Page 8 - The Mathematic Model - initial assumptions
I am not going to try to construct a model of the volumes of the entire left ventricle, the maths is beyond me. Instead I am going to make the same assumption as does Simpson's Rule, that is, a thin slice of the ventricle in the short axis can be treated as a disc of circular cross section. I will extend it to assume that the wall in the same slice can be treated as rectangular extensions of the disc. Thus, within this slice, the volumes of the cavity, and of the wall around it, and therefore their combined volume, can be very easily calculated. In this way, the changes in wall thickness can be examined in a projection familiar to imagers, particularly echocardiographers. The model will also be applicable to any other position along the long axis with the exception of the basal ring and of the apical tip.
The image above represents this situation. We will consider a short axis zone of 5mm thickness, at right angles to the long axis (blue line); the cavity is the yellow box and the walls are in orange.
The other initial assumption is that at the middle point of the ventricular wall, the myocytes are in the plane of the short axis and radially aligned. If so then we can regard them as a single ring of sarcomeres, and we can assume that this mid-wall ring will shorten by 15% in systole. Now I realise that this is not exactly true as the mid-wall ring will be subject to a small hydraulic effect from the outer half of the wall. However, I am going to work towards a more final accurate model that avoids this assumption: to do so needs guidance from an initial model which uses this assumption.
When we come to calculate the changes in dimension in systole, two adjustments to the volume of muscle within the 5mm zone have to be made. First, as we saw in page 5, the myocardium is repacked in systole into a shorter long axis, so it follows that the mass of muscle contained in the 5mm zone will increase. As we have seen that the long axis shortens in a very regular manner, the increment to be added can be adequately quantified as the same as the percentage shortening of the long axis. The pixel measurements in page 5 show that a shortening of 15% is achieved. This is no surprise, as tracts of myocytes that are helically or obliquely oriented around an axis will project a percentage shortening onto the axis that is the same as their cell shortening.
Secondly, the new volume needs to be diminished by 4% to allow for the expression of blood from the myocardial vascular bed in systole.
By positioning the slice at the position of the echocardiographic short axis, the short axis dimensions that we will arrive at can be crudely converted into ventricular volume and ejection fraction by the "cubes" method (i.e. volume of a normally shaped LV is equal to the cube of the short axis diameter).