Page 10 - The Mathematic Model - the initial model
This is as complex as it gets, the final model will be easier!
In the images above, the short axis dimensions of our 5mm slice of the LV are presented with the cavity in pink and the wall divided into two halves of equal diastolic thickness, the outer in dark blue, the inner in light blue, with the mid-wall ring between them. The values required to solve all the dimensions are the wall thickness in diastole (Td), the short axis diameter in diastole (SAd) and the degree of shortening of the mid-wall circumference, which we previously said would be 15%.
The solution to the diagram is in a spreadsheet "midwall.xls" which you can click to download here, and in its second sheet the logic is set out in its entirety. The following paragraphs summarise the sequence in prose.
From the diastolic diameters, the radius of the cavity (Rd), radius to the outer rim (ROd) and to the mid-wall (RMd) follow naturally, and the volumes of the separate wall layers (VId,VOd) and of the cavity (VCd) follow. As we have determined that the mid-wall circumference will diminish by 15%, the systolic circumference of the mid-wall can be calculated. From this comes the radius to the mid-wall in systole (RMs).
Now the volume of the space to the mid-wall (i.e. the cavity volume VCs plus the inner wall volume VIs) in systole can be derived (pi times the square of RMs). The volume of the inner wall in systole is, as we previously discussed, the diastolic volume plus 15% from long axis repacking, less 4% for blood expression. Taking this away from the combined volume yields the cavity volume in systole VCs and in turn the radius of the cavity in systole Rs. Now it is an easy task to subtract Rs from RMs to get the systolic inner wall thickness ITs, and to double Rs to get the systolic short axis diameter SAs.
Next we can adjust the volume of the outer wall in diastole VOd to its systolic value VOs by the 15% more and 4% less as for the inner. As we already know VIs and VCs, we can add them to VOd to get total volume from which the radius ROs and the circumference of the outer rim can be derived, and from that the radius of the outer rim ROs. Finally the outer systolic wall thickness OTs comes from OTs = ROs-ITs-Rs.