy for the 2 simulations under increasing the noise factor and the no. of modulated frequencies (n)

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The two simulations were then repeated with a variable noise level as well as a different number of measured modulation frequencies (Table 2). Both indicate when the D² model fails to reach a desired classification accuracy threshold. The first, by determining how much noise the D² model can tolerate under a given number of modulation frequencies. The latter through how much the D² model can optimize the measurement time under a given noise level. The noise level was increased by changing the noise factor between 1.00-1.50 (i.e., the latter indicates an increase of 50% in the noise level). The number of modulation frequencies was increased in the range of 2-20. Each time, the modulation frequencies values were linearly spaced between 10-120 MHz. Again, in each simulation, each class was tested 50 times. The overall D² accuracy was calculated by repeating the process per given noise factor and number of modulation frequencies 200 times.

The D² was found as an effective classification technique for the two simulations under relatively challenging experimental conditions. Despite increasing the noise level 50% above the theoretical values (noise factor =1.5 in Table 2), the overall accuracy of the D² classification was above 90% when 10 modulation frequencies were used.

Furthermore, the addition of measured frequencies compensates for noisy data. For example, for an accuracy of at least 95%, both simulations demand 10 modulation frequencies under the presence of a moderate noise factor of 1.20 or less. However, if the noise factor increases, the number of modulation frequencies should increase to maintain the same accuracy. On the other hand, if one needed to decrease the measurement time at the expense of a slightly reduced accuracy of 90%, the D² implies that measuring 6 modulation frequencies would be enough (n = 6).

In conclusion, the robustness of D² classification was demonstrated under suboptimal conditions. The D² can determine under a given noise constraint, the minimal number of modulation frequencies for a requested classification accuracy.

4.2 Experimental results

Next, the concepts and technique of D² method were demonstrated on a fluorescein-glycerol system as a model for isotropic rotational dynamics with increasing viscosity (section 3.1). First, the FRD of the FI and FA decays were measured in six modulated frequencies between 10 to 120 MHz. The procedure by which the FRD were measured and used to perform the D² analysis is elaborated in section 3.4.2. Briefly, the FRD of the four samples were measured twice. This created a database of four groups by performing FRD averaging of the two experiments.

 

Fig. 3. FD data for FA decay of a 50 µM fluorescein solution with different glycerol concentrations (section 3.1). Generally, both r_ω (a), which at lower frequency modulations approaches $\bar{r}$, and Δϕ (b), which generally increased with the modulation frequency, exhibited distinguished values between the 4 samples with the variable glycerol concentration. The D² classified the different glycerol concentrations with a total accuracy of 100% (c).

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Then, each sample was measured another five times. Hence, 20 tests were created that needed to be associated with their rightful four groups. The viscosity alteration of the four groups resulted in distinct r_ω and Δϕ [Fig. 3(a-b)]. Thus, the D² classification successfully classified the 20 tests according to their rightful four groups (minimal D² value is marked with bold in each row) with 100% accuracy (Table 3).

The steady state FA ($\bar{r}$) of single exponential FI and FA decays can be extracted by the Jablonski equation, which is the anisotropy analogue of Perrins polarization equation [65,66]:

The findings, as shown in Table 4, indicate a clear dependence of FLT and $\bar{r}$ on glycerol concentration, a decrease from 4.00 ns at 0% to 3.61 ns at 80% and an increase of calculated $\bar{r}$ value from 0.01 to 0.22, correspondingly. The latter resulted from the strong dependance between the viscosity and τ_r as the latter increased from 0.19 ns to 5.94 ns, respectively. These trends agree with a previous publication [14].

In summary, this section demonstrated the potential of the D² in simulated data of widely spaced τ_r components, which suits a common situation of segmental motion of the aromatic ring in a protein that also undergoes overall rotation (simulation I). In addition, the advantages of the D² were exhibited on closely spaced simulated FA decay data, which suits the presence of two fluorophores or the same fluorophore in two separate but close environments (simulation II). The D² was successful in discriminating between several close classes in the two simulations, even under suboptimal conditions. Furthermore, the D² approach was tested on a fluorescein-glycerol system with different viscosities and again the D² easily distinguished between the different systems.

Table 3. Classification of the 20 samples into their rightful 4 groups based on the minimal D² value

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Table 4. Dependence of FI and FA decays on viscosity

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5. Discussion

The primary advantage of the TRF technique rests in its ability to characterize and categorize heterogeneity resulting from the alterations in the excited states kinetics (FLT) and in the complex diffusional dynamics (τ_r) at the molecular and cellular level. Thus, TRF assay is a promising technique for differentiation between molecular structures in biological cells, and hence performing qualitative diagnostic classification.

Measuring the FLT and τ_r in the FD has the potential to shed light on distinct photophysical characteristics or microenvironments in a rapid and efficient manner. Nevertheless, this ability may be reduced due to two main key points: the FD apparatuses and the complexity of the FI and FA decay. The D² approach can be used to address both challenges as will be discussed hereinafter.

5.1 FD apparatuses and conditions

The FD instrumentation plays a significant role in the determination of the extracted FI and FA decay data, and hence the ability of TRF measurements to distinguish between different models.

Clearly, judicious determinations of the controllable experimental variables would significantly increase the correctness of the experiments for a given data set. These especially include sensible choice of the modulation frequencies (range, values, and number), the region of interest, the gain amplification factors and, if requested, the added data processing techniques (e.g., filtering operations).

This could be one of the potential uses of the D². By changing a requested experimental variable (any of the stated above), the D² may clarify in a quantitative manner the most efficient experimental setting to achieve the optimal sample characterization. This principle was explored through examining the D² classification accuracy for the two simulations with varying noise level and number of frequencies (Table 2). As demonstrated, the D² can serve as a robust and reliable method for classification even under these suboptimal conditions.

As opposed to the range of modulation frequencies that are typically limited by the FD apparatuses, their number and values (within a given range) are typically unlimited. Since measurement time is a vital resource in medical diagnostics, the D² can optimize the minimal number of modulation frequencies requested for a given specific D² classification accuracy by sensible choice of their values in a given noise constraint (Table 2). This notion is elaborated by exhibiting the total accuracy of the D² classifications as a function of modulation frequencies for different noise levels [Fig. 4]. Following the same steps at the end of section 4.1, the D² classification accuracy was observed while increasing the number of modulation frequencies between 1-20 as well as the noise level using a noise factor that was changed between 1.00-1.40.

 

Fig. 4. Using the D² approach to determine the minimal number of modulation frequencies needed to accomplish a requested overall accuracy threshold for different noise factors for the classification of simulation I (a) and II (b). In this example, a threshold of 95% was determined. For example, in noise factor of 1.40, this is achieved for doubling the number of frequencies in simulation I (from 9 to 18) and simulation II (from 7 to 14)).

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In both simulations, a situation where the noise factor increased by 40% required doubling the number of modulation frequencies to maintain an accuracy threshold of 95% (in simulation I from 9 to 18 and simulation II from 7 to 14).

Another use of the D² is to explore whether the obtained FRD are sufficiently distinct, and hence applicable at all for classifying cell activation measurements for the specific application using the extraction of the FI and FA decays parameters.

5.2 Correct mathematical model for complex FI and FA decays

Unfortunately, typically there are inherent uncertainties in the extraction of the parameters describing the FI and FA decay data. These uncertainties are the result of the disadvantages of the curve fitting or estimations methods (such as poor accuracy for low SNR data and the susceptibility to error from initial assumption of decay parameters or the type of distributed noise) as well as the high correlation between the free variables that need to be determined. As the mathematical models become more complex, the uncertainties increase. Generally, the resolution of τ_r or FLT value starts to pose a difficulty in values with a less than two-fold difference [24,27]. As τ_i or θ_j become closer, the parameter values become more highly correlated. It was previously noted that two decay times spaced by a factor of 1.40 denotes the practical temporal resolution limit for FD analysis of double exponential decay and values that differ by about 20% or less cannot usually be resolved using TD or FD measurements [24,27]. This results in a misinterpretation of the biochemical processes or molecular features in question and consequently reducing or even eliminate the advantages of implementing TRF measurements.

The D² analysis is particularly well suited for performing sample classification in challenging experimental conditions. These include complex biological structures with multi exponential FI and FA decays, especially those that are characterized by closely spaced FLT and τ_r. When a situation of closely spaced decay times occurs, the D² offers an alternative rapid method of discriminating between different samples based on their FRD, which as is known and applied today, has no practical and diagnostic value.

Therefore, the D² can significantly facilitate physicians by performing sample classifications in a fast and effective manner beyond the practical temporal resolution limit due to the added information gained by the FRD (such as between infected and non-infected patients [19,60]). The latter opens the door for further and significant learning in complex systems (e.g., a significant improvement in the ability of FD TRF measurements to probe biochemical processes). There are other conventional methods that reduce the uncertainties of the TRF analysis as they avoid the need of nonlinear fitting with respect to the multi-frequency approach. These include the polar plot, the CRPO, and the noise corrected principal component analysis (NC-PCA) approaches. However currently, these methods do not utilize most of the FRD, which as proven in this paper, contain valuable information. In addition, the NC-PCA is inherent to single-photon detection such as the time-correlated single photon counting (TCSPC) technique [51,67,68].

The work in this paper is limited to single exponential decay of FI and single and double exponential decay of FA (previously, the D² was implemented on complex FI decays [60]). Since the D² method was experimentally demonstrated on homogenous solutions, the need to maintain the spatial resolution was less significant. However, the following works would be applying the D² approach in situations involving more complex biological environments that lead to more complex FI and FA decay simultaneously. For samples with heterogeneous structures, each sample can be divided into numerous subsamples where each can be analyzed and compare independently. Depending on the heterogeneity of the sample one can choose the size of the subsample which gives sufficient results. In this work there was no requirement for spatial resolution, hence for each sample, a region of interest of 21X21 pixels was determined out of the full samples region which was about 210X210. This means that if needed, each of the 4 samples may be divided into 10 different subsamples. Examples for complex biological environments include complicated fluorophore cluster geometries or nonisotropic distributions of fluorophores. There, this method could match between individual samples to their preliminary physician diagnosis based on changes in the diffusional rotational rates and/or the excited states kinetics.

6 Conclusion

This paper presents the D² approach, a simplified, efficient, and rapid method to utilize FD TRF measurements with increased reliability.

The usage of the D² approach for FD TRF analysis is valuable for several reasons. First, the D² can imply whether the experimental conditions of the FD instrument are sufficient to extract the FI and FA decay data, or how one may vary one or more experimental factors to achieve the optimal FD settings and conditions. Second, the D² approach may imply whether FA FD measurements are suited for the desired experimental application. Third, by creating and applying a designated FRD database for both FI and FA decays, the D² may confirm the presence or absence of different diseases even in cases where conventional TRF techniques or other methods of analysis fail. The latter is achieved as the D² approach could distinguish between samples with minor changes in their FRD that prevent them from being distinguished by the classical apparent FI and FA decay data. Thus, it is instructive to first create the FRD database of the requested groups that will serve as a ground for comparison. Then, using a learning process, perform the D² analysis to classify the requested given samples without the extraction of the FI and FA decay data. As such, the D² bypasses the distortion of the data or the concealing of the existing but hidden differences that result from close FLT or τ_r components (due to well-known and used mechanisms such as fluorescence resonance energy transfer (FRET), quenching, interacting ligands, free and bound fluorophore, and small molecular assemblies’ variations) [3,16,20,69,70].

The challenge to implement TRF assays in complex biological structures has been a barrier to the widespread application of TRF assays in common desired situations. The D² approach can contribute to the importance of the TRF assays for the health care system by achieving better diagnostic efficiency and quality in a shorter time.