Implementation from Klotz's paper 'A computational method of prediction of the end-diastolic pressure-volume relationship by single beat'.
More...
Implementation from Klotz's paper 'A computational method of prediction of the end-diastolic pressure-volume relationship by single beat'.
Theory
From Klotz's 2007 Nature Protocols paper 'A computational method of prediction of the end-diastolic pressure-volume relationship by single beat', the following equations are taken.
Equations 2 and 5 seem to describe the end-diastolic pressure-volume relation (EDPVR) as:
.. math:
P = A_n V_n^{B_n}
where :math:P is cavity pressure, :math:V_n is normalised volume, and the constants :math:A_n and :math:B_n were determined emperically in their earlier study as 27.78 mmHg and 2.76 respectively.
Normalised volume :math:V_n is determined as explained in equation 4 as:
.. math:
V_n = \frac{V - V_0}{V_30 - V_0}
where :math:V is the cavity volume, :math:V_0 is the volume at zero pressure and :math:V_30 is the volume at a pressure of 30 mmHg.
The above can be arranged to equation 6, giving the :math:V_{30} constant:
.. math:
V_{30} = V_0 + \frac{V - V_0}{\left(\frac{P_m}{A_n}\right)^{(1/B_n)}}
:math:V_0 is determined using the emperical relation:
.. math:
V_0 = V_m (0.6 - 0.006 P_m)
where :math:V_m and :math:P_m are a pressure-volume pair measured.
The equation :math:P = \alpha V^\beta is then fitted to the above :math:(V_m, P_m) point, giving the following relations for the constants :math:\alpha and :math:\beta:
.. math:
\beta = \frac{\log\left(\tfrac{P_m}{30}\right)}
{\log\left(\tfrac{V_m}{V_{30}}\right)}
\alpha = \frac{30}{V_{30}^\beta}
To avoid a singularity in the above equations when :math:P_m \rightarrow 30, the above equations were reposed as:
.. math:
\beta = \frac{\log\left(\tfrac{P_m}{15}\right)}
{\log\left(\tfrac{V_m}{V_{15}}\right)}
\alpha = \frac{P_m}{V_m^\beta}
where :math:V_{15} is determined analytically as:
.. math:
V_{15} = 0.8 (V_{30} - V_0) + V_0
The paper advises using the first form of the :math:\alpha and :math:\beta equations (utilising :math:V_{30}) when using a :math:P_m value up to 22 mmHg and the second form (utilising :math:V_{15}) above 22 mmHg.
