**3. Results**

The approach of the cumulative sum curve analysis is intended to demonstrate that mean perfusion changes under thermal stimuli may have different characteristics from volunteer to volunteer in the context of power spectral distribution. Using the two parameters AbC and DBP as a feature space, two binary linear classifiers based on linear discriminant analysis (LDA) were implemented to distinguish the group of patients and the group of volunteers 2, (*Y*1) as well as the groups of volunteers 1 and 2 (*Y*2) (Figure 8).

The calculated discriminant functions have been identified to have the following form:

$$\begin{aligned} \Upsilon1 &= -5.0622 + 0.84 \cdot X1\_{2p} + 0.0177 \cdot X2\_{2p} \\ \Upsilon2 &= -4.28022 + 0.86 \cdot X1\_{12} + 0.0076 \cdot X2\_{12} \end{aligned}$$

Table 1 shows the area under curve (AUC) values for the classifiers based on the discriminant functions calculated using only one of the two input parameters and for their combined use.

**Figure 8.** Scatter plots for analysed groups of patients and volunteers with the dividing lines of the LDA classifiers.

The scatter plots for the collected data in Figure 8 are divided into three areas. The first one, the lower-left corner of Figure 8 is represented by the group of patients. Group 1 and group 2 occupied the intermediate position and the upper-right corner, respectively. Group 2 and some individuals from the patient group are characterised by a high DPB parameter. Other individuals showed a relatively high value of the AbC parameter.

## **4. Discussion**

The observed effects are potentially connected with the presence of two ways of mean blood perfusion increase. One of them occurs when the value of the AbC parameter is relatively high. Apparently, the increase in mean blood perfusion is connected with the redistribution of the spectrum amplitude to higher frequencies of Doppler broadening. Whereas, if the DBP parameter increases and the AbC parameter is small, then the blood perfusion changes due to a proportional amplification in the spectrum amplitudes, which will not drastically change the shape of the cumulative sum curves. Thus, the shape of the cumulative sum curve calculated during stage 3 is changed weakly. Consequently, the AbC parameter does not substantially increase. Group 1 is characterised by both mechanisms: a sufficient broadening of the spectrum, and a significant DBP parameter. In the case of group 2, the trend is not recognised. However, group 2 is characterised by reduced values of the DBP and significantly high scattering values relative to the AbC parameter. As for the patient group, both parameters are significantly reduced.

Variations in the AbC parameter can be associated with various factors. The article [49] presents the results of modelling the distribution of photons over the Doppler shift frequencies taking into account changes in both the speed of scattering particles and their concentration. The simulation results show that both of these parameters simultaneously influence the shape of the Doppler spectra. Indeed, during the thermal test, both an increase in the RBC concentration and a change in their average rate can occur. Moreover, article [30] described that the shape of power spectra depends on a distribution characteristic of RBC speed. It is difficult to say with complete certainty which of the described factors is decisive for a particular functional test in healthy volunteers.Further research and modelling is required since Figure 8 shows a substantial variation of the results from healthy volunteers. However, for patients with type 2 diabetes mellitus, it is important to note that changes in the shape of the power spectra in response to the heat test are weak in comparison with cohorts of healthy volunteers, which is demonstrated by the AbC.

Power spectrum shape-changing (consequently, changes in cumulative sum curves) can occur for several reasons. It was previously described that severe geometric shape

changing in capillary loops happens in patients with T2DM [50]. On the example of nailfold capillaroscopy, giant capillaries, and an increase of the distance between capillaries, the appearance of convoluted and intersecting capillaries are shown. The article [51] described the speed of distribution in a healthy capillary loop. To the best of our knowledge, such a numerical experiment for pathological capillaries did not perform. However, in the presence of sharp turns, bends and intersections in a capillary loop, the blood flow will differ substantially from laminar flow. The diameter of capillaries and its influence on power spectra is also considered by Fredriksson et al. [52]. Possibly, the cross-section of capillary loops in patients with T2DM is larger than in healthy volunteers. Thus, a change in the geometric shape of the capillaries, the cross-section and a decrease in their number [53] lead to phenomena where an increase in blood perfusion occurs only due to a proportional increase in the power spectrum amplitudes with relatively constant overall shape of the function. From a clinical point of view, the low-values of parameters AbC and DBP in the cohort of patients can be observed due to the decreasing of capillary loop quantity and pathological changes in the capillaries where the RBC speed profile changes slightly during the applied thermal stimuli.

Thus, introduced in this study, AbC parameter is a novel approach for LDF signal processing. As seen for group 1 and 2 in Table 2, combined use of the parameters significantly increases the prediction value in comparison with the classifiers built using the parameters separately.

To date, there are several methods for recording speed-resolved blood perfusion. For example, PeriFlux 6000 EPOS (Perimed, Järfälla, Sweden) enable simultaneously registering blood perfusion in the frame of three speed ranges. This approach is based on a mathematical model that takes into account the concentration of RBC, blood oxygenation, the potential geometric shapes of capillaries, as well as the optical properties of biological tissues under study. These capabilities are achieved by combining the LDF with the diffuse reflectance spectroscopy measuring channel. A series of studies were made using that technique, for example by Wang et al. [54]. Nevertheless, the current study approach can be implemented using hardware of both classical LDF devices and ones implementing more advanced techniques with recording speed-resolved blood perfusion, supplementing the present arsenal of the signal processing methods in the field.

Calculated AUC-scores of *v*-based classifiers and classifier with extended feature space for three involved parameters for group 2 vs. patients and group 1 vs. group 2 is shown in Table 2. The combined use of the three parameters augmented with parameter of blood perfusion improves the classification between the compared groups. The proposed estimates represent different aspects of blood perfusion and complement each other.

Thus, the introduced AbC parameter can be used for the interpretation of the origin of observing alterations in mean blood perfusion indicating whether the changes are due to the RBC speed redistribution, or due to the proportional increase in all components of blood perfusion. The AUC-score for the classifier between aged healthy volunteers and patients is of almost the same value as for using a single DBP parameter, which can also be explained by the limited size of the tested cohort of volunteers. Identification of factors, which affect the shape of perfusion distribution by Doppler frequencies, can be of particular interest in the diagnostics of disorders characterised by changes in blood perfusion. It is important to note that the cumulative sum-based estimations such as Area between Curves can be combined with conventional perfusion measurements (CMBC and blood cell speed *v*). As is shown on Table 2, the combination of the proposed and conventional parameters is better suited for classification for both considered pairs.

The proposed approach can find applications in clinical practice and diagnostics without significant modifications of the standard configuration of LDF channel, facilitating the translation of the study results in routine diagnostics procedures.


**Table 2.** AUC-scores for the tested classifiers for the groups of volunteers and patients.
