Method of residuals

Figure 3-7. The method of residuals applied to the system shown in Figs. 3-2 and 3-4. See the discussion of Eqs. (3-36) and (3-37).

Champoux, R. L., Analytical Experimental Methods of Residual Stress Effects in Fatigue, ASTM, STP 1004,1989. 

This method of calculation is often referred to as the method of residuals or feathering the curve. It is important to remember that the following assumptions were made ... 

Studies interested in the determination of macro pharmacokinetic parameters, such as total body clearance or the apparent volume of distribution, can be readily calculated from polyexponential equations such as Eq. (9) without assignment of a specific model structure. Parameters (i.e., Ah Xt) associated with such an equation are initially estimated by the method of residuals followed by nonlinear least squares regression analyses [30],... 

The integrals /1 and 12 are evaluated by the methods of residues. The standard calculation gives... 

When a compound is administered by a route other than intravenously, the plasma level profile will be different, as there will be an absorption phase, and so the profile will be a composite picture of absorption in addition to distribution and elimination (Fig. 3.26). Just as first-order elimination is defined by a rate constant, so also is absorption kab. This can be determined from the profile by the method of residuals. Thus, the straight portion of the semilog plot of plasma level against time is extrapolated to the y axis. Then each of the actual plasma level points, which deviate from this during the absorptive phase, are subtracted from the equivalent time point on the extrapolated line. The differences are then plotted, and a straight line should result. The slope of this line can be used to calculate the absorption rate constant kab (Fig. 3.26). The volume of distribution should not really be determined from the plasma level after oral administration (or other routes except intravenous) as the administered dose may not be the same as the absorbed dose. This may be because of first-pass metabolism (see above), or incomplete absorption, and will be apparent from a comparison of the plasma... 

Figure 3.28 Log10 plasma concentration against time profile for a foreign compound after intravenous administration. The distribution of this compound fits the two-compartment model. The dotted 1 line is determined by the method of residuals as described in the text. (Thus, R" = R - R, etc.)...

Equations (1) - (3) and (6) were simultaneously solved using the above boundary conditions by Laplace transformation and inversion by the method of residues to obtain the following analytical equations ... 

The identification and quantification (partial pressure measurement) of the gaseous components in vacuum systems is of increasing importance in vacuum technology. This is achieved by the widely applied method of residual gas analysis. The simple theory of this is stated and relevant examples illustrate the application. 

Magnetic moment, 153, 155, 160 Magnetic quantum number, 153 Magnetization, 160 Magnetogyric ratio, 153, 160 Main reaction, 237 Marcus equation, 227, 238, 314 Marcus plot, slope of, 227, 354 Marcus theory, applicability of, 358 reactivity-selectivity principle and, 375 Mass, reduced, 189, 294 Mass action law, 11, 60, 125, 428 Mass balance relationships, 19, 21, 34, 60, 64, 67, 89, 103, 140, 147 Maximum velocity, enzyme-catalyzed, 103 Mean, harmonic, 370 Mechanism classification of. 8 definition of, 3 study of, 6, 115 Medium effects, 385, 418, 420 physical theories of, 405 Meisenheimer eomplex, 129 Menschutkin reaction, 404, 407, 422 Mesomerism, 323 Method of residuals, 73 Michaelis constant, 103 Michaelis—Menten equation, 103 Microscopic reversibility, 125... 

The integrand of (4.107) has poles at a-Viejl and —a — iejl. We evaluate the integral by the method of residues. The result is translated into our original notation by inverting the transformation (4.106). It is... 

Other approaches have been used for more complex models. These include curve stripping or the method of residuals,either manually or using a computer program such as CSTRIP and ESTRIF. These techniques can separate a multiexponential curve into its component parts for initial estimates. Other techniques include deconvolution methods specific to the one and two compartment pharmacokinetic models. The objective of the deconvolution method is to mathematically subtract the results obtained after IV administration from the oral or extravascular data. This results in information about the input or absorption process alone. More general methods have been presented by various researchers that do not rely on a particular compartmental model. ... 

Fig. 2.2 Semilogarithmic graph of plasma drug concentration (o) after oral administration of a dose. The absorption phase is obtained by the method of residuals following back-extrapolation of the linear terminal (disposition) phase. Residual values are denoted by the dashed line. The zero-time intercept (in units of concentration) is log [FDosekaIV6 (ka-kd)]. (Reproduced with permission from Gibaldi Perrier (1982).)...

Several years later, the problem was again addressed using carbaryl radiolabeled with carbon-14 and liquid scintillation counting as the method of residue detection (2 results obtained... 

In the general method of residuals we desire to approximate (z) by a function ai, U2, as,an), which is a linear combination of the basis functions chosen from a linearly independent set. That is, the approximation of the solution is written as [62] ... 

When an extravascular dose is given, one-compartment-model serum concentrations rise during absorption, reach Cniax, and then decrease in a straight line with a slope equal to -k/2.303. The equation that describes the data is C = (FDka)/[VD( a -where F is the fraction of the dose absorbed into the systemic circulation. The absorption rate constant (ka) is obtained using the method of residuals. 

The method of residuals is used to obtain the individual rate constants (Fig. 5-6). A is determined by extrapolating the terminal slope to the y axis can be obtained by calculating the slope or 0/2 and using the formnlas given for the intravenons bolns case. At each time point in the absorption portion of the curve, the concentration value from the extrapolated line is noted and called the extrapolated concentration. For each point, the actual concentration is subtracted from the extrapolated concentration to compute the residual concentration. When the residual concentrations are plotted on semilogarithmic coordinates, a line with y intercept equal to A and slope equal to -dtJ2.303 is obtained. When these values are calculated, they can be placed into the equation (C = Ae - Ae <, where A = FDk l[V k - k)]) and used to compute the serum concentration at any time after the... 

The residual line is calculated as before using the method of residuals. The terminal line is extrapolated to the y axis, and extrapolated concentrations are determined for each time point. Because actual concentrations are greater in this case, residual concentrations are calculated by subtracting the extrapolated concentrations from the actual concentrations. When plotted on semilogarithmic paper, the residual line has a y intercept equal to A. The slope of the residual line is used to compute a (slope = -a/2.303). With the rate constants (a and /S) and the intercepts (A and B), concentrations can be calculated for any time after the intravenous bolus dose (C = Ae -E or pharmacokinetic constants can be computed Cl = D/[ A/a ) + (S//S)], Vo, = C1//3, Vd.ss = D[(A/ 2) -E (BZ/S )] /[(A/a) -E (B//S)]2. 

If serum concentrations of a drug given as a continuous intravenous infusion decline in a biphasic manner after the infusion is discontinued, a two-compartment model describes the data set (Fig. 5-9). In this instance, the postinfusion concentrations decrease according to the equation C = Re " + Se , where f is the postinfusion time (T = 0 when infusion is discontinued) and R, S, a, and /S are determined from the postinfusion concentrations using the method of residuals with the y axis set att = 0. R and S are used to compute... 

This section deals with current industrial methods of residue treatment. 

Peeling, curve stripping, or the method of residuals is an old technique advocated in some of the classic textbooks on pharmacokinetics (Gibaldi and Perrier, 1982) as a method to obtain initial estimates. The method will not be described in detail the reader is referred to the references for details. Its use will be briefly illustrated. Consider the biexponential model... 

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