**Appendix:**

846 Biomedical Science, Engineering and Technology

(1 / )tan( ) *a v k t* = − ω

[sin( ) cos( )] TV 1

*BC t k t*

ωω

and *tv* are known, we can evaluate *ka* from Equation (38-b).

<sup>−</sup> <sup>=</sup> <sup>+</sup>

For our COPD patients, the ranges of the computed values of these parameters are:

to develop an integrated index lung ventilatory incorporating these parameters.

**5.4 Formulating a Lung Ventilatory Index (***LVI***) incorporating** *Ra* **and** *Ca*

lung-ventilatory index (*LVI*) can be expressed, as given by Equation (30):

Since both

Since ω

amplitude *B*.

mechanical ventilator.

have

where *RF* is the respiratory-rate frequency.

ω

Now from Equation (38-b), we can put down:

Then knowing *ka* and *Ca*, we can determine

 ω

2 2

 ω

*R kC aaa* = / (40)

*k*

, *tv* and *ka* are known, we can now determine *Ca* in terms of TV and applied pressure

( ) ( ) <sup>1</sup> <sup>1</sup>

Now that we have determined the expressions for the parameters *Ra* and *Ca*, the next step is

We believe that the correlations between average airflow-resistance (*Ra*), average lungcompliance (*Ca*), tidal volume (TV), respiratory rate (*RF*), and maximum inspiratory pressure or peak pressure (*Pm*) can be used as a possible indicator for determining lungstatus in a mechanically ventilated COPD patient with acute respiratory failure. We hence propose that a composite index (*LVI*), incorporating these isolated parameters, can have a higher predictive power for assessing lung status and determining when a patient on a

For this purpose, we note that COPD patients have higher *Ra*, lower *Ca*, lower TV, higher *Pm* and higher respiratory rate (or breathing frequency) *RF*. If we want the non-dimensional lung-ventilatory index (*LVI*) to have a high value for a COPD patient, further increasing *LVI* for deteriorating lung-status and decreasing *LVI* for improving lung-status in a mechanically ventilated COPD patient in acute respiratory failure, then the non-dimensional

2

*a m*

Let us obtain the order-of-magnitude values of this *LVI*2 index for a mechanically ventilated COPD patient in acute respiratory failure (by using representative computed values of the parameters *Ra, Ca, RF,* TV*,* and *Pm*), in order to verify that the formula for *LVI*2 (given by Equation (42)) can enable distinct separation of COPD patients in acute respiratory failure from patients ready to be weaned off the respirator. For an intubated COPD patient, we

(TV) ( ) (60) ( )

2 2

*a*

*R RF LVI C P* ⎡ ⎤ = × ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

2

(42)

2 2 <sup>9</sup> –43 cmH 0 sL ; 0.020 –0.080 L cmH 0 *R C a a* <sup>−</sup> <sup>−</sup> = = (41)

*a va v a*

ω

(38-b)

(39)

**Solution procedure to obtain the best values of** *R***,** *C***,** *Pk* **and** *P***0 provided in Equation (29).**  For the four unknowns *R*, *C*, *Pk* and *P*0, where

 *R*: 2.1, 2.2, 2.3 (cmH2O)sL-1; *C*: 0.20, 0.21, 0.22 L(cmH2O)-1 ;  *Pk*: 0.4, 0.5, 0.6 (cmH2O);  *P0*: 2.4, 2.5, 2.6 (cmH2O);

we want to find the best values of *R*, *C*, *Pk* and *P*0 such that they satisfy the following equations:

$$R(0.48) + \frac{0.3}{C} = P\_0 \tag{24}$$

$$\frac{0.48}{C} = 3.73P\_k \tag{25}$$

$$0.33R + \frac{0.48}{C} = P\_k + P\_0 \tag{26}$$

$$-0.622R + \frac{0.33}{C} = 0\tag{27}$$

$$\frac{0.55}{C} = P\_0 \tag{28}$$

### **Solution:**

We can rewrite Equations (26) to (30) so that the terms are collected at the LHS, i.e.,

$$R(0.48) + \frac{0.3}{C} - P\_0 = 0\tag{A-1}$$

$$\frac{0.48}{C} - 3.73P\_k = 0\tag{A-2}$$

$$0.33R + \frac{0.48}{C} - P\_k - P\_0 = 0\tag{A-3}$$

$$-0.622R + \frac{0.33}{C} = 0\tag{A-4}$$

$$\frac{0.55}{C} - P\_0 = 0\tag{A-5}$$

Since we are trying to find the best values of *R*, *C*, *Pk* and *P*0, the RHS of Equations (A-1) to (A-5) can be replaced by an error term so that

$$R(0.48) + \frac{0.3}{C} - P\_0 = e\_1 \tag{A-6}$$

$$\frac{0.48}{C} - 3.73P\_k = e\_2\tag{A-7}$$

$$0.33R + \frac{0.48}{C} - P\_k - P\_0 = e\_3\tag{A-8}$$

$$-0.622R + \frac{0.33}{C} = e\_4 \tag{A-9}$$

$$\frac{0.55}{C} - P\_0 = e\_5 \tag{A-10}$$

In general, we can represent the overall error incurred at any values of *R*, *C*, *Pk* and *P*0 by an objective function *F* so that

$$F(R, C, P\_{k'}, P\_0) = \sum\_{i=1}^{5} \|e\_i\|\tag{A-11}$$

We can then find the best values of *R*, *C*, *Pk* and *P*0 but solving Equation (A-11) as an optimization problem with the aim to minimize *F*, subjected to the bounded constraints:

```
2.1 ≤ R ≤ 2.3 
        0.20 ≤ C ≤ 0.22 
 2.4 ≤ P0 ≤ 2.6 
 0.4 ≤ Pk ≤ 0.6
```
The optimal solution is obtained at *F* = 0.239539, and the associated best values of *R*, *C*, *Pk* and *P*0 are

 *R* = 2.29425 cmH2O)sL-1  *C* = 0.219758 L(cmH2O)-1  *Pk* = 0.57926 (cmH2O)  *P0* = 2.46702 (cmH2O)

(A-12)
