Input-function method

The prescribed input function method - is also called deconvolution through convolution. In this method, the U1R(0 is first obtained in the same way as in the direct method just described. A suitable functional is then chosen to represent the input rate e.g., this prescribed function may be a polyexponential expression, a polynomial function, a cubic spline function, or other empirical function. Let this... 

In the fitting of R to input-response data the UIR function is kept constant i.e., the above A and a parameters are fixed, whereas the B and P parameters of the prescribed biexponential input function are treated as fitting parameters. Most frequently, the prescribed input function method is a two-step procedure where in the first step the UIR is determined separately. The UIR is then subsequently used in... 

SE7 Mathematically inexact deconvolution. Numerical procedures such as numerical integration, numerical solution of differential equations, and some matrix-vector formulations of linear systems are numerical approximations and as such contain errors. This type of error is largely eliminated in the direct deconvolution method where the deconvolution is based on a mathematical exact deconvolution formula (see above). Similarly, the prescribed input function method ( deconvolution through convolution ) wiU largely eliminate this numerical type of error if the convolution can be done analytically so that numerical convolution is avoided. 

An IV reference is available. The absorption rate may be determined by deconvolution using a direct deconvolution method or a prescribed input function method as described above. 

System analysis techniques have been used to generate input functions for PB-PK models. Oral administration of carbon tetrachloride in different vehicles was successfully described by absorption input functions obtained by deconvolution and disposition decomposition methods [25,26],... 

Figure 11P.1 can be used to determine the dimensionless dispersion parmeter ( l/uL) for a system of interest. Use the transfer function method to evaluate the mean residence time and QjJuL) for a system subjected to the arbitrary input shown in the figure. Note that the output response has been shifted 62.5 sec to the left. Response values for the input and output streams were as follows. 

Harris et al. also employed a less-known CCK procedure, which Meehl and Yonce (1996) named SQUABAC, but the authors referred to as the Parabolic Function Method. Two SQUABAC analyses were performed, one with the PCL-R total score as the input variable and criminal recidivism as the output variable, and another with adult criminal history and recidivism as input and output variables, respectively. Recidivism history was paired with the two potential taxon indicators because it is a conceptually related but distinct variable. It is expected to be a valid indicator of the taxon, but it is not redundant with other indicators, thus nuisance correlations should not be a problem. 

Aris (A8), Bischoff (Bll), and Bischoff and Levenspiel (B14) have utilized a method that does not require a perfect delta-function input. The method involves taking concentration measurements at two points, both within the test section, rather than at only one as was previously done. The remaining sketches in Table II show the systems considered. The variances of the experimental concentration curves at the two points are calculated, and the difference between them found. This difference can be related to the parameter and thus to the dispersion coeflScient. It does not matter where the tracer is injected into the system as long as it is upstream of the two measurement points. The injection may be any type of pulse input, not necessarily a delta function, although this special case is also covered by the method. 

In fact, with simple input functions common in pharmacokinetic applications (e.g., impulse or step function), the columns of the observation matrix X created from the integrals in (5.69) tend to be linearly dependent, resulting in ill - conditioned estimation problems. As discussed in the next section, this method is, however, excellent for input identification. 

Since the convolution integral is symmetrical in u(t) and hit), this problem is similar to the one of system identification considered in the previous section. Nevertheless, it is usually easier to find the weighting function h(t) since its form is more - or - less known (e.g., as a sum of polyexponentials), and hence parametric methods apply, whereas the input function u(t) is a priori arbitrary. Therefore, the non - parametric point -area method is a popular way of performing numerical deconvolution. It is really simple evaluating the integral means h, f, . .., fy, of the weighting function over the subinterval [t. ., t ] we can easily solve the set (6.68)... 

D. P. Vaughan and M. Dennis, Mathematical basis for the point-are deconvolution method for determining in vivo input functions. 

More recently, Boley (B9) shows that the method can be extended to find minimum and maximum bounds for ablation rates. The proof is based on a uniqueness theorem deriving from a theorem due to Picone (P3) and leads to the physically obvious result that the higher the heat input rate, the higher the melting rate. The procedure consists of forming a lower bound for the known arbitrary heat input function in terms of a sequence of constant heat flux periods, for which, as noted above, the solution can be written in terms of integrals of the error function. Upper bounds are constructed in a similar manner. 

The describing function method is applicable to any non-linearity which has the characteristic that if the input is a sinusoidal signal then the output is a periodic function 38 401. Because of its simplicity and wide range of applicability, the describing function technique is one of the most versatile procedures for analysing non-linear effects. 

Also the method used for post-processing can influence the CBV The reliability and stability of the results depend among other things on the fact, whether or not curve fitting is used to fit the concentration time curve of tissue and arterial input function (Smith et al. 2000b). As long as there is no universally accepted procedure for PI measurement... 

Vaughan, D. Denis, M. Mathematical basis of the point area deconvolution method for determining the in vivo input functions. J. Pharm. Sci. 1978, 64, 663-665. 

A parallel development has taken place for related transfer-fimction methods. For electrochemical systems, impedance spectroscopy, which relies on measurement of current and potential, provides the general system response. As described in Chapters 14 and 15, transfer-function methods allow the experimentalist to isolate the portion of the response associated with specific inputs or outputs. 

The transport approach has been used very early, and most extensively, to calculate the chromatographic response to a given input function (injection condition). This approach is based on the use of an equation of motion. In this method, we search for the mathematical solution of the set of partial differential equations describing the chromatographic process, or rather the differential mass balance of the solute in a slice of column and its kinetics of mass transfer in the column. Various mathematical models have been developed to describe the chromatographic process. The most important of these models are the equilibrium-dispersive (ED) model, the lumped kinetic model, and the general rate model (GRM) of chromatography. We discuss these three models successively. 

Tlic second method is based on an idea of Ng and uses the history of the iteration procedure to construct the new input functions [341]. Specifically, one constructs new input by extrapolating the previous output from at least two or, in most cases, several previous iterations according to the prescription... 

Many methods for measurement of tumor blood flow are based on quantifying the tumor uptake of a systemically administered tracer. Such uptake will obviously be a function of regional blood flow, but it will also depend on the arterial input function and the efficiency with which the tracer is extracted from the blood during passage through the microvasculature. The latter parameter is defined as the extraction ratio (ER) ... 

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