**3. Respiratory system models**

The IOS and FOT impedance curves can be represented by equivalent electrical circuit models of the human respiratory system with components analogous to the resistances, compliances and inertances inherent in the characterization of this system. Respiratory system component values could be estimated using well-established parameter estimation methods. These could then be used to assist physicians in the diagnosis and treatment of different respiratory diseases (43).

In our research we have special interest in the small airways disease (SAD) or small airway impairment (SAI) and asthma. An effective means to characterize small airway dysfunction is by integrating realistic models of lung mechanics based on physiological measurements made by FOT or IOS and other techniques (74).

Impulse Oscillometric Features and Respiratory System Models Track Small Airway Function in Children 121

2

(6)

(7)

 

Rp

Cp

This model was proposed as an improvement to the eRIC model and it can be considered as a simplification of the Mead's model. aRIC is composed of central (large airway) Resistance (Rc), large airway Inertance (I), peripheral (small airway) Compliance (Cp), peripheral (small airway) Resistance (Rp) and an additional compliance Ce (see figure 4), representing extrathoracic compliance. Its additional capacitance Ce, representing extrathoracic compliance, is thought to increase the real part of the respiratory system's impedance at the higher frequencies due to upper

*A RcA R*

Ce

 

*A IC R CpC C RcA R*

<sup>2</sup> <sup>2</sup> 2 22

*e pe e p*

 

   

*p*

<sup>2</sup> <sup>222</sup>

<sup>2</sup> <sup>2</sup> 2 22

*e pe e p*

*p ep e p*

*I A R Cp A C I A R Cp C Rc A R*

*A IC R CpC C RcA R*

Lung properties of a subject can be characterized by determination of the parameters of a respiratory system model that best fits its behavior. This information can then be used, with comparison to Reference Values, to determine underdeveloped features or existence of

For this research the eRIC (Rc, I, Rp and Cp) and aRIC (Rc, I, Rp, Cp and Ce) model parameters were estimated using the average values of Resistances (R values) and Reactances (X values) from 5 to 25 Hz (R5, R10, R15, R20, R25, X5, X10, X15, X20 and X25)

*p p p p p*

*R Cp R Cp*

*R R C*

1 1

*Z Rc j I*

airways shunt effects as observed in IOS data (44).

**b. Augmented RIC Model** 

**Figure 4.** aRIC Model (47).

The aRIC impedance is given by:

*Z*

*j*

**3.1. Parameter estimation technique** 

pathological conditions (55).

for each child tested.

1

Rc I

1

2 2

Different equivalent electrical circuit models with lumped parameter components representing the resistances, inertances, and compliances of the respiratory system have been developed and analyzed over the years by different research groups (44).

Previous work by our research team has focused on development and analysis of six different equivalent electrical circuit models of human respiratory impedance. Our efforts to date, have demonstrated that the performance of the extended RIC (eRIC) and augmented RIC (aRIC) models rank in the middle of a series of conventional models developed over the past several decades in terms of total cumulative error. However, they provide parameter estimates that are physiologically more realistic and in line with expected values in healthy subjects and those suffering from pulmonary diseases than previous models (43-51).

In the following paragraphs we will introduce the two respiratory system models analyzed in this study: the Extended Resistance-Inertance-Compliance (eRIC) and the Augmented Resistance-Inertance-Compliance (aRIC) models.

#### **a. Extended RIC (eRIC) Model**

This model is proposed as an improvement of the RIC model, with an additional Peripheral resistance (Rp) connected in parallel with the capacitance. Therefore, the eRIC model is composed of central (large airway) Resistance (Rc), large airway Inertance (I), peripheral (small airway) Compliance (Cp) and peripheral (small airway) Resistance (Rp). This added Rp allows for the frequency-dependence of resistance observed in impedance data, which is not possible for the RIC model. Rp models the small airways resistance. On the other hand the eRIC model can be also considered as a simplification of either DuBois' (with It equal to zero and Ct equal to infinity) or the Mead's model (with Cl, Cw equal to infinity and Ce equal to zero) (43). The electrical equivalent circuit for the eRIC model is shown in Figure 3:

**Figure 3.** eRIC model (47).

The eRIC model impedance is calculated as follows:

Impulse Oscillometric Features and Respiratory System Models Track Small Airway Function in Children 121

$$Z = R\varepsilon + \frac{R\_p}{1 + \left(\rho \alpha \mathcal{R}\_p \mathcal{C} p\right)^2} + j \left(\rho I - \frac{\rho \alpha \mathcal{R}\_p^2 \mathcal{C}\_p}{1 + \left(\rho \alpha \mathcal{R}\_p \mathcal{C} p\right)^2}\right) \tag{6}$$

#### **b. Augmented RIC Model**

120 Practical Applications in Biomedical Engineering

made by FOT or IOS and other techniques (74).

Resistance-Inertance-Compliance (aRIC) models.

equivalent circuit for the eRIC model is shown in Figure 3:

The eRIC model impedance is calculated as follows:

I

**a. Extended RIC (eRIC) Model** 

**Figure 3.** eRIC model (47).

In our research we have special interest in the small airways disease (SAD) or small airway impairment (SAI) and asthma. An effective means to characterize small airway dysfunction is by integrating realistic models of lung mechanics based on physiological measurements

Different equivalent electrical circuit models with lumped parameter components representing the resistances, inertances, and compliances of the respiratory system have

Previous work by our research team has focused on development and analysis of six different equivalent electrical circuit models of human respiratory impedance. Our efforts to date, have demonstrated that the performance of the extended RIC (eRIC) and augmented RIC (aRIC) models rank in the middle of a series of conventional models developed over the past several decades in terms of total cumulative error. However, they provide parameter estimates that are physiologically more realistic and in line with expected values in healthy

In the following paragraphs we will introduce the two respiratory system models analyzed in this study: the Extended Resistance-Inertance-Compliance (eRIC) and the Augmented

This model is proposed as an improvement of the RIC model, with an additional Peripheral resistance (Rp) connected in parallel with the capacitance. Therefore, the eRIC model is composed of central (large airway) Resistance (Rc), large airway Inertance (I), peripheral (small airway) Compliance (Cp) and peripheral (small airway) Resistance (Rp). This added Rp allows for the frequency-dependence of resistance observed in impedance data, which is not possible for the RIC model. Rp models the small airways resistance. On the other hand the eRIC model can be also considered as a simplification of either DuBois' (with It equal to zero and Ct equal to infinity) or the Mead's model (with Cl, Cw equal to infinity and Ce equal to zero) (43). The electrical

Rc

Cp

Rp

been developed and analyzed over the years by different research groups (44).

subjects and those suffering from pulmonary diseases than previous models (43-51).

This model was proposed as an improvement to the eRIC model and it can be considered as a simplification of the Mead's model. aRIC is composed of central (large airway) Resistance (Rc), large airway Inertance (I), peripheral (small airway) Compliance (Cp), peripheral (small airway) Resistance (Rp) and an additional compliance Ce (see figure 4), representing extrathoracic compliance. Its additional capacitance Ce, representing extrathoracic compliance, is thought to increase the real part of the respiratory system's impedance at the higher frequencies due to upper airways shunt effects as observed in IOS data (44).

**Figure 4.** aRIC Model (47).

The aRIC impedance is given by:

$$\begin{aligned} Z &= \frac{A\left(RcA + R\_p\right)}{\left[A\left(1 - \alpha^2 I \mathcal{C}\_\epsilon\right) + \left(\alpha^2 R\_p^2 \mathcal{C} p \mathcal{C}\_\epsilon\right)\right]^2 + \left[\alpha \mathcal{C}\_\epsilon \left(RcA + R\_p\right)\right]^2} + \\\ j &\frac{\alpha\left(IA - R\_p^2 \mathcal{C} p\right) \left[A - \alpha^2 \mathcal{C}\_\epsilon \left(IA - R\_p^2 \mathcal{C} p\right)\right] - \alpha \mathcal{C}\_\epsilon \left(RcA + R\_p\right)^2}{\left[A\left(1 - \alpha^2 I \mathcal{C}\_\epsilon\right) + \left(\alpha^2 R\_p^2 \mathcal{C} p \mathcal{C}\_\epsilon\right)\right]^2 + \left[\alpha \mathcal{C}\_\epsilon \left(RcA + R\_p\right)\right]^2} \end{aligned} \tag{7}$$

#### **3.1. Parameter estimation technique**

Lung properties of a subject can be characterized by determination of the parameters of a respiratory system model that best fits its behavior. This information can then be used, with comparison to Reference Values, to determine underdeveloped features or existence of pathological conditions (55).

For this research the eRIC (Rc, I, Rp and Cp) and aRIC (Rc, I, Rp, Cp and Ce) model parameters were estimated using the average values of Resistances (R values) and Reactances (X values) from 5 to 25 Hz (R5, R10, R15, R20, R25, X5, X10, X15, X20 and X25) for each child tested.

In Table 5 an example of respiratory system's IOS-based Resistance and Reactance values for a healthy male child (15 years old, 181.6 cm height and 84.1 kg weight) are presented. These IOS data were recorded before and after the use of a bronchodilator (Pre-B and post-B). In Table 6 an example of the Model Parameters calculated for the same child is given.

Impulse Oscillometric Features and Respiratory System Models Track Small Airway Function in Children 123

For each child's averaged IOS data a total of 50 iterations were used to find parameter estimates minimizing the error function, with the Matlab program stopping each time when E (error value) changed by less than a factor of 10-9 from one iteration to the next one. Therefore, the LS error value at the end provided a measure of the goodness of fit to the

This study was developed as part of a NIH-funded research to perform IOS evaluation of lung function in Anglo and Hispanic children 5-19 years old living in the El Paso, Texas area. The data were collected at Western Sky Medical Research clinic and in a Health Fair held in a Socorro District school over a 3 years period (2006-2008). The IOS data collected for this research were quality-assured by our expert clinician and pulmonologist, the late Professor Michael Goldman and were then classified by him into four categories: Normal, Probable

The data presented in this study were collected in 2008. The data collected were Pre- and Post-bronchodilation data. The date collected analyzed in this study were acquired from 47 children in 2008. Three to five IOS test replicates were performed on each subject to ensure reproducible tests without artifacts caused by air leaks, swallowing, breath holding or vocalization. IOS data were carefully reviewed off line and quality-assured to reject segments affected by airflow leak or swallowing artifacts. A Jaeger MasterScreen IOS (Viasys Healthcare, Inc. Yorba Linda, CA, and USA) was used in this study. The system was calibrated every day before data collection using a 3-L syringe for volume calibrations and a reference resistance (0.2 KPa/L/s) for pressure calibrations. Children were asked to wear a nose clip, while breathing normally through a mouthpiece and were instructed to tightly close their lips around it to avoid air leakage. Children tested using a bronchodilation medicine, called Levalbuterol (Xopenex), were tested (pre-bronchodilation), performing 3 to 5 IOS tests. Then the medicine (Xopenex) was given to the children using a nebulizer for 6 minutes and after that the children were asked to rest for 10 minutes; and finally after this waiting period the children were again tested recording 3 to 5 IOS tests (post-

This research study was supported in part by NIH grant #1 S11 ES013339-01 A1: UTEP-UNM HSC ARCH Program on Border Asthma. The research protocol was approved by the Institutional Review Board of the University of Texas at El Paso. Informed consents were obtained from children over 18 years and the parents or care givers of children under 18 years. A questionnaire on asthma or allergy symptoms provided by the ARCH Program was

Statistical analyses of IOS measured and calculated parameters as well as the eRIC and aRIC model parameters between Pre- and Post-bronchodilation data were made using tdistribution (differences of the means) test and statistical significance was established at a p

Small Airway Impairment (PSAI), Small Airway Impairment (SAI) and Asthma (A).

given test data for each model (44).

bronchodilation).

< 0.05 level.

then completed by the participants or their parents.

**4. Research method design and statistical analysis** 


**Table 5.** Respiratory system's Resistance (kPa/l/s) and Reactance (kPa/l/s) values for a healthy male child.


**Table 6.** Model Parameters calculated for a healthy male child.

Estimating Model Parameters is comparable to curve-fitting. Consequently a suitable error criterion E has to be selected and minimized. For this research the least square (LS) criterion was selected as follows:

$$E = \sum \left[ ZR\left(f\right) - ZR, \text{est}\left(f\right) \right]^2 + \left[ Z\mathbf{x}\left(f\right) - Z\mathbf{x}, \text{est}\left(f\right) \right]^2 \tag{8}$$

where

*f* 5, 10, 15, 20, 25 *Hz*

This LS criterion was used to minimize the sum of the squared errors between the measured IOS ZR and Zx and the estimated resistive ZR,est and the estimated reactance Zx,est at frequencies between 5 to 25 Hz (at 5 Hz intervals). The LS criterion was selected due to its commonplace use, its relation with other system identification algorithms and its availability in different software packages (44).

Due to the nonlinear dependence of the aRIC and eRIC impedance functions on the model parameters, the Matlab lsq-nonlin (nonlinear LS) was used in both algorithms, which are based on Newton's Method. Each estimation run began with an initial random guess, a parameter estimate vector produced by a random number generator appropriately weighted. Random initial guesses ranging consistently from 0 to 5, 0 to 0.5, and 0 to 0.05 were used to estimate the values of resistances, capacitances and inductances, respectively. For each child's averaged IOS data a total of 50 iterations were used to find parameter estimates minimizing the error function, with the Matlab program stopping each time when E (error value) changed by less than a factor of 10-9 from one iteration to the next one. Therefore, the LS error value at the end provided a measure of the goodness of fit to the given test data for each model (44).

## **4. Research method design and statistical analysis**

122 Practical Applications in Biomedical Engineering

child.

where

**pre-B aRIC Rc** 

**post-B aRIC Rc** 

(kPa/l/s)

(kPa/l/s)

was selected as follows:

**Rp**  kPa/l/s)

**Rp**  (kPa/l/s)

availability in different software packages (44).

**I**  (kPa/l/s2)

**I**  (kPa/l/s2)

**Table 6.** Model Parameters calculated for a healthy male child.

In Table 5 an example of respiratory system's IOS-based Resistance and Reactance values for a healthy male child (15 years old, 181.6 cm height and 84.1 kg weight) are presented. These IOS data were recorded before and after the use of a bronchodilator (Pre-B and post-B). In

**Pre-B R5 R10 R15 R20 R25 X3 X5 X10 X15 X20 X25** 

**Post-B R5 R10 R15 R20 R25 X3 X5 X10 X15 X20 X25** 

**Table 5.** Respiratory system's Resistance (kPa/l/s) and Reactance (kPa/l/s) values for a healthy male

**Cp**  (l/kPa)

**Cp**  (l/kPa)

0.32 0.27 0.27 0.28 0.3 -0.11 -0.08 -0.01 0.03 0.06 0.08

0.3 0.26 0.26 0.28 0.3 -0.12 -0.08 0 0.04 0.08 0.11

(l/kPa) **eRIC Rc** 

(l/kPa) **eRIC Rc** 

(kPa/l/s)

(kPa/l/s)

**Rp**  kPa/l/s)

**Rp**  (kPa/l/s)

**I**  (kPa/l/s2)

**I**  (kPa/l/s2)

**Cp**  (l/kPa)

**Cp**  (l/kPa)

**Ce** 

0.256 0.209 0.001 0.233 0.004 0.277 0.314 0.0007 0.282

**Ce** 

0.245 0.239 0.001 0.241 0.004 0.274 0.58 0.0008 0.289

Estimating Model Parameters is comparable to curve-fitting. Consequently a suitable error criterion E has to be selected and minimized. For this research the least square (LS) criterion

*f* 5, 10, 15, 20, 25 *Hz*

This LS criterion was used to minimize the sum of the squared errors between the measured IOS ZR and Zx and the estimated resistive ZR,est and the estimated reactance Zx,est at frequencies between 5 to 25 Hz (at 5 Hz intervals). The LS criterion was selected due to its commonplace use, its relation with other system identification algorithms and its

Due to the nonlinear dependence of the aRIC and eRIC impedance functions on the model parameters, the Matlab lsq-nonlin (nonlinear LS) was used in both algorithms, which are based on Newton's Method. Each estimation run began with an initial random guess, a parameter estimate vector produced by a random number generator appropriately weighted. Random initial guesses ranging consistently from 0 to 5, 0 to 0.5, and 0 to 0.05 were used to estimate the values of resistances, capacitances and inductances, respectively.

2 2

*E ZR f ZR est f Zx f Zx est f* – , – , (8)

Table 6 an example of the Model Parameters calculated for the same child is given.

This study was developed as part of a NIH-funded research to perform IOS evaluation of lung function in Anglo and Hispanic children 5-19 years old living in the El Paso, Texas area. The data were collected at Western Sky Medical Research clinic and in a Health Fair held in a Socorro District school over a 3 years period (2006-2008). The IOS data collected for this research were quality-assured by our expert clinician and pulmonologist, the late Professor Michael Goldman and were then classified by him into four categories: Normal, Probable Small Airway Impairment (PSAI), Small Airway Impairment (SAI) and Asthma (A).

The data presented in this study were collected in 2008. The data collected were Pre- and Post-bronchodilation data. The date collected analyzed in this study were acquired from 47 children in 2008. Three to five IOS test replicates were performed on each subject to ensure reproducible tests without artifacts caused by air leaks, swallowing, breath holding or vocalization. IOS data were carefully reviewed off line and quality-assured to reject segments affected by airflow leak or swallowing artifacts. A Jaeger MasterScreen IOS (Viasys Healthcare, Inc. Yorba Linda, CA, and USA) was used in this study. The system was calibrated every day before data collection using a 3-L syringe for volume calibrations and a reference resistance (0.2 KPa/L/s) for pressure calibrations. Children were asked to wear a nose clip, while breathing normally through a mouthpiece and were instructed to tightly close their lips around it to avoid air leakage. Children tested using a bronchodilation medicine, called Levalbuterol (Xopenex), were tested (pre-bronchodilation), performing 3 to 5 IOS tests. Then the medicine (Xopenex) was given to the children using a nebulizer for 6 minutes and after that the children were asked to rest for 10 minutes; and finally after this waiting period the children were again tested recording 3 to 5 IOS tests (postbronchodilation).

This research study was supported in part by NIH grant #1 S11 ES013339-01 A1: UTEP-UNM HSC ARCH Program on Border Asthma. The research protocol was approved by the Institutional Review Board of the University of Texas at El Paso. Informed consents were obtained from children over 18 years and the parents or care givers of children under 18 years. A questionnaire on asthma or allergy symptoms provided by the ARCH Program was then completed by the participants or their parents.

Statistical analyses of IOS measured and calculated parameters as well as the eRIC and aRIC model parameters between Pre- and Post-bronchodilation data were made using tdistribution (differences of the means) test and statistical significance was established at a p < 0.05 level.
