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Kim H. Parker added subsection_fitting_the_proportionality_parameter__.tex
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\subsection{fitting the proportionality parameter $a$}
Given $b$ and $P_\infty$, it remains only to find the constant of proportionality $a$. In our previous work \cite{Aguado2007}, this was done by fitting the reservoir pressure at the end of systole to the reservoir pressure at the beginning of diastole. However, it was found that this method was very sensitive to the choice of the matching time. We now find $a$ by minimising the square error between the reservoir pressure calculated over the whole of diastole using the model based on eq. \ref{eqwkpressure}. In practice, this minimisation is done using the Matlab function \textit{fminsearch}.
The diastolic reservoir pressure obtained in this way deviates slightly from the exponential fit obtained in the previous step. However, the fitting process is much more robust than in the single point fitting used previously. It is possible that better fitting could be obtained by iterating to find small improvements in $a$, $b$ and $P_\infty$ simultaneously instead of finding the optimal value of $a$ for fixed values of $b$ and $P_\infty$. However, in the cases in which this was tried, we found no significant differences in the fitted parameters.
