In the methods described below, it is assumed that the volume of the heart is to be estimated from *N* SAX slices and one LAX slice. Our default LAX orientation is the four-chamber view of the heart; i.e. horizontal LAX. Nevertheless, the effect of changing this orientation will be studied as discussed in “Results and discussion” section. The proposed methodology is identical for calculating the volume enclosed by the epicardium and the volume enclosed by the endocardium at any timeframe. Therefore, for simplicity, we will use the general terms of myocardium contours and cardiac volume when discussing calculating the volume enclosed by a set of contours (epicardium or endocardium) at a specific timeframe.

### Problem formulation

Given a number, \( N \), of SAX slices and one LAX slice, the myocardium boundaries are delineated to obtain a set of *N* SAX and one LAX contour respectively. Ignoring delineation errors and misregistration due to different levels of breath-holds, these contours can be thought of as a coarse grid representing the intersection between the different image planes and the myocardium surface. It is therefore required to calculate the cardiac volume enclosed by the myocardium surface represented by these contours. As can be seen in Fig. 1, a number of *N* parallel SAX planes can virtually divide the heart into *N* chunks (ignoring the part above the most-basal plane). The plane of the LAX contour intersects with the contour of the upper and lower surfaces of the *i*th chunk and results in a line segments of length \( d^{i} \left( {h,0} \right) \) and *d*
^{i}(0, 0), respectively, where *h* is the height of the chunk from the lower surface to the upper.

In general, within the *i*th chunk, the diameter of the upper and lower surfaces at any given angle, \( \theta \), are denoted by *d*
^{i}(*h*, *θ*) and *d*
^{i}(0, *θ*), respectively, where *θ* is measured from the plane containing the LAX contour. To account for the unsymmetrical shape of the LAX contour, the right and left parts of the LAX contour within the *i*th chunk are denoted by, *C*
^{r}_{
i
}
and \( C_{i}^{l} \), respectively. We further define \(A_{LAX}^{i}\) (0) as the area enclosed by the curves *d*
^{i}(0, 0), *C*
^{r}_{
i
}
, *d*
^{i}(*h*, 0), and *C*
^{l}_{
i
}
. As can be shown in Fig. 1, the area below the most apical slice, \(A_{LAX}^{N}\) (0), is enclosed by two curves only: \( d^{N} \left( {0, 0} \right),C_{N}^{r} \), and \( C_{N}^{l} \). For all the myocardium chunks, \(A_{LAX}^{i}\) (0) is numerically calculated by computing the area of a polygon formed by the points on the surrounding curves.

Having defined the basic quantities that are used in the proposed method, the following section describes a simple geometric model that can be used to estimate the cardiac volume of the *i*th chunk from the contour areas, \(A_{LAX}^{i}\) (0), and diameters, *d*
^{i}(*h*, 0) and *d*
^{i}(0, 0). Adding the volumes of all chunks yields the required total cardiac volume.

### Cross-sectional modeling using equivalent trapezoids

To simplify the volume calculations, a simple trapezoid is used to approximate the shape of any given long-axial cross-section of an LV chunk. For a given chunk, *i*, all modeling trapezoids are assumed to have the same height, *h*
_{
i
}, but different lengths of the upper and lower sides depending on the orientation of the LAX plane. For a LAX plane making angle *θ,* with the acquired LAX image plane, upper, *d*
^{i}(*h*, *θ*) and lower, \( d^{i} \left( {0,\theta } \right) \), sides of its modeling trapezoid is calculated from the line segments representing the intersection between this LAX plane and the upper and lower SAX contours. The trapezoid height, *h*
_{
i
}, can be calculated by setting the trapezoid area equal to the cross-sectional area \(A_{LAX}^{i}\) (0) described above. That is,

$$ h_{i} = \frac{{2 A_{LAX}^{i} \left( 0 \right)}}{{d^{i} \left( {h,0} \right) + d^{i} \left( {0,0} \right)}} $$

(2)

For any virtual LAX plane intersecting the *i*th chunk and making an angle, *θ*, with the acquired LAX plane, the area of intersection, \(A_{LAX}^{i}\) (*θ*), may also be represented by a trapezoid of height, *h*
_{
i
}, and thus can be estimated by,

$$ A_{LAX}^{i} \left( \theta \right) = \frac{{d^{i} \left( {h,\theta } \right) + d^{i} \left( {0,\theta } \right)}}{2} h_{i} $$

(3)

Substituting from Eqs. (2) and (3), the area of the equivalent trapezoid at any angle *θ* can be written in terms of *A*
_{
LAX
}(0, *i*) as follows,

$$ A_{LAX}^{i} \left( \theta \right) = \frac{{d^{i} \left( {h,\theta } \right) + d^{i} \left( {0,\theta } \right)}}{{d^{i} \left( {h,0} \right) + d^{i} \left( {0,0} \right)}} A_{LAX}^{i} \left( 0 \right) $$

(4)

If the equivalent trapezoid is rotated with infinitesimal angle, *dθ*, a wedge-like structure is obtained (as shown in Fig. 2) with volume given by,

$$ V_{wedge} \left( {\theta ,i} \right) = \frac{{A_{LAX}^{i} \left( \theta \right)}}{2} \times \frac{{\left( {d^{i} \left( {h,\theta } \right) + d^{i} \left( {0,\theta } \right)} \right)/2}}{2} d\theta $$

(5)

That is, the volume of the *i*th chunk, \( V_{i} \), can be obtained by integrating Eq. (5) from *θ* equal zero to 2π. Substituting from Eq. (4) into (5), it can be shown that,

$$ V_{i} = \frac{{0.5 A_{LAX}^{i} \left( 0 \right)}}{{d^{i} \left( {h,0} \right) + d^{i} \left( {0,0} \right)}}\mathop \smallint \limits_{0}^{\pi } \left( {\frac{{d^{i} \left( {h,\theta } \right) + d^{i} \left( {0,\theta } \right)}}{2}} \right)^{2} d\theta $$

(6)

Since the SAX contours are available, the diameters *d*
^{i}(*h*, *θ*) and \( d^{i} \left( {0,\theta } \right) \) can be readily calculated and the integration in Eq. (6) can be numerically solved. Observing that the integration in Eq. (6) is done over the square of the mean diameter at angle, *θ*, i.e., \( d_{mean}^{i} \left( \theta \right) \equiv \frac{{d^{i} \left( {h,\theta } \right) + d^{i} \left( {0,\theta } \right)}}{2} \), then it can be approximated by double the area of a virtual SAX contour with diameter \(d_{mean}^{i}\) (*θ*). The area of this virtual contour can be further approximated by the average area of the upper and lower SAX contours; that is,

$$ V_{i} \cong \frac{{A_{LAX}^{i} \left( 0 \right) }}{{d^{i} \left( {h,0} \right) + d^{i} \left( {0,0} \right)}}\left( {A_{SAX}^{upper,i} + A_{SAX}^{lower,i} } \right) $$

(7)

It is worth noting that, in the most apical chunk (at *i* = *N*), the lower base of the chunk is a single point representing the cardiac apex. That is, the LAX cross-section is approximated by a triangle where the values of *d*
^{N}(0, 0) and \(A_{SAX}^{lower,N}\) are set to zero. That is, the volume of the most-apical chunk is calculated using the following equation,

$$ V_{N} = \frac{{A_{LAX}^{N} \left( 0 \right) \cdot A_{SAX}^{upper,N} }}{{2 d^{N} \left( {h,0} \right)}} $$

(8)

Equation (7) can also be used to calculate the LV volume represented by the LAX contour segments that extend above the most-basal SAX slice (as shown in Fig. 1). First, these free LAX contour segments are used to define a virtual chunk above the most basal SAX plane with volume, *V*
_{0}. Then, the volume of this virtual chunk is calculated by respectively setting the area *A*
^{upper,0}_{
SAX
}
and the diameter *d*
^{0}(*h*, 0) equal to \( A_{SAX}^{lower,0} \) and *d*
^{0}(0, 0). It can be shown that this approximation results in a volume of a virtual chunk with identical upper and lower surfaces and height equal to the average heights of the two LAX segments extending above the most basal plane. It is worth noting that this volume is excluded from the calculations because there is no reported standard method, and thus a ground truth, for calculating it. It is worth noting that the misregistration between SAX and LAX slices can be corrected by various intensity and contour based methods (as proposed by [16, 17]). Nevertheless, due to imperfect segmentation of the myocardium boundaries in both LAX and SAX images, slight misalignment of the contours causes the LAX contour not to be intersecting with each SAX contour in exactly two points. This gives two possible values for the LV diameter, *d*
^{i}(*h*, 0) and \( d^{i} \left( {0,0} \right) \). In this work, the diameters *d*
^{i}(*h*, 0) and *d*
^{i}(0, 0) are calculated from the LAX contours. This is because the LAX slices are less prone to the boundary blurring caused by the partial volume effects and thus the LAX contours are usually more accurate in delineating the LV especially at the apex. Having calculated the cardiac volume for every chunk, the total volume can be then calculated as,

$$ Vol = \mathop \sum \limits_{i = 1}^{N} V_{i} $$

(9)

### Oblique LAX

In practice, the plane of the LAX slice is not perfectly selected perpendicular to the acquired stack of the SAX slices (as shown in Fig. 3). This oblique orientation results in a larger apparent area of the LAX slice and thus the calculated area of the LAX contour, \(A_{LAX}^{i}\) (0), should be compensated to account for this factor. One simple solution is to replace \(A_{LAX}^{i}\) (0) with a corrected area, \(A_{LAX}^{\prime i}\) (0) given by,

$$A_{LAX}^{\prime i} (0) = A_{LAX}^{i} \left( 0 \right) \cos \left( {\varPhi_{i} } \right) $$

(10)

where *Φ*
_{
i
} is the angle between the line connecting the center-of-mass points of the SAX contours forming the chunk and the LAX image plane.

### Model validation using CT-based phantoms

In order to validate the developed model, the actual surface geometry of five human hearts have been constructed from data acquired using Computed Tomography (CT) as described in [18]. The dataset (publicly available on the internet [19]) contains single breath-hold cardiac-gated CT acquisitions with resolution 0.43 × 0.43 mm. Rendering of the 3D volume for each heart has been done and the volume is calculated and recorded as the ground truth. Then, each reconstructed volume was re-sliced to create cross-sectional images (matrix size: 512 × 512; voxel size: 0.43 × 0.43 × 3.5 mm) in the SAX and LAX directions as shown in Fig. 4. All processing was done using 3D-Slicer software tool [20]. First, a stack of twelve SAX slices covering the LV from base to apex was reconstructed. Secondly, a set of four LAX image slices with different orientations was reconstructed. The epicardium and endocardium contours of all acquired images have been manually delineated and used to calculate the difference LV volumes using the different methods.

Two sets of experiments have been done to test the performance and the robustness of the proposed method. The first experiment was done to quantify the error resulting from decreasing the number of SAX slices. In this experiment, the proposed model and mSimp method has been used to calculate the cardiac volume from one (4CH) LAX slice combined with different number of SAX slices (n = 4, 6, 8, 10, 12). The reduced set of SAX slices was selected such that we include the most basal slice in which the LV SAX contour appears as a complete ring. In addition, the set includes the most apical slice where the blood pool can barely be differentiated at end-systole phase. The remaining slices are selected to uniformly cover the distance between the already selected basal and apical slices. The volume estimated by each method was recorded and the mean and standard deviation of the error (relative to the ground truth) was calculated.

The second set of experiments was done to assess the robustness and reproducibility of the proposed method. First, the proposed method was tested to report its reliability in presence of misregistration among the LAX and SAX contours caused by respiratory motion. This was done by simulating different levels of breath-holds by randomly changing the location of the heart in the 3D space prior to the re-slicing operation described above. The breathing-induced motion was assumed to be in the superior-inferior direction with maximum displacement of 18 mm and in the anterior-posterior direction with maximum displacement of 2.5 mm [21]. The whole experiment is repeated 10 times with random displacement and the mean and standard deviation have been recorded for the different number of slices as above. Another experiment was done to test the reproducibility of the proposed model at different selections of LAX imaging planes. For this purpose, a set of LAX image planes was used to reconstruct: one horizontal LAX slice (i.e. 4-chamber view or 4CH); one vertical LAX slice (i.e. 2-chamber view or 2CH); and two rotated horizontal LAX slices (±20°) around the axis of the LV. Each of these four LAX images was combined with different numbers of SAX slices (*n* = 4, 6, 8, 10, 12) to calculate the volume.

### Model validation using real MRI data

A database of MRI images for 25 human subjects with symptoms of ischemic heart disease to test and evaluate the proposed model. Ten patients were scanned using 1.5T Siemens scanner, and 15 patients were scanned using 3T Philips scanner. The number of slices for each dataset was (9–12) SAX slices and one LAX slice. The pixel size was in the range of (1.116–1.406 mm) and the slice thickness ranges from 5 to 8 mm. Only the end-diastole and end-systole timeframes were considered for processing and analysis. In general, all slices are assumed to be acquired while the patient is holding his/her breath at the same level. To quantify the volume calculation error, the ground truth volume for a given heart was calculated by mSimp method applied to all available SAX slices. Then, the proposed model was applied to compute the volume using one LAX slice and different numbers of SAX slices: 1 (mid-cavity), 2 (most basal and most apical), 3, 5, 7, 9 and 11. For a number of slices >2, the slices are selected to include and uniformly cover the distance between the selected basal and apical slices. After calculating the volumes enclosed by the cardiac contours, two functional parameters, namely ejection fraction and stroke volume, have been estimated by the two methods and the error was calculated. Due to the anticipated inadequate performance of the mSimp method at very low number of SAX slices (<4), other model-based methods described in literature have been investigated and compared to the proposed method. These model-based methods approximate the shape of the heart using simple geometries such as single plane ellipsoid, Biplane ellipsoid, Teichholz model, Hemisphere cylinder (for more details about these models, please refer to [14]).