Bilinear Equation

2 Bilinear Equation. This equation is used, for example, to fit the Jaffe equation - Chapter 4. 

For triads of data y, Xi, and X2 the coefficients a, b, and c nvsy be obtained by substituting the following values into Equations (8) to (11)  

3 Global Analysis of Two Linear Equations. Plots of Pnuc versus 

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Note that the lipophilicity parameter log P is defined as a decimal logarithm. The parabolic equation is only non-linear in the variable log P, but is linear in the coefficients. Hence, it can be solved by multiple linear regression (see Section 10.8). The bilinear equation, however, is non-linear in both the variable P and the coefficients, and can only be solved by means of non-linear regression techniques (see Chapter 11). It is approximately linear with a positive slope (/ ,) for small values of log P, while it is also approximately linear with a negative slope b + b for large values of log P. The term bilinear is used in this context to indicate that the QSAR model can be resolved into two linear relations for small and for large values of P, respectively. This definition differs from the one which has been introduced in the context of principal components analysis in Chapter 17. 

Kubinyi [23] showed that the bilinear equation (7.43) can be approximated by a general form... 

It is curious that the log of the expression for R, Eq. (7.18), produces a Kubinyi-like bilinear equation... 

We ve looked at many other analyses of "simple" processes, but my favorite is a correlation for the absorption of an acid and a base in the same equation, — not only that, but with each one at six different pH s, (Figure 2). The data are from Schurmann Turner ( ) the base is propranolol and the acid, 4-n-hexylphenylacetic. Only a single parameter is required, log D, Equation 6. Kubinyi s bilinear equation gives an even better correlation ( ), Equation 7. (This is a special version of the bilinear model which sets the coefficients of each term equal.)... 

Quite generally, and for media ranging from the gas phase to CCI4, it is known547 that for thousands of (AH, B) systems, the bilinear equation 161b yields log10 fc values with a satisfactory degree of precision ... 

A bilinear equation applies in this special situation where phospholipid and glyco-calyx layers preceed the lipoidal membrane. As in colon and small intestine there... 

The micelle-solubilized fraction is not available for absorption. The bilinear equation is in this case applicable to correlate absorption rate constant values and lipophilicity parameters ... 

Figure 11 Extended Grunwald-Winstein plot for the solvolysis of isopropyl tosylate , plot of the data against , improved plot when bilinear equation is...

Figure 6 Illustration of the global fit of data to the bilinear equation...

In order to prove the canonicity of the transformation induced by the KS procedure, it is necessary to choose the initial conditions in order that the bilinear equation is satisfied. 6) The second derivative with respect to the fictitious time of the physical coordinates is given by... 

However, the parabolic model is still valuable for structure-activity analyses. It is the simpler model, easier to calculate, and most often a sufficient approximation of the true structure-activity relationship. The calculation of bilinear equations is relatively time-consuming, as compared to the parabolic model strange results may be obtained in ill-conditioned data sets. On the other hand, in many cases the, bilinear model gives a better description of the data, especially if additional physicochemical parameters are included in the regression equation. The lipophilicity optimum of symmetrical curves is precisely described by both, the parabolic model (optimum log P = — b/2a) and the bilinear model (optimum log P = — log P). In the case of unsymmetrical curves the site of the lipophilicity optimum is described much better by the bilinear model (optimum log P = log a — log p — log (b — a) eq. 93) than by the parabolic model. 

In addition, the biological data should cover a range of at least one, better two or even more logarithmic units they should be well distributed over the whole distance i.e., no clustering of activity values should occur, as discussed in chapter 2). Also the physicochemical parameters should be spread over a certain range and should be more or less evenly distributed if a certain parameter has identical values for all but one or two objects, then this parameter must be considered as a hidden indicator variable and should be replaced by such a term. In parabolic and especially in bilinear equations the nonlinear parameter should cover a range of at least two logarithmic units, in order to justify the presence of a nonlinear term. 

Nonspecific hemolytic, antibacterial, and antifungal in vitro activities generally follow linear [18, 182], parabolic [18, 19], and bilinear [23, 345] lipophilidty-activity relationships. Many of the bilinear equations (e.g. eq. 100, chapter 4.4) have... 

Model simulations (see chapter 4.4) substantiate that the lipophilicity dependence of the rate constants of drug transport should follow bilinear relationships [41,156, 175,345,440,442]. Indeed, bilinear equations have been derived for the rate constants of drug transport in n-octanol/water (eqs. 95 — 98, chapter 4.4) [444 —447] and for the rate constants of the transfer of various barbiturates (38) in a Sartorius absorption simulator from an aqueous phase (pH = 3) through an organic membrane to another aqueous phase (pH = 7.5), modeling the gastric absorption of these compounds (Figure 41) (eq. 162 recalculated optimum log P value) [442]. 

Also for side chain variations, clear - and in this case quantitative - structure-activity relationships have been established at the target level. In a series of oximino esters of type 4, we obtained a curve (Fig. 13.2.7) in accordance with a bilinear equation [7, 9]. Similar correlations have been deduced for enol ethers, oximino amides, crotonic esters and methoxycarbamates [66]. 

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