Forced oscillation technique for early detection of the effects of smoking and COPD: contribution of fractional-order modeling

Figures & data

Figure 1 Two-compartment integer-order model used to analyze respiratory impedance.

Notes: The resistance (R), inductance (I) and capacitance (C) are the analogs of respiratory resistance, inertance and compliance, respectively. Rp represents the peripheral resistance.

Figure 2 Two-compartment fractional-order model evaluated in this study.

Notes: The model includes a constant phase inertance (CPL) and a constant phase compliance (CPC) composed of a frequency-dependent fractional inertia (FrL) and a frequency-dependent fractional compliance (FrC). A simplified description of the ability of the fractional terms to represent the resistive and reactive respiratory properties, depending on the α and β values is also described. R is associated with the influence of the respective fractional-order term on the resistance values; L describes the influence of the FrL in the reactance, producing a more positive reactance in higher frequencies, while C describes the influence of the FrC in reactance, producing more negative reactance values in low frequency.

Table 1 Anthropometric, spirometric and traditional FOT measurements of the groups studied

	Group I control (n=40)	Group II smokers (n=40)	Group III mild COPD (n=40)	ANOVA
Age (years)	53.9±16.7	42.3±10.4b	61.8±12.4a	<0.0001
Weight (kg)	67.4±10.7	66.2±15.0	63.1±12.6	ns
Height (cm)	164.6±8.3	163.2±9.6	162.3±7.6	ns
BMI (kg/m^2)	24.7±2.5	24.7±4.5	23.9±4.1	ns
Pack-years	–	20.3±9.3	66.2±43.9	–
Spirometry
FEV1 (L)	3.02±0.8	3.01±0.7	2.19±0.4****	<0.0001
FEV1 (%)	106.8±17.6	101.3±16.4	83.4±10.2	<0.0001
FVC (L)	3.6±1.0	3.7±0.9	3.3±0.7	ns
FVC (%)	104.3±18.5	106.3±16.9	101.6±12.0	ns
FEV1/FVC	83.1±6.1	88.0±11.6	66.2±4.0****	<0.0001
Traditional forced oscillation
R0 (cmH2O/L/s)	2.4±0.66	3.2±0.9****	3.5±1.5c	<0.0001
Rm (cmH2O/L/s)	2.3±0.6	3.1±0.7****	3.2±1.2c	<0.0001
R4 (cmH2O/L/s)	2.3±0.7	3.1±0.81****	3.3±1.4c	<0.0001
S (cmH2O/L/s)	−10.6±18.5	−9.9±32.8	−25.5±38.0	ns
Xm (cmH2O/L/s)	0.3±0.5	0.2±0.4	0.1±0.6	ns
Ax (cmH2O/L)	8.2±5.4	12.7±6.8c	19.3±19.2c	<0.0004
Cdyn (L/cmH2O)	25.5±11.1	16.0±4.2****	16.1±6.5****	<0.0001
Fr (Hz)	12.5±4.4	13.0±2.7	15.3±6.2	ns
Z4 (cmH2O/L/s)	3.0±0.9	4.1±1.0****	4.4±1.8c	<0.0001

Notes:

a P<0.05.

b P<0.01.

c P<0.001,

**** P<0.0001, compared with the control group.

Abbreviations: FEV₁/FVC, Tiffeneau index; FOT, forced oscillation technique; n, number of evaluated patients; ns, not significant.

Figure 3 Typical results obtained during the eRIC (red) and FrOr (blue) models adjustment in the resistance (closed circles) and reactance (open circles) of normal subjects (A) normal subjects with increasing resistance (B), smokers (C) and mild COPD patients (D).

Figure 4 Changes in the eRIC model parameters in the studied groups.

Notes: Central airway resistance (R; A), peripheral resistance (Rp; B), total resistance (Rt; C), pulmonary inertance (I; D), and alveolar compliance (C; E). *P<0.05; **P<0.01; ****P<0.0001; in relation to control.
Abbreviations: COPD I, mild COPD; ns, not significant.

Figure 5 Changes in the fractional order model parameters in the studied groups.

Notes: Fractional inertance (FrL; A), Fractional inertance exponent (alpha; B), Fractional compliance (FrC; C), fractional compliance exponent (beta; D), Damping factor (G; E), Elastance (H; F). ***P<0.001; **P<0.01; ****P<0.0001; in relation to control.
Abbreviations: COPD I, mild COPD; ns, not significant.

Table 2 Errors in the integer and FrOr models used for control individuals, smokers and patients with mild COPD

	eRIC	FrOr
Control
MSEr (cmH2O/L/s)	0.081±0.075	0.054±0.050
MSEx (cmH2O/L/s)	0.047±0.047	0.046±0.048
MSEt (cmH2O/L/s)	0.0972±0.084	0.074±0.066
Rd (%)	5.135±2.285	4.772±2.669
Smokers
MSEr (cmH2O/L/s)	0.098±0.083	0.085±0.081
MSEx (cmH2O/L/s)	0.123±0.347	0.072±0.049
MSEt (cmH2O/L/s)	0.184±0.344	0.116±0.089
Rd (%)	3.265±1.279	3.300±1.542
Mild COPD
MSEr (cmH2O/L/s)	0.097±0.103	0.097±0.097
MSEx (cmH2O/L/s)	0.185±0.334	0.085±0.103
MSEt (cmH2O/L/s)	0.234±0.333	0.134±0.137
Rd (%)	4.597±2.908	3.711±1.788

Note: The smallest errors in the comparisons of the evaluated models are indicated in bold.

Abbreviations: eRIC, extended resistance-inertance-compliance model; FrOr, fractional-order model.

Table 3 Correlation analysis among traditional forced oscillation parameters and spirometric results

		FEV1 (L)	FEV1 (%)	FVC (L)	FVC (%)	FEV1/FVC	FEF25–75 (L)	FEF25–75 (%)
R0	r p	−0.35 <0.0001	−0.33 <0.0002	−0.32 <0.0003	−0.17 ns	−0.17 ns	−0.29 <0.001	−0.32 <0.0003
S	r p	0.39 <0.0001	0.38 <0.0001	0.39 <0.0001	0.36 <0.0001	0.17 ns	0.27 <0.003	0.23 <0.01
Rm	r p	−0.28 <0.002	−0.25 <0.005	−0.25 <0.006	−0.08 ns	−0.15 ns	−0.26 <0.004	−0.30 <0.0006
R	r p	−0.33 <0.0003	−0.30 <0.0009	−0.30 <0.0008	−0.14 ns	−0.13 ns	−0.26 <0.004	−0.31 <0.0005
Xm	r p	0.52 <0.0001	0.26 <0.004	0.55 <0.0001	0.23 <0.01	0.08 ns	0.29 <0.001	0.20 <0.03
Fr	r p	−0.46 <0.0001	−0.26 <0.004	−0.49 <0.0001	−0.24 <0.009	−0.06 ns	−0.27 <0.003	−0.19 <0.03
Ax	r p	−0.50 <0.0001	−0.32 <0.0003	−0.48 <0.0001	−0.19 <0.03	−0.15 ns	−0.38 <0.0001	−0.33 <0.0002
Cdyn	r p	0.42 <0.0001	0.32 <0.0004	0.39 <0.0001	0.14 ns	0.20 <0.03	0.38 <0.0001	0.37 <0.0001
Z4	r p	−0.38 <0.0001	−0.31 <0.0005	−0.35 <0.0001	−0.14 ns	−0.17 ns	−0.33 <0.0002	−0.35 <0.0001

Note: The two strongest correlations for each FOT parameter are indicated in bold.

Abbreviations: FEF, Forced expiratory flow; FEV₁/FVC, Tiffeneau index.

Table 4 Correlation analysis among eRIC model parameters and spirometric results

		FEV1 (L)	FEV1 (%)	FVC (L)	FVC (%)	FEV1/FVC	FEF25–75 (L)	FEF25–75 (%)
R	r p	−0.13 ns	−0.05 ns	−0.14 ns	0.04 ns	−0.01 ns	−0.08 ns	−0.14 ns
Rp	r p	−0.29 <0.002	−0.39 <0.0001	−0.14 ns	−0.20 <0.03	−0.35 <0.0001	−0.35 <0.0001	−0.36 <0.0001
Rt	r p	−0.29 <0.0009	−0.33 <0.0003	−0.21 <0.02	−0.11 ns	−0.21 <0.02	−0.30 <0.0009	−0.38 <0.0001
I	r p	0.33 <0.0003	0.09 ns	0.33 <0.0002	0.08 ns	0.11 ns	0.21 <0.03	0.09 ns
C	r p	0.39 <0.0001	0.31 <0.0006	0.37 <0.0001	0.18 <0.05	0.13 ns	0.32 <0.0003	0.34 <0.0002

Note: The two strongest correlations for each eRIC parameter are indicated in bold.

Abbreviations: eRIC, extended resistance-inertance-compliance model; FEF, Forced expiratory flow; FEV₁/FVC, Tiffeneau index; ns, not significant.

Table 5 Correlation analysis among FrOr model parameters and spirometric results

		FEV1 (L)	FEV1 (%)	FVC (L)	FVC (%)	FEV1/FVC	FEF25–75 (L)	FEF25–75 (%)
FrL	r p	−0.06 ns	−0.36 <0.0001	0.10 ns	−0.09 ns	−0.20 <0.03	−0.32 <0.0004	−0.47 <0.0001
α	r p	0.22 <0.02	0.18 <0.05	0.12 ns	0.06 ns	0.08 ns	0.23 <0.009	0.21 <0.02
FrC	r p	0.10 ns	0.00 ns	0.17 ns	0.01 ns	0.04 ns	0.00 ns	−0.06 ns
β	r p	0.31 <0.0004	0.43 <0.0001	0.18 <0.05	0.16 ns	0.27 <0.003	0.46 <0.0001	0.57 <0.0001
G	r p	−0.37 <0.0001	−0.34 <0.0002	−0.33 <0.0002	−0.17 ns	−0.18 <0.05	−0.36 <0.0001	−0.40 <0.0001
H	r p	0.01 ns	0.15 ns	−0.07 ns	0.07 ns	0.01 ns	0.11 ns	0.21 <0.02
η	r p	−0.31 <0.0004	−0.43 <0.0001	−0.18 <0.05	−0.16 ns	−0.27 <0.003	−0.46 <0.0001	−0.57 <0.0001

Note: The two strongest correlations for each FrOr parameter are indicated in bold.

Abbreviations: FEF, forced expiratory flow; FEV₁/FVC, Tiffeneau index; FRC, frequency-dependent fractional compliance; FrOr, fractional-order model; FrL, frequency-dependent fractional inertia; ns, not significant.

Table 6 Diagnostic accuracy, sensitivity, specificity and cut-off point for the traditional, eRIC and FrOr parameters in detecting respiratory alterations in smokers

	AUC	Se (%)	Sp (%)	Cut-off
Traditional
R0	0.768	70.0	67.50	2.6522
S	0.561	60.0	57.5	−12.158
Rm	0.821	77.5	75.0	2.5522
R4	0.774	70.0	70.0	2.5216
Xm	0.576	60.0	50.0	0.2551
Ax	0.737	72.5	67.5	8.7027
fr	0.584	65.0	52.5	11.397
Cdyn	0.786	72.5	67.5	17.488
Z4	0.799	75.0	75.0	3.5708
eRIC model
R	0.814	72.5	72.5	2.5491
Rp	0.545	62.5	62.5	0.0947
Rt	0.830	80.0	75.0	2.7968
I	0.562	47.5	47.5	0.0091
C	0.775	75.0	72.5	0.0159
FrOr model
FrL	0.798	70.0	72.5	0.1061
α	0.552	67.5	60.0	0.6036
FrC	0.505	47.5	45.0	0.0342
β	0.903	82.5	85	0.7028
G	0.811	80.0	77.5	13.1363
H	0.604	65.0	58.0	26.5456
η	0.903	82.5	85	0.4932

Note: Bold values indicate the highest accuracy for each class of FOT.

Abbreviations: AUCs, areas under the curve; FRC, frequency-dependent fractional compliance; FrL, frequency-dependent fractional inertia; FrOr, fractional-order model.

Figure 6 ROC curves for the most accurate traditional FOT, eRIC and FrOr parameter in smokers (A) and mild COPD patients (C). Associated comparative analysis of these AUCs and the standard errors in smokers (B) and mild COPD patients (D) are also described.

Abbreviations: AUCs, areas under the curve; FOT, forced oscillation technique; FrOr, fractional-order model; ns, not significant; ROC, receiver operating characteristic.

Table 7 Diagnostic accuracy, sensitivity, specificity and cut-off point for the traditional, eRIC and FrOr parameters in detecting respiratory alterations in patients with mild COPD

	AUC	Se (%)	Sp (%)	Cut-off
Traditional
R0	0.727	70.0	65.0	2.4613
S	0.603	55.0	40.0	−11.2899
Rm	0.748	72.5	60.0	2.276
R4	0.718	70.0	67.5	2.4303
Xm	0.597	52.5	52.5	0.2218
Ax	0.737	65.0	65.0	8.6576
fr	0.639	57.5	60.0	12.5102
Cdyn	0.763	67.5	65.0	17.9997
Z4	0.749	70.0	65.0	3.2327
eRIC model
R	0.652	67.5	65.0	2.1222
Rp	0.826	80.0	87.5	0.544
Rt	0.827	80.0	72.5	2.958
I	0.557	57.5	57.5	0.009
C	0.683	62.5	57.5	0.0175
FrOr model
FrL	0.953	92.5	92.5	0.118
α	0.678	82.5	67.5	0.5938
C	0.612	65.0	62.5	0.0358
β	0.945	95	97.5	0.6602
G	0.796	70.0	70.0	12.8706
H	0.736	82.5	70.0	22.1073
η	0.945	95.0	97.5	0.5477

Note: Bold values indicate the highest accuracy for each class of FOT.

Abbreviations: AUCs, areas under the curve; FrL, frequency-dependent fractional inertia; FrOr, fractional-order model.