Waveform of crackle sounds (frequency analyses)

Crackling lung sounds in different diseases have distinctive waveform features [6]. They are significantly different between diseases such as fibrosing alveolitis, bronchiectasis, chronic obstructive pulmonary disease (COPD), and heart failure, indicating that the waveform analysis has diagnostic value [20]. To quantitatively distinguish the frequency content of these wave forms, Fast Fourier Transform (FFT) [20,19,12,11,16] and time expanded waveform [17,1,16] have been extensively used. Theses studies agree that most of the crackle sound energy fluctuates between 160 to 230 Hz. It has also been speculated that the initial spike is determined by the transpulmonary pressure and the envelope of the wave decay as an exponential function [16].

Figure 4: (top) Example of a crackle waveform from measured crackle sound; (bottom) Isolated crackle and a optimal fit with Eq.5.

We used wavelet-based time series analysis to characterize the non-stationary behavior of the crackle time series, developing a dynamic crackle model which can mimic several features of the experimental data. A given isolated crackle $i$, with the initial spike at time $t_i$, recorded at the trachea can be fit using a combination of sinusoids with an exponential envelope,

$$S_i(t'=t-t_i)=a_1e^{-k(t')}(a_2\sin(f_1t'+p_1)+\sin(f_2t'+p_2))\,$$ (5)

where $a_1$, $a_2$, $k$, $f_1$, $f_2$, $p_1$ and $p_2$ are parameters to be fit. As a first approximation, we incorporated these oscillations with randomly fluctuating frequency, phase, and decay parameters in our model, using realistic ranges. We obtained time series from the model similar to the experimental time series of crackles We applied the CWT to both the experimental data and the model generated time series.

Figure 5: Expansion of crackle sound from experimental data (top) and model (bottom) - the gray scale represents the magnitude of the wavelet coefficient $F(a,b)$.

For each scale $a$ in the CWT we calculate the histogram of coefficients. After normalization all histograms for each scale collapse on to a single master curve , which means that fluctuations behave similarly for all scales, a signature of a fractal behavior [10]. From this study, we conclude that the crackle time series shows a scale-free behavior and this feature is related to the hierarchical tree structure of the airways [5,3].

Figure 6: Collapsed histograms of the wavelet coefficient from Fig.6 at different scales for the experimental data (left) and the model (right).