**4.1 Remodeling quantification in terms of local curvedness indices and regional curvedness index**

### **4.1.1 Overall approach**

The remodeling is characterized in terms of the *Regional Curvedness* index of the 16 segments of the LV endocardial surface. The method entails generation of the LV endocardial shape and its compartmentalization into 16 segments or regions (as explained later on). Each segment is discretized into triangular meshes, so that a point on the endocardial surface is a vertex of a triangular mesh. At each vertex point, we determine the *Local Curvedness* index [2], by employing its surrounding neighboring vertices in the form of *n*-rings around the local point. To ensure accurate curvedness computation without over smoothing, the optimal number of *n*-rings is determined to be 5. Then, for each segment, we determine the *Regional Curvedness* index, as the mean of the *Local Curvedness* indices in the segment.

The values of *Regional Curvedness* are determined for normal and MI patients, at enddiastolic and end-systolic instants. For normal patients, the *Regional Curvedness* index changes significantly from diastole to systole, as given by the *Diastole-to-Systole Change in Curvedness* (%∆C) [2]

$$\% \Delta \text{C} = \frac{\text{C}\_{ED} - \text{C}\_{ES}}{\text{C}\_{ED}} \times 100 \tag{1}$$

wherein C*ED* and C*ES* are end-diastolic and end-systolic curvedness, defined later on. The mean alteration in regional curvedness index value of %∆C is determined for all the segments, for normal subjects and MI patients (and logged in Table 3). It can be seen that for MI patients, the regional curvedness index or (%∆C) does not change significantly from end-diastole to end-systole, in comparison with normal subjects.

### **4.1.2 Clinical application methodology**

Our study involved 10 normal subjects and 11 patients after myocardial infarction (MI). The hemodynamic and volumetric parameters of the subjects are summarized in Table 2.

*Human Subjects and MRI Scans*: The study to characterize regional curvedness index or (%∆C) involved ten normal subjects and 11 patients after myocardial infarction [2]. All subjects underwent diagnostic MRI scans. For each subject, short-axis MRI images were taken along the plane which passes through the mitral and aortic valves of the heart at an interval of 8mm thickness. Each image has a spatial resolution of 1.5mm, acquired in a single breath hold, with 25 temporal phases per heart cycle. Of these images, the set of images corresponding to the cardiac cycle at end-diastole and end-systole are then used for the study.

*LV Endocardial Surface Reconstruction and Segmentation:* The MRI images were processed, by using a semi-automatic technique that is included in the CMRtools suite (Cardiovascular Solution, UK). The contours demarcating the myocardium and the LV chamber were defined by means of B-spline curves. The endocardial surface of the LV was reconstructed by joining the series of contours to form a triangle mesh. In order to facilitate quantification of the LV segmental regional curvedness, the endocardial surface was partitioned into 16 segments; the method of segmentation of the LV endocardial surface is provided in our paper [2].

*Left Ventricular Shape Analysis:* To quantify LV remodeling, we first define a measure known as the *Local Curvedness* index [2]. This is essentially a shape descriptor used to quantify how curved the surface is in the vicinity of a vertex on the LV endocardial surface. This is done by using the 3-d mesh of the LV endocardial surface as an input. Each vertex of the mesh is processed by fitting a quadric surface over a local region around the vertex as described in our paper [2]. The extent of this local region is determined by the *n*-ring parameter. Next, the *Local Curvedness* index of each vertex can be calculated from the coefficients of the fitted quadric surface by [2]:

$$C = \sqrt{\frac{\kappa\_1^2 + \kappa\_2^2}{2}} = \frac{1}{A^2} \sqrt{\frac{2B^2 + A^2 \left(4ac - b^2\right)}{A}}\tag{2}$$

such that

786 Biomedical Science, Engineering and Technology

[2], by employing its surrounding neighboring vertices in the form of *n*-rings around the local point. To ensure accurate curvedness computation without over smoothing, the optimal number of *n*-rings is determined to be 5. Then, for each segment, we determine the

> % 100 *ED ES ED*

wherein C*ED* and C*ES* are end-diastolic and end-systolic curvedness, defined later on. The mean alteration in regional curvedness index value of %∆C is determined for all the segments, for normal subjects and MI patients (and logged in Table 3). It can be seen that for MI patients, the regional curvedness index or (%∆C) does not change significantly from

Our study involved 10 normal subjects and 11 patients after myocardial infarction (MI). The

*Human Subjects and MRI Scans*: The study to characterize regional curvedness index or (%∆C) involved ten normal subjects and 11 patients after myocardial infarction [2]. All subjects underwent diagnostic MRI scans. For each subject, short-axis MRI images were taken along the plane which passes through the mitral and aortic valves of the heart at an interval of 8mm thickness. Each image has a spatial resolution of 1.5mm, acquired in a single breath hold, with 25 temporal phases per heart cycle. Of these images, the set of images corresponding to the

*LV Endocardial Surface Reconstruction and Segmentation:* The MRI images were processed, by using a semi-automatic technique that is included in the CMRtools suite (Cardiovascular Solution, UK). The contours demarcating the myocardium and the LV chamber were defined by means of B-spline curves. The endocardial surface of the LV was reconstructed by joining the series of contours to form a triangle mesh. In order to facilitate quantification of the LV segmental regional curvedness, the endocardial surface was partitioned into 16 segments; the method of segmentation of the LV endocardial surface is provided in our

*Left Ventricular Shape Analysis:* To quantify LV remodeling, we first define a measure known as the *Local Curvedness* index [2]. This is essentially a shape descriptor used to quantify how curved the surface is in the vicinity of a vertex on the LV endocardial surface. This is done by using the 3-d mesh of the LV endocardial surface as an input. Each vertex of the mesh is processed by fitting a quadric surface over a local region around the vertex as described in our paper [2]. The extent of this local region is determined by the *n*-ring parameter. Next, the *Local Curvedness* index of each vertex can be calculated from the coefficients of the fitted

( ) 22 2 2 2

*A A*

2 4 1

*B A ac b*

+ − <sup>+</sup> = = (2)

2

hemodynamic and volumetric parameters of the subjects are summarized in Table 2.

<sup>−</sup> Δ= × (1)

*C C <sup>C</sup> C*

end-diastole to end-systole, in comparison with normal subjects.

cardiac cycle at end-diastole and end-systole are then used for the study.

1 2

κ κ

*C*

2

**4.1.2 Clinical application methodology** 

*Regional Curvedness* index, as the mean of the *Local Curvedness* indices in the segment. The values of *Regional Curvedness* are determined for normal and MI patients, at enddiastolic and end-systolic instants. For normal patients, the *Regional Curvedness* index changes significantly from diastole to systole, as given by the *Diastole-to-Systole Change in* 

*Curvedness* (%∆C) [2]

paper [2].

quadric surface by [2]:

$$A = \sqrt{d^2 + c^2 + 1}$$

$$B = a + ac^2 + c + cd^2 + bde$$

where *a*, *b*, *c*, *d* and *e* are the coefficients of the fitted quadric surface at the vertex. In order to derive the *Regional Curvedness*, the endocardial surface is partitioned into 16 segments (Fig 3). The *Regional Curvedness* for the each segment is the mean of the *Local Curvedness* indices in the segment. The flowchart of the overall workflow for the regional LV shape analysis is shown in Fig. 4.

### **4.1.3 Clinical studies results**

In our clinical studies, it was found that (1) MI patients exhibit decreased curvedness and %Δ*C*, (2) MI patients exhibit increased variation of curvedness and variation of %Δ*C*, and (3) LV ejection fraction is positively correlated with curvedness and %Δ*C*, and inversely correlated with variation of %Δ*C*.

The *Diastole-to-Systole Change in Curvedness* (%Δ*C*), as defined by equation (1), is a measure of regional deformity due to contraction. Positive values of %Δ*C* indicate regions of increasing inward concavity of the LV wall during systole, while negative values of %Δ*C* indicate wall regions of decreasing inward concavity. The %Δ*C* measure can be employed to relate the regional differences in hypokinesis due to myocardial infarction.

*Variation of curvedness:* The extent of LV surface inhomogeneity is characterized by a coefficient of variation of curvedness at end-diastole (CV\_*C*ED) and end-systole (CV\_*C*ES):

$$\text{CC}\,\text{'}\,\text{C}\,\text{'} = \frac{\sigma(\text{C})}{\mu(\text{C})} \tag{3}$$

where σ(*C*) is the standard deviation of the regional curvedness and μ(*C*) is the mean of the regional curvedness of the segments of the LV mesh.

To evaluate the extent of functional non-uniformity of LV regions, the index CV\_D*C* was determined as [2]:

$$\text{CV\\_}\\_\text{\Delta C} = \frac{\text{CV\\_}\\_\text{C}\_{ED} - \text{CV\\_}\\_\text{C}\_{ES}}{\text{CV\\_}\\_\text{C}\_{ED}} \times 100\tag{4}$$

In general, the larger the values of CV\_*C*ED and CV\_*C*ES, the more inhomogeneous the LV endocardial surface appears. Hence, the larger the value of index CV\_Δ*C*, the more functionally non-uniform are the LV shape changes due to LV contraction.

*Curvedness, Variation of curvedness and Diastole-to-systole change %ΔC in MI patients compared to Normal subjects:* The hemodynamic and volumetric parameters of the subjects are summarized in Table 2. For patients after MI, the LV ejection fraction was significantly lower than that in the control subjects. In addition, their LV end-diastolic and end-systolic indexed volumes were greater than those in the control subjects.

The values of regional curvedness from apex to base in MI patients and normal subjects are given in Table 3 and Fig. 5, to highlight the regional variations of the LV curvature. In the normal group, there was a significant increase in the curvedness at the apex from diastole to systole. However, in MI patients, there was no significant difference in curvedness in all segments. Significant differences in end-diastolic curvedness *C*ED and end-systolic curvedness *C*ES were noted between MI and normal groups. Among the 16 segments of the LV, the variation coefficient of *C*ES (CV\_*C*ES) was significantly lower in MI patients than in the normal group (18±4% in MI vs 31±8% in normal, p<0.0001), indicating fair homogeneity of LV shape in MI at end-systole. Correspondingly, the diastole-to-systole change in curvedness (%Δ*C*) was significantly lower, and the variation of %Δ*C* was higher in MI patients compared to normal group, indicating ventricular functional non-uniformity due to the pathologic state.


Table 2. Characteristics of Normal Control and Patients after MI, involved in the study.


\*, p<0.05; #, p<0.01; ξ, p<0.001

Table 3. Left ventricular Regional curvedness, Diastole-to-systole change in curvedness (%Δ*C*), Variation of curvedness and %Δ*C* in MI compared to normal state, as defined by equation (1). This table is related to our work in Ref [2].

segments. Significant differences in end-diastolic curvedness *C*ED and end-systolic curvedness *C*ES were noted between MI and normal groups. Among the 16 segments of the LV, the variation coefficient of *C*ES (CV\_*C*ES) was significantly lower in MI patients than in the normal group (18±4% in MI vs 31±8% in normal, p<0.0001), indicating fair homogeneity of LV shape in MI at end-systole. Correspondingly, the diastole-to-systole change in curvedness (%Δ*C*) was significantly lower, and the variation of %Δ*C* was higher in MI patients compared to normal

Control (*n*=10) MI (*n*=11) p value

Controls (*n*=10) MI (*n*=11)

*C*ED(x10-2 mm-1)

*C*ES(x10-2 mm-1)

∆*C* (%)

∆*C*  (%)

group, indicating ventricular functional non-uniformity due to the pathologic state.

Age (years) 41 ± 16 60 ± 6 0.003 Weight (kg) 67 ± 15 65 ± 14 0.30 Height (cm) 169 ± 8 165 ± 10 0.86 Diastolic pressure (mmHg) 73 ± 12 74 ± 18 0.79 Systolic pressure (mmHg) 122 ± 17 116 ± 20 0.50 HR (beats/min) 70 ± 9 84 ± 13 0.012 CI (L/min/m2) 3.3 ± 0.4 2.2 ± 0.5 <0.001 EDVI (ml/m2) 73 ± 10 148 ± 40 <0.001 ESVI (ml/m2) 26 ± 6 122 ± 38 <0.001 EF (%) 65 ± 5 18 ± 5 <0.001 Sphericity index 0.52 ± 0.06 0.62 ± 0.08 0.01 LV mass index 56 ± 12 83 ± 13 0.004

Table 2. Characteristics of Normal Control and Patients after MI, involved in the study.

*C*ES(x10-2 mm-1)

1. Basal anterior 4.1 ± 0.8 5.6 ± 0.7 38 ± 22 3.4 ± 0.5\* 3.7 ± 0.5ξ 7 ± 17# 2. Basal anterior septal 3.4 ± 0.6 5.3 ± 1.1 57 ± 30 3.7 ± 1.0 3.7 ± 0.9\* 4 ± 26ξ 3. Basal inferior septal 3.1 ± 0.5 4.8 ± 0.6 61 ± 25 3.6 ± 0.9 4.0 ± 1.0\* 12 ± 23ξ 4. Basal inferior 4.0 ± 0.5 5.8 ± 1.0 45 ± 27 3.7 ± 1.0 4.1 ± 1.0\* 13 ± 21\* 5. Basal inferior lateral 3.5 ± 0.5 5.3 ± 0.9 50 ± 22 3.0 ± 0.6\* 3.3 ± 0.9ξ 13 ± 25\* 6. Basal anterior lateral 3.6 ± 0.8 5.0 ± 0.7 46 ± 26 3.1 ± 0.8 3.2 ± 0.8ξ 8 ± 24\* 7. Middle anterior 3.9 ± 0.5 6.0 ± 0.1 56 ± 20 3.4 ± 0.2\* 3.6 ± 0.3ξ 6 ± 12ξ 8. Middle anterior septal 3.9 ± 0.4 0.6 ± 1.4 55 ± 26 3.3 ± 0.6\* 3.4 ± 0.4ξ 5 ± 17ξ 9. Middle inferior septal 3.6 ± 0.6 5.2 ± 1.1 44 ± 18 3.5 ± 0.7 3.4 ± 0.5ξ 1 ± 16ξ 10. Middle inferior 4.1 ± 0.5 6.0 ± 1.2 51 ± 31 3.8 ± 0.7 4.0 ± 0.7ξ 7 ± 17\* 11. Middle inferior lateral 3.6 ± 0.3 5.5 ± 0.8 52 ± 25 3.1 ± 0.6\* 3.4 ± 0.6ξ 10 ± 14ξ 12. Middle anterior lateral 3.2 ± 0.3 4.7 ± 0.9 45 ± 19 2.9 ± 0.5 3.0 ± 0.3ξ 4 ± 14ξ 13. Apical anterior 4.8 ± 1.0 9.3 ± 2.0 96 ± 33 3.8 ± 0.7\* 4.1 ± 1.0ξ 6 ± 15ξ 14. Apical septal 4.9 ± 0.6 9.0 ± 1.9 83 ± 34 4.4 ± 0.7 4.6 ± 0.9ξ 4 ± 14ξ 15. Apical inferior 5.7 ± 0.9 11 ± 2.8 90 ± 44 4.7 ± 1.0\* 4.9 ± 1.0ξ 5 ± 14ξ 16. Apical lateral 4.4 ± 0.7 8.8 ± 2.2 103 ± 65 3.8 ± 0.4 4.0 ± 0.7ξ 2 ± 12ξ Mean 4.0 ± 0.4 6.5 ± 1.0 61 ± 18 3.6 ± 0.5\* 3.8 ± 0.5ξ 7 ± 9ξ Coefficient of variation (%) 21 ± 5 31 ± 8 51 ± 14 19 ± 5 18 ± 4ξ 392 ± 501\*

Table 3. Left ventricular Regional curvedness, Diastole-to-systole change in curvedness (%Δ*C*), Variation of curvedness and %Δ*C* in MI compared to normal state, as defined by

mm-1)

Segment *C*ED(x10-2

\*, p<0.05; #, p<0.01; ξ, p<0.001

equation (1). This table is related to our work in Ref [2].

Fig. 3. Partitioning of LV mesh into 16 segments; this figure is based on our work in Ref. [2].

Fig. 4. Flowchart of the overall workflow for the regional LV shape analysis; this figure is based on our work in Ref. [2].

Fig. 4. Flowchart of the overall workflow for the regional LV shape analysis; this figure is

based on our work in Ref. [2].

Fig. 5. Comparison of regional curvedness (5-ring selection) in normal subjects and patients after myocardial infarction; this figure is based on our work in Ref. [2].

**5. Employment of cardiac contractility Index d**σ**\*/dtmax to demonstrate (i) its excellent correlation with traditional contractility index dP/dtmax, in patients with varying ejection fractions, and (ii) its capacity to diagnose Heart failure with normal ejection fraction (HFNEF, or diastolic heart failure) and with reduced ejection fraction (HFREF, or systolic heart failure)** 

### **5.1 Cardiac contractility expressed in terms of maximum rate of change of pressurenormalized LV wall stress, d**σ**\*/dtmax**

Left Ventricular (LV) contractility can be termed as the capacity of the LV to develop intramyocardial stress, and thereby intra-cavitary pressure, to eject blood volume as rapidly as possible. It is hence rational and appropriate to formulate a LV contractility index on the basis of LV wall stress. The traditional *dP/dtmax* is based on left ventricular intra-cavitary pressure which is generated by an active myocardial stress. Hence, analogous to *dP/dtmax*, which is based on LV intra-cavitary pressure, we have formulated a novel LV contractility index based on LV wall stress, namely the maximum rate of change during systole of LV wall stress normalized to LV intra-cavitary pressure, *d(σ/P)/dtmax* or *dσ\*/dtmax ,* where *σ\** = *σ /P* [3].

*Model Analysis and Index Formulation:* For mathematically simplicity, we have approximated the LV as a thick-wall spherical shell consisting of incompressible, homogeneous, isotropic, elastic material. The maximum circumferential wall stress (σθ ) can be expressed at the endocardium, as:

$$\sigma\_{\theta} \left( r\_{i} \right) = P \left[ \frac{r\_{i}^{3} \;/ \; r\_{e}^{3} + 1 \;/ \; 2}{1 - r\_{i}^{3} \;/ \; r\_{e}^{3}} \right] \tag{5}$$

where *ri* and *re* are the inner and outer radii, *P* is LV intracavitary pressure. By normalizing wall stress to LV intra-cavitary pressure (*P*), we obtain:

$$\sigma^\*\left(r\_i\right) = \frac{\sigma\_\theta}{P} = \frac{r\_i^3}{r\_e^3 - r\_i^3} \left(1 + \frac{r\_e^3}{2r\_i^3}\right) \tag{6}$$

Since the maximum wall stress occurs at the inner endocardial wall, we have:

$$\sigma^{\ast\ast}\left(r=r\_i\right) = \left(\frac{V\,\left(V\_m + V\right) + 1\,/2}{1 - V\,\left(V\_m + V\right)}\right) = \left(\frac{3V + V\_m}{2V\_m}\right) = \left(\frac{3V}{2V\_m} + \frac{1}{2}\right) \tag{7}$$

where *P* is LV intra-cavitary pressure; σθ is the wall stress; *V* (= <sup>3</sup> 4 /3 *<sup>i</sup>* π*r* ) denotes LV volume; *Vm* (= ( ) 3 3 4 /3 *e i* π *r r* − ) denotes LV myocardial volume; *ri* and *re* are the inner and outer radii of the LV, respectively. Differentiating equation (7) with respect to time, we get:

$$d\sigma^\* \left/ dt\_{\text{max}} = \left| \frac{d\left(\sigma\_{\theta} \,'\, P\right)}{dt} \right|\_{\text{max}} = \frac{3}{2V\_m} \left| \left(\frac{dV}{dt}\right) \right|\_{\text{max}} \tag{8}$$

**5. Employment of cardiac contractility Index d**σ**\*/dtmax to demonstrate (i) its excellent correlation with traditional contractility index dP/dtmax, in patients with varying ejection fractions, and (ii) its capacity to diagnose Heart failure with normal ejection fraction (HFNEF, or diastolic heart failure) and with** 

**5.1 Cardiac contractility expressed in terms of maximum rate of change of pressure-**

Left Ventricular (LV) contractility can be termed as the capacity of the LV to develop intramyocardial stress, and thereby intra-cavitary pressure, to eject blood volume as rapidly as possible. It is hence rational and appropriate to formulate a LV contractility index on the basis of LV wall stress. The traditional *dP/dtmax* is based on left ventricular intra-cavitary pressure which is generated by an active myocardial stress. Hence, analogous to *dP/dtmax*, which is based on LV intra-cavitary pressure, we have formulated a novel LV contractility index based on LV wall stress, namely the maximum rate of change during systole of LV wall stress normalized to LV intra-cavitary pressure, *d(σ/P)/dtmax* or *dσ\*/dtmax ,* where

*Model Analysis and Index Formulation:* For mathematically simplicity, we have approximated the LV as a thick-wall spherical shell consisting of incompressible, homogeneous, isotropic,

3 3

33 3 \* 1

*r r r P*

3 3 / 1/2 1 / *i e*

3 3

*i e*

*ei i r r*

*P rr r*

*VV V V V V*

⎛ ⎞ + + ⎛ ⎞⎛ ⎞ <sup>+</sup> = = ⎜ ⎟ = =+ ⎜ ⎟⎜ ⎟

*VV V V V*

*dt V dt*

⎛ ⎞ = = ⎜ ⎟

== + ⎜ ⎟

2

*m mm*

*r r* − ) denotes LV myocardial volume; *ri* and *re* are the inner and

*max* <sup>2</sup> *<sup>m</sup> max*

− + ⎝ ⎠ ⎝ ⎠⎝ ⎠ (7)

is the wall stress; *V* (= <sup>3</sup> 4 /3 *<sup>i</sup>*

⎛ ⎞

*i e*

*r r*

σθ

<sup>⎡</sup> <sup>+</sup> <sup>⎤</sup> <sup>=</sup> <sup>⎢</sup> <sup>⎥</sup> ⎢⎣ <sup>−</sup> ⎥⎦ (5)

⎜ ⎟ <sup>−</sup> ⎝ ⎠ (6)

π

⎝ ⎠ (8)

*r* ) denotes LV

) can be expressed at the

**reduced ejection fraction (HFREF, or systolic heart failure)** 

elastic material. The maximum circumferential wall stress (

( )

where *ri* and *re* are the inner and outer radii, *P* is LV intracavitary pressure. By normalizing wall stress to LV intra-cavitary pressure (*P*), we obtain:

( )

*r*

σ

( ) ( )

max

σ

*r r*

where *P* is LV intra-cavitary pressure;

σ

volume; *Vm* (= ( ) 3 3 4 /3 *e i* π

*i*

σθ

Since the maximum wall stress occurs at the inner endocardial wall, we have:

( ) / 1/2 <sup>3</sup> 3 1 \* 1 / 2 22 *<sup>m</sup> <sup>m</sup> <sup>i</sup>*

> σθ

outer radii of the LV, respectively. Differentiating equation (7) with respect to time, we get:

( )

*d P dV d dt*

/ <sup>3</sup> \* /

σθ

σθ

*i*

**normalized LV wall stress, d**σ**\*/dtmax**

*σ\** = *σ /P* [3].

endocardium, as:

It can be thus noted that in contrast to the indices of *dP/dtmax*, *Ees* and *Ea,max*, our *dσ\*/dtmax* index can be determined solely from non-invasive assessment of LV geometry and flow. Normalizing LV wall stress to LV pressure obviates the need for invasive LV pressure measurement.

Our LV contractility formulation has been based on the premise that LV wall stress (due to LV myocardial sarcomere contraction) is responsible for the development of LV pressure. Hence, it is more rational to base LV contractile function on LV wall stress per pressure. Hence, analogus to *dP/dtmax*, this LV contractility index is formulated as the maximal rate of pressure-normalized wall stress (as given by the above equation 8), to represent the maximal flow rate out of the ventricle (*dV/dt*) normalized to myocardial volume (*Vm*). This is somewhat in keeping with cardiac output or maximal volume change having been used as a measure of myocardial contractility in rats or human, provided that the influence of afterload is taken into account.

### **5.2 Medical application to subjects with varying ejection fractions, to demonstrate excellent correlation of dσ\*/dtmax with dP/dtmax**

We have validated *dσ\*/dtmax* against *dP/dtmax* and *Ea,max* in 30 subjects with disparate ventricular function in Figure 6, and demonstrated the index's load independence, albeit under conditions of limited preload and after load manipulations [3].

For this study, thirty volunteers [mean 58.1 (range 48–77) yr of age, 13:2 male-to-female ratio] with diverse cardiac conditions were recruited. From their LV pressure-volume data, LVEF and *dP/dtmax* were computed directly from these traces. Active elastance *E*a at various times was also computed from the pressure-volume loops, from the data in Table 4, based on our earlier work on *E*a definition and determination [4]. *Ea,max* was extrapolated from the peak of the *E*a-time curve [4]. The single-beat estimation of endsystolic elastance *E*es (SB) was determined, using bilinearly approximated time-varying elastance [5].

The patients were divided into three groups on the basis of tertiles of LVEF, with 10 individuals in each group, as shown in Table 5. Intergroup comparisons show significant differences between the mean values of *dP/dtmax*, *Ea,max*, *E*es(SB), and *dσ\*/dtmax* in those in the highest tertile compared with those in lowest and middle tertiles. There is agreement with regard to the index *dσ\*/dtmax* with *dP/dtmax*, *Ea,max*, and *E*es(SB) across the three tertiles of ascending LVEF values, with statistically significant differences in LV contractility indexes among the three groups. Values of *dP/dtmax*, *E*es(SB), and *dσ\*/dtmax* were statistically significantly lower in patients in the lowest and middle tertiles had than those in the highest tertile.

Figure 6 summarizes the correlation between *dσ\*/dtmax*, *dP/dtmax*, and *Ea,max*, as well as *E*es(SB). Linear regression analysis revealed good correlation between *dσ\*/dtmax* and *dP/dtmax*, *Ea,max*, and *E*es(SB), with significant correlation coefficients in each case: *dσ\*/dtmax*=0.0075*dP/dtmax*- 4.70 (*r* = 0.88, *P* < 0.01), *dσ\*/dtmax*=1.20*Ea,max* + 1.40 (*r* = 0.89, *P*<0.01), and *dσ\*/dtmax*= 1.60*E*es(SB) + 1.20 (*r* = 0.88, *P* < 0.01). In contrast, the correlation between *dσ\*/dtmax*and LVEF is less strong (*r* = 0.71), as is the correlation between *E*es (SB) and LVEF (*r* = 0.78), underscoring the lack of specificity of LVEF as an index of myocardial contractility.


Table 4. Active Elastance *Ea computed at discrete time points during isovolumic contraction and relaxation in a sample subject: i*, time instant in the cardiac cycle (frame number from enddiastole); *t*, time from start of isovolumic contraction; *P*, measured left ventricular intracavitary pressure; *V*, measured left ventricular intracavitary volume; *Ea,i,* calculated active elastance at instant *i*. This table is related to our work in Ref [4].


Table 5. *LV contractility indexes classified into tertiles of LVEF:* Left ventricular contractility indices classified into tertiles of left ventricular ejection fraction. Values are expressed as mean ± standard deviation. Asterisks denote statistically significant difference (p<0.05) when compared with corresponding values in the highest tertile of left ventricular ejection fraction. *dP/dtmax*, peak first time-derivative of the ventricular pressure; *Ea,max*, maximum left ventricular elastance; *E*es(SB), single-beat LV end-systolic elastance; *dσ\*/dtmax*,left ventricular contractility index. This table is related to our work in Ref [3].

Frame No. *(i) t*, s P, mmHg V, ml *Ea*, mmHg/ml *Isovolumic contraction* 1 0 18 136.7 0 2 0.02 22 135.7 0.0295 3 0.04 32 134.6 0.1038 4 0.06 52 133.5 0.2536 5 0.08 80 132.5 0.4636 *Isovolumic relaxation* 18 0.34 74 85.0 0.0590 19 0.36 50 85.5 0.1778 20 0.38 30 86.4 0.3127 21 0.40 17 90.6 0.4636

Table 4. Active Elastance *Ea computed at discrete time points during isovolumic contraction and relaxation in a sample subject: i*, time instant in the cardiac cycle (frame number from enddiastole); *t*, time from start of isovolumic contraction; *P*, measured left ventricular intracavitary pressure; *V*, measured left ventricular intracavitary volume; *Ea,i,* calculated

Ejection fraction 0.38 ± 0.12\* 0.49 ± 0.13\* 0.63 ± 0.05 Age,yr 58.30 ± 8.86 56.10 ± 6.15 59.90 ± 6.17 Heart rate,beats/min 71.18 ± 10.72 71.77 ± 10.68 71.46 ± 9.09 *dP/dtmax*,mmHg/s 960 ± 115\* 1,121 ± 113\* 1,360 ± 97 *E*a,max, mmHg/ml 0.95 ± 0.32\* 1.85 ± 0.59\* 3.61 ± 0.62 *E*es(SB),mmHg/ml 0.72 ± 0.26\* 1.51 ± 0.20\* 2.81 ± 0.51 *dσ\*/dtmax*,s-1 2.30 ± 0.58\* 3.60 ± 1.06\* 5.64 ± 1.13

Table 5. *LV contractility indexes classified into tertiles of LVEF:* Left ventricular contractility indices classified into tertiles of left ventricular ejection fraction. Values are expressed as mean ± standard deviation. Asterisks denote statistically significant difference (p<0.05) when compared with corresponding values in the highest tertile of left ventricular ejection fraction. *dP/dtmax*, peak first time-derivative of the ventricular pressure; *Ea,max*, maximum left ventricular elastance; *E*es(SB), single-beat LV end-systolic elastance; *dσ\*/dtmax*,left ventricular

contractility index. This table is related to our work in Ref [3].

Lowest tertile Middle tertile Highest tertile

active elastance at instant *i*. This table is related to our work in Ref [4].

Fig. 6. Linear regression analysis demonstrates good correlation between dσ\*/dtmax and dP/dtmax : [dσ\*/dtmax = 0.0075dP/dtmax-4.70, r=0.88; top figure], between dσ\*/dtmax and Ea,max [dσ\*/dtmax = 1.20Ea,max+1.40, r=0.89; middle figure], and between dσ\*/dtmax and Ees(SB) [dσ\*/dtmax = 1.60Ees(SB)+1.20, r=0.88; bottom figure]. This figure is based on our work in Ref [3].

### **5.3 Use of cardiac contractility index** *dσ\*/dtmax* **to diagnose heart failure with normal ejection fraction (HFNEF) and with reduced ejection fraction (HFREF)**

In another study [6], we assessed the capacity of *dσ\*/dtmax* to diagnose heart failure in patients with normal ejection fraction (HFNEF) and with reduced ejection fraction (HFREF).

### **5.3.1 Introduction**

Heart failure (HF) is a major health care burden: it is the leading cause of hospitalization in persons older than 65 years, and confers an annual mortality of 10%. HF can occur with either normal or reduced LV ejection fraction (EF), depending on different degrees of ventricular remodeling. Both heart failure with normal ejection fraction (HFNEF) and heart failure with reduced ejection fraction (HFREF), also commonly known as diastolic and systolic heart failure respectively, have equally poor prognosis [7]. Medical therapy targets to reduce load, by using vasodilators and/or to alter contractile strength using inotropic agents. Alternatively, some therapies target to affect cardiac remodeling, such as passive cardiac support devices, surgical restoration of LV shape (i.e. the Dor procedure), and stem cells therapies.

Assessment of left ventricular (LV) contractility is important for HF management and evaluation of the heart's response to medical and surgical therapies. Although approaches based on pressure-volume analysis, stress-strain analysis, and *dP/dtmax*-EDV relations [8]can provide assessments of contractile function, these relations generally require invasive data measured at several chamber loads and thus are difficult to apply in routine or long-term clinical studies. This is an important limitation, because heart failure often requires longitudinal evaluation. The ideal measure of contractility should have the following characteristics: sensitivity to inotropic changes, independence from loading conditions as well as heart size and mass, ease of application, and proven usefulness in the clinical setting. LV ejection fraction (EF) is the index overwhelmingly used to assess cardiac function in both clinical and experimental studies, despite the fact that it is highly dependent upon preload and afterload. Based on the National Heart Lung and Blood Institute's Framingham Heart Study, an LVEF50% as cut-off for the presence of normal LVEF has been used in the present study [9].

**Usefulness of** *dσ\*/dtmax* **as Contractility Index:** During LV systole, LV wall stress is generated intrinsically by sarcomere contraction and results in the development of extrinsic LV pressure. We have shown earlier that our novel LV contractility index, *dσ\*/dtmax* (maximal change rate of pressure-normalized wall stress) correlates well with LV *dP/dtmax* [3]. We have proposed and validated a new LV contractility index, *dσ\*/dtmax* **,**based on the maximal rate of development of LV wall stress with respect to LV pressure. From the righthand side of equation (8), this index is also seen to represent the maximal flow rate from the ventricle (cardiac output) normalized to myocardial volume (or mass).

This index is easily measured non-invasively (i.e. from echocardiography or magnetic resonance imaging), is sensitive to LV inotropic changes, and has been demonstrated by us to be preload and afterload independent [3]. Importantly, it is measured at a single steadystate condition, as opposed to the multiple variably loaded cardiac cycles required for many of the other indices. Thus *dσ\*/dtmax*has several qualities that make it a useful LV contractility index. This study [6] has constituted an important step toward establishing the clinical utility of *dσ\*/dtmax* as a tool for diagnosis of HF (both HFNEF and HFREF) as well as for follow-up surveillance of LV function. Hence we see a great potential for application of this novel index to evaluate heart function in diverse heart conditions.
