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Kim H. Parker added subsection_exponential_fitting_during_diastole__.tex
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\subsection{exponential fitting during diastole}
Two of the parameters, $b$ and $P_\infty$ are determined by fitting the pressure waveform during diastole when the model predicts an exponential decrease in pressure. In the previous method, the fitting algorithms was not very robust and frequently gave unreasonable values for the fitted parameters. After considering many options, we suggest the use of a variant of the traditional method of moments which we call the method of exponential moments. Define $\theta = t-T_n$ as the time during diastole and $T_d=T-T_n$ as the duration of diastole. Rather than defining moments based on powers of $\theta$, we define moments $E_n$ based on powers of $e^{\theta/T_d}$
\[
E_n = \int\limits_0^{T_d} (P(\theta)-\langle P \rangle ) e^{n\theta /{T_d}} d\theta,
\]
where $\langle P \rangle$ is the mean of $P$. Assuming that our model has the general exponential form $P = \alpha e^{-\beta \theta}+\gamma$, where $\alpha$, $\beta$ and $\gamma$ are constants to be determined, the mean and first and second exponential moments are
\[
\langle P \rangle = \frac{\alpha}{1-\beta T_d}(e^{1-\beta T_d}-1) + \gamma,
\]
\[
E_1 = \alpha T_d \left(\frac{(e^{1-\beta T_d}-1)}{1-\beta T_d}
- \frac{(1-e^{-\beta T_d})(e-1)}{\beta T_d} \right),
\]
\[
E_2 = \alpha T_d \left(\frac{(e^{2-\beta T_d}-1)}{2-\beta T_d}
- \frac{(1-e^{-\beta T_d})(e^2-1)}{2\beta T_d} \right).
\]
From these expressions, we see that the ratio $R=\frac{E_2}{E_1}$ is a function of $\beta T_d$ only.
The fitting of real data is achieved by calculating $R(\beta T_d)$ for the measured $P$ and inverting the transcendental relationship between $R(\beta T_d)$ and $\beta T_d$ numerically to find the parameter $\beta$. Once $\beta$ is known, $\alpha$ is found using the expression for $E_1$
\begin{equation}
\label{eqratioofmoments}
\alpha = \frac{E_1}{T_d} \left(\frac{1}{1-\beta T_d}(e^{1-\beta T_d}-1)
- \frac{1}{\beta T_d}(1-\beta T_d)(e-1) \right)^{-1}.
\end{equation}
Given $\alpha$ and $\beta $, $\gamma $ is obtained from the mean,
\[
\gamma = \langle P \rangle - \frac{\alpha}{1-\beta T_d}(e^{1-\beta T_d}-1).
\]
When performing the numerical evaluations of the parameters, care must be taken for the particular values $\beta T_d=1$ and $\beta T_d=2$. These numerical singularities can be resolved by observing that $\lim_{x \rightarrow 0} \frac{e^x-1}{x} = 1$. There is also a numerical problem when $\beta T_d=0$, which is not important physically but can cause problems during iterative convergence steps. Since $E_1(0) = E_2(0) = 0$, the limit of the ratio requires the use of l'Hopital's rule with the result $R(0) = \frac{1}{(3-e^1)}$.
Finally, we set $b=\beta$ and $P_\infty=\gamma$.
