Numerical deconvolution using system identification methods

A deconvolution method is presented for use in pharmacokinetic applications involving continuous models and small samples of discrete observations. The method is based on the continuous-time counterpart of discrete-time least squares system identification, well established in control engineering. The same technique, requiring only the solution of a linear regression problem, is used both in system identification and input identification steps. The deconvolution requires no a prioriinformation, since the proposed procedure performs system identification (including optimal selection of model order), selects the form of the input function and calculates its parametric representation and its values at specified time points.On leave from Eötvös Lorand University, Budapest, Hungary.     

An inequality-constrained least-squares deconvolution method

The output-function (Y) of a linear system is the convolution of the input function (I) to the system with the disposition-function (H) of the system. Given Y and H deconvolution yields I. A non-parametric method for numerical deconvolution is described. The method is based on an inequality-constrained least-squares criterion and approximates I by a discontinuous function. No assumptions are made about the form of H or Y. Numerical stability and physical realism are obtained by constraining the estimated I to be nonnegative and piecewise-monotonic (nonincreasing, nondecreasing, or alternating segments of both). When I is constrained to be monotonic, the deconvolution yields a staircase function. The method can be used to calculate drug input rates. It is compared to previously published deconvolution methods for this purpose, using simulated data and real theophylline and pentobarbital data.     

An inequality-constrained least-squares deconvolution method

The output-function (Y) of a linear system is the convolution of the input function (I) to the system with the disposition-function (H) of the system. Given Y and H deconvolution yields I. A non-parametric method for numerical deconvolution is described. The method is based on an inequality-constrained least-squares criterion and approximates I by a discontinuous function. No assumptions are made about the form of H or Y. Numerical stability and physical realism are obtained by constraining the estimated I to be nonnegative and piecewise-monotonic (nonincreasing, nondecreasing, or alternating segments of both). When I is constrained to be monotonic, the deconvolution yields a staircase function. The method can be used to calculate drug input rates. It is compared to previously published deconvolution methods for this purpose, using simulated data and real theophylline and pentobarbital data.Work supported in part by U.S. Department of Health, Education and Welfare grants AG03104, GM26691.On leave from Mario Negri Institute of Pharmacological Research, Milan, Italy 20157.     

Comments on two recent deconvolution methods

In a recent paper Vajda et al. presented a deconvolution method based on the assumptions that the response of a system and the input function to a system are described by first-order linear processes. The method is similar to one proposed by Veng-Pedersen, and obtains similar results. In this article a simpler, not new, and now generally available method for this special use is considered to point out potential risks associated with all three deconvolution methods.     

An algorithm for constrained deconvolution based on reparameterization.

The method of deconvolution is the most general method of evaluating drug absorption. Deconvolution does not normally subscribe to a particular structured model for the input function that would ensure non-negativity of the calculated input rate. Unfortunately, the increased flexibility gained by this "model independence" sometimes results in a calculated input rate that becomes negative in the late absorption phase. A method for constrained deconvolution is proposed to overcome this deficiency. The method is based on a reparameterization of the input function in a "model-free" context. The reparameterization schemes proposed are optimal in the sense that they give the input function its maximum flexibility possible while at the same time ensuring non-negativity. The method makes use of "deconvolution through convolution." The basic procedure of this technique is, during the curve fitting, to iteratively adjust the input function so that when it is convolved with the unit impulse response function it results in a curve that best fits the data from the extravascular administration. Cimetidine data from oral and iv administrations are deconvolved by the proposed method to demonstrate the basic procedures involved. The proposed method is easy to implement since it relies on simple curve-fitting procedures as routinely performed in pharmacokinetics. The procedure can be carried out using any of the many curve-fitting programs available that allow the user to supply the function to be fitted and allow simple bounds to be specified for the fitted parameters. The convolution required by the method can be done analytically using a previously published computer program.     

Mathematical basis and generalization of the Loo-Riegelman method for the determination ofin vivo drug absorption

A general numerical deconvolution method is derived for the determination of in vivodrug input functions. The derivation is based on linear interpolation of observed drug concentrations and deconvolution of the resulting trapezoidal function. Derived in vivoinput functions are discontinuous. A general expression for the cumulative drug input is also derived. The latter expression is a generalization of the Loo-Riegelman equations. This deconvolution method gives similar results to the point-area deconvolution method when deriving in vivodrug input functions.     

Comments on two recent deconvolution methods

In a recent paper Vajda et al. presented a deconvolution method based on the assumptions that the response of a system and the input function to a system are described by first- order linear processes. The method is similar to one proposed by Veng- Pedersen, and obtains similar results. In this article a simpler, not new, and now generally available method for this special use is considered to point out potential risks associated with all three deconvolution methods.Work supported in part by U.S. Department of Health, Education and Welfare, grants AG03104, GM26691.On leave of absence from Mario Negri institute of Pharmacological Research, Milan, Italy 20157.     

A nonparametric subject-specific population method for deconvolution: I. Description,internal validation,and real data examples

In a pharmacokinetics context deconvolution facilitates the following: (i) Given data obtained after intravascular (generally intravenous) input one may estimate the disposition function; (ii) given the disposition function and data obtained after extravascular administration one may estimate the extravascular to vascular input rate function. In general if the data can be represented by the convolution of two functions, of which one is unknown, deconvolution allows the estimation of the unknown one. Attention has been given in the past to deconvolution and in particular to its nonparametric variants. However, in a population context (multiple observations collected in each of a group of subjects) the use of nonparametric deconvolution is limited to either analyzing each subject separately or to analyzing the aggregate response from the population without specifying subject-specific characteristics. To our knowledge a fully nonparametric deconvolution method in which subject specificity is explicitly taken into account has not been reported. To do so we use so-called “longitudinal splines”. A longitudinal spline is a nonparametric function composed of a template spline, in common to all subjects, and of a distortion spline representing the difference of the subject's function from the template. Using longitudinal splines for input rate or disposition function one obtains a solution to the problem of taking subject specificity into account in a nonparametric deconvolution context. To obtain estimates of longitudinal splines we consider three different methods: (1) parametric nonlinear mixed effect, (2) least squares, and (3) two-stage. Results obtained in one simulated and two real data analyses are shown.     

FIVE MODIFIED NUMERICAL DECONVOLUTION METHODS FOR BIOPHARMACEUTICS AND PHARMACOKINETICS STUDIES

Four improved finite-difference numerical deconvolution methods and one nonlinear regression numerical deconvolution method are proposed and implemented using IMSL/IDL^TM . These five numerical deconvolution methods are evaluated using simulated data generated with and without added noise under six different dosing cases. Comparisons between these methods are made in terms of the superimposability of the calculated cumulative amount of drug released or absorbed–time profiles with the theoretical data. The results indicate that the proposed fixed step number equal step length numerical deconvolution method is simple and accurate and therefore is appropriate for pharmacokinetic and biopharmaceutic studies. When an analytic function is legitimate to represent the drug input rate, the nonlinear regression numerical deconvolution method will yield enhanced numerical accuracy and stability.     

A polyexponential deconvolution method. Evaluation of the "gastrointestinal bioavailability" and mean in vivo dissolution time of some ibuprofen dosage forms

A new deconvolution algorithm (DCON) suitable for pharmacokinetic applications is presented. It requires that both the impulse and input responses, typically systemic drug levels, be well described by polyexponential equations. DCON has a wider range of applications than an earlier method (DECONV) from which it is derived. A FORTRAN program is provided, making implementation of the technique a simple matter. DCON is demonstrated to evaluate the "GI bioavailability," defined as the rate and the extent of gastrointestinal drug release, of various ibuprofen dosage forms. The GI drug release kinetics exemplifies a pharmacokinetic system which cannot be evaluated using the previous deconvolution algorithm (DECONV) because of an initial zero drug level response. This limitation is not found in DCON. It is also demonstrated how the mean in vivo dissolution time MDT can be evaluated by deconvolution.     

卷积法在体内外相关性研究中的应用

本文介绍了体内外相关性和卷积法/反卷积法的概念及原理, 阐述了采用卷积法使用Excel根据制剂的药代动力学数据计算其在体内释放行为的策略及方法, 并用于体内外相关性研究。该法用数学软件对药代动力学数据进行拟合以弥补缺失的数据点, 在假设输入函数基本符合威布尔规律的前提下, 在Excel中根据卷积法原理以试错法的方式确定拟合度最佳的输入函数(威布尔函数)参数, 最后以拟合度最佳的输入函数作为制剂的体内释放行为与体外释放数据进行相关性研究。在实例中不仅详细说明了该法的应用, 并且通过与隔室模型法和反卷积法的比较证明此方法简单有效, 可以作为体内外相关性研究的有力工具。     

Numerical deconvolution by least squares: Use of prescribed input functions

A new method for numerical deconvolution is described, for use in calculating drug input rates. The method is based on the least-squares criterion and is applicable when the input function can be assumed to take a prescribed form. In particular, an exponential input function and an input function derived from the cube-root dissolution law are considered. The stability of the method to data noise is shown by means of examples, using simulated data.     

Numerical deconvolution by least squares: Use of polynomials to represent the input function

A new method for numerical deconvolution is described, for use in calculating drug input rates. The method is based on the least-squares criterion and approximates the input rate by a polynomial function. Ill-conditioning of the normal equations is avoided by using orthogonal functions. The use of the method is illustrated by means of examples, using simulated data.     

Handling of computational in vitro/in vivo correlation problems by Microsoft Excel: III. Convolution and deconvolution.

Convolution and deconvolution are the classical in-vitro-in-vivo correlation tools to describe the relationship between input and weighting/response in a linear system, where input represents the drug release in vitro, weighting/response any body response in vivo. While functional treatment, e.g. in terms of polyexponential or Weibull distribution, is more appropriate for general survey or prediction, numerical algorithms are useful for treating actual experimental data. Deconvolution is not considered an algorithm by its own, but the inversion of a corresponding convolution. MS Excel is shown to be a useful tool for all these applications.     

A comparison of six deconvolution techniques

We present results for the comparison of six deconvolution techniques. The methods we consider are based on Fourier transforms, system identification, constrained optimization, the use of cubic spline basis functions, maximum entropy, and a genetic algorithm. We compare the performance of these techniques by applying them to simulated noisy data, in order to extract an input function when the unit impulse response is known. The simulated data are generated by convolving the known impulse response with each of five different input functions, and then adding noise of constant coefficient of variation. Each algorithm was tested on 500 data sets, and we define error measures in order to compare the performance of the different methods. The work described in this paper was carried out as part of Grant GR/J67130 “Identifiability and Indistinguishability of Nonlinear Dynamic Systems” from the U.K. Engineering and Physical Sciences Research Council.     

An algorithm and computer program for deconvolution in linear pharmacokinetics

The procedure of deconvolution to evaluate the rate and the extent of input from absorption data and data from intravenous administration is the most fundamental and least assumptive method of accurately evaluating drug absorption in linear pharmacokinetics. It is shown for linear systems that if the absorption response and the response from an intravenous infusion or bolus administration are both well approximated by a polyexponential function, then the rate of absorption can be expressed as a sum of exponentials. An algorithm and computer program are presented whereby the absorption function is uniquely defined from the model-independent parameters of the polyexponential expressions fitted to the absorption data and data from intravenous administration. Fitting a sum of exponentials to data has become a routine procedure in pharmacokinetics. The method presented therefore makes the previously complex task of deconvolution a simple procedure. The deconvolution approach is discussed in relation to conventional methods of evaluating drug absorption and appears to have some distinct advantages over these methods. The method is tested using simulated data and demonstrated using pentobarbital and cimetidine data from human subjects.     

Reversible Jump Markov Chain Monte Carlo for Deconvolution

To solve the problem of estimating an unknown input function to a linear time invariant system we propose an adaptive non-parametric method based on reversible jump Markov chain Monte Carlo (RJMCMC). We use piecewise polynomial functions (splines) to represent the input function. The RJMCMC algorithm allows the exploration of a large space of competing models, in our case the collection of splines corresponding to alternative positions of breakpoints, and it is based on the specification of transition probabilities between the models. RJMCMC determines: the number and the position of the breakpoints, and the coefficients determining the shape of the spline, as well as the corresponding posterior distribution of breakpoints, number of breakpoints, coefficients and arbitrary statistics of interest associated with the estimation problem. Simulation studies show that the RJMCMC method can obtain accurate reconstructions of complex input functions, and obtains better results compared with standard non-parametric deconvolution methods. Applications to real data are also reported.     

Metabolite Formation Pharmacokinetics: Rate and Extent of Metabolite Formation Determined by Deconvolution

A two-step analytic procedure to determine the rate and extent of metabolite production following administration of the parent compound is described. The procedure provides the rate and extent of metabolite production as a function of time by application of the general model independent approach of deconvolution. The metabolite unit impulse response function is obtained by implicit deconvolution of the metabolite data with a truncated constant-rate metabolite input function. Then the obtained unit impulse response function is used in an analytic deconvolution with metabolite data following administration of the parent compound to obtain the rate and extent of metabolite production. The input function is also deconvolved with metabolite data to obtain the unit impulse response function appropriate for prediction of metabolite levels given a selected input of parent compound. The expected profile following administration of the consecutive infusions of parent drug is shown for both parent and metabolite. The rationale for selection of deconvolution methods is discussed. The approach is applied to data for procainamide and N-acetylprocainamide from three human subjects. The results indicate that from 27 to 39% of the procainamide was converted to N-acetylprocainamide in these subjects.