Direct fitting

Ideally, pharmacological data should directly be fit to specific models and parameters derived from the direct fit. However, there are cases where the specific models predict surrogate parameters that can be derived without fitting data to the specific model. This can be an advantage. For example, the equation for simple competitive antagonism of receptors (see Section 6.3) is... 

Figure 2.10 (A) Direct fit of initial velocity as a function of substrate concentration to the...

The quantitative analysis of a multiphase topology comprises the formulation of structure models and the fitting of measured data. Fitting is discussed in Chap. 11. In this section the setup of topological models is discussed. The problem arises from the fact that most structural models of particle correlation are anisotropic and the visualization of structure in anisotropic materials by means of the CDF shows that suitable models must be rather complex. Thus a direct fit of anisotropic data would require fitting of a measured 3D or 2D function by a complex model. Both the effort to setup such models, and the computational effort to fit the data are very high. 

Practical Value. The presented analytical expressions are very useful, predominantly for the analysis of the scattering from weakly distorted nanostructures. Because of their detailed SAXS curves, direct fits to the measured data return highly significant results (cf. Sect. 8.8.3). Nevertheless, some important corrections have to be applied [84], They comprise deviations from the ideal multiphase structure as well as thorough consideration of the setup geometry and machine background correction (cf. Sect. 8.8). 

Larsson et al. 92) demonstrated also that the anisotropic pseudorotation model could be used for direct fitting of experimental PRE data for aniline protons in the complex between a Ni(II) chelate (Ni(II)(dpm)2) and ring-deuterated aniline in toluene-dg solution. The Ni(II) ion is in this case surrounded by four oxygen atoms and two nitrogen atoms, which should result in a sizable static ZFS, which indeed is confirmed by the fits. 

The recent versions of the slow motion approach were applied to direct fitting of experimental data for a series of Ni(II) complexes of varying symmetry (97). An example of an experimental data set and a fitted curve is shown in Fig. 9. Another application of the slow-motion approach is to provide benchmark calculations against which more approximate theoretical tools can be tested. As an example of work of this kind, we wish to mention the paper by Kowalewski et al. (98), studying the electron spin relaxation effects in the vicinity and beyond the Redfield limit. 

The concentration dependence of D on C may be obtained by directly fitting the diffusion profile if the functional form of D as a function of C is known for example. 

In addition to the direct fit using error function above, another popular fitting method is to use the inverse error function to fit. For example. Equation 3-124c may be written as... 

The expression in Equation 8.16 emphasizes the least accurate points, that is, those that are close to the end of the reaction. The preferred approach consists of direct fitting of the data to the expression for [A] in Equation 8.18 ... 

We note that the linear and volumetric CTEs in Refs.60 and 61 are not self-consistent in that the latter do not follow from the former according to the formal expression (for a monoclinic cell) Xv=Xa+Xb+Xc+(cosp/sinp)Pxp.. The reason for this is that xv was determined from a direct fit to the temperature dependent unit cell volume rather than from the linear CTEs (M. Herrmann, private communication). Although the same approach was used here, we obtained self-consistency to the percent level. 

A new accurate fitting of the calculated PES was realised by a Newton-Raphson procedure adopting the same potential form as used by Buck et al. [65]. The agreement between the MO-VB ab initio potential and the Buck potential, determined by direct fitting of the experimental data, is very satisfactory. 

The observations in kinetic studies are usually concentration or conversion measurements. CSTRs and differential PFRs yield rate data directly, so the experimentally observed rate can be directly fitted on the rate expression and its parameters estimated. 

Under a stream of nitrogen, replace the stopper on the third neck by a stop-cocked long glass tube. Connect it with flexible plastic hoses to the outlet port of a peristaltic pump whose inlet port is directly fitted to the top of the condenser by means of adapters and/or a length of plastic hose. Your apparatus should by now resemble the one shown in Figure 5.2. 

---