Preliminary to results

2.4.1 Determination of curve characteristics

The curve characteristics are obtained for both the Classical and RMS methods of

parameter estimation. Although curve characteristics are not required to fit the data in the RMS method, they are used as either inputs into the model or in an analysis of goodness of fit. Curve characteristics of the simulated data, where calculated, are done so using the same method as for the observed data.

The halftimes τwand τs are obtained from the slopes kwand ks of semilog plots of the

observed data. For τwin a HPP, semilog plots are taken during the initial portion of the

curve. For τs, semilog plots are taken over the region t>2tm.

Extreme values (tm, Pm) can be obtained by reading off the discrete data (t,P) values, or

by interpolating between the data values to give a more accurate value. Here interpolation was used by fitting a second degree polynomial to a window around the extremum. The initial equilibrium pressure P0and the final equilibrium pressure PE were obtained by

averaging 10 or more observed values of P.

The simulated relaxation curves are very sensitive to the time t0at which the perturbation

occurs. It is therefore important to determine an accurate value for t0, however this is not

always possible. For an HPP, P1 was taken to be the maximum change in P from P0. The

since there is often some noise around this point. For an OPP, there is some freedom in choosing t0 since it is unknown when exactly the perturbation pulse impacts on the cell.

For an OPP, theory dictates that P0 = PE. Therefore in Eqs. (2.14) & (2.16) either the

observed P0 or observed PE value can be used for P0 in the equations. However, in many

data sets there is some variation from this, as the experiment may not have been run for long enough for a steady equilibrium to be attained, a long-term pressure drift was present in the data, or changes in the external solution occurred around t0. Therefore,

using P0 in the equations will mean that PE may not be predicted well by the model, and

using PE will mean that P0 may not be predicted well by the model. Here it was chosen to

use the observed P0 value.

2.4.2 Error analysis

The error in the estimation of each parameter or curve characteristic must be calculated for the model fits. The error is comprised of: a) numerical error in the model or

optimization method, and b) experimental error in the observed data due to the

measurement precision of the CPP. The experimental noise in the CPP has been found to have a standard deviation of ± 0.0003 MPa and a maximum of 0.0008MPa, so the error in P is taken to be about ± 0.0005 MPa (M. Tyree, unpublished). The experimental error in t is taken to be the time-resolution of the data (about ± 0.05s in Henzler et al.2004). There may also be some error in the exact size of the pressure perturbation, however the

magnitude of this error is unknown and should not be large, so is ignored in the present study.

In the Classical method of parameter estimation, Eqs. (2.12)–(2.16) are used to determine the membrane parameters that reproduce the curve characteristics obtained from the observed data. The accuracy of this method depends on the accuracy of the analytical solution, and the accuracy to which the curve characteristics can be determined from observed data. Standard errors (SE) in the calculated parameters are derived from the errors in the curve characteristics of the observed data. The standard errors in the

estimated parameters were determined using a formula for propagation of errors (Young, 1962). If Q = f(a,b,c), then the error δQ in Q is:

2 2 2 Q 2 Q 2 _... Q a b a b δ =⎛_⎜∂ ⎞_⎟ δ +⎛_⎜∂ ⎞_⎟ δ + ∂ ∂ ⎝ ⎠ ⎝ ⎠ , (2.18)

where δa is the error in a, etc. For the HPP experiment, the error in the slope of the semilog plot of the observed data, kw, can be obtained from the linear regression. From

this, the error in Lp is calculated using Eq. (2.13), and the SE in τw calculated using Eq.

(2.18). The error in PEwas derived by calculating the standard deviation of the last 10

values of P of the observed data.

For the OPP experiment, the error in ks was obtained from the error in the regression of

the semilog plot of the solute phase. From the measurement error in tm and the error in ks,

the error in kw can be determined using Eqs. (2.15) & (2.18). The errors in Lp, ps and σ

can be determined using Eq. (2.18) and the previous parameter definitions given by Eqs. (2.10), (2.13) & (2.16).

Standard errors in optimized or fitted parameters using the RMS method were calculated using the NonlinXL toolbox in Excel (P. Sands, unpublished). For the HPP, the SE in τw

was also calculated from the SE in Lp by using Eqs. (2.7), (2.8) & (2.18). For the OPP,

the percentage departure of the model curve characteristics from the data curve characteristics were also calculated.