# Blood Pressure Estimation by Photoplethysmogram Decomposition into Hyperbolic Secant Waves
## 4 Discussion  
Here, we compared models of blood pressure description by sech PDA and by Gaussian PDA, and their characteristics. Gaussian PDA showed the best performance in the literature on PDA blood pressure measurement [8]; however, sech PDA has not been tested in the literature for blood pressure estimates. Our results showed that the use of sech PDA instead of Gaussian PDA could lead to more accurate and robust blood pressure measurement. Our main results are shown in Figures 7 and 8. Sech PDA better explains BP (78% vs. 58%, \(p\) = 0.0000) than Gaussian PDA, in our data. Our data is challenging because it presented a large variation in BP in 8 min (Figure 5). To prevent possible issues in the comparison between the sech and the Gaussian function PDA, in our tests, we did not complete a full prediction, but instead performed a Leave One Out model test, in 8-s intervals. The choice of Leave One Out and the choice of the 8-s intervals were arbitrary. Figure 10: Mean squared error between the original pulse wave and the reproduced wave based on Gaussian PDA or sech PDA in each subject. Figure 9: The multiple correlation coefficients for all the subjects’ data and the correlation coefficients for all the subjects’ data with Leave One Out analysis. The reduction in the coefficients compared to Figure 7 implies that PDA for each individual must be calibrated by an independent regression. Results of Figures 6 and 9 do not show statistically significant differences between sech PDA and Gaussian PDA for timeframes (averages) of the order of 80 minutes, with averages also between individuals. This was expected since Gaussian PDA is a good method for BP estimates. In addition, our results of Figures 7 and 8 imply that sech PDA correlated at least as well with blood pressure as Gaussian PDA. We also found that multiple regression between PDA features and blood pressure, using all features, achieved higher accuracy compared to simply using one feature--which was not surprising. We had a higher accuracy with all features than the other combinations of features in the multiple correlation coefficients. Arithmetic operations (such as the ratio index a2/a1 used in [13]) gave a lower (worse) multiple correlation coefficient for all subject data. We believe this behavior was caused by the dimensionality loss in arithmetically combining multiple features, worsening the pulse contour description. Figure 10 shows that the mean square error of the PDA fits was slightly lower for Gaussian PDA than for sech PDA. Because sech PDA gave a better correlation with blood pressure, we speculated that if sech did not provide a better fit to the pulse than the Gaussian function, this was due to the difference in the importance of different regions of pulse wave contour for blood pressure estimates [16]. Pulse wave contours reproduced by Gaussian PDA and sech PDA fit different contours, despite the same goodness of fit. Such analysis is beyond the scope of the present paper. An important early study based on Gaussian PDA [7] found R\({}^{2}\) correlations with central line catheter SBP and DBP to be 0.92 and 0.78, respectively. By comparison, our study shows the R\({}^{2}\) was 0.58 for SBP based on the Gaussian PDA (Figure 8a), and 0.76 for SBP based on the sech PDA (Figure 8b), with ground truth reference BP peripherally measured in the finger. While [7] used data from 38 male and 25 female patients of 62.7 \(\pm\) 11.5 years old, our study used data of healthy volunteers 22.4 \(\pm\) 1.7 years old (8 males and 2 females) whose blood pressures varied by up to 60% during data collection, which made the prediction in our case more challenging. More detailed tests on clinical datasets will be the subject of future work. Among the limitations in our study was the windowing process, which averaged the feature values and blood pressure in time intervals of 8 s. It is possible that the mean value of the parameters in this window could not accommodate for a wide variation of blood pressure.