# The Generic Relation Between Atrial Volume (Pre-A Volume, Taken Before the Onset of Atrial

Appendix

The generic relation between atrial volume (pre-A volume, taken before the onset of atrial contraction), the global atrial longitudinal strain during active contraction and the SV generated by the contraction were investigated using a geometrical mathematical model. In this model, the atria were approximated by ellipsoids with two of the three radii being equal. The following relation between the atrial volume () and its dimensions have been used:

, / (1)where is the average long axis to short axis ratio of an atrium and is the atrial short axis radius. The values of were estimated from our datasets and are: 1.26 for the left and 1.16 for the right atrium.

In order to simulate the potential range in which the atria of the subjects of our study could function, we calculated the volume-strain-SV relations for pre-A volumes from 18 to 95 ml (for the LA) or 19 to 105 ml (for the RA) and for a SV from 3.5 to 10 ml (for both atria). Using the eq. (1) the volumes have been mapped to the ellipsoids with the three radii equal to . Given the ellipsoids corresponding to pre-A and end of the atrial systole, the global atrial longitudinal strain was estimated as the change in its largest circumference given, in pre-A, by the perimeter of an ellipse with radii and . The perimeter was approximated by the following equation, due to Ramanujan [Weisstein, Eric W. “Ellipse”. From MathWorld-A Wolfram Web Resource.

, / (2)The results of these calculations can be seen as level curves in Fig. 5. In order to relate these volume-strain-SV data to the measurements from our population, the latter were represented by dots in Fig. 5.

The SV reserve has been estimated as , where is the SV at rest (measured) and is the SV that would be generated if the atrium would contract with a maximum of 40% longitudinal strain. has been measured by echocardiography. In order to estimate the following equation has been solved for (the short radius of the atrium contracted with 40% strain):

, or simply / (3)Finally, the atrial wall stress was calculated for each individual using [Falsetti et al. 1970] :

, / (4)Where P is the average pressure (estimated as 10 mmHg for the left atrium and 5 mmHg for the right one), and W is the wall thickness (estimated as 1mm). is the short radius of the ellipsoid that corresponds to the maximum measured atrial volume.