Leakage Estimation Using Kalman Filtering in ... - IEEE Xplore

1234

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 60, NO. 5, MAY 2013

Leakage Estimation Using Kalman Filtering in Noninvasive Mechanical Ventilation G. G. Rodrigues, U. S. Freitas, D. Bounoiare, L. A. Aguirre∗ , and C. Letellier

Abstract—Noninvasive mechanical ventilation is today often used to assist patient with chronic respiratory failure. One of the main reasons evoked to explain asynchrony events, discomfort, unwillingness to be treated, etc., is the occurrence of nonintentional leaks in the ventilation circuit, which are difficult to account for because they are not measured. This paper describes a solution to the problem of variable leakage estimation based on a Kalman filter driven by airflow and the pressure signals, both of which are available in the ventilation circuit. The filter was validated by showing that based on the attained leakage estimates, practically all the untriggered cycles can be explained. Index Terms—Kalman filter (KF), leakage estimation, noninvasive mechanical ventilation (NIV).

I. INTRODUCTION ONINVASIVE mechanical ventilation (NIV) is a useful procedure in clinical practice [1]. NIV is mainly applied to decrease patient’s work of breathing and to keep blood gas concentrations at appropriate levels. This type of ventilation support is less aggressive than invasive ventilation since it does not require intubation or tracheotomy. Besides, NIV can be used outside ICU in hospitals or even at home care. This procedure is recommended when a patient is unable to sustain adequate ventilation. This situation is usually accompanied by increased dyspnea, high ventilation rates, use of accessory ventilation musculature, and abnormalities in gas exchange. Some respiratory failures that are treated using noninvasive ventilation include chronic obstructive pulmonary disease (COPD), obesity hypoventilation syndrome (OHS), and sleep-related breathing disorders. In NIV, the ventilation circuit is not perfectly sealed and, although leakage is an intrinsic part of noninvasive

N

Manuscript received June 25, 2012; revised October 13, 2012; accepted November 21, 2012. Date of publication December 3, 2012; date of current version April 15, 2013. This work was supported by Coordenaça˜ o de Aperfeiçoamento de Pessoal de N´ıvel Superior (CAPES), in part by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico (CNPq) Bras´ılia, Brazil, and in part by the Centre National de la Recherche Scientifique, Paris, France. The work of D. Bounoiare was supported by ADIR Association. Asterisk indicates corresponding author. G. G. Rodrigues is with Centro Federal de Educaça˜ o Tecnológica de Minas Gerais, Belo Horizonte 30510-000, MG, Brazil (e-mail: giovani@ des.cefetmg.br). U. S. Freitas is with ADIR Association, - H. Bois Guillaume 147 Avenue du Maréchal Juin 76031 Rouen Cedex, France (e-mail: [email protected]). D. Bounoiare and C. Letellier are with the CORIA UMR 6614— Université de Rouen, Site Universitaire du Madrillet BP 12 76801 Saint Etienne du Rouvray Cedex, France (e-mail: [email protected]; [email protected]). ∗ L. A. Aguirre is with the Universidade Federal de Minas Gerais, Belo Horizonte 31270-901, MG, Brazil (e-mail: [email protected]). Digital Object Identifier 10.1109/TBME.2012.2230630

ventilation since intentional leaks are used to enable CO2 elimination, this turns ventilation management rather difficult. Critical leakages are those resulting from nonintentional contributions due to mask misadjustments, mouth opening, and so on. Minimizing leaks is, therefore, a major challenge to improve alveolar ventilation and reduce asynchrony between the patient and the ventilator; otherwise, sleep quality during NIV can be affected [2]. An alternative to reducing leaks is to compensate for it and such a feature is present in most ventilators nowadays. Unfortunately, the performance of such compensation schemes greatly differ from one ventilator to the next [3], [4]. Nonintentional leaks decrease the airflow actually delivered to the patient and, may significantly deteriorate ventilator performance [3]. Besides, there are evidences of correlation between leakage and patient ventilator asynchrony events [5]. During noninvasive ventilation, pressure and airflow at the ventilator output are routinely monitored variables. Patient breathing efforts can be estimated using an esophageal catheter but this intrusive technique is very uncomfortable, especially during the night, and may demand readjustments to obtain reliable measurements. For these reasons, the esophageal pressure is not often routinely monitored. The measured airflow includes both patient airflow and leakages (intentional and nonintentional); hence, it is difficult to have an estimate of each component. Difficult though as it is, the estimation of nonintentional leakages is critical because a good estimation of such a flow corresponds to having a good estimate of the patient flow. Thus, it comes as no surprise that some alternatives to estimate leakage in NIV are available in the technical literature. Some authors assume that the mean patient airflow over a large number of breathing cycles is null [2]. Consequently, a non-null component in the mean value of measured airflow is associated with leakage. This hypothesis is useful to derive a “mean” approximation of leakage but is inadequate to estimate non-constant leakage on a cycle-to-cycle basis. The other option is to evaluate the leak-associated volume as the difference between inhaled and exhaled volumes [6]. Nevertheless, this may not perform well [7], [8], and cannot be used with home ventilators. A general aim of this paper is to compare some leakage estimating procedure and, in particular, to investigate the use of a Kalman filter (KF) [9] for this purpose. Such a class of filters has already been applied in noninvasive ventilation problems. An unscented KF was used in [10] to estimate parameters of a nonlinear model for respiratory mechanics. The simulated and clinical data supplied to the filter came from the mask pressure, airflow, and integrated volume. However, leakage was not estimated in that work. A linear KF with Rauch–Tung–Striebel smoother was employed in [11] to estimate the pressure produced by the ventilation muscles. The corresponding signal was

0018-9294/$31.00 © 2012 IEEE

RODRIGUES et al.: LEAKAGE ESTIMATION USING KALMAN FILTERING IN NONINVASIVE MECHANICAL VENTILATION

supposed to vary slowly. Mask and esophageal pressures and airflow were acquired in COPD during spontaneous breathing. The ventilation system compliance and resistance were assumed to be known a priori. To the best of the authors knowledge, this study is the first study to investigate KF techniques to estimate nonconstant leakage from routinely measured signals in NIV. Our aim is to evaluate whether it is possible to improve leakage estimation by coupling a leakage model and information about the patient ventilation. In order to do so, a discrete-time model used to describe the ventilation mechanics was derived from first principles and subsequently approximated to yield a mono-compartment first order linear model. This approximation is widely used to simplify the dynamical representation when assessing clinical relevant information from acquired signals [12]–[17]. In this study, the measured pressure and airflow signals serve as inputs to the KF. The subsequent part of this paper is organized as follows. Section II is devoted to a brief description of the measurements. In Section III, three different strategies to estimate leakage are described, the third one being based on a KF which is the main contribution of the present paper. In Section IV, the three techniques are compared using ventilation signals from COPD and OHS patients during nocturnal ventilation. Section V gives a discussion and some conclusions. II. PATIENTS AND MEASUREMENTS Data investigated in this study were already analyzed with a different focus in [5] and [18]. In an observational study, a set of 35 patients was monitored during nocturnal ventilation assistance at the Rouen University Hospital (France); see [5, Tables 1 and 2] and [18, Tables 5.1 and 5.2], respectively. These patients suffered from two different respiratory dysfunctions: 15 patients had COPD, and 20 patients had OHS. Patients were noninvasively ventilated in the pressure support mode with no backup frequency. NIV was applied through nasal (n = 24) and facial (n = 11) masks according to patient uses. Ventilator settings were adjusted by physicians according to each patient clinical status. A series of physiological signals were acquired using a polysomnography (CID102L8, CIDELEC St Gemmessur-Loire, France). Mask pressure and ventilator airflow were acquired at 128 Hz using a differential pressure transducer and a Fleisch pneumotachometer, respectively. The only preprocessing applied was a low-pass filtering to smooth the signals (fc = 5 Hz). Data were processed according to their original sampling frequency. Full night monitoring produce very long time series made of about 6000 ventilation cycles. The ventilation signals display various patient-ventilator asynchronies, which may complicate the leakage estimation (see [5], [18] for details). III. METHODS Leakage estimation during NIV is an important problem to solve since it would help to reduce asynchrony events. If only constant leakages are taken into account, the estimation problem can be addressed as in [19]. However, in the presence of nonconstant leakages it is often hard for the ventilators to achieve the

1235

preset quantities (inspiratory positive airway pressure (IPAP), volume, etc.) [20], [21]. A very general approach is to consider that the relation among flows can be expressed as q(t) = qp (t) + qL (t)

(1)

where q(t) is the total flow measured at ventilator output, and qp (t) and qL (t) are the patient airflow and leakage, respectively. Assuming that leakage flow is turbulent (see H2), we have qL (t) = G paw (t).

(2)

where paw (t) is the airway pressure and G is an appropriate parameter. Substituting (2) into (1) yields q(t) = qp (t) + G paw (t).

(3)

The most straightforward procedure is to define a way to estimate adequately parameter G. In order to estimate the leakage qL , some hypotheses about the airflow in the ventilatory circuit must be done to develop an estimator. For instance, the following hypotheses were proposed in [22]: H1) the patient’s average airflow must be zero, that is, all the air entering patient’s lungs must actually be exhaled; H2) leakage is assumed to be turbulent, that is, pressure drop is proportional to the square of the airflow; H3) leakage does not vary quickly. A. Constant Leakage A simple approach to leakage estimation is to assume that G is a constant and that the total leakage is only due to intentional leakage, thus disregarding unintentional leakages. In that case patient airflow becomes qp (t) = q(t) − G paw (t) where G is a constant (see Appendix A). Such a constant G will be later designated as Gintentional . This approach does not consider hypothesis H1. B. Mean Leakage In order to circumvent the problem of a fast-diverging volume, the total flow in (3) can be low-pass filtered to yield q(t) = qp (t) + G paw (t) = qp (t) + G paw (t) = qp (t) + G paw (t)

(4)

where · indicates low-pass filtering. To reach (4) G was assumed not to vary significantly within a breathing cycle (see H3). In order to guarantee H1, qp (t) = 0 must hold; hence q(t) = G paw (t)

(5)

1236


ˆ and the slow-varying leak parameter G(t) can finally be estimated using ˆ = q(t) G(t) paw (t)

(6)

where q(t) and paw (t) are the measured variables. Instead of assuming a constant G as in Section III-A, the total leakage ˆ can be estimated using (2) with G(t) in (6), which will be designated as Gm ean and should now account for intentional and unintentional leakages. Patient flow can be then obtained from (1). C. Kalman Filter for the Estimation of Instantaneous Leakage A KF is an algorithm which combines in an optimal way measurements and model information in order to provide estimates for unmeasured quantities (e.g., states). The equations for such a filter are briefly reviewed in Appendix B. A KF requires the use of a model. This section describes the development of such a model. Because the ventilator is connected to the patient, the observed global behavior depends on features of both subsystems. Moreover, inspiratory and expiratory triggering must be adjusted to patients muscular power and other features (as inertance I, elastance E, and resistance R). In addition to that, the ventilation system is spatially distributed, displays time delays and nonlinear relationships between the variables. It is thus a difficult task to derive a detailed model able to reproduce so many individual features [23]–[25]. The KF combines in an optimal way the available measurements and the output of a mathematical model. It is as if the measurements were used to correct the model or vice versa. Therefore, even models which could turn out to be oversimplified in other applications can still be helpful in the context of the KF. The respiratory mechanics is therefore represented using a unique compartment and assuming lumped parameters [13], [16], [17], [26], [27]. Parameters of this model must be tuned to capture the main behavior of the patient-ventilator system. A simplified equation for a single compartment structure is ˙ + Ev(t) + p0 + pmus (t) paw (t) = I v¨ (t) + Rv(t)

(7)

where v is the volume, v˙ and v¨ the airflow (at patient airways input) and its derivative, respectively and p0 is the airway pressure when volume, airflow and its derivative are zero for null pmus . The pressure paw is measured at the airway opening and pmus quantifies the mobilization of the inspiratory musculature. Several works in the literature advocate that this equation is a reasonable approximation to study dynamical aspects associated with respiratory mechanics [10]–[13], [15], [16], [28]. The problem of estimating pmus can be addressed using unknown-input observers [29], [30]. Such techniques, however, are based on assumptions which are not verified in our case. Fortunately, when the ventilator is well set it delivers a major part of the breathing work and patient ventilation musculature is unloaded [12] and pmus can be neglected as a first approximation.

Setting p0 = 0, (7) is reduced to x˙ 1 = x2 x˙ 2 =

−Rx2 − Ex1 + paw I

(8)

where x1 = v and x2 = v˙ are the state variables and indicate the patient volume and airflow, respectively, and paw is the exogenous input. The resulting model represents the patient as a passive system driven by pressure u = paw imposed by the ventilator. The values of elastance and resistance may differ from what is expected from a physiological point of view because the model characterizes the patient–ventilator system and not only the patient. Ideally, such parameters, that cannot be known a priori, should be estimated. In this study, the only parameter to be estimated recursively from data is G. This can be done by defining G as an extended state variable. When the inertance is neglected, model (7) is reduced to x˙ 1 =

1 [−E x1 + paw ] R

(9)

where x1 = v. The explicit Euler method was used to discretize (9) with discretization step equal to the original sampling time, that is Ts = 1/128 s, x1 (k) designates x1 (kTs ), and defining G as the extended state variable x2 (k), the corresponding state equations become Ts E Ts x1 (k) = 1 − x1 (k − 1) + u(k − 1) + ξ1 (k) R R x2 (k) = x2 (k − 1) + ξ2 (k)

(10)

where ξ2 is the uncertainty associated with G and u = paw is the model input. Measured airflow is now −E T u(k−1) y(k) = + ν(k) u(k−1) x1 (k) x2 (k) + R R where the additive dynamical noise ξ = [ξ1 ξ2 ]T and observation noise ν are assumed Gaussian white processes with covariance matrix θ and covariance ρ, respectively. Therefore, the system matrices are 1 − TRs E 0 A= (11) 0 1 B = [Ts /R 0]T −E C(k) = u(k−1) R D = 1/R.

(12) (13) (14)

The KF (see Appendix B) uses matrices (11)–(14), the measured airway pressure u, the measured total flow q = y, and the user-defined constants θ and ρ to estimate the state vector x = [v G]T . Finally, from (3), qp (t) is determined. D. Validation of Each Technique As leakage is not directly measured, an indirect strategy must be used to validate the method. Ineffective triggering efforts (ITE) for which the ventilator fails to pressurize at IPAP level

ITE

20

OP

ITE

OP

ITE

1237

TABLE I RESULTS FOR LEAKAGE ESTIMATION

ITE

OP

15 10

method

5

(N ITE )qmax 0, none of the cycles shown in Fig. 1 would have been triggered without leakage estimation. The quality of the leakage estimator can thus be estimated by addressing the following question: what percentage of ITE can be explained using the estimated flow? If a large percentage, say X%, of ITE can be explained, the estimation procedure is considered satisfactory. The reason for this is that if such an estimation method were used to determine ventilator triggering, then the ITE would be (100 − X)%. So, in fact, the figure of merit reported is the percentage of ITE, ρITE , that would remain if the corresponding leakage estimator were used. To compute ρITE , the ventilatory cycles in the dataset for each patient were delimited according to an automatic procedure discussed in [33]. After that, patient airflow was estimated using each one the three methods described in Section III. In the estimated airflow, the cycles that do not become positive are marked as nontriggered. The number of ITE, NITE |q m a x < 0 , was thus estimated, leading to the rate of ineffective triggerings ρITE (the number of ITE counted according to patient’s estimated airflow such as qm ax < 0 divided by the number of patient’s breathing cycles). We thus computed 1) the mean number of ventilation cycles N cycles recorded over each night and 2) the mean number of ITE with negative patient’s estimated maximum

Estimated patient flow (l/s)

⎯ρITE is the percentage of ITE which would remain undetected. MVD stands for the mean volume drift (×10–5).

Esophageal Pressure (cmH2O)

Flow (l/s)

Pressure (cmH2O)


2 1,5 1 0,5 0 -0,5 -1

0

2

4

6

8

10

12

14 16 Time (s)

18

20

22

24

26

28

30

Fig. 2. Patient flow estimated by removing the leakage using (3) with constant G = G inte ntio n a l = 0.20 l· s−1 ·(cmH2 O)−1 / 2 , from the total flow (the middle tracing shown in Fig. 1).

airflow N ITE |q m a x < 0 , where qm ax is the maximum flow value in the corresponding cycle. IV. RESULTS The three different techniques were used to estimate the leakage and, therefore, the patient flow. Table I summarizes the results for 34 patient data recordings (data of one patient was left out). Using Gintentional and the total flow, q(t), shown in Fig. 1, the estimated patient flow, qp (t), (see Fig. 2) is less than q(t): in particular, it is now negative during expiration phases. Assuming that an inspiratory effort can be detected by qp = 0 associated with q˙p > 0, the ventilator would have supported all patient cycles shown in Fig. 2 with this simple leakage estimation. Unfortunately, when Gintentional is used, the mean volume drift is at least two order of magnitude larger than with the two other techniques (see Table I). Since hypothesis H1 must be true, assuming G constant does not lead to a correct volume estimation. In order to compute Gm ean , a third-order Butterworth lowpass filter with a cutoff frequency at 0.0833 Hz (5 cycles per min) was used to obtain the low-passed signals in (6). As expected, the calculated volume does not diverge as before, however, the estimated patient airflow does not become positive (as in Fig. 2) during ITE (see Fig. 3), which is incorrect because breathing effort means inspiration and, therefore, positive patient airflow. With a ventilator assumed to trigger the insufflation when patient airflow obeys qp = 0 and q˙p > 0, such a leakage estimator, if used, would have not triggered at every inspiratory effort (as actually observed in the measured data). In fact, in the measured data, the mean number of cycles per night was 4096 ± 668 and the mean number of ITEs was 793 ± 682, that is, significantly more than with Gm ean (199 ± 201). Since Gm ean was developed by the manufacturer of the ventilator used in our protocol, this ventilator should work more or less in agreement with our results obtained with Gm ean . The departure must result from the

without an antibacterial filter with an antibacterial filter

1,5 1 0,5 0 -0,5 -1 0

2

4

6

8

10

12

14 16 Times (s)

18

20

22

24

26

28

30


ˆ Fig. 3. Patient flow estimated by removing the leakage, (3) with G(t) = G m e a n in (6), from the total flow (middle time series shown in Fig. 1). The thin curve corresponds to patient airflow estimated taking into account the antibacterial filter. 1 0,75 0,5 0,25 0 -0,25 -0,5 -0,75 -1

Pressure (cmH2O)

2 1,5 1 0,5 0 -0,5 -1 -1,5 -2

ITE

ITE

ITE

0

5

10

15

20

25

30

0

5

10

15 Time (s)

20

25

30

20 15 10 5 0

Fig. 5. Patient airflow estimated using G m e a n , and measured airway pressure. The airflow does not become positive during nontriggered cycles (ITE). 0

5

10

15

20

25

30

0

5

10

15 Time (s)

20

25

30

20 15 10 5 0



Pressure (cmH2O)


1238

Fig. 4. Patient airflow estimated using a KF with E = 50 cmH2 O/L and R = 8 cmH2 O·s/l for a OHS patient under nocturnal pressure support ventilation with a high rate of asynchronisms.

antibacterial filter placed between the ventilator and the pneumotachograph. It was reported [18] that the pressure actually delivered to the patient was reduced by at least 4 cmH2 O compared to the pressure measured at the ventilator. With IPAP = 14 cmH2 O as used in the example shown in Fig. 3, the patient airflow estimated by the ventilator would be reduced by 0.15 l·s−1 , thus deteriorating the detection of inspiratory efforts as shown in Fig. 3. In the case of the KF, the initial state vector was x(0) = [0.2 0.1]T and the initial covariance matrix (see appendix B) was P (0) = 1 × 105 I2 , where I2 is a 2 × 2 identity matrix. The covariance matrix for dynamical noise was chosen as θ = 0.1 I2 and for the measurement noise the following covariance was used ρ = 0.05. Because there is no prior information about the dynamical and measurement noise, the covariance matrices were tuned by trial and error based on the filter performance. With such values, the filter converges in about 200–250 iterations, which corresponds to about 2 s of data. The convergence rate and the estimated values are robust to the choice of the initial state x(0). The patient airflow estimated using the KF with E = 50 cmH2 O/l and R = 8 cmH2 O·s/l for one patient is shown in Fig. 4. The corresponding airway pressure delivered by the ventilator (see Fig. 5) is also presented to characterize the patient-ventilator interaction during this data window. Fig. 4 shows an alternating pattern of triggered and non-triggered

cycles. The estimated airflow becomes positive during the non-triggered cycles, indicating that the patient sustained the breath and is zero at the end of inspiration. We found that the amount of such ITE was significantly (p < 0.01) smaller using the intentional leaks parameter Gintentional (see Section III-A and Appendix A) than using the mean leaks parameter Gm ean , estimated using (6). Using Gintentional , if we only consider the patients (n = 11) who have a rate of ITE less than the rate of ITE obtained with Gm ean , the mean rate ρITE was significantly (according to the Mann-Whitney U test) greater (p < 0.001) for these patients (46.4 ± 28 %) than for the others (13.6 ± 13.7 %). This means that, in some patients, using Gintentional , therefore, allows us to strongly improve the triggering strategy than using Gm ean . Nevertheless, Gintentional leads to a poor volume estimation as already mentioned. Using the leakage flow estimated with the KF, the resulting patient flow had a positive maximum in about 99.6% of the patient’s breathing cycles during which the insufflation was not triggered by the ventilator. This means that if the estimated flow were used to trigger the insufflation, 99.6% of the ITE actually observed would have not happened. Compared to the use of Gm ean , this is 97% of the ITEs which would have been removed. Compared to the use of Gintentional , this is 94.6% of the cycles with a negative maximum airflow which have now a positive maximum. Hence, a Kalman-based procedure would provide more appropriate conditions to trigger the switch from EPAP to IPAP. Moreover, the KF-based estimation leads to a stable volume (the drift is at least as low as the one observed with Gm ean ). Hence, the KF scheme developed in this paper compares very favorably to the other two techniques. V. DISCUSSION AND CONCLUSION The fact that Gm ean does not provide very good results leads to the conclusion that hypothesis H1 is not supported by such a method. For instance, it is reasonable to consider that during the expiratory phase intentional leaks are dominant whereas significant nonintentional leakage may occur during the inspiratory phase. The fact that nonintentional leakage could vary within a ventilation cycle was shown in [7] where a mannequin was


ventilated via a mask with an inflatable low-pressure air cushion. For this reason, Lyazidi and co-workers [21], [34] introduced a variable nonintentional leakage in their bench studies to evaluate ventilator performances. If leakage varies during the ventilation cycle, the mean leakage approach (Gm ean ) leads to underestimated leakage during insufflation and overestimated otherwise. If the ventilator triggers insufflation according to the estimated patient flow, it will not detect the end of patient inspiration since it will misinterpret leakage as a patient continuing inspiration effort. This explains the occurrence of OP cycles shown in Fig. 1. Conversely, during expiration, the ventilator may erroneously estimate a very negative flow for the patient, even during efforts, which may lead to ITEs. The problem of nonintentional leaks is often reported as one of the main reasons for causing asynchrony events. This is mostly because the strategy used by ventilators to trigger a new cycle requires some sort of leakage estimation. Three strategies to estimate nonintentional leaks were compared: constant leakage approach (based on the value of an intentional leak parameter analytically estimated), averaged leakage approach (over a sliding window, as patented by RESMED), and a new technique based on Kalman filtering. The number of inspiratory efforts for which there is no positive peak airflow (after a negative expiratory airflow) is nearly zero with the proposed KF. The latter technique is, therefore, the most reliable among those investigated to estimate nonintentional leaks that vary within breathing cycles. A ventilator using such a technique should be able to detect most of the inspiratory efforts.

1239

TABLE II NUMERICAL VALUES OF GEOMETRIC PARAMETERS OF THE VENTILATION CIRCUIT Air density Air kinematic viscosity Bend cœfficient Enlargement cœfficient Shrinkage cœfficient Tube length Tube diameter Leak edge diameter Side length of the triangular section

ρ = 1.17 kg.m3 ν = 16.84 · 10−6 m2 .s τ2 = 0.2 τ3 = 0.89 τ4 = 0.5 L = 1.6 m D = 1.7 · 10−2 m df = 2 · 10−3 m l = 9 cm.

given by Vv D/νair , where νair is the kinematic viscosity. After some manipulation, we get ρ 1 1 + τ4 2 paw = − 2 (1 − τ2 − τ3 ) Qf 2 Sf2 S1 1/4 0.316L St νair 7/4 + Qf . (18) St2 D This expression was plotted against the pressure paw using numerical values provided in Table II. Our measurements and the data provided by RESMED are in a very good agreement with our analytical expression (not shown). A quadratic regression qL = α pβaw lead to α = 0.192 and β = 0.505, which is quite √ close to qL = G paw [see (2)], with G = 0.2, which was used as Gintentional . APPENDIX B

APPENDIX A

STATE AND PARAMETERS ESTIMATION

INTENTIONAL LEAKS PARAMETER Intentional leak parameter is estimated when the patient airflow is supposed to be null. Applying the conservation of volume energy (the so-called Bernoulli theorem) leads to ρVv2

Vf =

2Qf 4Qf and Vv = 2 3πdf πD2

(16)

where df is the diameter of the six holes placed on a full face mask manufactured by RESMED (used in our measurements) and D is the diameter of the tube between the mask and the ventilator. The pressure drop due to the different elements of the circuits is ⎤ ⎡ Δp =

x(k) = A(k−1)x(k−1)+B(k−1)u(k−1)+G(k − 1)ξ(k−1)

ρVf2

≈ pf + + Δpr (15) 2 2 where paw and Vv are, respectively, the pressure and air velocity at the ventilator output and Vf is the air velocity of the intentional leaks. According to the airflow conservation, we have paw +

The KF [9] allows us to estimate state variables of a linear system corrupted by Gaussian dynamical and observational noise represented by the model

⎥ ρair ⎢ ⎢ 0.316L + τ2 + τ3 + τ4 ⎥ Vf2 0,25 ⎦ ⎣ 2 Re 2D b end enlarg leaks

(19) y(k) = C(k)x(k) + D(k)u(k) + H(k)ν(k)

(20)

where x(k) ≡ x(kTs ) ∈ R is the state space vector and Ts is the sampling time. Vector y(k) ∈ Rm is the output and u(k) ∈ Rr the input. The random variables ξ and ν are assumed Gaussian with zero mean and covariance matrices Q(k) e R(k), respectively. A, B, C and D can be time-variant. The state vector at time k − 1 is propagated using the model which yields x ˆ(k|k − 1) and yˆ(k|k − 1). The propagated state, x ˆ(k|k − 1), is corrected using the current measurement y(k); thus, yielding the posterior estimation is x ˆ(k|k) indicating that information up to instant k was used to obtain the estimate at time k. The KF is given by n

x ˆ(k|k−1) = A(k−1) x ˆ(k−1|k−1) + B(k−1) u(k−1) (17)

tub e

where τ4 = ((Sf − Se )/Se )2 , where Sf is the total leak section and Se , the contracted section. Sf /Se was obtained from an abacus, which leads to τ4 = 0.5. Re is the Reynolds number

(21) yˆ(k|k−1) = C(k) x ˆ(k|k−1) + D(k) u(k).

(22)

P (k|k−1) = A(k−1) P (k−1|k−1) A(k−1) + Q(k − 1) T

(23)

1240


Py y (k|k−1) = C(k) P (k|k−1) C(k)T + R(k)

(24)

Pxy (k|k−1) = P (k|k−1) C(k)T

(25)

K(k) = Pxy (k|k−1) Py−1y (k|k−1)

(26)

x ˆ(k|k) = x ˆ(k|k−1) + K(k) (y(k)− yˆ(k|k−1))

(27)

T

(28)

P (k|k) = P (k|k−1)−K(k) Py y (k|k−1) K(k)

where P , Pxy , and Py y are covariance matrices and K(k) in (26) is named the Kalman gain. The KF cannot be directly applied to nonlinear systems, in which case the extended Kalman filter (EKF) is a standard alternative approach. The EKF provides a competitive suboptimal solution to the state estimation problem by using a linearized version of system equations in the data-assimilation step while using the full nonlinear model for propagating the state vector. By representing some parameters as additional states, the KF or EKF may be extended to simultaneously estimate both states and parameters. In such a case one should define a dynamical law to propagate the parameters. When parameters vary slowly, î (k − 1|k − 1) is a convenient choice [this was x î (k|k − 1) = x done in (10)]. The data-assimilation stage is the same. In order to estimate states and parameters from measured data, the augmented state space must be observable from the measurements. ACKNOWLEDGMENT The data used in this paper come from a study which was reviewed and authorized by the Institutional Review Board of the Rouen Hospital. REFERENCES [1] S. Mehta and N.-S. Hill, “Noninvasive ventilation,” Amer. J. Respir. Crit. Care Med., vol. 163, pp. 540–577, 2001. [2] H. Teschler, J. Stampa, R. Ragette, N. Konietzko, and M. Berthon-Jones, “Effect of mouth leak on effectiveness of nasal bilevel ventilation assistance and sleep architecture,” Eur. Respir. J., vol. 14, pp. 1251–1257, 1999. [3] S. Mehta, F.-D. McCool, and N.-S. Hill, “Leak compensation in positive pressure ventilators: A lung model study,” Eur. Respir. J., vol. 17, pp. 259– 267, 2001. [4] L. Vignaux, D. Tassaux, and P. Jolliet, “Performance of noninvasive ventilation modes on ICU ventilators during pressure support: A bench model study,” Intens. Care Med., vol. 33, pp. 1444–1451, 2007. [5] R. Naeck, D. Bounoiare, U. S. Freitas, H. Rabarimanantsoa, A. Portmann, F. Portier, A. Cuvelier, J.-F. Muir, and C. Letellier, “Dynamics underlying patient-ventilator interactions during nocturnal noninvasive ventilation,” Int. J. Bifurcat. Chaos., vol. 22, 1250030 (17 pp.), 2012. [6] A. Sabil, G. Mroue, H. Prigent, D. Orlikowski, M. Bohic, P. Baconnier, F. Lofaso, and G. Benchetrit, “Air leakage during nocturnal mechanical ventilation in patients with neuromuscular disease,” ITBM-RBM, vol. 27, pp. 227–232, 2006. [7] G. P. P. Schettino, M. R. Tucci., R. Souza, C. S. V. Barbas, M. B. P. Amato, and C. R. R. Carvalho, “Mask mechanics and leak dynamics during noninvasive pressure support ventilation: A bench study,” Intens. Care Med., vol. 27, pp. 1887–1891, 2001. [8] G. Schmalisch, H. Fischer, G. Roehr, and H. Proquitté, “Comparison of different techniques to measure air leaks during CPAP treatment in neonates,” Med. Eng. Phys., vol. 31, pp. 124–130, 2009. [9] R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME-J. Basic Eng. D, vol. 82, pp. 35–45, 1960 [10] E. Saatçi and A. Akan, “Lung model parameter estimation by unscented kalman filter,” in Proc. IEEE Conf. Eng. Med. Biol. Soc., Aug. 2007, vol. 2007, pp. 2556–2559. [11] Z. Zhao, K. Möller, and J. Guttmann, “Non-invasive assessment of pressure generated by respiratory muscles using the Kalman filter,” IFMBE Proc., vol. 25, no. 4, pp. 1407–1410, 2010.

[12] G. A. Iotti, A. Braschi, J. X. Brunner, A. Palo, and M. C. Olivei, “Noninvasive evaluation of instantaneoustotal mechanical activity of the respiratory muscles during pressure support ventilation,” Chest, vol. 108, no. 1, pp. 208–215, 1995. [13] G. Avanzolini, P. Barbini, A. Cappello, G. Cevenini, and L. Chiari, “A new approach for tracking respiratory mechanical parameters in real-time,” Ann. Biomed. Eng., vol. 25, pp. 154–163, 1997. [14] G. Nucci, M. Mergoni, C. Bricchi, G. Polese, C. Cobelli, and A. Rossi, “On-line monitoring of intrinsic PEEP in ventilator-dependent patients,” J. Appl. Physiol., vol. 89, pp. 985–995, 2000. [15] A. Jensen, H. Atileh, B. Suki, E. P. Ingenito, and K. R. Lutchen, “Signal transduction in smooth muscle selected contribution: Airway caliber in healthy and asthmatic subjects: Effects of bronchial challenge and deep inspirations,” J. Appl. Physiol., vol 91, no. 1, pp. 506–515, 2001 [16] F. C. Jandre, A. V. Pino, I. Lacorte, J. H. S. Neves, and A. Giannella-Neto, “A closed-loop mechanical ventilation controller with explicit objective functions,” IEEE Trans. Biomed. Eng., vol. 51, no. 5, pp. 823–831, May 2004. [17] F. C. Jandre, F. C. Modesto, A. R. C. Carvalho, and A. Giannella-Neto, “The endotracheal tube biases the estimates of pulmonary recruitment and overdistension,” Med. Biol. Eng. Comput., vol. 46, pp. 69–73, 2008. [18] H. Rabarimanantsoa-Jamous, “Qualité Des Interactions PatientVentilateur en Ventilation Non Invasive Nocturne,” Ph.D dissertation, Rouen Univ., Mont-Saint-Aignan, France, 2008 [19] C. Rabec, M. Georges, N. K. Kabeya, N. Baudouin, F. Massin, O. ReybetDegat, and P. Camus, “Evaluating noninvasive ventilation using a monitoring system coupled to a ventilator: A bench-to-bedside study,” Eur. Respir. J., vol. 34, pp. 902–913, 2009. [20] A. Lyazidi, A. W. Thille, J.-C. Richard, F. Templier, O. Besson, and L. Brochard, “Bench comparison of transport and NIV ventilators: Impact of leaks,” Intens. Care Med., vol. 35, no. 1, p. A0919, 2009. [21] A. Lyazidi, “Evaluation Des Performances et Des Limitations Des Ventilateurs Sur Banc D’essai,” Ph.D dissertation, Université Paris-Est, Paris, France, 2010 [22] M. Berthon-Jones, “Determination of leak and respiratory airflow,” U.S. Patent 7661428, Feb. 2010 [23] S. P. Tomlinson, D. G. Tilley, and C. R. Burrows, “Computer simulation of the human breathing process,” IEEE Eng. Med. Biol. Mag., vol. 13, no. 1, pp. 115–124, Feb./Mar. 1994. [24] M. H. Tawhai and A. Ben-Tal, “Multiscale modeling for the lung physiome,” Cardiovasc. Eng., vol. 4, no. 1, pp. 19–26, 2004. [25] A. G. Polak and J. Mroczka, “Nonlinear model for mechanical ventilation of human lungs,” Comput. Biol. Med., vol. 36, pp. 41–58, 2006. [26] F. C. Jandre, A. R. S. Carvalho, A. V. Pino, and A. Giannella-Neto, “Effects of filtering and delays on the estimates of a nonlinear respiratory mechanics model,” Respir. Physiol. Neurobiol., vol. 148, pp. 309–314, 2005. [27] B. Diong, A. Rajagiri, M. Goldman, and H. Nazeran, “The augmented RIC model of the human respiratory system,” Med. Biol. Eng. Comput., vol. 47, pp. 395–404, 2009. [28] A. G. Polak, “Optimization of recursive algorithms for respiratory mechanics monitoring during artificial ventilation,” IFMBE Proc., vol. 25, no. 4, pp. 1916–1919, 2010. [29] M. Witczak and P. Pretki, “Design of an extended unknown input observer with stochastic robustness techniques and evolutionary algorithms,” Int. J. Contr. vol. 80, no. 5, pp. 749–762, 2007 [30] S. Pan, H. Su, J. Chu, and H. Wang, “Applying a novel extended kalman filter to missile-target interception with APN guidance law: A benchmark case study,” Control Eng. Pract., vol. 18, pp. 159–167, 2010. [31] P. Leung, A. Jubran, and M. J. Tobin, “Comparison of assisted ventilator modes on triggering, patient effort, and dyspnea,” Amer. J. Respir. Crit. Care Med., vol. 155, pp. 1940–1948, 2007. [32] D. Tassaux, M. Gainnier, A. Battisti, and P. Jolliet, “Impact of expiratory trigger setting on delayed cycling and inspiratory muscle workload,” Amer. J. Respir. Crit. Care Med. vol. 172, pp. 1283–1289, 2005 [33] L. Achour, C. Letellier, A. Cuvelier, E. Vérin, and J.-F. Muir, “Asynchrony and cyclic variability in pressure support noninvasive ventilation,” Comput. Biol. Med., vol. 37, pp. 1308–1320, 2007. [34] G. Carteaux, A. Lyazidi, A. Corboda-Izquierdon, L. Vignaux, P. Jolliet, A. W. Thille, J. C. Richard, and L. Brochard, “Patient-ventilator asynchrony during noninvasive ventilation: A bench and clinical study,” Chest, vol. 10, pp. 367–376, 2012.

Authors’ photographs and biographies not available at the time of publication.