Evaluation of Respiratory System Models Based on Parameter Estimates from Impulse Oscillometry Data. S. Baswa1, B. Diong2, H. Nazeran1, P. Nava1 and M.
Proceedings of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference Shanghai, China, September 1-4, 2005
Evaluation of Respiratory System Models Based on Parameter Estimates from Impulse Oscillometry Data S. Baswa1, B. Diong2, H. Nazeran1, P. Nava1 and M. Goldman3 1
Department of Electrical and Computer Engineering, The University of Texas at El Paso, El Paso, TX 79968, USA 2 Texas Christian University, Fort Worth, TX 76129, USA 3 Geffen School of Medicine, University of California at Los Angeles, Los Angeles, CA 90034, USA
Abstract—Impulse oscillometry offers advantages over spirometry because it requires minimal patient cooperation, it yields pulmonary function data in a form that is readily amenable to engineering analysis. In particular, the data can be used to obtain parameter estimates for electric circuit-based models of the respiratory system, which in turn may assist the detection and diagnosis of various diseases/pathologies. Of the six models analyzed during this study, Mead’s model seems to provide the most robust and accurate parameter estimates for our data set of 5 subjects with airflow obstruction including asthma and chronic obstructive pulmonary disease and another 5 normal subjects with no identifiable respiratory disease. Such a diagnostic approach, relying on estimated parameter values from a respiratory system model estimate and the degree of their deviation from the normal range, may require additional measures to ensure proper identification of diseases/pathologies but the preliminary results are promising. Keywords—Respiratory impedance, respiratory resistance, respiratory reactance
I. INTRODUCTION Lung function is most commonly assessed by standard spirometric pulmonary function tests. However, spirometric measurements require maximal coordinated inspiratory and expiratory efforts. The considerable degree of cooperation required from the patient makes spirometry inappropriate for young children and older adults. In contrast, respiratory function assessment by the method of forced oscillation [1] requires minimal patient cooperation, and only ‘passive cooperation,’ while wearing a nose clip to close the nares, and breathing normally. Air pressure and rate of air flow at the entrance to the respiratory system are measured, thereby defining its mechanical impedance. In particular, the Impulse Oscillometry System (IOS) is a commercially available product for measuring forced oscillatory impedance by employing brief 60-70 millisecond pulses of pressure. IOS measurements yield frequency-dependent impedance values which may be correlated with models consisting of electrical components that are analogous to the resistances, compliances and inertances inherent in the respiratory system. Consequently, parameter estimates for such respiratory system models [2] can then serve as quantitative measures for better detection and diagnosis of various diseases/pathologies. This paper describes work to identify the most appropriate model(s) to use for respiratory system disease detection and diagnosis. Using IOS data for adults with
0-7803-8740-6/05/$20.00 ©2005 IEEE.
respiratory disease, we have examined the performance of six respiratory models by estimating their parameters and calculating the corresponding estimation error. Although this work is similar in some respects to one described earlier [3], there are the following significant differences: x data for the present work include responses of normal adults in addition to adults with respiratory ailments, x data for the present work was collected recently in the U.S. and Australia, while data for the previous work was collected a few years ago in Australia, so differences exist in equipment, technical personnel, measurement technique, x data for the present work resulted from several visits and multiple tests of the same patient (with a series of tests performed during each visit) while data for the previous work consisted of only one test result per patient, x data for the present work includes responses to inhalation of dry powdered mannitol in addition to baseline measurements, while data for the previous work did not include such responses, and x a subtle but important change has been made to the implementation of the analytical technique, which is detailed in Section III. II. DESCRIPTION OF THE MODELS Of the six models used to fit the data for each patient, considerable work has previously been done on the DuBois model [1], [4], [5], and [6] while Schmidt et al. [2] analyzed three of these models (RC, RIC and Mead’s) with respect to infant data. Another of these models (extended RIC) has been introduced only recently [3]. A. RC model The resistance of the airways and the compliance of the alveoli are modeled as a simple RC circuit (Fig. 1 with R in cmH2O/L/s, C in L/cmH2O) with impedance [2] given by j (1) Z R Z C
R
C
Fig. 1. RC model.
where Z is the angular frequency in radians/second. This model can be used with tidal breathing measurements, but is insufficient for higher frequency analysis [2].
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B. RIC model
Rp
Rc
Lung inertance I, is included in the RIC model (Fig. 2 with I in cmH2O/L/s2) with impedance 1 · § Z R j ¨ ZI ¸ (2) ZC ¹ © I
R
Fig. 4. Mead’s model.
E. DuBois’ model
Fig. 2. RIC model.
C. Extended RIC model This model is proposed as an improvement to the previous model. Specifically, the additional peripheral resistance associated with the compliance (see Fig. 3) allows for the frequency dependence observed of typical real impedance data, which is beyond the RIC model’s capability. The extended RIC model’s impedance is Z
R
Rp 1 ZR p C
2
§ ZR p2 C j ¨ ZI 2 ¨ 1 ZR p C ©
· ¸ ¸ ¹
This model was proposed by DuBois et al [1]. It divides the airway, tissue, and alveolar properties into different compartments. The parameters are airway and tissue resistance (Raw, Rt), airway and tissue inertance (Iaw, It), and tissue and alveolar compliance (Ct, Cg). Its impedance is Z
(5)
Raw jZI aw 2
(3)
Rt Ct Ȧ 2 2 2 Ct It Ȧ6 Cg Ct Cg Ct Rt 2 It Cg + Ct Ȧ4 Cg + Ct Ȧ2
g
>
2
t
2
Rp
2
@ jZ>C C I Ȧ C 2 C I + C I C C R Ȧ C + C @ C C I Ȧ >C C C C R 2 I C + C @Ȧ C + C Ȧ C
g
I
Cw
Cb
Ce
C
Cl
g
R
t
2
2
4
t
t
g t
t
Raw
g
t
g
t
g
Iaw
t
2
t
2
6
t
t
g
t
g
t
2
4
t
t
Rt
g
t
It
2
Ct
C Fig. 3. Extended RIC model.
Cg
D. Mead’s model
Fig. 5. DuBois’s model.
Mead’s model simulates different mechanics in the lung and chest wall [2]. Its seven parameters are inertance (I), central and peripheral resistances (Rc and Rp), and lung, chest wall, bronchial tube, and extrathoracic compliances (Cl, Cw, Cb, Ce) as shown in Fig. 4. These result in the total impedance [2] j (4a) Z Zm ZC e where
Zm j (
R p Cl
F. Viscoelastic model The viscoelastic model parameterizes the respiratory system based on overall resistance (Raw), static compliance (Cs), and viscoelastic tissue resistance and compliance (Rve, Cve) [5]. It differs from the extended RIC model because system inertance is not included (see Fig. 6). Its impedance is
Z
2
Z 2 R p 2 C b 2 C l 2 (C b C l ) 2
RC
Z 2 R p 2 Cb Cl 2 Cb Cl Z[Z 2 R p 2 C b 2 C l 2 (C b C l ) 2 ]
Raw
§ 1 Rve ZRve2 Cve · (6) ¨ j ¨ ZC 1 (ZR C ) 2 ¸¸ 1 (ZRveCve ) 2 ve ve ¹ © s Rv e
(4b)
ZI
Raw
1 ) ZC w
Cs
Cv e
Rm jX m
Fig. 6. Viscoelastic model.
and
Re(Z)
III. PARAMETER ESTIMATION TECHNIQUES
Rm 2
2
2
Im(Z)
2
1 2ZCe Xm Z2Ce (Rm Xm ) (4c)
2
Xm ZCe (Rm Xm ) 2
2
2
1 2ZCe Xm Z2Ce (Rm Xm )
Parameter estimation is similar in concept to curvefitting. Therefore, it is necessary to first select a suitable error criterion E that is to be minimized, where (7) E g{ f 1 (x), f 2 (x),..., f m (x)} in which f1 (x), f 2 (x),..., f m (x) are functions involving the nvector x of parameters x1 , x2 ,..., xn and the independent
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variables, e.g., frequency, of the m data samples [6]. Error criteria that are commonly used in parameter estimation problems include least absolute value (LAV), least squares (LS), minimax, and maximum likelihood. The LAV criterion is effective in dealing with data outliers and is nearly as accurate as LS for data with normally distributed errors, while the minimax function minimizes the maximum element [6]. But the LS criterion is by far the most commonly used for curve fitting and parameter estimation. In its generalized form, the LS criterion
ª min«E ¬
m
º
i1
¼
¦{wi fi x }2»
(8)
minimizes the weighted (by wi) sum of the squared errors (differences from the m data samples). It was chosen for this work due to its commonplace use, its relation with other system identification algorithms [7, 8], and its availability in different software packages. A linear LS algorithm and a nonlinear LS algorithm (both are descent-based) were used to estimate the parameters of the various respiratory models. The former can be applied to relatively simple functions and was used for the RC and RIC models. The latter was necessary for the other models because of the nonlinear dependence of their impedance functions on the parameters. Unlike the linear LS algorithm, the nonlinear LS algorithm may produce parameter estimates corresponding to a local error minimum rather than a global minimum. In order to circumvent this problem, a procedure was used whereby each estimation run began with an initial guess, i.e., a parameter estimate vector generated by a uniform random number generator. This was then repeated twenty-four more times per model (extended RIC, Mead’s, DuBois’, viscoelastic) for each patient’s data to find the parameter estimates yielding the least error. Note that the number of iterations performed here was greater than the ten iterations that were carried out on each of the 105 patients’ data for our previous study [3]. IV. RESULTS AND DISCUSSION The IOS data for each patient belonging to a sample of 5 adults diagnosed with mild bronchiectasis (airway obstruction) was separated into two groups: real impedance (ZR), and imaginary impedance (ZX). The data samples were at 5, 10, 15, 20, 25, 30 and 35 Hz for both ZR and ZX. For the RC model, the linear LS algorithm was used to obtain the parameter estimates for R and C and their associated error values. The R parameter estimates did not seem to have major outliers and were therefore considered acceptable. On the other hand, there were negative C estimates for several patients. This indicated an inherent problem with the proposed solution, consistent with current opinion that this model is too simplistic in nature. For the RIC model, estimation of I and C was a twodimensional optimization problem. While estimates of R were the same as for the RC model, the linear LS algorithm was used to determine the optimal I and C values. The added inertance rectified the problem of negative C estimates derived from the RC model.
To determine parameter estimates for the extended RIC model, DuBois’ model, Mead’s model, and the viscoelastic model, it was necessary to use the nonlinear LS algorithm instead. The initial guesses of the parameter values were random, with a uniform distribution over the range of numbers between 0 and 5. A total of twenty-five guesses and estimation runs were performed for each patient’s data. Table I shows the estimation errors obtained for each model for one patient’s data where the total estimation error is defined as the root of the sum of the least square real and imaginary impedance errors. In the case of this patient, it is seen that Mead’s model provides the best fit, and the viscoelastic model the worst fit. As an illustration, Fig. 7 compares the respiratory impedance (ZR and ZX) for this patient as measured by IOS to the impedance estimated from Mead’s model (with estimates of Rp = 1.6580, Rc = 3.2617, Cl = 1116.4, Cb = 0.0076877, Cw = 0.047647, Ce = 0.00032009 and I = 0.014593). For the entire data set of 5 patients with mild airflow obstruction both at baseline, and after provocation by inhalation of dry powdered mannitol, Mead’s model again yielded the best fit (see Fig. 8), while the viscoelastic model still provided the worst fit. However, three issues should be mentioned regarding the results. First, the static compliance for the viscoelastic model is relatively higher than the other compliance estimates obtained (see Table II showing the mean and standard deviation of error values for each model); and the lung compliance required by Mead’s model is unrealistically large for these patients with mild airflow obstruction. For the viscoelastic model, large changes in static compliance resulted in only minimal changes to the associated error, which implies that this parameter does not have a significant impact on the model’s (in)accuracy. Secondly, a significant proportion of the estimates, for each particular model and patient, converged to the same values although their initial guesses were different. This suggests a global minimum was reached in these cases. Thirdly, while Mead’s model yields minimal errors, it demands unrealistic values of Cl and Cw. In contrast, the Rp and R values from the extended RIC model (where R is analogous to central airway resistance), are more in line with what is expected in these patients with mild airflow obstruction. Additionally, the values for C estimated from the extended RIC model is roughly comparable to what would be expected of lung (or respiratory system) compliance in these patients. A data set of 5 normal adults with no known respiratory disease was also analyzed. In this case, Mead’s model again yielded the lowest total error values followed by Dubois’ model (Fig. 9). The viscoelastic model again provided the worst fit; Table III shows the mean and standard deviation of total error values for each model. It may also be noted that while the Mead model provided minimal total error, the importance of reactance should not be overlooked, as current clinical research has shown the pre-eminence of reactance parameters as being most sensitive to small airway obstruction in patients with chronic airflow obstruction. Thus, while the extended RIC model’s total error was slightly greater than the DuBois model, the reactance error was much less in extended RIC, which is to its advantage.
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V. CONCLUSION
Model RIC model Ext. RIC model Mead's model DuBois’ model Viscoelastic model
Mean total error
Total error std dev
1.0826842 0.6914813 0.3030142 0.6038563 2.3439133
0.5907650 0.0874103 0.1771912 0.2590051 0.3577037
Comparison of Mean Total Errors 4.5 4 3.5 Total Error
Based on a comparison of the parameter estimation errors for five commonly-used respiratory system models and one recently introduced model, it appears that Mead’s model yielded the least estimation errors across the given data set of both normal adults and adult patients with mild chronic obstructive pulmonary disease. However as noted, some of the Mead’s model parameter estimates for the patient and normal subject data sets are not realistic, while those of the extended RIC model are more so. This conclusion differs from the one obtained in our previous study [3], which was that the DuBois and extended RIC models yielded the most robust and accurate estimates. The cause of this discrepancy is most probably because not as many iterations were performed for the previous study, particularly in optimizing the Mead’s model parameter estimates. This hypothesis is now under investigation and the results will be duly reported.
Table II Comparison of parameter estimate errors for each model using the nonlinear least squares algorithm for the entire ill-patient data set. Mean ZX LS error Model Mean ZR LS error RIC model 1.273322 0.2456910 Ext.RIC model 0.4138234 0.1266643 Mead's model 0.0385869 0.0844299 DuBois’ model 0.2212519 0.2100522 Viscoelastic model 0.5939614 5.0271157
Table I Comparison of parameter estimation errors for each model using the nonlinear least squares algorithm on data from one patient.
3 2.5 2 1.5 1
ZX LS error 0.511495 0.1595595 0.0667966 0.2547768 3.9771314
Total error 1.419121 0.6769738 0.3210018 0.6007860 2.1294725
0.5 0 RIC model
10
15
20
25
30
Viscoelastic model
35
ACKNOWLEDGEMENT The support of Professor Alan Crockett at Flinders Medical Centre with the Australian IOS data is gratefully acknowledged.
2 1
REFERENCES
0
[1] Patient data Reactance estimate
-1 5
10
15
20
25
30
[2]
35
Frequency (Hz)
Fig. 7. Resistance and reactance plots for the patient data in Table 1 using parameter estimates for Mead’s model.
[3]
Comparison of Mean Total Errors
[4]
2.5
2
Total Error
Dubois' model
Table III Mean and standard deviation of total error values for each model for the entire normal subject data set. Model Mean total error Total error std dev RIC 0.4747891 0.1201609 Ext. RIC 0.4051607 0.0778816 Mead’s 0.2374095 0.0670969 DuBois’ 0.3240069 0.0958537 Viscoelastic 3.8746582 0.5371220
Patient data Resistance estimate 5
Mead's model
Fig. 9. Comparison of the models based on the normal subject data set.
4
3.5
Ext RIC model
Respiratory Model
4.5
Reactance (cm-H2O/L/s)
Resistance (cm-H2O/L/s)
RIC model Ext. RIC model Mead's model DuBois’ model Viscoelastic model
ZR LS error 1.919586 0.3122282 0.0473943 0.1184898 0.5947824
[5]
1.5
1
0.5
[6]
0 RIC m odel
Ext RIC m odel
Mead's m odel
Dubois ' m odel
Vis coelas tic m odel
Respiratory Model
[7] [8]
Fig. 8. Comparison of the models based on the entire patient data set.
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A. B. DuBois, A. W. Brody, D. H. Lewis, and B. F. Burgess, “Oscillation mechanics of lungs and chest in man,” Journal of Applied Physiology, vol. 8, pp. 587-594, 1956. M. Schmidt, B. Foitzik, O. Hochmuth, and G. Schmalisch, “Computer simulation of the measured respiratory impedance in newborn infants and the effect of the measurement equipment,” Medical Engineering & Physics, vol. 20, pp. 220-228, 1998. T. Woo, B. Diong, L. Mansfield, M. Goldman, P. Nava and H. Nazeran, “A comparison of various respiratory system models based on parameter estimates from Impulse Oscillometry data,” Proc. IEEE Engineering Medicine Biology Conf., San Francisco, CA, Sep 2004. K. R. Lutchen, “Optimal selection of frequencies for estimating parameters from respiratory impedance data,” IEEE Transactions on Biomedical Engineering, vol. 35, no. 8, pp. 607-617, 1988. K. R. Lutchen and K. D. Costa, “Physiological interpretations based on lumped element models fit to respiratory impedance data: use of forward-inverse modeling,” IEEE Transactions on Biomedical Engineering, vol. 37, no. 11, pp. 1076-1086, 1990. P. R. Adby, and M. A. H. Dempster, Introduction to Optimization Methods. London: Chapman and Hall, 1974. T. C. Hsia, System Identification. Lexington: Lexington Books, 1977. N. K. Sinha, and B. Kuszta, Modeling and Identification of Dynamic Systems. New York: Van Nostrand Reinhold Co., 1983.
