**5. Medical test: Cardiac fitness index based on treadmill HR variation**

In this procedure, the cardiac fitness model consists of a first-order differential-equation system model, describing the heart rate (HR) response (y) to exertion (exercise, jogging etc) monitored in terms of the work-load, where y is defined as follows:

$$y = \frac{HR(t) - HR(rest)}{HR(rest)}\tag{1}$$

The subject is exercised on the treadmill for a period of time *te* (minutes) at a constant work load (W), while the HR(t) (and hence y) is monitored, as displayed in fig 15. Now we develop a model to simulate (i) the y(t) response during exercise, i.e. during *t ≤ te*, and (ii) thereafter for y(t) decay, after the termination of exercise. In a way, *te* represents the exercise endurance of the subject [6].

For a person, the model equation for HR response is represented by:

$$\frac{dy}{dt} + k\_1 y = \mathbb{C}\_0 \mathcal{W} \tag{2}$$

For *t ≤ te*, the solution is given by:

$$y = \frac{y\_e(1 - e^{-k\_1 t})}{(1 - e^{-k\_1 t\_s})} \tag{3}$$

For the recovery period *(t ≥ te )*, the solution of eqn. (2) is :

$$y = y\_{\epsilon}e^{-k\_2(t-t\_{\epsilon})} \tag{4}$$

where *k1* and *k2*, are the model parameters, which can serve as cardiac-fitness parameters (in min-1).

### **Non-dimensional Cardiac Fitness Index:**

A typical y(t) response is illustrated in Fig 15.

Fig. 15. Graph of y (the computed HR response) vs. t during treadmill exercise *t=te* and during the recover of period *t=te.*

The parameters *k1* and *k2* can be continued into a single nondimensional cardiac-fitness index (CFI):

$$\text{CFI} = k\_1 k\_2 t\_e \,\text{ ${}^2$ }\tag{5}$$

According to this formulation of CFI, a healthier subject has (i) greater *k1* (i.e., slower rate-ofincrease of HR during exercise (ii) greater *k2* (i.e., faster rate-of-decrease of HR after exercise) (iii) greater *te* (i.e., exercise endurance), and hence (iv) higher value of CFI.


Table 1. CFI values for athletic, fit and unfit subjects.

where *k1* and *k2*, are the model parameters, which can serve as cardiac-fitness parameters

Fig. 15. Graph of y (the computed HR response) vs. t during treadmill exercise *t=te* and

The parameters *k1* and *k2* can be continued into a single nondimensional cardiac-fitness

According to this formulation of CFI, a healthier subject has (i) greater *k1* (i.e., slower rate-ofincrease of HR during exercise (ii) greater *k2* (i.e., faster rate-of-decrease of HR after exercise)

(iii) greater *te* (i.e., exercise endurance), and hence (iv) higher value of CFI.

Table 1. CFI values for athletic, fit and unfit subjects.

*y e <sup>y</sup> e*

For the recovery period *(t ≥ te )*, the solution of eqn. (2) is :

**Non-dimensional Cardiac Fitness Index:**  A typical y(t) response is illustrated in Fig 15.

during the recover of period *t=te.*

index (CFI):

(in min-1).

1 1 (1 ) (1 )*<sup>e</sup> k t <sup>e</sup> k t*

<sup>−</sup> <sup>=</sup> − (3)

<sup>2</sup> ( )*<sup>e</sup> k tt <sup>e</sup> y ye*− − <sup>=</sup> (4)

<sup>2</sup> *CFI k k t* <sup>=</sup> 1 2 *<sup>e</sup>* (5)

− −

Now, we need to evaluate CFI for a big spectrum of patients, and then compute its distribution curve, to determine the efficiency of this index, in order to yield distinct separation of CFI ranges for healthy subjects and unfit patients. This CFI can also be employed to assess improvement in cardiac fitness following cardiac rehabilitation regime. This CFI is non-dimensional, and it can be useful to clinicians as they are able to predict the heart condition or fitness performance of a person by referring to the value of a single index value.
