AutoPEEP can be visually observed and detected through flow signal. Figure 1 shows an example of flow signal with AutoPEEP captured during mechanical ventilation on a patient. Let f _t be the clean flow signal. AutoPEEP can be regarded as the non-return of the flow signal at the end of each expiratory phase to the null value. In practice, during the observation of the air flow, various factors might get involved, including the mechanical vibration of the air tube, the patient movement, the electro-magnetic interference, etc. Therefore, the flow signal at the end of the expiratory phase will never be exactly zero, even in absence of noise. Testing directly the hypothesis f t k ≠ 0 , where t _k is the end-expiration instant of the considered breath, might thus not be realistic. A tolerance τ > 0 is then introduced to take into account possible distortions on the signal under consideration. Given τ, the problem is then the testing of f t k ≤ τ versus f t k > τ based on the flow signal observation in presence of noise. This tolerance τ is specified by the clinician. Its value is usually derived from his/her expertise of the domain. Other technical factors could also be taken into account, such as: the flow sensor precision, the dynamic range of the signal, etc. Multiple values of τ could also be employed to provide a semi-quantitative evaluation of persisted AutoPEEP on patient.

With respect to the discussion above, a platform for automatic detection of AutoPEEP based on a noisy observation of the flow signal can be developed. Figure 2 depicts such a platform. The main processing components include: the data acquisition and Serial/Parallel conversion, the phase-change detector, the estimator and the AutoPEEP detector. These components are briefly presented as follows before being detailed in the sequel.

This very-first module acquires the discrete flow signal y _n provided by the ventilator or by an independent flow sensor installed inside the air-tube during the mechanical ventilation. Although every flow datum is acquired, only end-expiration flow data of each breath is useful for the detection of AutoPEEP. When the end-expiration instant t _k of the k-th breath is provided by the phase change detector, the Data Acquisition and Serial/Parallel conversion module will log L samples at the end of the expiratory phase to form the observation vector Y k = [ y t k − L + 1 , y t k − L + 2 , … , y t k ] T for the k-th breath. This output observation vector Y _k is finally injected into the AutoPEEP detector module.

The main role of this module is to detect the end-expiration of each breath and provide this instant to trigger the data logging process and the Serial/Parallel conversion described above. This can also be regarded as a breath detector, which separates the continuous flow signal into different breaths.

This module consists of two estimators, which estimate necessary parameters for the AutoPEEP detection algorithms. These parameters are the so-called waveform vector (p _k for the k-th breath) and the noise standard deviation estimate ( σ ̂ ). The waveform vector will be used to aggregate multi-samples at the end of the expiratory phase of a breath into a decision (cf. Section Single-breath detector), while the noise standard deviation estimate will be provided to adjust the AutoPEEP detector.

The AutoPEEP detector is the main core of the whole platform. Given a specified tolerance τ and the desired maximum false-alarm rate (level) γ, the AutoPEEP detector will decide whether an AutoPEEP is present or not for a given breath, on the basis of its observation Y _k and estimated parameters p k , σ ̂ .

Although the definition of AutoPEEP is based solely on the final sample of the expiratory phase of each breath, it is expected that taking multiple samples into account will improve the detection performance. By introducing the waveform vector, namely p _k , with dimension L, one can aggregate L samples at the end of the expiration to carry out a single decision for the breath under consideration. Let Y _k be the observation vector containing the last L samples of the expiratory phase of the k-th breath under consideration. Y _k is modeled as:

Y k = f k + X k

where f k = f t k − L + 1 … f t k − 1 f t k T is the flow signal vector and X k ∼ N ( 0 , σ 2 I L ) is additive gaussian noise with standard deviation σ. Vector f _k can be factorized as:

f k = p k f t k

where p k = p 1 ( k ) p 2 ( k ) … p L ( k ) T is the waveform vector. It should be noted that p L ( k ) = 1 . This vector p _k corresponds to the local form of the flow signal near the end of the expiratory phase. It is also worth mentioning that this local waveform vector p _k depends mainly on the configuration of the interface, including the patient condition and the ventilator settings. As long as the interface stays unchanged, the waveform vector remains almost the same regardless whether or not an AutoPEEP might occur. In practice, either p _k is known prior to the detection or it can be estimated from the observation using one of the methods proposed in Section Waveform regression to compute p _k .

To aggregate L observed samples into one decision for the considered breath, Y _k is projected onto the direction generated by p _k . We thus have:

u k = f t k + w k

where u k = p k T Y k / ∥ p k ∥ 2 , w k = p k T X k / ∥ p k ∥ 2 and ∥ p k ∥ 2 = p k T p k is the L ₂ norm of waveform vector p _k . By such proceeding, noise w _k follows normal distribution with zero mean and variance σ w 2 = σ 2 / ∥ p k ∥ 2 . According to 9, Theorem 27.4, p. 362 and equation (14), it can be proved that, even when the original noise is not gaussian, the resulting noise w _k tends to a normally distributed random variable, as long as L is large enough and the original noise samples are i.i.d (independent and identically distributed). In practice, the i.i.d condition can be significantly relaxed. The problem in (4) is the same as that in previous section, except that the noise level is reduced (σ _w ≤ σ). Moreover, no information on the correlation among samples of noise vector X _k is required. The two hypotheses are unchanged: h 0 : | f t k | ≤ τ and h 1 : | f t k | > τ . The detection is thus carried out as follows. We decide that there is an AutoPEEP if | u k | > σ w λ γ ( τ σ w ) , where λ _γ (.) is calculated as in (3). Otherwise, the considered breath is labeled with Non-AutoPEEP.

It should be noted that ∥p _k ∥ increases with respect to the number L of samples. The noise standard deviation σ _w will thus decreases when more samples are taken into account. By reducing the noise standard deviation, the detection probability is improved while the false-alarm rate is still limited to the specified level γ. Theoretically, L is only limited by the length of expiratory phase. However, L must not be too long so that the local waveform vector can be considered stable and stays almost unchanged for a large number of breaths.

Let us consider K consecutive breaths. Using the same aggregation scheme as in Section Single-breath detector for each breath, we have:

u k = f t k + w k for k = 1 , 2 , … , K

and w k ∼ iid N ( 0 , σ w 2 ) . By averaging over the K breaths, we obtain:

u 1 : K = f t 1 : K + w 1 : K

where:

u 1 : K = 1 K ∑ k = 1 K u k , f t 1 : K = 1 K ∑ k = 1 K f t k , w 1 : K = 1 K ∑ k = 1 K w k .

It is worth mentioning that w 1 : K ∼ N ( 0 , σ w , K 2 ) and that σ w , K = σ w K is strictly decreasing with the number K of breaths used.

Assuming that the true hypothesis (AutoPEEP/NON-AutoPEEP) remains the same for K consecutive breaths, the AutoPEEP detection for these breaths amounts to determining whether or not the average end-expiration flow f t 1 : K exceeds the specified tolerance τ. Given level γ, the from-above test for this problem is:

T λ 1 : K ( h ) ( u 1 : K ) = 1 if | u 1 : K | > λ 1 : K ( h ) 0 if | u 1 : K | ≤ λ 1 : K ( h )

for testing h 0 : | f t 1 : K | ≤ τ against h 1 : | f t 1 : K | > τ and the from-below test is:

T λ 1 : K ( ℓ ) ( u 1 : K ) = 0 if | u 1 : K | > λ 1 : K ( ℓ ) 1 if | u 1 : K | ≤ λ 1 : K ( ℓ )

for testing h 0 ′ : | f t 1 : K | > τ against h 1 ′ : | f t 1 : K | ≤ τ . The two associated thresholds are thus:

λ 1 : K ( h ) = σ w , K λ γ ( τ / σ w , K )

and

λ 1 : K ( ℓ ) = σ w , K λ 1 − γ ( τ / σ w , K )

where λ 1 : K ( h ) > λ 1 : K ( ℓ ) for any 0<γ<0.5.

Summarizing, in a sequential decision framework, the AutoPEEP detection is carried out as follows. Firstly, the detector tries to make a decision based solely on the observation of the first breath, using the test:

T ∗ ( u 1 : 1 ) = 1 (AutoPEEP) if | u 1 : 1 | > λ 1 : 1 ( h ) 0 (NON-AutoPEEP) if | u 1 : 1 | ≤ λ 1 : 1 ( ℓ ) ? (not decided yet) otherwise

If the decision cannot be made yet (i.e. λ 1 : 1 ( ℓ ) ≤ | u 1 : 1 | ≤ λ 1 : 1 ( h ) ), it will be delayed until the next observation (i.e. u ₂) is obtained and the test is performed based on u _1:2 using T ∗ ( u 1 : 2 ) . If the decision still cannot be performed, it will be delayed again until the next observation, where the test T ∗ ( u 1 : 3 ) is used. The process is iterated until the decision is made. Then the process is restarted for a new sequence of observations.

As shown in Figure 3, both λ 1 : K ( h ) and λ 1 : K ( ℓ ) tend to tolerance τ when σ w , K → 0 (the proof will be carried out in a future work which covers all theoretical aspects of Sequential SNT). It should also be noted that σ w , K → K → ∞ 0 . Therefore, it could be expected that the probability to make a decision is one when K → ∞ . However, if K is too high, the assumption that the same state remains for the considered K consecutive breaths might no longer hold. Moreover, a high number of breaths to be acquired may yield an unacceptable delay-to-decision. One simple solution is then to limit the number of breaths to some value M. If M breaths have been observed but no decision has been made, a hard decision is then performed. To assure the false-alarm rate, threshold λ 1 : M ( h ) is used. The hard decision is carried out by:

T λ 1 : M ( h ) ( u 1 : M ) = 1 (AutoPEEP) if | u 1 : M | > λ 1 : M ( h ) 0 (NON-AutoPEEP) if | u 1 : M | ≤ λ 1 : M ( h )

The value M must be chosen so that the assumption mentioned above holds valid and the delay-to-decision is still in an acceptable range. In our experimental settings, we use M=10, which corresponds to about 30 seconds of observation in the usual case with a breathing frequency of 20 [breaths/min].

With regard to Section Single-breath detector, the waveform vector p _k is the key which makes it possible to aggregate multiple end-expiration flow samples into one decision. This vector p _k can be calculated from the regression of the flow signal at the end of the expiration. Indeed, during the expiratory phase of a breath, the mechanical ventilation system works based solely on the passive response of the patient lung. Due to the resistance of the airways and the elasticity of human lung, the flow signal during the expiratory phase of a breath can be modeled by:

y ( t ) = C − ϕ e − μt ,

with ϕ > 0 and μ > 0, even in presence of AutoPEEP. This model is used to estimate the referenced waveform at the end of the expiration using a nonlinear robust regression method. Given a set of N data points {(t _i, y(t _i )),i = 1..N} where y(t _i ) is the observation at instant t _i , the non-linear robust regression aims at solving the least square problem:

( C , ϕ , μ ) ∗ = arg min C , ϕ , μ ∑ i = 1 N ξ i y ( t i ) − ( C − ϕ e − μ t i ) 2

where the introduction of weight vector [ξ ₁,ξ ₂,.. ξ _N ] makes it possible to reduce the influence of outliers onto the final result. The MATLAB routine nlinfit performs such a regression task. This routine uses a weighted version of the Levenberg-Marquardt algorithm 14 to solve the non-linear least squares problem (13). The weights are iteratively updated with respect to corresponding residues | y ( t i ) − ( C − ϕ e − μ t i ) | , i = 1 ..n to downweight the outliers and therefore reduce their effects on the final regression curve. Figure 6 shows an example of the flow signal at the end of the expiratory phase and the regression resulting from the aforementioned non-linear robust method. The signal is shown to be well-fitted by the model function (12).

Even though only L samples are required to calculate the L-dimensional vector p _k , more samples should be used to achieve a better regression curve. Let L _ext(L _ext ≥ L) be the number of samples to be used. L _ext is only limited by the length, namely T _e (in samples), of the expiratory phase, i.e. L _ext ≤ T _e . Regarding the transition between different respiratory phases, samples at the beginning of the expiration are very sensitive to transition and may bias the regression. Therefore, only a proportion of the T _e samples of the expiratory phase should be taken into account:

L ext = α T e

where 0 < α < 1 is the proportion and ⌊.⌋ is the floor function. Proportion α must be chosen so that L ext = α T e ≥ L . Additionally, to avoid the border effect, one might consider the weighted regression with a weighting scheme that puts more weight on the middle samples than on the side ones. Figure 7a shows the regression on end-expiration samples of a flow signal recorded from a patient. In this example, αis set to 0.75 to avoid the transition effects at the beginning of the expiratory phase. Since it is not so crucial in the situation experienced in this work, an unweighted regression was employed.

Given the regression at the end of expiration, namely [ ŷ t k − L ext + 1 , ŷ t k − L ext + 2 , … , ŷ t k ] , the last L values are used to calculate the estimate p ̂ k for the considered breath:

p ̂ k = ŷ t k − L + 1 , ŷ t k − L + 2 , … , ŷ t k T / ŷ t k

According to Section Single-breath detector, waveform vector p _k concerns the current (k-th) breath. However, as aforementioned, this waveform vector does not vary much. It is then sensible to use estimates from previous breaths so as to improve the estimation of p _k . In this respect, the following strategies can be considered to compute the waveform vector estimate to be used in AutoPEEP detectors:

"Static waveform vector: The waveform vector is computed based on the first N _ref breaths right after a verification/tuning session of the clinician. These N _ref breaths are used as reference after validation by the clinician.

p k = 1 N ref ∑ k = 1 N ref p ̂ k

This waveform vector will be updated each time the machine is tuned or after a verification session by the clinician. One may also want to update the estimation on a regular time basis.

Dynamic waveform vector: The waveform vector to be used is the one estimated from the current breath:

p k = p ̂ k

Adaptive waveform vector: In this strategy, the waveform vector is updated every time a new breath is observed. Previous estimates are taken into account with a forgetting factor μ such that 0 < μ< 1:

p k = 1 − μ 1 − μ k ∑ i = 1 k μ k − i p ̂ i

Noise is unknown in practice. As long as the noise standard deviation in concerned, it must be estimated from the observation. In this work, we propose two solutions: one based directly on the result obtained by waveform regression, whereas the other is based on an estimation from the wavelet coefficients of the flow signal.

By using the regression, the residue can be considered as noise. Therefore, the noise standard deviation can be estimated directly from this residue. For the k-th breath, we have:

σ ̂ k = 1 L ext − 1 ∑ i = t k − L ext + 1 t k y i − ŷ i 2

To aggregate σ ̂ from σ ̂ k , the same strategies as those proposed for the waveform vector can be considered.

Studies on nonparametric estimation based on Wavelet Shrinkage have shown that most of the wavelet coefficients obtained from the first level wavelet decomposition of a piecewise smooth signal are of very small amplitude. Only a small number of these wavelet coefficients, which correspond to signal, are of higher amplitude 12 . This fact allows the use of robust estimators on the wavelet coefficients to provide noise estimation. One can consider the MAD (median absolute deviation) 15 16 to accomplish such a task. The method is usual 12 15 16 and we recall it for readiness sake. Let c ₁,c ₂,…c _N be the wavelet coefficients obtained from the first level discrete wavelet decomposition of an N-sample segment of the flow signal y. The estimate σ ̂ MAD of σ is then provided by:

σ ̂ MAD = b × med i | c i − med j c j |

where b≈1.4826. Since the noise is central, white and gaussian, the formula is simplified to:

σ ̂ MAD = b × med i | c i |

knowing that med _i c _i = 0.

In 17 , another robust estimator was proposed, namely the d-dimensional adaptive trimming estimator (DATE). The method is summarized as follows. Let c ₍₁₎ c ₍₂₎,…,c _(N) be sequence of wavelet coefficients c ₁,c ₂,…c _N sorted by increasing magnitude. Put m min = N 2 − N 4 ( 1 − Q ) where Q = 0.95. Let m be the smallest integer, m _min≤m≤N such that:

| c ( m ) | ≤ 2 . 7238 × 1 m ∑ k = 1 m | c ( k ) | < | c ( m + 1 ) |

If such an integer m does not exist, set m = m _min. The estimate σ ̂ DATE of σ is then provided by:

σ ̂ DATE = 1 . 2533 × 1 m ∑ k = 1 m | c ( k ) |

It has been shown in 17 that this estimator outperforms the MAD when the number of outliers increases. The DATE can thus be employed as an alternative to the MAD mentioned above in such situation. For the cases considered in this work, because the number of large wavelet coefficients pertaining to signal remains small, the two estimators yield similar performance. The MAD estimator is thus adopted for its lower complexity and higher rapidity.

To illustrate the detection performance of the proposed algorithms, the flow signal was first synthesized on computer. For each breath, L end-expiration samples were generated. The waveform vector was supposed to be known and set to p k = p = [ 1 , 1 , … , 1 ] T . It is worth mentioning that, by construction, |p ₁|≥|p ₂|≥…≥|p _L |=1 and, as a result, σ w = σ ∥ p k ∥ ≤ σ L . The equality happens when and only when p _i =1 for all i=1..L. With regard to noise level σ _w , by setting p k = p = [ 1 , 1 , … , 1 ] T , we considered the worst case where ∥ p k ∥ 2 = L and σ w = σ L . For sequential SNT-based detector, M was set to 10 [breaths], which corresponds to about 30 seconds of observation. The tolerance was empirically set to τ=2 [l/min] by clinician. The values of f t k were randomly and uniformly generated between 0 and − τ 1 − Π , where Πis the proportion of positive cases (AutoPEEP). Since the false-alarm rate P _FA is always restricted to the specified value γ, it is more meaningful to plot the detection rate P _D versus different values of Π, namely the detection curve, than to present the usual ROC (Receiver Operating Curve). Figure 8 shows detection curves for different noise levels and different values of L. The detection rate is significantly improved when more samples are aggregated. Of course, the lower the noise level, the better the detection. In this respect, the Sequential SNT-based detector also showed higher detection rate while still keeping the false-alarm rate below the specified value γ.

The proposed AutoPEEP detectors were also tested in a more realistic setting in which the interface between a ventilator and a lung model was established. In these experiments, the respiratory system analog was constituted by a G5 ventilator (Hamilton Medical, Bonaduz, Switzerland) connected to the ASL5000 computerized lung model (Ingmar Medical Ltd., Pittsburgh, PA, USA), making it possible to modify respiratory mechanics. Thirteen sets of parameters (cf. Table 1) for both the lung emulator and the ventilator, which correspond to various practical situations, were carried out. The tolerance τ = 2 [l/min] was employed again. With respect to this tolerance, among the 13 settings, 7 cases were reported as AutoPEEP and the other 6 cases were labeled as NON-AutoPEEP, thanks to an independent clinical analysis from the Intensive Care unit of Brest University Hospital, Brest, France. The detection was performed on the basis of the flow signal captured by the sensor integrated in the ASL5000 lung model. For each case, about 1.5 minute of the signal flow was recorded. The corresponding number of breaths varied from 13 to 34, depending on the setting. In total, 323 breaths were recorded. For both the proposed detectors, the dynamic waveform vector was employed for its simplicity. Level γ was set to 0.01. The detection results are reported in Table 1. All the 13 cases were successfully detected by the two proposed methods: the Single-breath SNT-based detector and the Sequential SNT-based detector. Moreover, in each case, all the breaths were precisely classified. No detection error was found among the 323 breaths analyzed.

^aVentilator parameters include: Positive Expiratory Pressure PEP [cmH ₂ O], air volume Vt [ml], frequency f [breaths/min], pause time P [%], Inspiratory to expiratory time ratio I:E.

^bLung model parameters include: compliance C [ml/cmH ₂ O] and resistance R [cmH ₂ O/l/s].

^cFor each of the experiments, the AutoPEEP detection provides: the number of breaths detected as AutoPEEP (denoted as P for Positive), the number of breaths detected as NON-AutoPEEP (denoted as N for Negative) and the overall label for the considered setting.

For further evaluation, the AutoPEEP detectors were tested ex-vivo on various patient curves. These curves were retrospectively extracted from data files issued from the Medical Intensive Care Unit of Brest University Hospital, France and from the Institut Universitaire de Cardiologie et de Pneumologie de Québec, Canada. For each patient undergoing mechanical ventilation, the flow signal was recorded. All these data were then mixed up to form a unique dataset. In total, the final dataset contains 1998 breaths from 15 patients with different health conditions and different treatments. The parameters of the ventilator also varied depending on the situation. According to the retrospective aspect of the study and to the fact that the files were anonymized, the study was considered to be in accordance with French legislation by our local ethics committee.

The analysis was performed both manually by a set of experts and automatically by the proposed methods. On the one hand, each breath was carefully screened by two experts of the domain. They performed a dual analysis, separately, before confronting their points of view and delivering a final assessment of the data. For each breath of the dataset, their decision was then regarded as the ground-truth label (AutoPEEP/NON-AutoPEEP). On the other hand, the proposed detectors were used to predict the label of every breath of the dataset. The two analyses were carried out independently and anonymously. The results were then compared together to evaluate the detection performance of the proposed methods.

In these experiments, the tolerance was set to τ = 2 [l/min] as before. In this respect, the dataset includes 1383 breaths with AutoPEEP and 615 breaths with NON-AutoPEEP. The dataset is somehow unbalanced with the presence of AutoPEEP in 69% of the cases. For the proposed detectors, level γ was set to 0.01 as usual. Figure 7 presents a typical case with the regression at end-expiration and the corresponding detection. It can be seen that the detection algorithm can precisely reveal the true label for all the breaths.

To quantitatively assess the detection performance of the proposed methods, we considered four usual evaluation measures: Accuracy, Precision, Recall (Sensitivity) and Specificity. These measures are defined as follows:

Accuracy = T P + T N T P + T N + F P + F N

Precision = T P T P + F P

" Recall (Sensitivity) = T P T P + F N

" Specificity = T N T N + F P

where: T P (resp. T N) is the number of true positives (resp. true negatives), defined as the number of breaths with (resp. without) AutoPEEP that are correctly predicted; F P (false positive) is the number of breaths without AutoPEEP that are falsely predicted as AutoPEEP, and F N(false negative) is the number of breath with AutoPEEP that are not detected. These four values T P, F P, T N, and F N form the so-call confusion matrix of the detection. In terms of the four aforementioned evaluation measures, the performance results for the two proposed detectors are reported in Table 2. The results show that both the detectors worked very well on patient data with an accuracy higher than 93%, a precision higher than 99%, a recall (sensitivity) higher than 90% and a specificity higher than 98%. For the considered dataset, the two proposed AutoPEEP detectors provided similar results. It is worth mentioning that, by reducing the noise impact, the Sequential SNT-based detector is aimed at improving the detection performance of the Single-breath detector in case the latter fails to reveal ‘twilight region’ AutoPEEP, i.e. AutoPEEP with an end-expiration flow value near the given tolerance τ. Thence, the higher the number of twilight-region AutoPEEPs in the dataset, the more significant the performance improvement can be observed. However, in the considered clinical dataset, the number of twilight region AutoPEEPs, which are also difficult for the clinician to analyze, was very limited. Therefore, no significant difference in detection performance could be seen. However, the use of the Sequential SNT-based detector is recommended for better performance and robustness.

The experiments were carried out with τ = 2[l/min]. For both the detectors, the level was set to γ = 0.01, which corresponds to an average of 1 false-alarm per 5 minutes (with the usual breathing frequency of 20 [breaths/min]).