Time-varying signal analysis to detect high-altitude periodic breathing in climbers ascending to extreme altitude

Abstract

This work investigates the performance of cardiorespiratory analysis detecting periodic breathing (PB) in chest wall recordings in mountaineers climbing to extreme altitude. The breathing patterns of 34 mountaineers were monitored unobtrusively by inductance plethysmography, ECG and pulse oximetry using a portable recorder during climbs at altitudes between 4497 and 7546 m on Mt. Muztagh Ata. The minute ventilation (V_E) and heart rate (HR) signals were studied, to identify visually scored PB, applying time-varying spectral, coherence and entropy analysis. In 411 climbing periods, 30–120 min in duration, high values of mean power (MP^VE) and slope (MSlope^VE) of the modulation frequency band of V_E, accurately identified PB, with an area under the ROC curve of 88 and 89 %, respectively. Prolonged stay at altitude was associated with an increase in PB. During PB episodes, higher peak power of ventilatory (MP^VE) and cardiac (MP ^HR_LF ) oscillations and cardiorespiratory coherence (MP ^Coher_LF ), but reduced ventilation entropy (SampEn^VE), was observed. Therefore, the characterization of cardiorespiratory dynamics by the analysis of V_E and HR signals accurately identifies PB and effects of altitude acclimatization, providing promising tools for investigating physiologic effects of environmental exposures and diseases.

Appendix: Methods used for time series analysis

1.1 Power spectral density

The signal x(n) is modeled through an autoregressive model by

$$x(n) = - \sum\limits_{k = 1}^{p} {a\left[ k \right] \cdot x\left( {n - k} \right)} + e(n)$$

where e(n) denotes zero-mean white noise with variance σ ² _e , a[k] the AR coefficients and p the model order. Once the autoregressive coefficients and the variance σ ² _e have been estimated, the PSD of an autoregressive process is computed by means of

$$\hat{P}_{{x {\text{AR}}}} (f) = \frac{{\sigma_{e}^{2} }}{{\left| {1 + \sum\nolimits_{k = 1}^{p} {a[k] \cdot e^{ - j2\pi fkT} } } \right|^{2} }}$$

being T the sampling period.

The advantage of model-based frequency estimation is its capacity to predict future samples outside of the observation interval, instead of assuming zero as occurs with conventional nonparametric or Fourier-based spectral analysis [34]. The accuracy of the AR model was evaluated through the mean square prediction error. The optimum model order (ranging from 2 to 50) was selected for each signal according to the criterion proposed by Rissanen [30], based on selecting the model order that minimizes the description length.

1.2 Coherence

The coherence function measures the strength of the linear interaction between two time series at each frequency. Its value ranges from 0, implying no temporal correlation between the two signals to 1 implying maximum correlation. It is described as

$$C_{xy} (f) = \frac{{\left| {P_{xy} (f)} \right|}}{{\sqrt {P_{x} (f) \cdot P_{y} (f)} }}$$

where P _x (f) and P _y (f) are the PSD of both signals x(n) and y(n), respectively, and P _xy (f) is the cross-PSD between them. In order to get higher frequency resolution, a parametric coherence is implemented in this study [19].

In parametric coherence analysis, a bivariate autoregressive model is applied to calculate the parametric cross-PSD. The relationship between both signals is described by

$${\mathbf{S}}(n) = - \sum\limits_{k = 1}^{p} {{\mathbf{A}}[k]S(n - k)} + {\mathbf{E}}(n)$$

where the matrix S contains both signals, matrix A autoregressive coefficients and matrix E zero-mean white noise inputs.

$$\left[ {\begin{array}{*{20}c} {x(n)} \\ {y(n)} \\ \end{array} } \right] = \sum\limits_{k = 1}^{p} {\left[ {\begin{array}{*{20}l} {a_{xx} (k)} \hfill & {a_{xy} (k)} \hfill \\ {a_{yx} (k)} \hfill & {a_{yy} (k)} \hfill \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {x\left( {n - k} \right)} \\ {y\left( {n - k} \right)} \\ \end{array} } \right]} + \left[ {\begin{array}{*{20}c} {e_{x} (n)} \\ {e_{y} (n)} \\ \end{array} } \right]$$

The matrix of PSD for this bivariate autoregressive model is obtained by

$${\mathbf{\rm P}}(f) = \left[ {\begin{array}{*{20}l} {{\rm P}_{xx} } \hfill & {{\rm P}_{xy} } \hfill \\ {{\rm P}_{yx} } \hfill & {{\rm P}_{yy} } \hfill \\ \end{array} } \right] = {\mathbf{\rm A}}(f)\left[ {\begin{array}{*{20}l} {\sigma_{xx}^{2} } \hfill & {\sigma_{xy}^{2} } \hfill \\ {\sigma_{yx}^{2} } \hfill & {\sigma_{yy}^{2} } \hfill \\ \end{array} } \right]{\mathbf{\rm A}}^{*} (f)$$

where * denotes the conjugate transpose of the matrix. The matrix A is obtained through

$${\mathbf{\rm A}}(f) = {\mathbf{\rm I}} + \sum\limits_{k = 1}^{p} {{\mathbf{\rm A}}\left[ k \right] \cdot e^{ - j2\pi fkT} }$$

1.3 Complexity analysis

Approximate entropy (ApEn) and sample entropy (SampEn) provide quantitative information about the complexity of the signals. ApEn is approximately equal to the negative average natural logarithm of the conditional probability that two sequences that are similar for m points remain similar, that is, within a tolerance r, at the next point. In order to avoid the occurrence of ln(0) in the calculations, ApEn algorithm counts each sequence as matching itself. ApEn is therefore heavily dependent on the record length and lacks relative consistency. SampEn is the negative natural logarithm of the conditional probability that two sequences similar for m points remain similar at the next point, where self-matches are not included in calculating the probability [8].

Considering the signal x(n) and the parameters m and r, the ApEn is computed by

• 1. Form the m-vectors

  $$\varvec{X}(i) = \left[ {x(i), x(i + 1), \ldots ,x\left( {i + m - 1} \right)} \right];\quad 1 \le i \le N - m + 1$$

  being N the length of the signal.

• 2. Calculate distances between vectors

  $$d[X(i),X(j)] = \max_{k = 1, \ldots m} \left( {\left| {x\left( {i + k} \right) - x(j + k)} \right|} \right)$$

• 3. Compute frequency or regularity of similar patterns

  $$C_{r}^{m} (i) = \frac{{N^{m} (i)}}{N - m + 1}$$

  where N ^m(i) is the number of $j \left( {1 \le j \le N - m + 1} \right)$ so that $d\left[ {X(i),X(j)} \right] \le r.$

• 4. Take the natural logarithm of each C ^m _r and average it over

  $$\emptyset^{m} (r) = \frac{1}{N - m + 1}\sum\limits_{i = 1}^{N - m + 1} {\ln \left( {C_{r}^{m} (i)} \right)}$$

• 5. Repeat this process for m + 1 and the ApEn is finally computed as follows

  $$ApEn\left( {m,r,N} \right) = \emptyset^{m} (r) - \emptyset^{m + 1} (r)$$

Considering the signal x(n) and the parameters m and r, the SampEn is computed by

• 1. Form the m-vectors

  $$\varvec{X}(i) = \left[ {x(i), x(i + 1), \ldots ,x(i + m - 1)} \right];\quad 1 \le i \le N - m + 1$$

  being N the length of the signal.

• 2. Calculate distances between vectors

  $$d[X(i),X(j)] = \max_{k = 1, \ldots m} \left( {\left| {x\left( {i + k} \right) - x(j + k)} \right|} \right)$$

• 3. Define for each of $i\left( {1 \le i \le N - m } \right)$

  $$\begin{aligned} B_{i}^{m} (r) & = \frac{{N^{m} (i)}}{N - m + 1} \\ A_{i}^{m} (r) & = \frac{{N^{m + 1} (i)}}{N - m + 1} \\ \end{aligned}$$

  where N ^m(i) is the number of $j \left( {1 \le j \le N - m + 1} \right),\;j \ne i$ so that $d\left[ {X(i),X(j)} \right]_{m} \le r,$ and N ^m+1(i) is the number of $j\left( {1 \le j \le N - m + 1} \right),\;j \ne i$ so that $d\left[ {X(i),X(j)} \right]_{m + 1} \le r$.

• 4. Compute

  $$\begin{aligned} B^{m} (r) & = \frac{1}{N - m }\sum\limits_{i = 1}^{N - m} {B_{i}^{m} (r)} \\ A^{m} (r) & = \frac{1}{N - m }\sum\limits_{i = 1}^{N - m} {A_{i}^{m} (r)} \\ \end{aligned}$$

• 5. SampEn is finally computed as follows

  $$SampEn\left( {m,r,N} \right) = - \ln (A^{m} (r)/B^{m} (r))$$

The complexity of the V_E and HR signals was evaluated with tolerance values ranging from 0.05 to 0.5 and m = 2 (see Fig. 7). Sample entropy and approximate entropy of V_E and HR for PB and nPB periods with different tolerance values are plotted. According to this analysis and previous studies, the tolerance value of 0.15 was selected for further processing [2].

Fig. 7

Approximate entropy (a, b) and Sample entropy (c, d) of V_E and HR for periods with and without PB (PB and nPB, respectively) with different tolerance (r) values, being m = 2. Dotted and solid lines represent PB and nPB periods, respectively