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Introduction
The Massardo method is one of a number of approaches for estimating the volume of the ventricles of the heart. It is employed as part of a MUGA (multigated acquisition) scan, used to evaluate the functioning of the heart by calculating the ejection fraction of the left ventricle. Briefly, a MUGA scan is a nuclear imaging method involving the injection of a radioactive isotope (Tc-99m) that acquires gated 2D images of the heart using a SPECT scanner. The pixel values in such an image represent the number of counts (nuclear decays) detected from within that region in a given time interval. The Massardo method enables a 3D volume to be calculated from each 2D image of counts via:

$$ V = 1.38 M^3 r^{\frac{3}{2}} $$ ,

where $$ M $$ is the pixel dimension and $$ r $$ is the ratio of total counts within the ventricle to the number of counts within the brightest (hottest) pixel. The Massardo method relies on two assumptions: (i) the left ventricle is spherical and (ii) the radioactivity is homogeneously distributed.

The ejection fraction, $$ E_f $$, can then be calculated:

$$ E_f(\%) = \frac{ \text{EDV - ESV} }{ \text{EDV} } \times 100 $$,

where the EDV (end-diastolic volume) is the volume of blood within the ventricle immediately before a contraction and the ESV (end-systolic volume) is the volume of blood remaining in the ventricle at the end of a contraction. The ejection fraction is hence the fraction of the end-diastolic volume that is ejected with each beat.

The Siemens Intevo SPECT scanners employ the Massardo method in their MUGA scans.

Derivation
Define the ratio $$r$$ as the ratio of counts within the chamber of the heart to the counts in the hottest pixel:

$$ r = \frac{ \text{Total counts within the chamber} }{ \text{Total counts in the hottest pixel} }   = \frac{ N_t }{N_m}   $$.

Assuming that the activity is homogeneously distributed, the total count is proportional to the volume. The maximum pixel count is thus proportional to the length of the longest axis perpendicular to the collimator, $$D_m$$, times the cross-sectional area of a pixel, $$M^2$$. We can thus write:

$$ N_m = K M^2 D_m $$ ,

where $$K$$ is some constant of proportionality with units counts/cm$$^3$$. The total counts, $$N_t$$, can be written $$N_t = K V_t $$ where $$V_t$$ is the volume of the ventricle and $$K$$ is the same constant of proportionality since we are assuming a homogeneous distribution of activity. The Massardo method now makes the simplification that the ventricle is spherical in shape, giving:

$$ N_t = K \left( \frac{\pi}{6} \right) D^3 $$ ,

where $$D$$ is the diameter of the sphere and is thus equivalent to $$D_m$$ above. This allows us to express the ratio $$r$$ as:

$$ r = \frac{N_t}{N_m} = \frac{\pi D^2}{6 M^2} $$ ,

finally giving the diameter of the ventricle in terms of $$r$$, i.e. counts, alone:

$$ D^2 = \left( \frac{6}{\pi} \right) M^2 r $$.

From this, the volume of the ventricle in terms of counts alone is simply:

$$ V_t = \sqrt{ \frac{6}{\pi} } M^3 r^{\frac{3}{2}} \approx 1.38 M^3 r^{\frac{3}{2}} $$.