**2. Lung ventilation model**

### **2.1 Scope**

In this section, we have developed a lung ventilation model by modeling the lung volume response to mouth minus pleural driving pressure (by means of a first order differential equation) in terms of resistance-to-airflow (*R*) and the lung compliance (*C*). The lung volume solution of the differential equation is matched with the clinical volume data, to evaluate the parameters, *R* and *C*. These parameters' values can help us to distinguish lung disease states, such as obstructive lung and lung with stiffened parenchyma, asthma and emphysema.

### **2.2 Role of lung ventilation**

Lung ventilation constitutes inhalation of appropriate air volume under driving pressure (= mouth pressure – pleuralpressure), so as to: (i) provide adequate alveolar *O2* amount at 

Fig. 1. Lumped Lobule Lung Model. In the figure, *Pa* is the alveolar pressure; *Pmo* is the pressure at the mouth; *Pp* is the pleural pressure; *Pel = Pa – Pp = 2h*σ*/r = 2T/r*, the lung elastic recoil pressure; *r* is the radius of the alveolar chamber and *h* is its wall thickness; *T* is the wall tension in the alveolar chamber; *V* is the lung volume; *R* is the resistance to airflow; and *C* is the lung compliance. This figure is adopted from our work in Ref [1].

designate a specific lung disease for the patient. The *LVI* concept for detecting lung disease is more convenient to adopt in clinical practice, because it enables detection of lung disease

Now, in this methodology, we need to monitor (i) lung volume, by means of a spirometer, and (ii) lung pressure (*PL*) equal to *Pmo* (pressure at mouth) minus pleural pressure (*Pp*). The pleural pressure measurement involves placing a balloon catheter transducer through the nose into the esophagus, whereby the esophageal tube pressure is assumed to be equal to the pressure in the pleural space surrounding it. Now this procedure cannot be carried out non-traumatically and routinely in patients. Hence, for routineand noninvasive assessment of lung ventilation for detection of lung disease states, it is necessary to have a method for determining *R* and *C* from only lung volume data. So, then, we have shown how we can compute *R*, *C* and lung pressure values non-invasively

Finally, we have presented how the lung ventilation modeling can be applied to study the lung ventilation dynamics of COPD patients on mechanical ventilation. We have shown how a COPD patient's lung *C* and *R* can be evaluated in terms of the monitored lung volume and applied ventilatory pressure. We have also formulated another lung ventilator index to study and assess the lung status improvement of COPD patients on mechanical

In this section, we have developed a lung ventilation model by modeling the lung volume response to mouth minus pleural driving pressure (by means of a first order differential equation) in terms of resistance-to-airflow (*R*) and the lung compliance (*C*). The lung volume solution of the differential equation is matched with the clinical volume data, to evaluate the parameters, *R* and *C*. These parameters' values can help us to distinguish lung disease states, such as obstructive lung and lung with stiffened parenchyma, asthma and emphysema.

Lung ventilation constitutes inhalation of appropriate air volume under driving pressure (= mouth pressure – pleuralpressure), so as to: (i) provide adequate alveolar *O2* amount at 

Fig. 1. Lumped Lobule Lung Model. In the figure, *Pa* is the alveolar pressure; *Pmo* is the

*T* 

*Pel*

*V, Pa*

recoil pressure; *r* is the radius of the alveolar chamber and *h* is its wall thickness; *T* is the wall tension in the alveolar chamber; *V* is the lung volume; *R* is the resistance to airflow; and

σ

*Pp*: pleural pressure

*R Pmo*

*/r = 2T/r*, the lung elastic

pressure at the mouth; *Pp* is the pleural pressure; *Pel = Pa – Pp = 2h*

Air

*C* is the lung compliance. This figure is adopted from our work in Ref [1].

Diaphragm *T* 

ventilation, and to decide when they can be weaned off mechanical ventilation.

states in the form of just one lung-ventilation number.

from just lung volume measurement.

**2. Lung ventilation model** 

**2.2 Role of lung ventilation** 

**2.1 Scope** 

appropriate partial pressure, (ii) oxygenate the pulmonary blood, and (iii) thereby provide adequate metabolic oxygen to the cells. Hence, ventilatory function and performance assessment entails determining how much air volume is provided to the alveoli, to make available adequate alveolar oxygen for blood oxygenation and cellular respiration.

In this lumped lobule lung model [1], we have (i) a lumped alveolar chamber of volume *V* and pressure *Pa*, and (ii) lumped airway having airflow resistance *R*. In this airway, the pressure varies from *Pmo* at the mouth to *Pa* in the alveolar chamber. The pleural pressure is *Pp*.

### **2.3 Lung ventilation analysis (using a linear first-order differential equation model)**

We first analyze Lung Ventilation function by means of a model represented by a first-order differential equation (*Deq*) in lung-volume (*V*) dynamics in response to the driving pressure *PL* (= mouth pressure − pleural pressure). In this model [2]**,** the lung lobes and the alveoli are lumped into one lung lobule, as depicted in Figure 1. Figure 2 displays typical data of lung volume and flow, alveolar and pleural pressure.

Fig. 2. Lung ventilatory model and lung-volume and pleural-pressure data. In the bottom figure, Curve 1 represents the negative of *Pel*, the pressure required to overcome lung elastance (=1/*C*) plus elastic recoil pressure at the end of expiration. Curve 2 represents *Pp* = – *Pel* + *Pa*. Now, as can be noted from Figure 1, *Pa – Pp = Pel*. The lung driving pressure *PL* = *Pmo* – *Pp*, and the net driving pressure *PN*(*t*) in Equation (1-b) equals *PL* minus *Pel* at endexpiration. We define resistance-to-airflow (*R*) as ( )/ *P PV mo a* <sup>−</sup> . We define lung compliance *C* = *V*/(*Pel* – *Pel*0) = *V*/(*Pa* – *Pp*) – *Pel*0. This figure is adopted from our work in Ref [1].

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪

(1-a)

⎭

Based on Figures 1 and 2, we can put:

(i) (– ) – 0 *a p el PP P* =

$$\text{(iii)}\qquad P\_{\text{el}} = \text{(2}\sigma\text{h)}\text{ / }\text{R}\quad = \text{2 T}\text{ / }\text{R}\quad = V\text{/C}\text{ + }P\_{\text{el}0}\text{(at end}-\text{expplication)}$$


The lung ventilation model governing equation is hence formulated as:

$$R\,\dot{V} + \frac{V}{C} = P\_L(t) - P\_{el0} = P\_N(t) \tag{1-b}$$

wherein:


$$P\_{\rm el0} = P\_{\rm el} - \frac{V}{C} \tag{1\text{-}c}$$

v. At end-expiration when ω *t* = ω *T, PL = Pel0*

Now, in order to evaluate the lung model parameters *C* and *R*, we need to simulate this governing equation to lung volume (*V*) and pressure (*PN*) data. This clinical data is shown in Figure 3. The lung volume is measured by integrating the airflow velocity-time curve, where the airflow velocity can be measured by means of a ventilator pneumatograph; the lung volume can also be measured by means of a spirometer. Inhalation and exhalation pressures are measured by means of a pressure transducer connected to the ventilatory tubing; likewise, a pressure transducer can also be similarly connected to the spirometer tubing. The pleural pressure is measured by placing a balloon catheter transducer through the nose into the esophagus; it is assumed that the esophageal tube pressure equals the pressure in the pleural space surrounding it.

In Equation (1-a), we have put 3 1 *N i ii* ( ) sin( ) *i Pt P tc* ω = = + ∑ , expressed as a Fourier series. The

governing equation (1-b) now becomes:

$$R\,\dot{V} + \frac{V}{C} = P\_N(t) = \sum\_{i=1}^{3} P\_i \sin(a\phi\_i t + c\_i) \tag{2-a}$$

where the right-hand side represents the net driving pressure minus pleural pressure: *PN =* (*Pmo* – *Pp*) – *Pel0*. This *PN* is in fact the driving pressure (*Pmo* – *Pp*) normalized with respect to its value at end-expiration. Equation (2-a) can be rewritten as follows:

$$\dot{V} + \frac{V}{RC} = \frac{1}{R} \sum\_{i=1}^{3} P\_i \sin(a\phi\_i t + c\_i) \tag{2-b}$$

wherein the *P*(*t*) clinical data (displayed in Figure 3**)** is represented by:

834 Biomedical Science, Engineering and Technology

(ii) (2 ) / 2 / / at end expirati

The lung ventilation model governing equation is hence formulated as:

*C*

iv. *Pel0* is the lung elastic-recoil pressure at the end of expiration, and

ω *t* = ω

= = =+ − ( )

(iii) ( – ) /

 (v) / / – (lung elastic recoil pressure at end expiration) *R dV dt V C P P L el*

*<sup>V</sup> RV P t P P t*

i. the values of pressure*PN*(*t*) are obtained from the *PL* (= *Pmo* – *Pp* ) data relative to *Pel*<sup>0</sup> ii. the parameters of this Governing Deq are lung compliance (*C*) and airflow-

iii. *V* = *V*(*t*) −*V*0 (the lung air volume at the end-expiration = lung air volume inspired

+= − =

resistance (*R*); in the equation both *R* and *C* are instantaneous values

*el el* 0 *<sup>V</sup> P P*

3

1 *N i ii* ( ) sin( )

ω

3

1 *N i ii* ( ) sin( )

ω

*i <sup>V</sup> RV P t P t c*

where the right-hand side represents the net driving pressure minus pleural pressure: *PN =* (*Pmo* – *Pp*) – *Pel0*. This *PN* is in fact the driving pressure (*Pmo* – *Pp*) normalized with respect

=

*i Pt P tc*

*C*

to its value at end-expiration. Equation (2-a) can be rewritten as follows:

=

 *T, PL = Pel0* Now, in order to evaluate the lung model parameters *C* and *R*, we need to simulate this governing equation to lung volume (*V*) and pressure (*PN*) data. This clinical data is shown in Figure 3. The lung volume is measured by integrating the airflow velocity-time curve, where the airflow velocity can be measured by means of a ventilator pneumatograph; the lung volume can also be measured by means of a spirometer. Inhalation and exhalation pressures are measured by means of a pressure transducer connected to the ventilatory tubing; likewise, a pressure transducer can also be similarly connected to the spirometer tubing. The pleural pressure is measured by placing a balloon catheter transducer through the nose into the esophagus; it is assumed that the esophageal tube pressure equals the pressure in the

*el el*

*P h R T R VC P*

( ) ( )

*L mo p mo a a p*

and expired during a single breath)

*P P P P P PP*

0

+ = − ⎪

<sup>0</sup> () () *L el N*

on

*<sup>C</sup>* = − (1-c)

= + ∑ , expressed as a Fourier series. The

+= = + ∑ (2-a)

(1-b)

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪

(1-a)

⎭

Based on Figures 1 and 2, we can put:

=

(iv) – – ( – )

== + ( ) <sup>0</sup>

*P P R dV dt*

=

v. At end-expiration when

pleural space surrounding it.

In Equation (1-a), we have put

governing equation (1-b) now becomes:

(i) (– ) – 0

*mo a*

wherein:

*a p el*

σ

*PP P*

$$P(t) = \sum\_{i=1}^{3} P\_i \sin(a\varrho\_i t + c\_i) \tag{3}$$

$$\begin{array}{llll} \text{P}\_1 = 1.581 \text{ cm} \text{H}\_2\text{O} & \text{P}\_2 = -5.534 \text{ cm} \text{H}\_2\text{O} & \text{P}\_3 = 0.5523 \text{ cm} \text{H}\_2\text{O} \\\ o\_1 = 1.214 \text{ rad/s} & o\_2 = 0.001414 \text{ rad/s} & o\_3 = 2.401 \text{ rad/s} \\\ c\_1 = -0.3132 \text{ rad} & c\_2 = 3.297 \text{ rad} & c\_3 = \text{-}2.381 \text{ rad} \\\ \end{array}$$

The pressure curve (in Figure 3) represented by the above Equation (3) closely matches the pressure data of Figure 3. If, in Equation (1), we designate *Ra* and *Ca* as the average values (*R*  and *C*) for the ventilatory cycle, then the solution of Equation (2) is given by:

$$V(t) = \sum\_{i=1}^{3} \frac{P\_i \mathbf{C}\_a \left[ \sin(a\mathbf{\hat{q}} \, t + c\_i) - a\mathbf{\hat{q}} \, \mathbf{R}\_a \mathbf{C}\_a \cos(a\mathbf{\hat{q}} \, t + c\_i) \right]}{1 + a\mathbf{\hat{q}}^2 \left( \mathbf{R}\_a \mathbf{C}\_a \right)^2} - H e^{-\frac{t}{\mathbf{R}\_a \mathbf{C}\_a}} \tag{4}$$

wherein the term (*RaCa*) is denoted by τ*<sup>a</sup>*. We need to have *V* = 0 at *t* = 0. Hence, putting *V* (at *t* = 0) = 0*,* gives us:

$$H = \sum\_{i=1}^{3} \frac{P\_i \mathbb{C}\_a[\sin(\mathbf{c}\_i) - a\_i R\_a \mathbb{C}\_a \cos(\mathbf{c}\_i)]}{1 + a\_i^{\uparrow} \left(R\_a \mathbb{C}\_a\right)^2} \tag{5}$$

Then from Equations (4) and (5), the overall expressions for *V*(*t*) becomes

$$V(t) = \sum\_{i=1}^{3} \frac{P\_i \mathbb{C}\_a[\sin(a\sharp t + c\_i) - a\sharp \tau\_a \cos(a\sharp t + c\_i)]}{1 + a\varrho\_i^2 \tau\_a^2} - \sum\_{i=1}^{3} \frac{P\_i \mathbb{C}\_a[\sin(c\_i) - a\sharp \tau\_a \cos(c\_i)]}{1 + a\varrho\_i^2 \tau\_a^2} e^{-\frac{t}{\tau\_a}}\tag{6}$$

We also want that *dV*/*dt* = 0 at *t* =0, implying no air-flow at the start of inspiration. So then by differentiating Equation (6), we get the expression for air-flow (*V* ), as:

$$\dot{V} = \sum\_{i=1}^{3} \frac{P\_i \mathbb{C}\_a [a\rho \cos(a\varrho \, t + c\_i) + a\varrho^2 \tau\_a \sin(a\varrho \, t + c\_i)]}{1 + a\varrho^2 \tau\_a^2} + \sum\_{i=1}^{3} \frac{P\_i \mathbb{C}\_a [\sin(c\_i) - a\varrho \tau\_a \cos(c\_i)]}{(1 + a\varrho^2 \tau\_a^2)\tau\_a} e^{-\frac{t}{\tau\_a}} \tag{7}$$

For the above values of τ*<sup>a</sup>* = 0.485 s and for ω*<sup>i</sup>* and *ci*given by Equation (3), we get 3 1 ( 0) ( / )sin( ) 0 *ia i i Vt P R c* = = = ∑ <sup>≈</sup> , to satisfy the initial condition.

Now by matching the above *V*(*t*) expression in Equation (6) with the *V*(*t*) data in Figure 3, and carrying out parameter-identification, we can determine the *in vivo* values of *Ca*, *Ra* and τ*<sup>a</sup>* to be:

() () <sup>1</sup> <sup>1</sup> 2 2 0.218 L cmH O , 2.275 cmH O sL , 0.485 s *C R a aa* τ<sup>−</sup> <sup>−</sup> = == (8)

The computed *V*(*t*) curve, represented by Equation (6) for the above values of *Ca* and *Ra*, is shown in Figure 3.

Fig. 3. (a)The pressure curve represented by Equation (3) matched against the pressure data (represented by dots). (b) The volume curve represented by Equation (6), for *Ca* = 0.2132 L(cmH2O)-1 and *Ra* = 2.275 (cmH2O)sL-1, matched against the volume data represented by dots. In Figure 3(a), the terms *P0*, *Pm* and *Pk* refer to Equation (11). At *t* = *tv*, *V* is maximum and *V* is zero. This figure is adopted from our work in Ref [1].

Let us have some validation of the average values of *C* and *R* obtained by parameteridentification scheme, by determining the values of *C* and *R* at some specific time instants. For that purpose, we can put down from Equation (1-b),

$$
\vec{R}\,\vec{V} + \frac{\dot{V}}{C} = \dot{P}\_N(t) \tag{9}
$$

Now the volume (*V*) curve in Figure 3(b) has an inflection point at *t* = *ti* = 1.18 s, at which *V* = 0 . At *t* = 1.18 s, *V* = 0.29 L, *V* = 0.48 Ls-1, *PN* = 2.53 cmH2O, and <sup>1</sup> <sup>2</sup> *PN* 1.66 (cmH O)s<sup>−</sup> <sup>=</sup> . Upon substituting these values into Equation (9), we get *C* = 0.289 L(cmH2O)-1. Then substituting this value of *C* along with the values of *V*, *V* and *PN* into Equation (1-a), we get *R* = 3.18 (cmH2O)sL-1. These values of *C* and *R* at *t* = *ti* = 1.18 s are of the same order of magnitude as the average values of *C* and *R* given by Equation (8). This provides us a measure of confidence to our parameter-identification scheme for obtaining the average values *Ca* and *Ra*.

Now since Lung disease will influence the values of *R* and *C*, these parameters can be employed to diagnose lung diseases. For instance in the case of emphysema, the destruction of lung tissue between the alveoli produces a more compliant lung, and hence results in a larger value of *C*. In asthma, there is increased airway resistance (*R*) due to contraction of the smooth muscle around the airways. In fibrosis of the lung, the membranes between the alveoli thicken and hence lung compliance (*C*) decreases. Thus by determining the normal and diseased ranges of the parameters *R* and *C*, we can employ this simple Lung-ventilation model for differential diagnosis.
