**7. Cardiology: LV contractility index based on normalized wall-stress**

The traditional cardiac contractility index *(dP/dt)max* requires cardiac catheterization.

Fig. 18. Sample cyclic variation of LV pressure and volume.

Since LV Pressure is developed by LV wall stress σθ(based on sarcomere contraction), we have developed a contractility index based on σθ (normalized with respect to LV pressure) [9].

For a thick walled spherical model of LV, the circumferential wall stress:

$$\sigma\_{\theta}(r\_{i}) = P \frac{\left(r\_{i}^{3} \;/\; r\_{\text{e}}^{3} + 0.5\right)}{\left(1 - r\_{i}^{3} \;/\; r\_{\text{e}}^{3}\right)} \tag{1}$$

Fig. 19. Thick walled Spherical Model of the LV.

$$\text{The normalized Stress} = \frac{\sigma\_{\theta}(r\_i)}{P} = \sigma^\* = \frac{3V}{2MV} + \frac{1}{2} \tag{2}$$

 We define the contractility index as:

26 Biomedical Science, Engineering and Technology

As can be seen from figure 17, normal (i.e., non-diabetic) subjects with no risk of becoming diabetic, will have DIN value less than 0.4, or be in the 1 – 2 range. Distinctly diabetic

Supposedly, clinically-identified normal subjects who have DIN values between 0.4 and 1.0,

On the other hand, clinically-identified diabetic subjects with DIN value between 0.4 – 1.2, or in the 3 – 6 range category are border-line diabetics, who can become normal (with diet

**Distribution P lot**

diabetes ov erdamped diabetes c ritic al normal c ritic al normal underdamped

secs

secs

diabetic (overdamped) diabetic (critically damped) normal (critically damped) normal (underdamped)

Fig. 17. DIN distribution plot of all the four categories subjects [8]. In the figure, diabetic (critically damped) category of subjects are designated as border-line diabetic; normal (critically damped) category of subjects are designated as normal subjects at risk of

1 2 3 4 5 6 7 8 9 10 11 12 **Range**

**7. Cardiology: LV contractility index based on normalized wall-stress**  The traditional cardiac contractility index *(dP/dt)max* requires cardiac catheterization.

Fig. 18. Sample cyclic variation of LV pressure and volume.

subjects will have DIN value greater than 1.2, or be in the 7 – 12 range categories.

or are in the 3 – 5 range, are at risk of becoming diabetic.

control and treatment).

becoming diabetic.

**frequency**

$$\text{Contrracibility Index (CONT1)} \left| \frac{\text{d}\,\sigma^{\star}}{\text{d}t} \right|\_{\text{max}} = \frac{3}{2MV} (\frac{dV}{dt})\_{\text{max}} \tag{3}$$

For the data shown in the figure 18, we have:

$$CONT1 = \frac{3}{200cc} (224cc \cdot s^{-1}) = 3.3s^{-1}$$

Now we formulate a non-dimensional cardiac contractility index,

$$\begin{aligned} \text{CONT2} &= \left| \frac{d\sigma^\*}{dt} \right|\_{\text{max}} \times \text{ ejection period} (= 0.3second) \times 100 \\ &\approx 100 \end{aligned} $$

Our new contractility indices do not require measurement of LV pressure, and can hence be evaluated noninvasively. In fig 20, we can see how well our contractility index *CONT*1 *(dσ\*/dt)max* correlates with the traditional contractility index *(dP/dt)max*. This provides a measure of confidence for clinical usage of this index.

Fig. 20. Correlation of our new contractility index *(dσ\*/dt)max* with the traditional contractility index(dP/dt)max

### **8. Diagnostics: LV contractility index based on LV shape-factor**

Cardiologists have observed shape changes taking place in an impaired LV. We have investigated the effect of ventricular shape on contractility and ejection function. In this study, a new LV contractility index is developed in terms of the wall-stress (σ\*, normalized with respect to LV pressure) of a LV ellipsoidal model (Fig. 21) [10, 11].

Using cine-ventriculography data of LV volume (V) and myocardial volume MV, the LV ellipsoidal model (LVEM) major (B) and minor axes (A) are derived for the entire cardiac cycle. Thereafter, our new contractility index *(dσ\*/dt)max* is derived in terms of the LV ellipsoidal shape factor *(s=B/A).* 

For the LV model (Fig 21) of a prolate spheroid truncated 50% of the distance from the equator to the base, we first put down the for σ\* ( = σθ /P ), and then determine dσ\*/dt[2.10]. Thereby, we obtain the following expression for the contractility index:

$$\left| \begin{array}{c} \text{Contractality index} \end{array} - I \text{ ( $CONTI$ )} \right|\_{\text{max}} $$

$$\mathbf{H} = \left| \frac{\dot{V}(2+s) + V\dot{s}}{MV} - \frac{V^2 s}{MV(4V + 2Vs + MV)^2} \right|\_{\text{+Và}} \frac{\text{s} \dot{V}(8 + 4MV / V + (8 + 2MV / V)\mathbf{s} + 2s^2)}{\text{+Và}(16 + 4MV / V + (16 + 3MV / V)\mathbf{s} + 4s^2)} \Bigg|\_{\text{max}} = F(s, \dot{s}, V, \dot{V}, MV)$$

where s is B/A, ś is first-time derivative of s; V and MV are LV volume and myocardial wall volume, Ṽ is the first-time derivative of V.

This index is analogous to the traditional employed index *(dP/dt)max*, but does not involve determination of the intra-LV pressure by catheterization. For patient A (with myocardial infarct and double vessel disease) and B (with double vessel disease and hypertension), the values of CONT1 are obtained to be 3.84 and 6.90 s-1, whereas the corresponding values of *(dP/dt)max* are obtained to be 985 and 1475 mmHg/s.

Fig. 20. Correlation of our new contractility index *(dσ\*/dt)max* with the traditional contractility

700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800

dP/dtmax (mmHg/s)

Cardiologists have observed shape changes taking place in an impaired LV. We have investigated the effect of ventricular shape on contractility and ejection function. In this study, a new LV contractility index is developed in terms of the wall-stress (σ\*, normalized

Using cine-ventriculography data of LV volume (V) and myocardial volume MV, the LV ellipsoidal model (LVEM) major (B) and minor axes (A) are derived for the entire cardiac cycle. Thereafter, our new contractility index *(dσ\*/dt)max* is derived in terms of the LV

For the LV model (Fig 21) of a prolate spheroid truncated 50% of the distance from the equator to the base, we first put down the for σ\* ( = σθ /P ), and then determine dσ\*/dt[2.10]. Thereby, we obtain the following expression for the contractility index:

where s is B/A, ś is first-time derivative of s; V and MV are LV volume and myocardial

2 2

(2 ) (8 4 / (8 2 / ) 2 ) (, , , ) (4 2 ) (16 4 / (16 3 / ) 4 )

*F s s V, V MV MV MV V Vs MV Vs MV V MV V s s*

2 2

max

This index is analogous to the traditional employed index *(dP/dt)max*, but does not involve determination of the intra-LV pressure by catheterization. For patient A (with myocardial infarct and double vessel disease) and B (with double vessel disease and hypertension), the values of CONT1 are obtained to be 3.84 and 6.90 s-1, whereas the corresponding values of

**8. Diagnostics: LV contractility index based on LV shape-factor** 

with respect to LV pressure) of a LV ellipsoidal model (Fig. 21) [10, 11].

\*

σ

/

*V s Vs V s sV MV V MV V s s*

max

+ + ⎛ ⎞ + ++ + = − ⎜ ⎟ <sup>=</sup> + + ⎝ ⎠ + + ++ +

index(dP/dt)max

ellipsoidal shape factor *(s=B/A).* 

*Contractility index 1 (CONT1) (d dt)*

− =

dσ\*/dt (s-1)

wall volume, Ṽ is the first-time derivative of V.

*(dP/dt)max* are obtained to be 985 and 1475 mmHg/s.

Fig. 21. Cineangiography imaged LV geometry and the corresponding constructed LV ellipsoidal model: (a) at an instant during the diastolic filling stage, (b) at an instant during the systolic ejection stage, (c) measured LV dimensions, (d) LV ellipsoidal model, depicting its geometrical parameters.

Fig. 22. Patient A: Cyclic variation of LV pressure-volume data, LV model shape factor s, computed σ\*, computed index *(dσ\*/dt)max*.

Fig. 22. Patient A: Cyclic variation of LV pressure-volume data, LV model shape factor s,

Patient A (with myocardial infarct and double vessel disease,

CONT1 = 3.84 s-1

computed σ\*, computed index *(dσ\*/dt)max*.

Fig. 23. Patient B: Cyclic variation of LV pressure-volume data, LV model shape factor s, computed σ\*, computed index *(dσ\*/dt)max*.

Fig. 24. Correlationships between (i) CONT1 and EF, and (ii) CONT1 and *(dP/dt)max*.

Fig. 24. Correlationships between (i) CONT1 and EF, and (ii) CONT1 and *(dP/dt)max*.

For our patient group, we have computed and plotted CONT1 vs EF and *(dP/dt)max* in Fig. 24. We can note the good level of correlation with the traditional contractility index of *(dP/dt)max.* Additionally, our new index can be determined noninvasively, and hence be more conducive for clinical use.

From these results, we can also infer that a non-optimal less-ellipsoidal shape (or a more spherical shape, having a greater value of *S = B/A*) is associated with decreased contractility (and poor systolic function) of the LV, associated with a failing heart. This has an important bearing on a quick assessment of a failing heart based on the values of *S* and *(dσ\*/dt)max*
