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Comparisons of deconvolution algorithms in pharmacokinetic analysis

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Comparisons of deconvolution algorithms in pharmacokinetic analysis

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Content COMPARISONS OF DECONVOLUTION ALGORITHMS IN PHARMACOKINETIC ANALYSIS by Joy C. Hsu A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (BIOMEDICAL ENGINEERING) August 2003 Copyright 2003 Joy C. Hsu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 1420370 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a com plete manuscript and there are m issing pages, th ese will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 1420370 Copyright 2004 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQ uest Information and Learning Company 300 North Z eeb Road P.O. Box 1346 Ann Arbor, Ml 4 8 1 06-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CA LIFORNIA THEGRADUATE SCHOOL UNIVERSITY PARK LO S A NG ELES, CALIFORNIA 90089-1695 This thesis, written by under the direction of thesis committee, and approved by all its members, has been presented to and accepted by the Director of Graduate and Professional Programs, in partial fulfillment of the requirements fo r the degree of J o y C h ia -y u . Hsu B io m e d ic a l E n g in e e r in g Director Date D ecem ber 1 7 . 2003 Thesis Committee Chair Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments I would like to thank Dr. David Z. D ’Argenio for his assistance, advice, and support throughout the course of this research. I would also like to thank Drs. Michael C. K. Khoo and Vasilis Z. Marmarelis for their inputs and encouragements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iii Contents Acknowledgements ii List of Tables v List of Figures vi A bstract vii 1 Introduction 1 1.1 Motivation .......................... 1 1.2 Review of Methods .............................. 2 1.3 Limitations ....................... 5 1.4 Aim of Study ................. 6 2 M ethods 8 2.1 Deconvolution Methods to Be Evaluated .............................. 8 2.1.1 The Extended Point-Area Deconvolution Approach . . . . . . 8 Piecewise Cubic Interpolation................... 8 Determining the Optimal Step-Size . . . . . . . . . . . . . . . 9 Step Functions for the Input R a te ................... 9 Impulse Response Evaluation . ............................. 10 Cumulative Absorption ............................. 10 Input Rate E stim a tio n .......................................... 11 2.1.2 The Regression-Splines Method ................................ 12 Analytical Deconvolution . . . . .. .. ... .. .. .. .. . 12 Numerical Deconvolution ................ 14 2.1.3 Nonparametric Deterministic Regularization M ethod 15 Discrete Convolution and Deconvolution .....................15 Unit Impulse Response ..... ... .. .. .. ... .. .. 16 Determining the Step-Size . . . . . .. ... .. .. .. .. .. 16 Regularization Method of Deconvolution Constrained to Non-Negative Values ................. 16 Quadratic Forms ....................... 17 Determining the Proportionality C onstant. . . . . . . . . . . . 19 Input Rate Estimation . ... .. .. .. ... .. .. .. .. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iv 2.1.4 Nonparametric Stochastic Regularization Method . . . . . . . 20 Discrete Deconvolution . .......................... 20 Determining the Step-Size . ................ 21 Determining the Regularization Parameter . . . . . . . . . . 21 The Discrepancy Criterion ................ 21 The Maximum Likelihood Criteria . . . . . . . . . . . . 21 Input Rate Estimation .. .. .. .. .. ............ 22 2.2 Test Input Functions ... ... .. .. .. ............... 22 2.2.1 Impulse Response ....................22 2.2.2 Input Functions ....................22 2.3 Obtaining Simulated Data ................ 24 2.4 Evaluation of Methods ............................................. 24 2.5 Implementation of Algorithms ...... .. .. .. .. .. ... .. 25 3 Results 26 3.1 The Extended Point-Area Deconvolution Approach .................... 26 3.2 The Regression-Splines Method ..................................................29 3.3 Nonparametric Deterministic Regularization Method ..............................31 3.4 Nonparametric Stochastic Regularization Method ................ 33 4 Discussion 39 4.1 The Extended Point-Area Deconvolution A p p ro a c h ............................. 39 4.2 The Regression-Splines Method ..................................................40 4.3 Nonparametric Deterministic Regularization Method ............................. 41 4.4 Nonparametric Stochastic Regularization Method . ......................... 42 4.5 Summary ................... 43 5 Conclusion 51 References 55 Glossary 59 Appendix 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V List of Tables 2.1 Defined Input functions .............................. 23 3.1 Performance measures for the extended point-area deconvolution approach . ................ 28 3.2 Performance measures for the regression-splines method . . . . . . . . . 28 3.3 Performance measures for the nonparametric deterministic regularization method ................ .. .. .. .. .. .. 31 3.4 Parameter values used for the nonparametric stochastic regularization method ................... 33 3.5 Performance measures for the nonparametric stochastic regularization method .......... .. .. .. .. .. ... .. .. . 33 4.1 Performance measures for all four deconvolution algorithms ..................... 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi List of Figures 2.1 The five input functions used ........................ 23 3.1 Estimates from the extended point-area deconvolution approach........ 27 3.2 Estimates from the regression-splines method ................... 30 3.3 Estimates from the nonparametric deterministic regularization method . . 32 3.4 Estimates from the nonparametric stochastic regularization method . . . . 34 3.5 Mean estimates for each input function from the extended point-area deconvolution app ro ach............................. 35 3.6 Mean estimates for each input function from the regression-splines method .......... .. .. .. .. .. ... .. . 36 3.7 Mean estimates for each input function from the nonparametric deterministic regularization method ............................. 37 3.8 Mean estimates for each input function from the nonparametric stochastic regularization method .. .. .. .. ............... 38 4.1 Mean estimates of input function 1 from all deconvolution methods .. . . 46 4.2 Mean estimates of input function 2 from all deconvolution methods . . . . 47 4.3 Mean estimates of input function 3 from all deconvolution methods . . . . 48 4.4 Mean estimates of input function 4 from all deconvolution methods .. . . 49 4.5 Mean estimates of input function 5 from all deconvolution methods . . . . 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vii Abstract Performances of four deconvolution algorithms that are considered representative of the variety of approaches proposed in the pharmacokinetic literatures are evaluated with five proposed input functions. The methods studied here are the extended point-area method, the regression-splines method, the nonparametric deterministic regularization method, and the nonparametric stochastic regularization method. Thirty data sets are generated for each input function by adding a Gaussian measurement error of 10% coefficient of variation. In order to assess the overall performance of the chosen deconvolution algorithms, the percent root mean square and the mean error are taken for the estimated input functions as measures of accuracy and bias, respectively. Results demonstrate that the regression-splines method provides the most accurate and least biased mean input function estimations for all the proposed input functions. On the other hand, the extended point-area method gives the worst performance with unreliable results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 Chapter 1 Introduction 1.1 Motivation In recent years, various biopharmaceutical companies have formulated extended-release drug systems that are capable of delivering long-acting dosage of drugs in treating chronic diseases. The formulation o f the extended-release drug systems is favored because it allows convenient dosing and minimal injections for the patients. Usually, the drug is encapsulated into biodegradable polymers. Upon oral ingestion or injection into the subcutaneous tissue, the drug is then released as the encapsulated polymers slowly degrade. The recombinant human growth hormone Nutropin Depot®, formulated by Genentech Inc., is an example of the extended-release drug system designed to treat pediatric growth hormone deficiency. Instead of getting daily growth hormone injections, patients with pediatric growth hormone deficiency only need to be subjected to once- or twice-a-month subcutaneous injections of Nutropin Depot®. Hence, the in vivo input rate of a pharmacokinetic system is critical in designing drug delivery systems. Unfortunately, this input rate is not directly measurable and can only be observed through its causally related effects on plasma Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 concentration. By assuming the pharmacokinetic system is linear and time-invariant, reconstructing its input from its measured output is made possible by deconvolution. The convolution integral used to model the response of the body to an input of a particular drug is defined as t C (t) = ^ R { t ) C 5 ( t - r ) d t (1.1) o where C(t) is the input response from measured plasma concentrations, R(t) is the input rate of the drug, and Cg(t) is the unit impulse response of the body, determined by the plasma concentrations following an intravenous bolus administration of a unit dose. In this context, deconvolution refers to solving Equation (1.1) for an input rate R(t), given the input response C(t) and the unit impulse response Cg(t). Deconvolution is thus a powerful tool, but it is also known to be a difficult technique in the sparse data environment o f pharmacokinetic studies. 1.2 Review of Methods One of the earliest methods proposed for estimation of the input rate of a drug was published by Wagner and Nelson [32] in 1963. In their method, the disposition model is assumed to be a single compartment from which the drug is eliminated by first-order kinetics. Estimation of the input rate can then be derived from the cumulative amount absorbed, and it can be done without the need of a separate intravenous injection study. Later in 1968, Loo and Riegelman [11] proposed using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 a two-compartmental open-system model, along with first-order kinetics, to estimate the input rate. However, for the Loo-Riegelman method, the disposition compartmental model parameters need to be obtained through a separate intravenous injection study. Both of the Wagner-Nelson method and the Loo-Riegelman method are model-dependent in which a specific model for drug input kinetics and a specific compartmental model for drug disposition are assumed. However, these model- dependent methods often do not give reliable estimates of the input rates because the drug disposition models are only approximate, and the simple mathematical terms used to describe their input kinetics can not fully portray the complexity of the system [22]. Many model-independent methods that assume no specific compartmental model structure or specific kinetic model were later proposed. One o f these methods was proposed by Vaughan and Dennis [20] in 1978 where they derived staircase input functions in their point-area deconvolution method. The input rate estimated by this method is assumed to be piecewise constant within each sampled interval. Re-convolving the staircase input function and the system’s impulse response will then result in an input response function in registration with the observed input response data points. However, in practice, the sampled intervals of the input response data are likely to be relatively large. Thus, the input rate might vary substantially within each interval, leading to large errors that are not representative of the system [33]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 In 1977, Cutler [3, 4] published a nonlinear least squares numerical deconvolution method in which he applied orthogonal polynomials to the direct fitting o f input response data. The input rate is then approximated by a polynomial function based on the least-squares criterion. Yeng-Pedersen [22, 23] has also proposed an analogous method to estimate the input rate using a sum of exponentials in 1979. By using the model-independent parameters o f the polyexponential expressions fitted to the input response data and the impulse response o f the system, Veng-Pedersen ’ s method is able to define a unique input function in the form of a sum o f exponentials. Although both Cutler and Veng-Pedersen’s methods have the advantage of being model-independent and do not assume any specific kinetic model, their input rates are limited to the form of one single polynomial function or a sum of exponentials. In addition, their methods do not impose any non-negativity constraints to their input rate estimation. Piecewise polynomial B-spline functions has been used by Verotta [25, 28, 29] in 1989 in his numerical deconvolution method to estimate the input rate of a drug. Verotta’s method is based on an inequality-constrained least-squares criterion which limits the estimated input rate to be nonnegative and piecewise-monotonic. However, when working with his constrained regression criterion, the issues of choosing the maximum number o f knots and the positions of these knots directly affect the accuracy of the input rate estimation [28]. Later in 1994, Gillespie [7] has proposed using explicit curve-fitting methods to solve the deconvolution problem, as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 implemented in his software PCDCON. In his method, the input response and the impulse response data are fitted by continuous poly-exponentials or cubic splines. Deconvolution is then applied to the fitted functions analytically using the Veng- Pederson method or numerically by the Richardson’s extrapolation [10]. There is no constraint on non-negativity imposed on the input rate estimation in this method. In recent years, different regularization methods have been proposed to solve the deconvolution problem. Hovorka et al. [9] have published a nonparametric deterministic regularization method in 1995 which constrains the estimate of the input function to non-negative values. Based on the amount o f measurement error expected in the input response data, a discrepancy criterion is used to determine the amount of regularization needed for smoothing the estimated input functions. One year after Hovorka et al. published their nonparametric deterministic regularization method, Sparacino and Cobelli [13-17] proposed a stochastic regularization method which uses the maximum likelihood principle for the expected measurement error in determining the regularization criteria. Non-negativity constraints are included in their method, and confidence intervals accounting for biased error are provided with the estimated input functions. 1.3 Limitations Deconvolution in pharmacokinetics is well known to be challenging in many different aspects. Typically, only a small number o f noisy observed input response Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 data is available, usually less than twenty. O f those input response data observed, they are usually unequally spaced in time. Therefore, standard techniques of numerical deconvolution like discrete Fourier transformation are not feasible since they require large samples of equally spaced data. Also, since noise is present in the data observed, there is a trade-off between input response data fit and the smoothness in estimated input function [17]. In addition, numerical deconvolution algorithms often result in negative values or oscillatory patterns in the estimation for input functions [9], Even though those negative values or oscillatory patterns can be explained numerically, they are physiologically unfeasible, hence meaningless. 1A Aim of Study A number of deconvolution techniques have been proposed in the literature over the past five decades. The aim of this study is to evaluate the performance of four different deconvolution algorithms that are considered representative of the variety o f approaches proposed in the pharmacokinetic literatures. The method studied are the extended point-area method [33] along with the regression splines method [7], the nonparametric deterministic regularization method [9], and the nonparametric stochastic regularization method [17] as implemented in the software PCDCON, CODE, and WINSTODEC, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 The unit impulse response and five input functions suggested by Hovorka et al [9] are used in this study. Those five input functions can be informally described as a mono-exponential decay input function, a quadratic decay input function, a single peak input function, a double-peak input function, and a controlled-release input function. For each input function selected, 30 time-concentration profiles are constructed by adding a Gaussian measurement error of 10% coefficient of variation to its noise-free profile. In order to assess the performance of those four chosen deconvolution algorithms, the percent root mean square and the mean error are taken for the estimated input functions. The percent root mean square is taken as a measure of accuracy, and the mean error is taken as a measure o f bias. These performance measures should be able to provide a good assessment of the performance of the deconvolution algorithms chosen for this study. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 Chapter 2 Methods 2.1 Deconvolution Methods to Be Evaluated 2.1.1 The Extended Polnt-Area Deconvolution Approach Piecewise Cubic Interpolation In the extended point-area deconvolution approach [33], piecewise cubic polynomials are used to obtain interpolations c ( t ) for the input response data (cj, cn)- The piecewise cubic polynomial algorithm, adapted from that originally developed by Fritsch-Butland [6], provides interpolated functions that are smooth and constrained to be differentiable only once at the points where the experimental input response data points are located. Based on the interpolation algorithm used in this approach, the function ci between two input response data points (t„-i, cn.j) and (tn, cn) is defined by the following cubic equation: C i = c(tn_ I) + c(tn_{)(f, - tn_x) + p n(t, - tn_x)2 + qn - tn _x ) (2.1) where n > 1, tx = 0 , c(tx ) = 0 and K-i —K dn =K~h-X „ c{tn)-c(tn-x) ti ~ , (2.2a) (2.2b) (2.2c) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 Pn = ^ - 0 S n -2 c { tn_l)~ c{tn)) (2.2d) < 7 „ =-^{c{tn _x) + c{tn) - 2 s n) (2.2e) c(tn _x ) , c(f„) = slopes of the curve evaluated at f( -./ and U, respectively; = the measurement time grid, and tt = the optimal time grid. Determine the Optimal Step-size To maximize the retrievable kinetic information from the input response data, the optimal step-size v in between the unequal sampling intervals is determined by the ratio o f the largest sampled interval to the smallest sampled interval in the study: U = (2.3) Thus, the optimal number of subintervals M used in this approach is equal to the product of v and the number of input response data points N: Step Functions for the Input Rate The input function is assumed to be step functions within each subinterval. Under such approximation conditions, Equation (1.1) can be written as: M = u x N (2.4) (2.5) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 When solved recursively, Equation (2.5) can be written as an expression for the average input rate r(tt) : i - l V,)- z y-2 / \ j c s ( ^ - T ) d T X \cs{ti-r)dT (2.6) h-i Impulse Response Evaluation The impulse response is evaluated by using a modified trapezoidal method for numerical integration: l j c s (ti - r } d T = *»' ■ ~,r l.[cs (tt - t j_l ) + c s (tj - tf_2)] 1 + -<«)] (2-7) Cumulative Absorption An integration of r(tt) yields the cumulative absorption a(tf) : a(*i) = E - f*~i)] k=2 (2.8) Based on the cumulative absorption a(tt) , the cumulative absorption at each sampled time tn when U - tn is: An = a(ti), n= 2, 3, N-l (2.9) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 Piecewise quadratic equations are fitted to the nearest three points An.j), (tn, A„), and (t„+i, A„+j) for the interior points (n = 2, 3 , N-l). A system of 3 linear equations with 3 unknowns is formed: 4,-1 = «» + + wnx 2 n^ (2.1 Oa) An=un+vnxn +wnx z n (2.10b) A + l = U n + + W nX2 n + l (2.10C) By solving the above equations (2.10a-c) for u„, v„, and w„, the slope for A„ can be written as follows: An =v„+2w„xn (2.11) If An is found to be negative, then it is estimated to be zero. Input Rate Estimation The slope at the first sampled time point (n = 1) A1 is approximated by the input rate of the first optimal subinterval ry. Similarly, the slope at the last sampled time point (n = N) AN is approximated by the input rate of the final optimal subinterval rM. The input rate r(t„) at the sampled time tn is then estimated by the moving average o f the 3 contiguous slope values: f ( 0 = ! [ ! A , + A ) + ^ i - + ' O ] . » = 2. 3, N-l (2.12) If the input rate r(t„) calculated at the sampled time point tn is negative, then it is estimated to be zero. No other constraints are imposed on r(tn ). As indicated in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 Equation (2.12), the input rates are not computed at the first sampled time point tj and the last sampled time point h v - In addition, input rates at the interpolated time points are not calculated due to limitations of the numerical algorithm and the piecewise cubic interpolation. 2.1.2 The Regression-Splines Method The software PCDCON, written by Gillespie [7], is used to implement the regression-splines method. In this software, the deconvolution problem is solved by explicit curve-fitting methods. First, an interpolating or a smoothing spline is used to fit the input response data. Then based on the form of the impulse response provided, deconvolution is carried out analytically or numerically to the fitted functions. Analytical Deconvolution If the impulse response provided is a poly-exponential function, the analytical deconvolution algorithm proposed by Veng-Pedersen [22, 23] is used to estimate the input rate. Given the impulse response Cs and input response C: P c , « ) = a, > 0 (2.13) t> t0 (2.14) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 Some parameters are defined as follows: I a,, Q ( 0 ) * 0 A, = { ' s (2.15a) 1 - Q(0) = 0 K , = 1 / £ a , (2.15b) K, - K,’ £ A ,a i (2.15c) ( = 1 1 th From the above parameters, roots y. are obtained from the following (p-1) -degree polynomial: 0(*) = t 4 I<«, + x) (2.15d) (=1 M M Subsequently, g. may be evaluated from roots yt based on Si = Ki\\{ r i+aj)lY [ ( J i - r j) (2.i5e) M The input rate can then be estimated from the following equation: D L~ia m = — X ui( - v i)e~V l< -t~‘o), t >t0 (2.16) 1U O i=i where D = input dose Di = impulse dose L = p + q - l (2.17a) C' (0 )* ° (2.17b) " [1, Q (0 ) = 0 = I ^ ’ i-l,2,...,q (2.17c) I-Ti-m* i = q + l,q + 2,...,L-i0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 C ,(0 )* 0 C,(0) = 0 (2.17d) (2.17e) X4 = 1 0 0 ^ /1 ) (2 -17 g) Numerical Deconvolution If the impulse response provided is in the form of raw data, then the deconvolution is solved numerically using the midpoint method with Richardson’s extrapolation [10] for error estimation and correction. Using the Richardson’s extrapolation, the estimated input rate using step size h can be written as: where r(t) is the true input rate, e(t) is a function independent o f step size h, 0(hp) is the magnitude of error with an accuracy of order p, and q is unknown. Then, the input rate estimation with step size h/2 and h/4 can be calculated as follows: r(t,h) = r(t) + e(t)h9 + 0 (h p), p> q (2.18) f h') ( h Y r = r(t) + e(t) - + 0(h p) (2.19a) I 4J (2.19b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 By ignoring the 0(h p) terms and eliminating the r(t) and e(t) from Equations (18), (19a), and (19b), q can be derived q n log f r(t,h/2)-r(t,h) ^ r(t,h/4)-r(t,h/2) (2.20) After estimating q, Richardson’s extrapolation can then be applied estimation of the input rate r{t) « r + - 1 2q -1 ' h ' t, - v 4 j - r f,2j v • + o(hp) 2.13 Nonparametric Deterministic Regularization Method Discrete Convolution & Deconvolution The convolution integral, Equation (1.1), can be rewritten in the discrete form: C = AR + e (2.22) where C = (c(tj) , ...., c(tff))T input response data vector; e = (e(tj), e(tM ))T error vector for the input response data, assumed to be normal, white, and with zero mean; R = (r(ti), ...., r(tM)) vector o f the input rate on a M-size optimal, virtual grid arbitrarily chosen by the user. A is an N x M convolution matrix, and elements a„i of matrix A are defined as an i = \cs {tn -t)dt (2.23) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Given the impulse response cs » which is used to construct the convolution matrix A, and given input response measurements C, the input function R can be solved by discrete deconvolution: R = A~lC (2.24) It is assumed that r(t() is a piecewise constant function. Unit Impulse Response In this regularization method, the unit impulse response cs is described by a sum of exponentials < d O = i > - * ' p - 25) 1 = 1 Determine the Step-size Any step-size can be used in this deconvolution method. Regularization Method of Deconvolution Constrained to Non- Negative Values The regularization method of deconvolution constrained to non-negative values avoids ill conditioning by introducing a regularization factor [9]. This method prefers smooth estimates of the input functions with the extent o f smoothing depending on the amount of noise expected to be present in measurements. For an input function estimate r , a two-component penalty function j\r'\ is defined by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 j{f] = a[f ] + A/?[r] (2.26) where /?[r] is a certain quadratic form of the regularization factor and X {X > 0) is a proportionality constant. a[f] evaluates the least-square fit to the input response measurements using the % 2 method: (2.27) -1 Or(hJ where c = A r . Given a proportionality constant X , standard deviations <j{tn) , and input response measurements c(tn), an input function r (q > 0) that minimizes Equation (2.26) is defined to be the estimate o f the input function by the regularization method of deconvolution constrained to non-negative values. The input function so determined is devoted by rx. Quadratic Forms Four different types o f the quadratic forms can be implemented: Type 1 Norm: The Norm o f First Differences This form is defined as / t l = X W ) ! (2-28> /= 2 where Ar. = HXlItL (2.29) ti-h- 1 This norm works best for constant input functions r(t) = a0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 Type 2 Norm : The Norm o f Second Differences The second type of quadratic form is designed for linear input functions r(f) = ax i + a0. This form is defined as the square of the norm of second differences (2-30) 1=3 where A2/; . =Ar( -Arw (2.31) Type 3 Norm: The Norm o f Third Differences This type of quadratic form is intended for quadratic input functions r(f) = a2t 2 +axt + a0. It is defined as the square of the norm of third differences (2-32) 1=4 where A ^ aV a2^,, (2.33) Type 4 Norm: The Norm fo r Exponential Inputs The fourth type of quadratic form is for input function of the form r(t) = a0 te~a '‘ . It is defined as i=3 where M f a2_ \ (2.34) t.e 1 1 - t . .e~ A z t =-------------^ -------- (2.35a) h ~ h '- l A2z; . = Az( . - AzlM (2.35b) The constant a? is calculated from the time-to-peak value tm a x as ax=Htmm (2.35c) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 It is proven that type 2 norm always provides reasonable results if no prior knowledge of the input function is available. Hence, the type 2 norm is adopted here. Determine the Proportionality Constant The proportionality constant X is used to obtain a balance between the measures of misfit and the smoothness. The value o f A , which results in (2.36) where s is the measurement error provided, should be selected. Input Rate Estimation A two-step algorithm is used to estimate the input function in the nonparametric deterministic regularization method. The first step is to search for a value of A . such that Equation (2.36) is satisfied. Then the second step is to minimize Equation (2.26). An input function rx is sought to be an estimate of the input function by the regularization method o f deconvolution constrained to non-negative values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 2.1.4 Nonparametric Stochastic Regularization Method Discrete Deconvolution In the nonparametric stochastic regularization method [13-17], the input rate estimation is also reconstructed from the discrete input response C, as illustrated in Equation (2.22). Assume that the covariance matrix of the input response measurement error s and the covariance matrix of the input rate r are Var[s]= fi2Y,s =B (2.37) and Var[f]=p2 (2.38) respectively. In these two equations, matrices and . are positive and definite while n2 and qr are positive scalars. The stochastically regularized input rate estimate is then defined as f = arg min |c - A r f B~x(c - Ar)+ yfT F r F r } (2.39) where F is the M x M penalty matrix, and y {y > 0) is the regularization parameter. The penalty matrix F is defined such that F r is the vector of the gth time differences of r , with g being a positive integer parameter. Therefore, F can be written as F = As (2.40) where A is the square iV-dimension lower triangular Toeplitz matrix, with the first column as [1, -1, 0, 0]T , and g can be chosen to equal to 1 or 2, depending on the particular problem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 Determine th e Step-size Any step-size can be used in this deconvolution method, given that M » N . Determine the Regularization Parameter It is defined that the residual vector, the weighted sum of squared residuals, and the weighted sum of squared estimates are as follows: rv(y) = c - Ar(y) (2.41) WSSR(y) = rv(y) ^ " W (2.42) WSSE{y)=r{y)Y~r(y) (2.43) respectively. The equivalent degrees of freedom of the input rate estimate is defined as qiy) = trace[B"1 a {a T B~l A + yFr f } 1 At ] (2.44) The regularization parameter y can be determined using any of the following 4 criterions described depending on the condition of a particular problem. The Discrepancy Criterion If the discrepancy criterion is used, y is chosen so that WSSR(y) = Nju2. The Maximum Likelihood Criteria 7 7 If both p , c / > are unknown, choose y such that VSSRir) . MSSEb) 4„ [N-q(yfi q(y) 7 7 If p is known and f is unknown, choose y such that WSSE(y) = ^ - ^ (2.46) r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 9 9 If jj, is unknown but (p is known, choose y such that WSSR(y) = y < p 2 [N - q{y)] (2.47) Input Rate Estimation In the nonparametric stochastic regularization method, a value of the regularization parameter y is chosen, depending on the criterion o f choice. Then the input rate can be estimated by minimizing Equation (2.38) with a nonnegative constraint. 2.2 Test Input Functions 2.2.1 Impulse Response The unit impulse response used is noise-free and is described by a sum of two exponentials cs(t) = e-‘ +e~5 ‘ (2.48) It is the same as that used originally by Cutler [3,4]. 2.2.2 Input Functions Five input functions described as mono-exponential decay, quadratic decay, single-peak, double-peak, and control-release are chosen to test the performance of different deconvolution methods under a wide set of realistic scenarios (Figure 2.1). They are identical to those suggested by Hovorka et al [9], and they are shown in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 Table 2.1. Symbolic units (0 < t < 4 for time, r(t) < 2 for all the input functions) are adopted. x Input Function 1 + Input Function 2 • input Function 3 O Input Function 4 + Input Function 5 1.2:- 3 g- 0.8 0.6 0.4 0.2 0.5 2.5 3 .5 Time Figure 2.1 The five input functions used These five input functions are adopted from Hovorka et al [9] to evaluate the performance of each deconvolution method. Table 2.1 Defined Input Functions Here are the analytical formulas and their informal descriptions for the five input functions Function Number Analytical Formula Description 1 r(f)= 1.2e"2 < Mono-exponential decay 2 r{t)- \ 0 £>1.15 Quadratic decay 3 r( f)= 20 te ~ 5 ' Single-peak 4 r{t)= 15£e'9 '1 7 + t^e"0'5 1 ' Double-peak '(0 = 0.5(1 - (1 - / / 0.4)2) 0.5 0.5(l - (l - (2 - / ) / 0 .4 )2 ) £<0.4 0.4 <£<1.6 1.6<£<2 t> 2 Controlled-release Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 2.3 Obtaining Simulated Data For each of the five input function selected, a noise-free time-concentration profile is generated using an analytical solution of Equation (1.1) at 15 time points: 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.2, 1.5, 1.8, 2.2, 2.7, and 4.0. Thirty time- concentration profiles are constructed for each input function by adding a measurement error of 10% coefficient of variation, using Equation (2.49), to the noise-free data. e, ~ a ( o ,( 0 .1 c J ) (2.49) All of the simulated data sets, along with the true input rate, were generated using Matlab 6.5 (Appendix A). 2.4 Evaluation of Methods Two measures are employed to assess the difference between the actual and the estimated input functions. The percent root mean square {% RMS) is taken as a measure of accuracy. % RM S = / f M f t )X . dt (2.50) Vo r(t)2 The mean error (ME) is taken as a measure of bias. M E - | (r (/) - r(t J)dt (2.51) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 For each method evaluated in this study, the mean estimates for all input functions are used to compute the percent root mean square and mean error. These two measures are used in assessing the performance for the four chosen deconvolution algorithms. 2.5 Implementation of Algorithm In this study, Matlab 6.5 was used to implement the extended point-area algorithms described by Yeh et al [33]. The regression-splines method, the nonparametric deterministic regularization method, and the nonparametric stochastic regularization method were carried out using software PCDCON (1994), CODE (1998), and WINSTODEC (2001), respectively. The software PCDCON, CODE, and WINSTODEC used in this study are the ones originally developed by their authors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 Chapter 3 Results 3.1 The Extended Point-Area Deconvolution Approach Based on the extended point-area deconvolution approach proposed by Yeh et al [33], the input functions are estimated at the time points that coincided with the observed input response data points. However, because of the moving average algorithm, input rates are not computed at the first sampled time point and the last sampled time point. Given there are 15 data points in each simulated input response profile used in this study, only 13 estimated points can be obtained for each input function estimation. In order to obtain input rate estimates that are more comparable with the ones obtained by the other three methods evaluated in this study, cubic splines are used to interpolate those 13 estimated input function points. Equidistantly spaced time-step o f 0.02 is then used to obtain input rate estimates at the interpolated time points. Therefore, a total of 131 points on the estimated input function are calculated for each set of simulated data. Two examples of input function 4 estimations are shown in Figure 3.1. The %RMS and M E values calculated for all five input functions are shown in Table 3.1, and mean estimates for each input function calculated using the extended point-area deconvolution approach is given in Figure 3.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 Exact input rate Estimate of Input Function 4 1.6 1.4 1.2 %RMS: 44.06 M E: - 23.7314 o.a 0.6 0.4 0.2 0 0.5 2.5 3.5 Time 1.8 Exact input rate Estimate of Input Function 4 1.6 1.4 1.2 %RMS: 54.38 M E: - 31.2208 0.8 0.6 0.4 0.2 0 0 0.5 1.5 2 2.5 3 3.5 4 Time Figure 3.1 Estimates from the extended point-area deconvolution approach. Here are two examples of input function 4 estimations from using the extended point-area deconvolution approach (Top: data set 16, Bottom: dataset 30). The input function estimations are always underestimated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 Table 3.1 Performance measures for the extended point- area deconvolution approach. The results have shown that the extended point-area deconvolution approach does not yield reliable results, and it always underestimates input functions as % RMS ME Input Function #1 48.72 -13.5365 Input Function #2 39.53 -9.4699 Input Function #3 36.25 -13.4152 Input Function #4 49.36 -28.5979 Input Function #5 47.56 -3.6440 Table 3.2 Performance measures for the regression-splines method. Three different values of concentration standard deviation are used in estimating each input function for the regression-splines method. The results have shown that this method provides the most accurate and the least biased results. Cone. SD Estimate % RMS ME 0.005 6.09 0.2366 Input Function #1 0.01 5.94 0.2258 0.05 9.29 0.0954 0.05 7.55 0.0807 Input Function #2 0.1 17.27 0.2126 0.15 14.46 0.1608 0.01 10.69 -0.0395 Input Function #3 0.05 18.77 0.3490 0.2 24.72 0.4809 0.005 6.53 -0.2218 Input Function #4 0.01 6.73 -0.2086 0.05 18.35 0.2346 0.05 14.19 0.1414 Input Function #5 0.1 12.14 -0.2297 0.15 11.18 -0.2130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 3.2 The Regression-Splines Method The software PCDCON written by Gillespie [7] is used to implement the regression-splines method in this study. In order to perform a deconvolution operation, a concentration standard deviation estimate is required for fitting smoothing cubic splines to the input response data. According to Gillespie, the estimate of the standard deviation o f input response can be interpreted literally or can be viewed as an empirical smoothing parameter. However, since the criteria for choosing a smoothing parameter value is not well characterized, three different values o f standard deviation are used to estimate each input function in this part of the study. All of the values used are listed in Table 3.2. Input function estimates are calculated at a default of 201 equidistantly spaced time points in PCDCON. Since deconvolution is done between time 0 and 4 throughout this study, there are a total of 201 points (with step size 0.02) calculated for each set of simulated data. Figure 3.2 shows two examples o f input function 4 estimations done by PCDCON. The %RMS and ME values calculated for all input functions are summarized in Table 3.2. Also, mean estimates that have the lowest %RMS value for each input function are presented in Figure 3.6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 Exact input rate Estimate of Input Function 4 %RMS: 22.45 M E: -3.0113 q : 3 § ■ 0 .6 0.4 0.2 -0 .2 0.5 2.5 3.5 Time Exact input rate Estimate of Input Function 4 %RMS: 46.80 M E: -1.6143 0.5 -0.5 0.5 2.5 3.5 Time Figure 3.2 Estimates from the regression-splines method. Here are two examples of input function 4 estimations from using the regression-splines method (Top: data set 29, Bottom: dataset 14). Oscillatory patterns are prominent in most of the input function estimates. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 3.3 Nonparametric Deterministic Regularization Method The nonparametric deterministic regularization method is carried out in this study using the software CODE, written by Hovorka et al [9]. Deconvolution operations are initially performed using an input function time grid of 200 equidistantly spaced steps (step size 0.02). However, CODE is not able to perform deconvolution under this condition due to excessive number of iterations done in the nonlinear regression analysis. Therefore, a step size of 0.05 is used instead in estimating the input function in this part of the study, and a total of 81 estimated input function points are calculated for each set of simulated data. In addition, a 10% coefficient of variation in measurement error (of input response data) is used as a parameter during calculations. Two examples of input function 4 estimations are shown in Figure 3.3. The mean estimates for each input function calculated using the nonparametric deterministic regularization method is presented in Figure 3.7, and Table 3.3 lists all %RMS and M E values calculated for the 5 input functions. Table 3.3 Performance measures for the nonparametric deterministic regularization method. The nonparametric deterministic regularization method provides fairly accurate estimates for each proposed input functions, except for the exponential and quadratic decay scenarios. In addition, this % RMS ME Input Function #1 43.72 -0.6016 Input Function #2 44.68 -1.0332 Input Function #3 19.78 -0.4845 Input Function #4 24.67 -0.5645 Input Function #5 16.55 -0.4105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 Exact input rate Estimate of Input Function 4 %RMS: 19.26 ME: - 0.1293 0.6 0.4 0 .2 3.5 0.5 2.5 Time Exact input rate Estimate of Input Function 4 %RMS: 33.36 M E: - 1.2736 0 .6 0.4 0 .2 3.5 0.5 2.5 Time Figure 3.3 Estimates from the nonparametric deterministic regularization method. Here are two examples of input function 4 estimations from using the nonparametric deterministic regularization method (Top: data set 20, Bottom: dataset 7). Only a minimal number of oscillations are present in its input function estimations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 3.4 Nonparametric Stochastic Regularization Method The nonparametric stochastic regularization method is implemented by the software WINSTODEC, written by Sparacino et al [17], in this study. A list of parameter settings and values used in running WINSTODEC is summarized in Table 3.4. W ith a step size of 0.02, 201 equidistantly spaced estimated input function points are calculated for each set of simulated data. Figure 3.4 shows two examples of input function 4 estimations. All %RMS and M E values calculated for the input functions are given in Table 3.5, and Figure 3.8 presents the mean estimates for each input function calculated using the nonparametric stochastic regularization method. Table 3.4 Parameter values used for the nonparametric stochastic regularization method. Here is a list of criteria and parameter values used to run the software WINSTODEC, which implements the nonparametric stochastic regularization method in this study. Measurement Error Statistics CV constant (10%) Virtual Grid Period 0.02 Penalty Function Order 2 Regularization Parameter Criteria Maximum likelihood Constraint Nonnegativity Table 3.5 Performance measures for the nonparametric stochastic regularization method. The results have shown that the nonparametric stochastic regularization method generally overestimate input functions. It provides very accurate and relatively unbiased results, with the exceptions to % RMS ME Input Function #1 41.73 -0.1913 Input Function #2 43.64 1.3680 Input Function #3 10.63 2.1140 Input Function #4 9.20 1.7727 Input Function #5 10.38 2.2432 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 — - Exact input rate Estim ate of Input Function %RMS: 17.45 M E: 1.2949 0 .6 0.4 0.2 -0.2 0.5 2.5 Time Exact input rate Estimate of Input Function %RMS: 28.61 M E: 8.3780 §- 0.8 0.6 0.4 0.2 0.5 2.5 3.5 Time Figure 3.4 Estimates from the nonparametric stochastic regularization method Here are two examples of input function 4 estimations from using the nonparametric stochastic regularization method (Top: data set 5, Bottom: dataset 20). Oscillations are present in almost all input function estimations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 1.2 input Function 1 — - Exact input rate Mean estim ate 0.5 1 1.5 2 2.5 Time input Function 3 3.5 Exact input rate • Mean estim ate T n — $-----*---- --- 1 ____i ____i ____i_ 1.5 2 2.5 Time input Function 5 0.9 — - Exact input rate —— Mean estim ate 0.7 0.6 0.5 0.4 3 0.3 0.2 0.5 2.5 3.5 Time 1.8 1.4 1.2 1 0.8 • 0.6 0.4 0.2 0 -0 .2, input Function 2 Exact input rate • Mean estimate 0.5 1 1.5 2 2.5 3 3.5 4 Time Input Function 4 — - Exact input rate Mean estim ate 0.4 0.2 0 0.5 1.5 2 2.5 3 3.5 4 1 Figure 3.5 Mean estimates for each input function from the extended point-area deconvolution approach. The extended point-area deconvolution approach gives the worst performance with the largest %RMS and ME. Although this method is able to estimate fairly accurately during the early stage of estimation, the mean estimated input rate drops to zero quickly. It provides unreliable results and underestimates input functions under all conditions. Mean estimates at the time points that coincided with the observed input response data points are labeled with one standard deviation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 Input Function 1 Input Function 2 — - Exact input rate — Mean estim ate _ J ____i ____I ____ L — 0.5 1 1.5 2 2.5 3 3.5 Time input Function 3 - Exact input rate — Mean estim ate 1.6 1.2 I 1 % 0 8 0.6 , 0.4, Q.2; 0 0.5 1 1.5 2 2.5 3 3.5 4 Time Input Function 5 0.6 - - Exact input rate — Mean estim ate 0.5 0.4 I 0.1-/ -0.1 0.5 2.5 3.5 Time 1.8 : - - Exact input rats — Mean estim ate 1.2 I I 0.4 0.2 0 0.5 1.5 2 2.5 3 3.5 4 1 Time Input Function 4 - - Exact input rate — - Mean estim ate 0.8 0.4 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 3.6 Mean estimates for each input function from the regression-splines method. The regression-splines method is the most accurate and least biased among the four deconvolution algorithms studied. Although many oscillatory patterns are present in each of its input function estimations, the mean estimates obtained by this method still provide very accurate results. In addition, since there is no constraint on the non-negativity in its input function estimates, negative input rates are present. Mean estimates at the time points that coincided with the observed input response data points are labeled with one standard deviation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 input Function 1 Input Function 2 - - Exact input rats Mean sstim ate 0.4 0.2 0.5 2.5 3.5 Time Input Function 3 — - Exact input rate — — Mean estim ate 3 C 0.4 0.2 0 - • -0 .2, 0.5 1.5 2.5 3.5 Time Input Function 5 0.6 — * Exact input rate — Mean estim ate 0.5 0.4 “ 0.3 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 1.6 Exact input rate Mean estim ate 1.2 I 5 c 0.6 0.4 0.2 2.5 3.5 0.5 Time Input Function 4 Exact input rate Mean estim ate 1.2 a i c 0.6 0.4 0.2 2.5 3.5 0.5 Time Figure 3.7 Mean estimates for each input function from the nonparametric deterministic regularization method. The nonparametric deterministic regularization method can provide fairly accurate estimations and minimal oscillations for the single-peak, double-peak, and controlled-release input functions. In the cases of exponential and quadratic decays, the input rate estimates at time zero is constrained to zero, despite the fact that relatively accurate estimations can be obtained for the rest of the input functions. Mean estimates at the time points that coincided with the observed input response data points are labeled with one standard deviation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 Input Function 1 Exact input rats Mean estim ate J= 0.6 0.4 0.2 1.5 2 2.5 Time Input Function 3 0.5 3.5 — • Exact input rate - Mean estim ate 1.6 1.4 1 0.6 0.4 0.2 0.5 3.5 2.5 Time Input Function 5 0.6 Exact input rate Mean estim ate 0.5 0.4 G > 3 0-3 I 0.2 0.1 0.5 2.5 3.5 Time input Function 2 - - Exact input rate — — Mean estim ate 1.8 1.6 I I 0.8 0.6 0.4 0.2 0.5 2.5 3.5 Time input Function 4 Exact input rate Mean estim ate 1 1 I 0.8 0.6 0.4 0.2 0.5 2.5 3.5 Time Figure 3.8 Mean estimates for each input function from the nonparametric stochastic regularization method. The nonparametric stochastic regularization method can provide relatively accurate and unbiased input function estimations comparable to that of the regression-splines method. However, for the exponential and quadratic decay input functions, its initial input rate estimates are constrained to zero. Mean estimates at the time points that coincided with the observed input response data points are labeled with one standard deviation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 Chapter 4 Discussion The aim of this study was to evaluate the overall performance of four different deconvolution algorithms under five realistic conditions. Therefore, an equidistant time step of 0.02 was used to estimate all the input functions, except those done by the nonparametric deterministic regularization method due to software limitations. A step size of 0.05 was used instead in estimating the input function in that part of the study. The performance measures of %RMS and ME were compared and contrasted for the five proposed input functions for all algorithms. In addition, mean estimates calculated for each input function were examined closely. Table 4.1 summarizes the performance measures for all deconvolution algorithms analyzed, and the mean estimates calculated from all methods are plotted for each proposed input function in Figures 4.1 - 4.5. For the regression splines method, results with the least %RMS value are shown in this chapter. 4.1 The Extended Point-Area Deconvolution Approach In the extended point-area deconvolution approach, the input functions are only estimated at the time points that coincided with the observed input response data Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 points. In practice, the sampled intervals for the data are likely to be relatively large, and it may lead to erroneous results. Cubic splines were used in this study to interpolate estimated input function points obtained by this approach. Therefore, negative input rates are resulted from the curve fitting instead o f from the numerical deconvolution. In addition, this deconvolution approach always underestimates input functions as indicated by its very negative values of ME. For each proposed input function, the extended point-area approach is able to estimate fairly accurately only during the early stage o f estimation. However, the mean estimated input rate drops steeply and dies off quickly due to the nature of the moving average algorithm. In general, the extended point-area deconvolution approach does not yield reliable results. It is considered to provide the worst input function estimations among the four deconvolution methods analyzed in this study. 4.2 The Regressiou-Splines Method The software PCDCON which implements the regression-splines method discussed in this study does not put any nonnegative constraints on its input function estimations. Although almost all input function estimates are of positive values, a few negative values are still present. The input function estimates have shown that the regression-splines method tends to generate many artificial oscillatory patterns. In the case of input function 3 (single-peak) and input function 4 (double-peak), oscillatory patterns are very Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 prominent in the input function estimations. However, in the meantime, this method is able to accurately estimate the peaks of the proposed input functions. On the other hand, for input function 2 (quadratic decay) and input function 5 (controlled-release), fairly accurate estimates are obtained with minimal oscillations observed. Although many oscillatory patterns are present in each of its input function estimations, the mean estimates obtained by the regression-splines method still provide very accurate results. Based on the %RMS and M E values summarized in Table 4.1 and Figures 4.1 - 4.5, this method is the most accurate and least biased among the four deconvolution algorithms studied. 43 Nonparametric Deterministic Regularization Method Although the non-negativity constraint is imposed in the nonparametric deterministic regularization algorithm, a minimal number o f negative values are still present in the input function estimations. Also, the negative values of M E indicate that this deconvolution technique generally underestimates input functions. The mean estimates of the input functions have shown that the nonparametric deterministic regularization method produces minimal oscillations in its estimations. However, in a few cases of the simulated data analyzed in this study, numerous unrealisticaliy huge oscillations are present in the input function estimations. In those cases, the erroneous estimates are probably contributed by the inadequate choice in the time step selection which directly affects the quadratic form of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 regularization factor used in this method. This problem can be solved by choosing a larger time step in doing the deconvolution. For input function 3 (single-peak), input function 4 (double-peak), and input function 5 (controlled-release), the nonparametric deterministic regularization method is able to provide fair input function estimations. In the cases of exponential and quadratic decays (input functions 1 and 2), the input rate estimates at time zero is constrained to a value of zero, despite the fact that relatively accurate estimations can be obtained for the rest of the input functions. Aside from exponential and quadratic decay equations, this method generally provides fairly good estimates of the input functions with minimal oscillations. 4.4 Nonparametric Stochastic Regularization Method Generally, the nonparametric stochastic regularization method tends to overestimate input functions, as indicated by its positive values of M E shown in Table 4.1. Even with the nonnegativity constraint implemented in this method, a minimal number of negative values can still be found in the input function estimations. Similar to the nonparametric deterministic regularization method, this nonparametric stochastic regularization algorithm also constrains its initial input rate estimation to be zero while estimating exponential and quadratic decay input functions. In addition, oscillations are present in all input function estimations, with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 even higher degrees of oscillation present in the exponential and quadratic decay scenarios. The mean estimates obtained by the nonparametric stochastic regularization method prove to have very accurate and relatively unbiased results, even though some oscillatory patterns are present in its input function estimations. Based on the %RMS and M E results provided in Table 4.1 and Figures 4.1 - 4.5, this method is one of the best among the four deconvolution algorithms studied in terms of accuracy and bias. 4.5 Summary All o f the results have shown that the regression-splines method, implemented by the software PCDCON, is able to provide the most accurate and least biased input function estimations. The nonparametric stochastic regularization method can also provide accurate and unbiased estimations comparable to that of the regression- splines method, but the former has a disadvantage dealing with exponential and quadratic decay input functions. For the nonparametric deterministic regularization method, the results have shown that relatively accurate input function estimations can be obtained. However, it shares the same problem as the nonparametric stochastic regularization method with exponential and quadratic decay scenarios. As for the extended point-area approach, the results have been proved to be unreliable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 This method is considered to provide the worst input function estimations among the four deconvolution methods analyzed in this study. For the exponential and quadratic decay input functions, the regression-splines method can provide the most accurate input function estimations. The nonparametric deterministic regularization and the nonparametric stochastic regularization methods also give fairly accurate estimations, even though much more oscillations are present in the latter. In addition, both of the regularization methods are unable to estimate the initial conditions of the exponential and quadratic decay input function due to their constraints at time zero. The nonparametric deterministic regularization method, the nonparametric stochastic regularization method, and the regression-splines method are able to provide relatively accurate input function estimations for the single-peak, double peak, and controlled-release scenarios, with the latter two methods having more advantages. However, even though the nonparametric stochastic regularization and the regression-splines method have comparable accuracy in their input function estimates, more oscillations are observed in the regression-splines estimations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 Table 4.1 Performance measures for all four deconvolution algorithms. The results have shown that the regression-splines method provides the most accurate and least biased estimations for all proposed input functions. On the other hand, the extended point-area approach gives the worst performance with the largest %RMS and ME. Both of the regularization methods can provide relatively accurate estimates for the single-peak, double peak, and controlled-release scenarios (input function 3, 4, and 5, respectively), with the nonparametric stochastic regularization method having more advantage. However, neither of the regularization methods can provide accurate input function estimations for exponential Extended Point- Area Regression-Splines Nonparametric Deterministic Regularization Nonparametric Stochastic Regularization % RMS ME % RMS ME % RMS ME % RMS ME Input Function 48.72 -13.5365 5.94 0.2258 43.72 -0.6016 41.73 -0.1913 #1 Input Function 39.53 -9.4699 7.55 0.0807 44.68 -1.0332 43.64 1.3680 #2 Input Function 36.25 -13.4152 10.69 -0.0395 19.78 -0.4845 10.63 2.1140 #3 Input Function 49.36 -28.5979 6.53 -0.2218 24.67 -0.5645 9.20 1.7727 #4 Input Function 47.56 -3.6440 11.18 -0.2130 16.55 -0.4105 10.38 2.2432 #5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 Input Function 1 | • Exact input rate j -j- E xtended point-area 4 f Regression-splines o N onparam etric determ inistic regularization x N onparam etric s to c h a stic regularization 1 .2 - 0.8 I o: I 0.6 0.4 0.2 3.5 0.5 2.5 Time Figure 4.1 Mean estimates of input function 1 from all deconvolution methods. The regression-splines method provides the best estimation for the exponential decay input function. For both of the regularization methods, they are unable to accurately estimate the initial input rates due to their constraints at time zero. However, they are both able to provide fairly accurate estimates for the rest of the exponential decay function. For the extended point-area method, the input function is greatly underestimated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 Input Function 2 # E x ac t input rate + E xtended point-area Regression-splines o N onparam etric determ inistic regularization x N onparam etric sto c h a stic regularization I 0.6 0.4 0.2 ■ 0.1 3.5 0.5 2.5 Figure 4.2 Mean estimates of input function 2 from all deconvolution methods. For the quadratic decay input function, neither o f the regularization method is able to accurately estimate the initial input rates again due to their constraints at time zero. However, for the rest of the input function estimates, they are able to provide relatively accurate estimations comparable to that of the regression-splines method. The extended point-area method provides the worst estimation as expected. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 Input Function 3 ♦ E xact input rate -f E xtended point-area # R egression-splines o Nonparametric deterministic regularization x N onparam etric sto c h a s tic regularization I E K I 0.6 0.4 0.2 0.5 2.5 1.5 3.5 Tima Figure 43 Mean estimates of input function 3 from all deconvolution methods. The regression-splines method and the nonparametric stochastic regularization methods are the two methods that can best estimate the single-peak input function, although more oscillations are present in the estimations from the regression-splines method. For the nonparametric deterministic regularization method, it lacks smoothness in its mean input function estimation. The worst estimation is again provided by the extended point-area method. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 input Function 4 • E xact input rate • f E xten d ed point-area ■ 4 Regression-splines o : N onparam etric deterministic regularization x Ngnparamelric s to c h a stic regularization I ; cc !_ c 0.6 0.4 0.2 0.5 2.5 3.5 Time Figure 4.4 Mean estimates of input function 4 from all deconvolution methods. The double-peak input function can be best estimated by using the regression-splines method and the nonparametric stochastic regularization method. Although the nonparametric deterministic regularization method can also provide fairly accurate estimations, they are not comparable to that obtained by the two methods mentioned earlier. As expected, the extended point-area method gives the worst estimation with the least accurate and most biased results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 Input Function 5 0.7 # E xact input rate + Extended pomt-ares 4 Ragressson-spfinss o Nonparafftelric deterministic regularization x N onparam etric sto c h a stic regularization 0.6 0.5 0.4 0.3 0.2 -0.05, 0.5 2.5 3.5 Time Figure 4.5 Mean estimates of input function 5 from all deconvolution methods. Both regularization methods and the regression-splines method can provide comparably accurate estimations for the controlled-release input function. As for the extended point-area method, it again provides the least accurate and most biased mean estimation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Chapter 5 Conclusion The deconvolution of pharmacokinetic data is a difficult task, and numerous algorithms have been proposed in the pharmacokinetic literatures over the past five decades. It is known to be challenging due to the fact that usually only a small number of noisy input response data is available. O f those input response data observed, they are usually unequally spaced in time. In addition, since noise is always present in the data, there is also a trade-off between input response data fit and the smoothness in estimated input function [17]. Moreover, numerical deconvolution algorithms often result in negative values or oscillatory patterns in the estimation for input function, which are physiologically meaningless [9]. In this study, performances of four different deconvolution algorithms that are considered representative of the variety of approaches proposed in the pharmacokinetic literatures were evaluated with five proposed input functions. The methods studied are the extended point-area method [33], the regression-splines method (as implemented in the software PCDCON) [7], the nonparametric deterministic regularization method (as implemented in the software CODE) [9], and the nonparametric stochastic regularization method (as implemented in the software WINSTODEC) [17]. In addition, the noise-free unit impulse response and five Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 proposed input functions used in this study are those suggested by Hovorka et al. For the five input functions selected, they can be informally described as a mono exponential decay, a quadratic decay, a single-peak, a double-peak, and a controlled- release input function. Thirty time-concentration profiles are then constructed for each input function by adding a Gaussian measurement error o f 10% coefficient of variation to its noise-free profile. In order to assess the averaged overall performance o f the chosen deconvolution algorithms, the percent root mean square and the mean error are taken for the estimated input functions as measures of accuracy and bias, respectively. The results have shown that the regression-splines method, implemented by the software PCDCON, provides the most accurate and least biased mean input function estimations for all the proposed input functions, even though oscillations are present in individual input function estimates. On the other hand, the extended point-area approach gives the worst performance with the largest percent root mean square and the mean error. It not only provides unreliable results but also underestimate input functions under all conditions. For the nonparametric stochastic regularization method, accurate and unbiased mean input function estimations comparable to that of the regression-splines method can be obtained, although it has a disadvantage dealing with exponential and quadratic decay input functions. The results have also shown that the nonparametric deterministic regularization method can provide relatively accurate input function Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 estimations, even though it shares the same problem as the nonparametric stochastic regularization method with exponential and quadratic decay scenarios. However, since the scope of this study is only limited to analyzing simulated data with a Gaussian error o f 10% coefficient o f variation, the performances of each deconvolution method might differ when different levels of noises are present in the data. Therefore, in future research, the sensitivity of each deconvolution method in response to different levels of noises in the simulated data can be analyzed. If the form o f an input function is known in a given situation, an appropriate deconvolution algorithm can then be chosen to solve the problem. In cases where the input function is in the form of an exponential or quadratic decay, the regression- splines method is able to provide the most accurate input function estimations. The nonparametric deterministic regularization and the nonparametric stochastic regularization methods can also give fairly accurate estimates, even though both of the regularization methods are unable to estimate the initial input rate estimation due to their constraints qt time zero. In cases where the input function is in the form of a single-peak, double-peak, or controlled-release, the nonparametric deterministic regularization method, the nonparametric stochastic regularization method, and the regression-splines method are all able to provide relatively accurate input function estimations with the latter two methods having more advantages. However, even though the nonparametric stochastic regularization and the regression-splines method Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 have comparable accuracy in their input function estimates, more oscillations are present in the regression-splines estimations. Unfortunately, the form of the actual input function is usually unknown in the deconvolution analysis in practice. The regression-splines method, the nonparametric deterministic regularization method, and the nonparametric stochastic regularization method are all capable of providing relatively accurate input function estimations, although the regression-splines method is preferred. One major drawback for the regression-splines method is determining the smoothing parameter value used in fitting the input response data. When a proper value for the smoothing parameter is chosen, deconvolution can be carried out accurately. However, with inadequate smoothing parameter values, the regression-splines method provides relatively worse performances. Nonetheless, as long as the criteria for choosing a smoothing parameter value can be adequately characterized, the regressions-splines method can become one of the best deconvolution algorithms used in pharmacokinetic analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 References 1. M. K. Charter, S. F. Gull. Maximum entropy and its application to the calculation of drug absorption rates. Journal o f Pharmacokinetics and Biopharmaceutics. 15:645-655 (1987). 2. D. J. Cutler. Linear systems analysis in pharmacokinetics. Journal o f Pharmacokinetics and Biopharmaceutics. 6:265-282 (1978). 3. D. J. Cutler. Numerical deconvolution by least squares: use of prescribed input functions. Journal o f Pharmacokinetics and Biopharmaceutics. 6:227-241 (1978). 4. D. J. Cutler. Numerical deconvolution by least squares: use o f polynomials to represent the input function. Journal o f Pharmacokinetics and Biopharmaceutics. 6:243-263 (1978). 5. G. De Nicolao, G. Sparacino, C. Cobelli. Nonparametric input estimation in physiological systems: problems, methods, and case studies. Automatica. 33:851-870(1997). 6. F. N. Fritsch, I. Butland. A method for constructing local monotone piecewise cubic interpolants. SIAM J. Sci. Stat. Comput. 5:300-304 (1984). 7. W. R. Gillespie. 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Journal o f Pharmaceutical Sciences. 57:918-928 (1968). 12. F. N. Madden, K. R. Godfrey, M. J. Chappell, R. Hovorka, R. A. Bates. A comparison of six deconvolution techniques. Journal o f Pharmacokinetics and Biopharmaceutics. 24:283-299 (1996). 13. G. Pillonetto, G. Sparacino, C. Cobelli. Handling non-negativity in deconvolution of physiological signals: a nonlinear stochastic approach. Annals o f Biomedical Engineering. 30:1077-1087 (2002). 14. G. Pillonetto, G. Sparacino, C. Cobelli. Reconstructing insulin secretion rate after a glucose stimulus by an improved stochastic deconvolution method. IEEE Transactions on Biomedical Engineering. 48:1352-1354 (2001). 15. G. Sparacino, C. Cobelli. A stochastic deconvolution method to reconstruct insulin secretion rate after a glucose stimulus. IEEE Transactions on Biomedical Engineering. 43:512-529 (1996). 16. G. Sparacino, C. Cobelli. Deconvolution of physiological and pharmacokinetic data: comparison of algorithms on benchmark problems. Modelling and Control in Biomedical Systems 1997; A Proceedings volume from the IF AC Symposium, Warwick, UK. 153-155 (1997). 17. G. Sparacino, G. Pillonetto, M. Capello, G. De Nicolao, C. Cobelli. WINSTODEC: A stochastic deconvolution interactive program for physiological and pharmacokinetics systems. Computer Methods and Programs in Biomedicine. 67:67-77 (2002). 18. S. Yajda, K. Godfry, P. Valko. Numerical deconvolution using system identification methods. Journal o f Pharmacokinetics and Biopharmaceutics. 16:85-107 (1988). 19. D. P. Vaughan. Approximation in point-area deconvolution algorithm as mathematical basis of empirical instantaneous mid-point-input deconvolution method. Journal o f Pharmaceutical Sciences. 70:831-832 (1981). 20. D. P. Vaughan, M. Dennis. Mathematical basis of point-area deconvolution method for determining in vivo input functions. Journal o f Pharmaceutical Sciences. 67:663-665 (1978). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 21. P. Veng-Pedersen. An algorithm and computer program for deconvolution in linear pharmacokinetics. Journal o f Pharmaceutical Sciences. 8:463-481 (1980). 22. P. Veng-Pedersen. Model-independent method of analyzing input in linear pharmacokinetic systems having polyexponential impulse response I: theoretical analysis. Journal o f Pharmaceutical Sciences. 69:298-305 (1980). 23. P. Veng-Pedersen. Model-independent method of analyzing input in linear pharmacokinetic systems having polyexponential impulse response II: numerical evaluation. Journal o f Pharmaceutical Sciences. 69:305-312 (1980). 24. P. Veng-Pedersen. Noncompartmentally-based pharmacokinetic modeling. Advanced Drug Delivery Reviews. 48:265-300 (2001). 25. D. Verotta. An inequality-constrained least-squares deconvolution method. Journal o f Pharmacokinetics and Biopharmaceutics. 17:269-289 (1989). 26. D. Verotta. Comments on two recent deconvolution methods. Journal o f Pharmacokinetics and Biopharmaceutics. 18:483-499 (1990). 27. D. Verotta. Concepts, properties, and applications of linear systems to describe distribution, identify input, and control endogenous substances and drugs in biological systems. Critical Reviews in Biomedical Engineering. 24:73-139 (1996). 28. D. Verotta. Estimation and model selection in constrained deconvolution. Annals o f Biomedical Engineering. 21:605-620 (1993). 29. D. Verotta. Two constrained deconvolution methods using spline functions. Journal o f Pharmacokinetics and Biopharmaceutics. 21:609-636 (1993). 30. P. Vicini, G. Sparacino, A. Caumo, C. Cobelli. Estimation of endogenous glucose production after a glucose perturbation by nonparametric stochastic deconvolution. Computer Methods and Programs in Biomedicine. 52:147-156 (1997). 31. I. G. Wagner. Pharmacokinetic absorption plots from oral data alone or oral/intravenous data and an exact Loo-Riegelman equation. Journal o f Pharmaceutical Sciences. 72:838-842 (1983). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 32. J . G. Wagner, E. Nelson. Percent absorbed time plots derived from blood level and/or urinary excretion data. Journal o f Pharmaceutical Sciences. 52:610- 611 (1963). 33. K. C. Yeh, D. J. Holder, G. A. Winchell, L. A. Wenning, T. Prueksaritanont. An extended point-area deconvolution approach for assessing drug input rates. Pharmaceutical Research. 18:1426-1434 (2001). 34. K. C. Yeh, R. D. Small. Pharmacokinetic evaluation of stable piecewise cubic polynomials as numerical integration functions. Journal o f Pharmacokinetics and Biopharmaceutics. 17:721-740 (1989). 35. Z. Yu, S. Hwang, S. Gupta. DEMONS— A new deconvolution method for estimating drug absorbed at different time intervals and/or drug disposition model parameters using a monotonic cubic spline. Biopharmaceutics & Drug Disposition. 18:475-487 (1997). 36. Z. Yu, J. B. Schwartz, E. T. Sugita, H. C. Foehl. Five modified numerical deconvolution methods for Biopharmaceutics and pharmacokinetics studies. Biopharmaceutics & Drug Disposition. 17:521-540 (1996). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 Glossary c(t) Input response from measured plasma concentrations cs(t) unit Impulse response of the body r(t) input rate of drug C(tj) concentration at time ti Cg (ti) unit impulse response time t; r(ti) input rate at time t. c(t) interpolation for input response r input rate estimation tn measurement time grid t; optimal time grid N total number of experimentally measured data points M total number of optimal number of subintervals Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 Appendix Table A-t Simulated data for input function 1 (data set 1— 10) Time Noise- free #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1884 0.1873 0.1644 0.2339 0.1581 0.1930 0.1492 0.1882 0.1902 0.1650 0.2132 0.2 0.2993 0.2661 0.3509 0.2900 0.3364 0.3029 0.3027 0.3181 0.3079 0.3212 0.3286 0.3 0.3610 0.4384 0.3647 0.3678 0.3857 0.3749 0.3218 0.3666 0.3942 0.3632 0.4232 0.4 0.3911 0.3382 0.3597 0.4282 0.3606 0.3884 0.3302 0.4920 0.4111 0.3969 0.3750 0.5 0.4010 0.4193 0.3508 0.3799 0.4223 0.3778 0.4266 0.3486 0.4108 0.4650 0.4237 0.6 0.3980 0.4136 0.4473 0.3536 0.4830 0.4167 0.3819 0.4387 0.3423 0.3671 0.4275 0.8 0.3707 0.4476 0.4274 0.3117 0.3838 0.4189 0.3558 0.3995 0.4066 0.4282 0.3789 1 0.3308 0.3201 0.3896 0.3697 0.3385 0.3849 0.3336 0.3032 0.3835 0.3657 0.3742 1.2 0.2881 0.3304 0.3063 0.3021 0.3252 0.2679 0.2756 0.2712 0.2467 0.2833 0.3062 1.5 0.2279 0.2165 0.2298 0.2654 0.2276 0.2463 0.2151 0.2294 0.1929 0.2351 0.2027 1.8 0.1766 0.1803 0.2144 0.1686 0.1531 0.1585 0.1921 0.1764 0.1753 0.2017 0.1687 2.2 0.1232 0.1325 0.1388 0.1356 0.1164 0.1392 0.1132 0.1223 0.1242 0.1272 0.1143 2.7 0.0771 0.0698 0.0636 0.0929 0.0829 0.0772 0.0751 0.0651 0.0706 0.0808 0.0798 4 0.0217 0.0230 0.0217 0.0230 0.0197 0.0222 0.0228 0.0254 0.0205 0.0226 0.0206 Table A-2 Simulated data for input function 1 (data set 11— 20) Time Noise- free #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1884 0.2116 0.1864 0.2219 0.1937 0.2056 0.2087 0.2153 0.1941 0.1758 0.1945 0.2 0.2993 0.2918 0.2785 0.2712 0.3239 0.2952 0.3222 0.2588 0.2748 0.3071 0.2431 0.3 0.3610 0.3355 0.3470 0.3279 0.4053 0.4084 0.3134 0.3936 0.3433 0.3476 0.3856 0.4 0.3911 0.3679 0.3917 0.4058 0.3886 0.4037 0.3712 0.4123 0.4250 0.4427 0.4002 0.5 0.4010 0.3905 0.4559 0.4375 0.4269 0.3819 0.4233 0.3824 0.4155 0.3748 0.4507 0.6 0.3980 0.4475 0.4146 0.4041 0.3275 0.4010 0.3201 0.4947 0.3948 0.4005 0.4030 0.8 0.3707 0.3133 0.3733 0.3632 0.3729 0.3668 0.3425 0.4456 0.3985 0.3434 0.3774 1 0.3308 0.3180 0.3405 0.3801 0.3057 0.3777 0.2500 0.3572 0.2715 0.3249 0.3108 1.2 0.2881 0.2960 0.3017 0.2702 0.2394 0.3085 0.2691 0.2890 0.3231 0.3194 0.2582 1.5 0.2279 0.2467 0.2685 0.2464 0.2531 0.2363 0.2253 0.2402 0.2265 0.2310 0.2212 1.8 0.1766 0.1593 0.1812 0.2107 0.2024 0.1660 0.1819 0.1887 0.1697 0.1763 0.2023 2.2 0.1232 0.1219 0.1418 0.1281 0.1261 0.1127 0.1160 0.1159 0.1307 0.1116 0.1454 2.7 0.0771 0.0781 0.0795 0.0705 0.0686 0.0822 0.0534 0.0751 0.0821 0.0657 0.0668 4 0.0217 0.0218 0.0234 0.0270 0.0269 0.0249 0.0251 0.0250 0.0239 0.0225 0.0225 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 Table A-3 Simulated data for input function 1 (data set 21—30) Time Noise- free #21 #22 #23 #24 #25 #26 #27 #28 #29 #30 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1884 0.1905 0.1878 0.1760 0.1734 0.1974 0.1722 0.2070 0.1865 0.1852 0.2104 0.2 0.2993 0.3236 0.2879 0.3398 0.3200 0.2859 0.2625 0.3186 0.3292 0.2787 0.2821 0.3 0.3610 0.3246 0.3410 0.4381 0.3595 0.3750 0.4564 0.4664 0.3767 0.3252 0.3514 0.4 0.3911 0.3518 0.4276 0.3976 0.4038 0.3932 0.3557 0.3423 0.3901 0.3892 0.3913 0.5 0.4010 0.4124 0.3482 0.3326 0.4213 0.3815 0.3519 0.4073 0.3858 0.4298 0.3994 0.6 0.3980 0.4095 0.3860 0.4090 0.3571 0.4077 0.4101 0.4294 0.3884 0.3867 0.3779 0.8 0.3707 0.3615 0.2743 0.3853 0.3744 0.3973 0.3436 0.3493 0.3275 0.3179 0.3278 1 0.3308 0.3236 0.3566 0.3275 0.3270 0.3359 0.2930 0.3483 0.3558 0.3461 0.3543 1.2 0.2881 0.3140 0.3055 0.2932 0.3080 0.2920 0.2474 0.3363 0.3216 0.3196 0.3091 1.5 0.2279 0.2133 0.2494 0.1860 0.2708 0.2050 0.2144 0.2461 0.2046 0.2041 0.2477 1.8 0.1766 0.1734 0.1585 0.1501 0.1824 0.1997 0.1858 0.1922 0.2067 0.1264 0.2195 2.2 0.1232 0.1320 0.1224 0.1333 0.1220 0.1380 0.1048 0.1204 0.1171 0.1355 0.1246 2.7 0.0771 0.0794 0.0777 0.0753 0.0773 0.0572 0.0764 0.0794 0.0784 0.0777 0.0773 4 0.0217 0.0251 0.0179 0.0182 0.0204 0.0172 0.0217 0.0206 0.0225 0.0176 0.0238 Table^^^^im uIateddatoforingutfim cttonl^atas^l^lO ^ Noise- Time free #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2482 0.2257 0.2664 0.2825 0.2294 0.2434 0.2307 0.1971 0.2349 0.2861 0.2130 0.2 0.3962 0.4246 0.4182 0.4513 0.4024 0.4352 0.3667 0.3038 0.4837 0.4811 0.3598 0.3 0.4767 0.4764 0.4379 0.4551 0.4947 0.4990 0.4944 0.4590 0.4478 0.4411 0.4077 0.4 0.5112 0.3673 0.4837 0.4791 0.5203 0.4414 0.5847 0.5769 0.5846 0.4329 0.4355 0.5 0.5142 0.5065 0.5815 0.5668 0.4832 0.5283 0.5379 0.5428 0.5756 0.5032 0.5192 0.6 0.4959 0.5245 0.4720 0.4717 0.4466 0.6272 0.5287 0.4870 0.5328 0.5203 0.4847 0.8 0.4239 0.4558 0.3496 0.4068 0.4743 0.4216 0.4716 0.4224 0.4188 0.4102 0.4105 1 0.3366 0.3310 0.3054 0.3837 0.4170 0.3525 0.3696 0.4017 0.3356 0.3643 0.3579 1.2 0.2614 0.2462 0.2855 0.2687 0.3206 0.2070 0.2270 0.2955 0.2352 0.2244 0.2624 1.5 0.1865 0.1933 0.1873 0.2216 0.2294 0.1715 0.1861 0.1799 0.1671 0.2078 0.1862 1.8 0.1366 0.1291 0.1423 0.1257 0.1329 0.1304 0.1384 0.1321 0.1452 0.1500 0.1427 2.2 0.0912 0.0965 0.0966 0.0861 0.0958 0.0930 0.1132 0.0920 0.0991 0.0957 0.0864 2.7 0.0553 0.0445 0.0659 0.0648 0.0546 0.0602 0.0603 0.0577 0.0668 0.0623 0.0495 4 0.0151 0.0159 0.0162 0.0141 0.0151 0.0127 0.0152 0.0133 0.0142 0.0160 0.0150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 Table A-5 Simulated data for input function 2 (data set 11— 20) Time Noise- free #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2482 0.2204 0.2760 0.2937 0.2624 0.2053 0.2886 0.2464 0.2620 0.2483 0.2923 0.2 0.3962 0.4541 0.3685 0.3580 0.4285 0.4246 0.3867 0.3818 0.4263 0.3880 0.3776 0.3 0.4767 0.4962 0.4600 0.4578 0.4253 0.4785 0.4309 0.4397 0.4494 0.4791 0.4326 0.4 0.5112 0.5691 0.6041 0.5158 0.4816 0.5018 0.5034 0.4984 0.5704 0.5585 0.5113 0.5 0.5142 0.5435 0.4299 0.5278 0.4178 0.4723 0.6007 0.4927 0.4984 0.5322 0.5470 0.6 0.4959 0.4504 0.5096 0.5104 0.5414 0.5450 0.5730 0.5661 0.4961 0.4564 0.4181 0.8 0.4239 0.4960 0.3953 0.4570 0.4007 0.3484 0.3794 0.3880 0.3643 0.3975 0.4536 0.3366 0.3358 0.3435 0.3502 0.3162 0.3351 0.3399 0.3422 0.3913 0.3847 0.3694 1.2 0.2614 0.2413 0.2956 0.2529 0.2621 0.2847 0.2680 0.2601 0.2651 0.2069 0.2596 1.5 0.1865 0.1709 0.1951 0.1795 0.1914 0.1780 0.2284 0.1310 0.1526 0.1871 0.1859 1.8 0.1366 0.1354 0.1542 0.1326 0.1468 0.1106 0.1408 0.1504 0.0985 0.1364 0.1407 2.2 0.0912 0.0969 0.0825 0.0755 0.0897 0.0876 0.1094 0.0993 0.0936 0.0861 0.0852 2.7 0.0553 0.0574 0.0619 0.0550 0.0577 0.0492 0.0546 0.0531 0.0560 0.0489 0.0550 4 0.0151 0.0142 0.0145 0.0154 0.0164 0.0178 0.0147 0.0111 0.0156 0.0153 0.0163 Time Noise- free #21 #22 #23 #24 #25 #26 #27 #28 #29 #30 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2482 0.2427 0.2500 0.2407 0.2735 0.2189 0.3040 0.2361 0.2561 0.2474 0.2705 0.2 0.3962 0.4190 0.4390 0.3420 0.4655 0.3737 0.3956 0.3831 0.4748 0.3748 0.3475 0.3 0.4767 0.4848 0.4451 0.5119 0.5238 0.5328 0.5118 0.5137 0.4460 0.4735 0.4827 0.4 0.5112 0.4859 0.5355 0.4905 0.4897 0.5227 0.4784 0.4942 0.5272 0.6265 0.5525 0.5 0.5142 0.5760 0.5782 0.5583 0.4970 0.5532 0.5885 0.6220 0.5961 0.5251 0.4594 0.6 0.4959 0.4903 0.4974 0.6617 0.4534 0.4538 0.4384 0.5152 0.4476 0.5052 0.3963 0.8 0.4239 0.4478 0.4532 0.4505 0.3912 0.5027 0.4530 0.3962 0.3753 0.4360 0.4342 1 0.3366 0.3356 0.3135 0.3797 0.3859 0.3354 0.3612 0.3347 0.2866 0.2851 0.3416 1.2 0.2614 0.2268 0.2603 0.2420 0.2899 0.3291 0.2749 0.2725 0.2362 0.1994 0.3296 1.5 0.1865 0.1663 0.1576 0.2058 0.1825 0.1736 0.2181 0.1851 0.1858 0.1486 0.1855 1.8 0.1366 0.1388 0.1220 0.1282 0.1560 0.1151 0.1274 0.1474 0.1076 0.1533 0.1364 2.2 0.0912 0.0778 0.0892 0.0951 0.0969 0.0856 0.1223 0.0900 0.0745 0.0879 0.0865 2.7 0.0553 0.0590 0.0460 0.0602 0.0676 0.0574 0.0531 0.0588 0.0468 0.0518 0.0620 4 0.0151 0.0156 0.0125 0.0147 0.0131 0.0170 0.0169 0.0175 0.0147 0.0148 0.0153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 T^ible^7< iii S ^ ^ a t e d d a t a J b r m £ u t ^ ^ ^ n 3 f ^ ^ s e t l— 10) Time Noise- free #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1253 0.1310 0.1110 0.1389 0.1451 0.1076 0.1338 0.1484 0.1106 0.1069 0.1238 0.2 0.3369 0.3729 0.4059 0.3311 0.3269 0.3013 0.3388 0.3466 0.2960 0.3102 0.3410 0.3 0.5077 0.5012 0.5081 0.4666 0.4899 0.5974 0.5749 0.4940 0.6101 0.4134 0.4930 0.4 0.6100 0.6215 0.6556 0.6097 0.6632 0.7386 0.5958 0.5989 0.7750 0.5938 0.5761 0.5 0.6518 0.6223 0.6870 0.5760 0.7053 0.5852 0.6854 0.6731 0.7115 0.5916 0.6689 0.6 0.6506 0.6103 0.6056 0.6085 0.6895 0.6433 0.6082 0.6502 0.6136 0.6889 0.6879 0.8 0.5807 0.4689 0.6437 0.6720 0.5830 0.5844 0.5815 0.5844 0.6150 0.5401 0.5687 1 0.4836 0.4147 0.5140 0.4899 0.4291 0.4624 0.5423 0.4755 0.4929 0.5119 0.3535 1.2 0.3930 0.4035 0.3633 0.3614 0.4198 0.4651 0.4485 0.3552 0.3117 0.3384 0.4777 1.5 0.2857 0.2753 0.2925 0.2828 0.2779 0.3017 0.2882 0.2849 0.2877 0.2311 0.2584 1.8 0.2087 0.1882 0.1836 0.1948 0.2121 0.1958 0.2618 0.1765 0.2062 0.1880 0.2488 2.2 0.1387 0.1576 0.1344 0.1289 0.1321 0.1214 0.1375 0.1365 0.1450 0.1472 0.1528 2.7 0.0838 0.0911 0.0709 0.0820 0.0876 0.0846 0.0946 0.0846 0.0846 0.0934 0.0614 4 0.0228 0.0218 0.0214 0.0218 0.0222 0.0230 0.0234 0.0236 0.0148 0.0214 0.0228 Time Noise- free #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1253 0.1265 0.1282 0.1332 0.1024 0.0969 0.1146 0.1380 0.1401 0.1262 0.1199 0.2 0.3369 0.3353 0.3399 0.3608 0.3031 0.3588 0.3376 0.3918 0.3441 0.2840 0.3401 0.3 0.5077 0.4877 0.5538 0.5185 0.4384 0.5240 0.4770 0.4995 0.4840 0.4162 0.5361 0.4 0.6100 0.5357 0.4963 0.7278 0.5638 0.5747 0.5906 0.6325 0.5674 0.6885 0.6112 0.5 0.6518 0.6969 0.6885 0.7477 0.6270 0.6912 0.6895 0.7403 0.6819 0.5905 0.7096 0.6 0.6506 0.6606 0.6818 0.5993 0.6053 0.6751 0.6906 0.6178 0.7119 0.6696 0.6342 0.8 0.5807 0.6176 0.6651 0.6585 0.4849 0.6143 0.5055 0.5399 0.6050 0.6349 0.6544 1 0.4836 0.5481 0.4817 0.5101 0.4660 0.4165 0.5297 0.5136 0.5109 0.5473 0.4339 1.2 0.3930 0.3443 0.4000 0.4686 0.4262 0.3682 0.3687 0.3891 0.3630 0.4061 0.3976 1.5 0.2857 0.2933 0.2861 0.3032 0.2763 0.2675 0.2511 0.2742 0.2612 0.3003 0.2471 1.8 0.2087 0.1832 0.1892 0.2097 0.1856 0.1885 0.1539 0.2007 0.1614 0.2098 0.1920 2.2 0.1387 0.1347 0.1451 0.1448 0.1406 0.1372 0.1196 0.1405 0.1617 0.1410 0.1253 2.7 0.0838 0.0847 0.0714 0.1014 0.0813 0.0894 0.0840 0.0930 0.0755 0.0801 0.0877 4 0.0228 0.0196 0.0208 0.0229 0.0232 0.0254 0.0212 0.0239 0.0236 0.0216 0.0221 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 Table A-9 Simulated data for input function 3 (data set 21— 30) Time Noise- free #21 #22 #23 #24 #25 #26 #27 #28 #29 #30 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1253 0.1263 0.1110 0.1015 0.1250 0.1240 0.1243 0.1189 0.1308 0.1132 0.1161 0.2 0.3369 0.2984 0.2965 0.3241 0.3634 0.4074 0.3494 0.3032 0.3359 0.2926 0.3383 0.3 0.5077 0.4973 0.4515 0.4672 0.6195 0.5499 0.5667 0.5259 0.5050 0.5543 0.5266 0.4 0.6100 0.6342 0.6740 0.5374 0.6946 0.6007 0.6414 0.5836 0.5213 0.5498 0.4970 0.5 0.6518 0.6537 0.7016 0.6967 0.7261 0.7117 0.6493 0.6728 0.6055 0.6809 0.6638 0.6 0.6506 0.6566 0.6019 0.6002 0.5812 0.6578 0.7744 0.7040 0.5757 0.6477 0.6596 0.8 0.5807 0.5315 0.6015 0.5931 0.6254 0.5447 0.6316 0.4720 0.5869 0.7046 0.5771 1 0.4836 0.4007 0.4208 0.5037 0.5412 0.4650 0.4616 0.4044 0.4676 0.3855 0.4751 1.2 0.3930 0.3855 0.3868 0.3813 0.3864 0.3865 0.4115 0.4439 0.3742 0.4146 0.3730 1.5 0.2857 0.3069 0.3466 0.2967 0.3059 0.3017 0.2811 0.2703 0.2679 0.3026 0.3310 1.8 0.2087 0.2090 0.2046 0.2430 0.1767 0.2147 0.2675 0.1890 0.2023 0.2241 0.2433 2.2 0.1387 0.1457 0.1478 0.1184 0.1508 0.1458 0.1279 0.1332 0.1259 0.1393 0.1269 2.7 0.0838 0.0833 0.0869 0.0787 0.0687 0.0894 0.0977 0.0852 0.0807 0.0830 0.0870 4 0.0228 0.0242 0.0223 0.0216 0.0192 0.0179 0.0213 0.0200 0.0195 0.0254 0.0233 Table A-10 Sim ulated data for input function 4 (data set 1—10) Noise- Time free #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1213 0.1011 0.1342 0.1297 0.1240 0.1174 0.1227 0.1224 0.1216 0.1293 0.1351 0.2 0.3592 0.3637 0.3613 0.3885 0.3261 0.3985 0.3705 0.3364 0.3231 0.3173 0.3346 0.3 0.5534 0.5693 0.5481 0.5928 0.4333 0.4497 0.6333 0.5224 0.5010 0.5279 0.4819 0.4 0.6424 0.5688 0.5889 0.7253 0.6386 0.6699 0.6199 0.6709 0.6183 0.6255 0.6377 0.5 0.6458 0.7227 0.6648 0.6890 0.5805 0.7036 0.6860 0.5845 0.5692 0.5675 0.6244 0.6 0.6101 0.6827 0.5286 0.6828 0.6476 0.6547 0.6588 0.6578 0.5457 0.5296 0.5586 0.8 0.5520 0.5499 0.5914 0.4856 0.5800 0.5839 0.6039 0.5834 0.6333 0.6034 0.5795 1 0.5683 0.5869 0.6606 0.5672 0.6645 0.5706 0.5119 0.5216 0.5715 0.5689 0.6529 1.2 0.6100 0.6207 0.5678 0.6004 0.6461 0.6513 0.6229 0.5938 0.5357 0.5706 0.5767 1.5 0.5708 0.5601 0.6198 0.4792 0.5341 0.6033 0.5844 0.5030 0.5684 0.6168 0.5225 1.8 0.4215 0.4521 0.4744 0.4323 0.4375 0.4107 0.3790 0.3287 0.3739 0.4313 0.4111 2.2 0.2661 0.2504 0.2237 0.2380 0.2392 0.2561 0.2464 0.2923 0.2302 0.2398 0.2837 2.7 0.1586 0.1932 0.1357 0.1810 0.1583 0.1539 0.1758 0.1504 0.1545 0.1798 0.1451 4 0.0431 0.0425 0.0456 0.0396 0.0429 0.0367 0.0425 0.0445 0.0472 0.0443 0.0379 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 Table A - ll Simulated data for input function 4 (data set 11— 20) Time Noise- free #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1213 0.1132 0.1252 0.1226 0.1209 0.1133 0.1116 0.1271 0.1109 0.1333 0.1201 0.2 0.3592 0.3685 0.2918 0.3884 0.3455 0.4077 0.3840 0.3431 0.3150 0.3823 0.3950 0.3 0.5534 0.5328 0.5911 0.4976 0.5227 0.6716 0.5511 0.5748 0.6996 0.7150 0.5774 0.4 0.6424 0.7271 0.6574 0.5779 0.7023 0.6531 0.6632 0.6458 0.5843 0.5622 0.6407 0.5 0.6458 0.6036 0.7258 0.6641 0.5608 0.5357 0.6785 0.6144 0.5668 0.6559 0.6213 0.6 0.6101 0.6139 0.6178 0.6278 0.5917 0.6270 0.5473 0.6250 0.6286 0.6582 0.5953 0.8 0.5520 0.5114 0.5619 0.5383 0.4085 0.5738 0.5575 0.5917 0.5117 0.5201 0.4877 0.5683 0.5581 0.5339 0.5559 0.6126 0.5627 0.5617 0.5770 0.5033 0.5983 0.6113 1.2 0.6100 0.6762 0.5468 0.6648 0.6468 0.6208 0.6520 0.6182 0.5238 0.7120 0.6810 1.5 0.5708 0.5786 0.5539 0.5341 0.6246 0.4659 0.6783 0.5134 0.5370 0.6165 0.5124 1.8 0.4215 0.4208 0.4829 0.4139 0.3783 0.3582 0.4352 0.4766 0.4434 0.4588 0.4932 2.2 0.2661 0.2411 0.3141 0.2852 0.2643 0.2879 0.2636 0.2980 0.2264 0.2601 0.2530 2.7 0.1586 0.1352 0.1374 0.1634 0.1599 0.1549 0.1591 0.1177 0.1572 0.1633 0.1613 4 0.0431 0.0447 0.0448 0.0498 0.0355 0.0361 0.0405 0.0341 0.0430 0.0409 0.0446 Table A-12 Simulated data for input function 4 (data set 21— 30) Time Noise- free #21 #22 #23 #24 #25 #26 #27 #28 #29 #30 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1213 0.1110 0.1336 0.1356 0.1221 0.1161 0.1224 0.1241 0.1289 0.0991 0.0938 0.2 0.3592 0.3600 0.4178 0.3669 0.3028 0.3627 0.3575 0.3624 0.3847 0.3232 0.3826 0.3 0.5534 0.5200 0.5445 0.5276 0.4537 0.5843 0.5316 0.6037 0.5652 0.4778 0.5711 0.4 0.6424 0.6220 0.6660 0.5975 0.7250 0.6437 0.5641 0.5226 0.7664 0.5937 0.6052 0.5 0.6458 0.6831 0.7335 0.6756 0.5850 0.7031 0.6905 0.6822 0.7408 0.6213 0.6849 0.6 0.6101 0.6476 0.5793 0.6676 0.6279 0.5947 0.6194 0.6394 0.5620 0.5676 0.6331 0.8 0.5520 0.4805 0.5132 0.5751 0.6035 0.6221 0.5871 0.6323 0.6259 0.4610 0.5839 1 0.5683 0.6224 0.6036 0.6003 0.6431 0.5099 0.6441 0.5660 0.5995 0.5476 0.4895 1.2 0.6100 0.5723 0.6039 0.5634 0.6303 0.6171 0.5343 0.6209 0.7274 0.6616 0.5715 1.5 0.5708 0.5017 0.5478 0.5218 0.6000 0.4938 0.5861 0.5715 0.6058 0.5519 0.5343 1.8 0.4215 0.3108 0.4054 0.3259 0.4236 0.3877 0.3700 0.3822 0.4235 0.3747 0.3806 2.2 0.2661 0.2295 0.2695 0.3102 0.2706 0.2404 0.2585 0.2784 0.2778 0.2697 0.2631 2.7 0.1586 0.1590 0.1759 0.1428 0.1515 0.1660 0.1603 0.1352 0.1918 0.1539 0.1692 4 0.0431 0.0400 0.0451 0.0446 0.0408 0.0417 0.0371 0.0394 0.0433 0.0438 0.0481 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 Time Noise- free #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.0206 0.0202 0.0184 0.0194 0.0161 0.0211 0.0198 0.0198 0.0222 0.0205 0.0192 0.2 0.0687 0.0790 0.0668 0.0674 0.0787 0.0733 0.0528 0.0645 0.0610 0.0682 0.0728 0.3 0.1276 0.1144 0.1181 0.1070 0.1256 0.1426 0.1180 0.1341 0.1241 0.1255 0.1439 0.4 0.1850 0.1647 0.1797 0.1875 0.1907 0.1728 0.1824 0.2039 0.1797 0.2085 0.1953 0.5 0.2334 0.2546 0.2525 0.2299 0.2571 0.2229 0.2226 0.2252 0.2424 0.2573 0.2435 0.6 0.2737 0.2946 0.2907 0.2020 0.2849 0.2710 0.2947 0.3124 0.2780 0.2705 0.2696 0.8 0.3378 0.3405 0.3186 0.2516 0.3764 0.3610 0.3200 0.3487 0.3405 0.3554 0.3713 0.3872 0.4196 0.3409 0.4156 0.3542 0.4284 0.4120 0.4267 0.4630 0.4474 0.4096 1.2 0.4264 0.5001 0.5163 0.3936 0.3781 0.4431 0.4086 0.4039 0.5100 0.4170 0.3351 1.5 0.4717 0.4463 0.4860 0.4638 0.4878 0.4691 0.4198 0.4974 0.5775 0.3366 0.4883 1.8 0.4901 0.4483 0.4168 0.4332 0.4925 0.5465 0.5451 0.4017 0.5459 0.5424 0.5139 2.2 0.3166 0.3899 0.3350 0.3128 0.2865 0.3039 0.3291 0.3861 0.2885 0.2202 0.3012 2.7 0.1811 0.1716 0.1827 0.1630 0.1799 0.2070 0.1627 0.1773 0.1494 0.1845 0.1800 4 0.0489 0.0374 0.0477 0.0561 0.0407 0.0431 0.0502 0.0524 0.0452 0.0485 0.0456 Table A-14 Simulated data for input function 5 (data set 11— 20) Time Noise- free #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.0206 0.0209 0.0194 0.0185 0.0200 0.0205 0.0233 0.0201 0.0221 0.0224 0.0201 0.2 0.0687 0.0643 0.0706 0.0704 0.0632 0.0631 0.0826 0.0646 0.0774 0.0709 0.0754 0.3 0.1276 0.1426 0.1028 0.1208 0.1315 0.1286 0.1240 0.1571 0.1004 0.1157 0.1286 0.4 0.1850 0.1943 0.1604 0.2038 0.1821 0.2005 0.1413 0.1808 0.1472 0.1648 0.1752 0.5 0.2334 0.2060 0.2512 0.2280 0.2433 0.2159 0.2175 0.2170 0.2190 0.1972 0.2071 0.6 0.2737 0.2396 0.2438 0.2901 0.2947 0.3061 0.3243 0.2946 0.2789 0.2390 0.2980 0.8 0.3378 0.3592 0.3307 0.4008 0.3403 0.3258 0.3267 0.3128 0.2998 0.3705 0.3583 1 0.3872 0.3194 0.3767 0.3130 0.3567 0.3196 0.3915 0.3452 0.3680 0.3856 0.2890 1.2 0.4264 0.4283 0.4483 0.3746 0.4757 0.4157 0.4175 0.4676 0.4616 0.4392 0.4289 1.5 0.4717 0.4353 0.4226 0.5326 0.3930 0.5412 0.4854 0.5313 0.4880 0.4494 0.4531 1.8 0.4901 0.5995 0.5240 0.4323 0.5110 0.5998 0.4386 0.5081 0.4591 0.4470 0.5034 2.2 0.3166 0.3136 0.2811 0.3144 0.3307 0.3084 0.3327 0.3095 0.3017 0.3113 0.3338 2.7 0.1811 0.2017 0.1560 0.1854 0.1883 0.2027 0.1951 0.1763 0.1877 0.1759 0.1670 4 0.0489 0.0457 0.0485 0.0530 0.0489 0.0446 0.0435 0.0455 0.0610 0.0492 0.0471 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Table A-15 Simulated data for input function 5 (data set 21— 30) Time Noise- free #21 #22 #23 #24 #25 #26 #27 #28 #29 #30 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.0206 0.0187 0.0207 0.0202 0.0213 0.0185 0.0227 0.0172 0.0218 0.0189 0.0208 0.2 0.0687 0.0748 0.0743 0.0665 0.0633 0.0664 0.0702 0.0694 0.0697 0.0649 0.0817 0.3 0.1276 0.1342 0.1454 0.1242 0.1234 0.1265 0.1159 0.1396 0.1091 0.1182 0.1361 0.4 0.1850 0.2322 0.1922 0.1889 0.1910 0.2011 0.2013 0.1764 0.1529 0.1623 0.1910 0.5 0.2334 0.2530 0.2427 0.2032 0.2710 0.2488 0.2035 0.2851 0.2285 0.2081 0.2232 0.6 0.2737 0.2829 0.3366 0.2701 0.2824 0.2663 0.2633 0.2695 0.3062 0.3115 0.2867 0.8 0.3378 0.3225 0.3622 0.3110 0.2702 0.3405 0.3558 0.3653 0.3444 0.3353 0.3302 1 0.3872 0.3787 0.3551 0.4163 0.4792 0.3851 0.3873 0.3947 0.3965 0.3996 0.2987 1.2 0.4264 0.4330 0.4235 0.4172 0.4293 0.4273 0.4943 0.5025 0.4521 0.4396 0.3997 1.5 0.4717 0.4290 0.4743 0.4873 0.4562 0.4607 0.4394 0.4613 0.5000 0.4290 0.4796 1.8 0.4901 0.4401 0.4930 0.4940 0.5223 0.5208 0.4933 0.4983 0.5284 0.5262 0.4891 2.2 0.3166 0.3028 0.3274 0.3098 0.3660 0.3543 0.2822 0.3422 0.2776 0.2960 0.2847 2.7 0.1811 0.2053 0.1516 0.1927 0.1805 0.1751 0.1471 0.2064 0.1678 0.1957 0.1798 4 0.0489 0.0541 0.0516 0.0544 0.0558 0.0528 0.0511 0.0456 0.0500 0.0518 0.0519 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Asset Metadata

Creator Hsu, Joy C. (author)

Core Title Comparisons of deconvolution algorithms in pharmacokinetic analysis

School Graduate School

Degree Master of Science

Degree Program Biomedical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, biomedical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor D'Argenio, David (committee chair), Khoo, Michael C.K. (committee member), Marmarelis, Vasilis (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-311073

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Document Type Thesis

Rights Hsu, Joy C.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

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