Software for experimental design: the computer program EXCAD

Computer Methods and Programs in Biomedicine 26 (1988) 159-166 Elsevier

159

CPB 00877 Section II. Systems and programs

Software for experimental design: the computer program EXCAD D. V e r o t t a Biostatistics Unit, Istituto di Ricerche Farmacologiche 'Mario Negri; Milan, Italy

A computer program (EXCAD) dedicated to the optimization of experimental designs to estimate parameters of a mathematical model, is presented. EXCAD computes D-optimal designs and sequentially augmented designs. D-optimal designs minimize the determinant of the variance-covariance matrix and parameters, thus obtaining the average most accurate estimate of parameters. D-optimal designs have generally as many support points as the number of parameters in the mathematical model, so sample scheduling is minimal, not extensive. Augmented designs add to an original design the point that maximizes the decrement of the determinant of the variance-covariance matrix. The general model, linearly or not linearly parametrized Y= F(X,P), that relates two independent variables and P parameters to different responses may be written in the program, while a set of prewritten models is provided. Computer-aided design; D-optimal design; Sequential optimal design; Minicomputer

1. Introduction A mathematical regression model relates one or more observations (Y) (in the following Y will be considered to be a scalar value) to a number of predictor variables ( X ) and model parameters (P), by means of a function (F). F may be seen as a rule to calculate values of Y from different values of P and X: Y= F(X, P)

(1)

Whatever the aim of the regression model is, two problems arise related to P and X. With P the problem is to estimate model parameters given X and Y. With X the design problem consists of selecting n values X~, from a region where it is experimentally feasible to make observations. To optimize the experimental design means choosing X~ points to improve the accuracy in estimating parameters P and facilitate data analysis, or reCorrespondence: D. Verotta, Department of Laboratory Medicine, School of Medicine C-255, University of California, San Francisco, CA 94143, U.S.A.

duce costs and work. Given a particular criterion for comparing experimental designs, based on the Eq. 1, the optimization problem can be solved by an algorithm for optimizing the criterion over the set of possible experimental designs. There is a demand for software for experimental designs [1] both because of the practical impossibility of solving the optimization design problem analytically and because of the benefit that interactive computer programs will lead to better experiments set up from statistical designs. The aim of a program to compute experimental design is outlined in Fig. 1, which sets out the i n p u t / o u t p u t relationships between the experiment, the parameter-estimation program and the design-computing program. In a previous paper [2] we presented a program to compute optimized design for model discrimination. In this paper we present the computer program EXCAD, which is specially dedicated to the optimization of experimental design for selected mathematical models. E X C A D offers an instrument to compute optimized designs, and to study and define a user-chosen experimental design. From the many optimized designs proposed in the literature [3] to estimate parameters,

0169-2607/88/$03.50 © 1988 Elsevier Science Publishers B.V. (Biomedical Division)

160 .... osE A,o. . . . . . .VARIABLE RELATE(X) S. . .AND . ~ER VED..... ~LE,Y, MODEL PARAMETERS {P) ........

TO SOME INDEPENDENT

,I fl ....

YO = F(X, J') + e

1 ~ [ EXP~,~,~NT ] ~

/

VAR(E)=a+b. FC+d.F e

[o]D~VAT~ON VARIANCE / l /

l/

OS~E~ ~R~ANEE ~

.....

[~

(21

E here, and in the program, is the variance model, while F is the structural model [6]. E is assumed to have mean value of zero and variance (VAR(E)) given by the following polynomial:

~oo~, [ o~e.... ~ON~ I

tout, I PARAMETER E. . . . . . .

model that takes account of errors so that the response observed (Yo) becomes:

~"E~OL~------~]

Fig. 1.

only two are implemented in the EXCAD program. The first is D-optimal design. A D-optimal design offers the average most accurate estimate of model parameters, and is also a minimal sample scheduling design: generally a D-optimal design has as many support points as the number of parameters in the mathematical model. Farreaching research has been done on D-optimal design [3-5]. The second design is a sequential optimized design. This finds the best point to add to a predefined design (see Section 2.2). These two designs help to define better experimental designs than empirical ones. The EXCAD program runs on a DEC VAX 750 and handles mathematical models mapping from R 2 to R ~ (i.e. two independent variables and n different sets of observations).

(3)

The variance of E is assumed to be dependent on F (and thus is also calculated from X and P). Consider now the covariance matrix of parameters, particularly its Cramer-Rao lower bound, the inverse of Fischer information matrix (J): J = (St. W - S )

(4)

where S is an n × p (sensitivity) matrix with row s ( X k) containing elements of the form D[F(Xk)/Pj] (F(Xk) is the model equation evaluated for the kth sample point, Pj is the j t h parameter of the model, D[a/b] is the symbol of the derivative of a in relation to b, t is the symbol of matrix transpose). W is the (diagonal) (n × n)dimensional weighting matrix and has elements (Wk) inversely proportional to the variances of the responses. (The symbol x is also adopted, instead of Xk, to indicate one point in XR; F(x), s(x) and W(x) are model equation, sensitivity vector and weight evaluated for this point.) Elements of J have the form:

J u = S U M ( k = 1, n) X(D[

F( Xk)/Pj]" D[ F( Xk)/Pg]" Wk)

(5) 2. Theory Eq. 1 relates the response observed at n design points to some independent variables X, constrained in an observation interval X R, and a set of model parameters P (P = t'1. . . . . Pp). Consider now the fact that responses have some degree of uncertainty. Eq. 1 becomes part of a more detailed

For a linear parametrized model J does not depend on the model parameter values (as S does not), while for non-linearly parametrized models J does depend. In this last case every optimal design criteria based on J requires a preliminary estimate of the nominal values of parameters. This estimate may be obtained from preliminary experiments,

161 literature, or independent measurements [7]. The predicted responses (Yc) for a point Xi that belongs to X R is given by Eq. 1. In accordance with Fedorov [3], we introduced the following notation for the weighted variance in correspondence with the sample point X~:

A3.2. In design H replace point X/ with point Xe that satisfies the condition, and update J ( H ) . A4. Repeat step 3 until A is sufficiently close to zero and a try is done on each support point (Xi) unsuccessfully.

d(X,) = ( s ( X ~ ) . j - 1 .s(X~)t). W~

Fedorov also presented a numberical procedure for updating J and J-1 that considerably reduces the computing time needed to calculate the D-optimal design. When the region X R of interest is continuous, points X e maximizing A may be found by some sort of procedure. We adopted a gridsearch procedure because: (1) the search procedure can be programmed very easily, (2) a search on a list of candidate points or an exhaustive search, or both, can be made, and (3) computing time can be cut because the elements needed by the algorithm can be stored without calculating them after each iteration [9]. It must be noted that this exchange exact algorithm, like all others, is not guaranteed to give a D-optimal n-point design [10]. At present, above all for complicated models, D-optimal n-points can only be guaranteed by an exhaustive search: the user is advised to make more than one try starting from (random) different designs. We also suggest that, to always find the minimal sample D-optimal designs, the numbet of points (n) must be set equal to the number of model parameters (p).

(6)

2.1. D-optimal design The D-optimal criterion minimizes the determinant (det) of the variance-covariance matrix which tends to produce the most precise simultaneous estimates of all parameters. The algorithmic problem is to find the n-points design with maximum value of det(J) (the minimum value of the determinant of the variance-covariance matrix). A number of different procedures have been proposed in order to achieve this. We adopted Fedorov's modified algorithm to compute D-optimal designs, as it saves computing time while producing high-efficiency D-optimal designs cornpared to the original Fedorov algorithm [8]. Suppose we have an arbitrary n-point design, and we wish to find a D-optimal design. Fedorov suggested replacing one of the points in the design, say X/, with a point from the design space X R, say Xe, such that the increase in the value of det(J) is maximum. Fedorov demonstrated ([3] p. 175) that the delta value (A) of the determinant of the information matrix J for the exchange is given by: A = d(Xe) - d(X/)

-(d(Xe).d(Xi)-(d(Xe,

X~))2)

(7)

where d ( X e, Xi) 2 = (s(Xe)" j - a . s(Xi)t)2. We" W~. The algorithm (A) to compute a n-point Doptimal design, is as follows: A1. Begin with an arbitrary, non-degenerate, npoint design H. A2. Calculate J ( H ) and J - I ( H ) A3. For i equals 1 to n: A3.1. Fix X~ and find Xe which obtains the maximum value of A.

2.2. Sequential design Sequential design experiments are often used in scientific research. In the case of mathematical modelling, a sequential design may be viewed as a sequence of steps (see Fig. 1), where experiments are repeated until the research goal is reached. D-optimal design can be inserted in this sequence. An alternative optimized design determines (after each experiment) the point (x) in X R that, added to the original design, results in maximum increase of det(J) [9]. Given an original, arbitrary, non degenerate, n-point design H, the algorithm (B) to find x is as follows: B1. Calculate J ( H ) and J - l ( H ) . B2. Find x in X R such that d(x) is maximum (update H and J ( H ) by adding the point x).

162 Similarly we can proceed when more than one point at a time is collected. If we want to take m other observations, we simply add m points to the original design, iterating algorithm B m times. The resulting design is no longer D-optimal but cornputation takes much less time on the computer.

EXCA0: ,~:p.ri,eot~l co,,poter ~ided desigo mo,Jel De,i~n ~As~c ~NPUTS

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<-::~ ,fS>

Exit SELECT



MODEL DATA X HELP START NEXT

,~H~> < .> ,:C ~:.

3. The program

~p~o9 ..... model p~rameter/s var iance r ~r,9e/s

{progr

am)

{menu}

3.1. Interface Input

To facilitate the use of the program a series of menus is provided. The first menu (see Fig. 2) gives the user access to the basic inputs (mathematical model (see Fig. 3), number of parameters, variance and structural model parameter values, independent variable range (XR)). Then a design is set (logarithmic equispaced, arithmetically equispaced, random from a uniform distribution, user-chosen). This design can be inspected by itself; otherwise it represents a first estimate from which algorithm A finds the D-optimal design, or the design to which a number of points are added by means of algorithm B. The last menu (Fig. 4) offers various options for inspecting a design: (1) Design print: print out the running experimental design, (2) Varcovar matrix: print out the approximate asymptotic variance-covariance matrix of model parameters and the correlation matrix of parameters. (3) Model parameters; running model parameter values, the a priori estimate of the approximate asymptotic standard deviation, coefficient of variation (CV) for each parameter. This option is very useful in practice as an easy-to-read index to determine the appropriateness of the design [11]. (4) Informational analysis: for every point x in the design the difference is computed between the information content of the full design (I, the logarithm (log2) of det(J)) and that of the design minus point x (i.e., I - log 2 (s(x). IV(x)). These values are of interest to establish points in a schedule with low information content [12]. (5) Graph point/plot: gives a graph of the model with scheduling points. (6) Weighted variance plot: gives a graph of the


the

selected

option,

ple~!,e

Fig. 2.

weighted variance (see Eq. 6). D-optimal design points have the maximum value of the weighted variance. From a descriptive point of view the D-optimal criterion tells where to collect samples to make the greatest contribution to the weighted variance. (7) Plot sensitivity: gives a graph of the (weighted or unweighted) sensitivity equations (see Eq. 5). Sensitivity equations are frequently used to aid in model building and particularly in parameter identifiability analysis [13]. The weighted sensitivity (plot) is of interest for designing experiments. For a single-parameter model or when only one parameter in the model is of interest, the optimal (single) point maximizes the weighted sensitivity equation of the parameter.

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OPTIONS FOR INSPECTING A DESIGN

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Table 1 reports D-optimal and sequentially optimized designs obtained with the program and D-optimal design obtained by analytic solutions.

W IGH ETRIIaLOT............ ser, slt ivltyp~O'

The following structural models are reported: s I : s t r a i g h t l i n e : Y = P t + P2" X

~n0,~t .~P~o~ I.ETr~R in ~. . . . ~o~t~,~ GOt..... ~ ......

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S2: M i c h a e l i s - M e n t e n :

Fig. 4.

= Vmax' P2 = K m ) Ss: l i g a n d - b i n d i n g i s o t h e r m :

Y = (a i - SQR(a 2 +

a 2 ) ) / 2 / P 2 ( a t = X . P2 + I + Pt " P2, a 2 = - - 4" X . P22 . PI ; Pl = Bmax, P2 = Ka)

3.2. Program structure T h e p r o g r a m is o r g a n i z e d i n a n u m b e r o f F O R TRAN subroutines and functions. The mathematical models are written in the subroutine

ss:

FUNCT.

exp(- X. P2))

The FORTRAN

(Pa

code to select a model

d u r i n g t h e e x e c u t i o n o f t h e p r o g r a m is w r i t t e n i n the subroutine SELECT MODEL. To write user-supplied models needs knowledge of basic FORTRAN. I t is a l s o p o s s i b l e t o w r i t e a p e r s o n a l library of models (FUNCT) and a corresponding menu (SELECT MODEL). A fixed batch output is p r o v i d e d ,

s4: m o n o - e x p o n e n t i a l mono-exponential

accumulation:

P~

(1-

F o r e v e r y m o d e l p a r a m e t e r v a l u e s a r e P I = 1, P2 -- 1. X R is t h e c l o s e d i n t e r v a l 0.1 - 10.0. O n l y t w o variance models are considered (constant variance and variance proportional to the squared value of the observation). The (two-point) starting design ( x t = 0.026, x2 = 7.27) w a s e x t r a c t e d f r o m a u n i f o r m d i s t r i b u t i o n o n X R. A l g o r i t h m A c h a n g e s t h e

M a t h e m a t l c a l model -

TABLE I

decay: V= PI" exp(-X-P2)

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.............................................................. SEARCH FOR A D-OPTIMAL DESIGN.

Theoretical D-optimal, computed D-optimal and sequential

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ITERATION Suppott point New point Determinant %dec~'ement ............................................................. i

sI

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1

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1

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1

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1

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=

................................................... CONVERGENCE

For all models parameter values are PI =1, /°2 =1, X R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1-10.0. The starting design is x 1 = 0.026, x 2 = 7.270. Fig. 5.

0.00

164

starting design until no further i m p r o v e m e n t in det(J) is reached. Algorithm B adds two new points to the starting design so that the increase of det(J) is m a x i m u m . D - o p t i m a l design for all the reported models is constituted by only two points (see [14] for model sl, [15] for models s2-s3, [16] for models s4-s5). Table 1 shows that D - o p t i m a l designs obtained b y the p r o g r a m agree well with the analytical ones, any discrepancy apparently depending only on the grid density.

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4.2. A simple model used in a real experimental situation

Fig. 7.

A simple model was adopted to describe the binding of the enkephaline analog [ 3 H ] D A G O , which is reported to bind at least one specific site as well as non-specific ones [17]. The model is:

[Bt] = (a 1 - S Q R ( a 2 + a 2 ) ) / ( 2 ( k + 1 ) ) / K a (8) where:

al = [T](Ek + l ) K a + l + Ka" Bm~x + k, a2=-4(k+

1)[T]

×Ka([T].k.Ka+k+Ka.

...... ~.~ ......

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O. 1250000E÷01 O. 141&IE+O0 O. 5654000E-O 1 0. 48675E-02 O. 4750000E-02 0 • 2&O33E-03 ........................................................

k

[Bt] is the total concentration (nmole) of radioligand bound, [T] is the concentration (nmole) of radioligand added, K~ is the association constant ( 1 / n m o l e ) , Bm~x the m a x i m u m b o u n d to the specific sites (nmole), k is the non-specific constant (adimensional). N o m i n a l p a r a m e t e r values were observed to be: Bm~x = 0.0565, K~ = 1.250, k = 0.00475, the variance model was VAR([Bt]) = 0.002 [Bt] 2. T h e analytic solution to obtain D - o p timal designs for this model is unknown. Fig. 5 reports details of the search for the D-optimal scheduling. Fig. 6 shows the v a r i a n c e - c o v a r i a n c e matrix, and p a r a m e t e r statistics. Figs. 7 and 8 report respectively the scheduling plot, and the plot of d ( X ) corresponding to the D-optimized v a r i a n c e - c o v a r i a n c e matrix. A simulation study on this mathematical model is reported, to show the ability of D - o p t i m a l design to find nominal

I I • 33 8,608 5. 480

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165

Acknowledgements

TABLE 2 Parameters estimated by D-optimal design (3 points) and arithmetically equispaced design (12 points) Parameter

Estimate

CV (%)

Bias (%)

Absolute

D-optimal Ka Bmax k

1.2403 0.0569 0.0047

13.60 10.02 6.86

0.78 0.66 0.67

10.79 7.87 5.60

Arithmetically equispaced Ka 1.2895 15.60 Bmax 0.0561 11.91 k 0.0047 5.75

3.16 0.77 1.01

11.39 9.51 3.16

bias (%)

Results are from 150 simulated experiments. Parameter values used to generate noisy data were: K~ = 1.25, Bma,,= 0.0565. k = 0.00475.

parameter values correctly, compared with a (12point) arithmetically equispaced design ('intuitive' design). Data for the D-optimal design and for the 'intuitive' design were generated by means of standard MonteCarlo methods [2,14] (by the reported structural and variance models, and parameters). 150 data sets w e r e generated and for each set parameter values were estimated by means of a non-linear fitting program [2]. Mean, variance, bias and absolute bias were computed for each estimated parameter (see Table 2). D-optimal design obtained better estimates than the 'intuitive' design with only three support points, The applicability of D-optimal design in real situations is also documented elsewhere [17-20]. I n conclusion, the computer program EXCAD should be a useful program for the biologist and in general for researchers using mathematical models. 5. Hardware and software specification The program currently runs on a DEC VAX 750 (VMS) and is written in F O R T R A N 77. The advanced graphics use the REGIS library on a DEC VT240 Graphic terminal. Screen copies are provided by means of an on-line DEC letterwriter 100.

The authors are indebted to Prof. S. Garattini and A. Tavani for their constructive advice, and to Judy Baggott for revising the manuscript.

References [1] D.M. Steinberg and W.G. Hunter, Experimental design: review and comment, Technometrics 26 (1984) 71-130. [2] D. Verotta, M. Recchia and R. Urso, MODDIS: a microcomputer program for model discrimination, Comput. Methods Programs Biomed. 22 (1986) 209-219. [3] V.V. Fedorov, Theory of Optimal Experiments (Academic Press, New York, 1972). [4] E.M. Landaw, Optimal experimental design for biologic compartmental systems with application to pharmacokinetic, Ph.D. Thesis, Department of Biomathematics, UCLA (University Microfilms International, 1980). [5] A.C. Atkinson and W.G. Hunter, The design of experiments for parameter estimation, Technometrics 10 (1968) 271-290. [6] L.B. Sheiner, Analysis of pharmacokinetic data using parametric models. I. Regression models, J. Pharmacokin.

Biopharm. 12 (1984)93-117. [7] N.R. Draper and W.G. Hunter, The use of prior distilbutions in the design of experiments for parameter estimation in non-linear situations, Biometrika 54 (1967) 147-153. [8] R.D. Cook and C.J. Nachtsheim, A comparison of algorithms for constructing exact D-optimal designs, Technometrics 22 (1980) 315-324. [9] Z. Galil and J. Kiefer, Time- and space-saving computer methods, related to Mitchell's DETMAX, for finding Doptimum designs, Technometrics 22 (1980)301-313. [10] R.C. St. John and N.R. Draper, D-optimality for regression designs: a review, Technometrics 17 (1975) 15-23. [11] L.E. Carson, C. Cobelli and L. Finkelstain, The Mathematical Modeling of Metabolic and Endocrine Systems (Wiley, New York, 1983). [12] Y. Bard, Non-Linear Parameter Estimation (Academic Press, New York, 1974). [13] R.J. Bogumil, Sensitivity analysis of biosystem models, Fed. Proc. 39 (1980) 97-103. [14] J.V. Beck and K.J. Arnold, Parameter Estimation in Engineering and Science (Wiley, New York 1977). [15] L. Endrenyi and F-Y. Chan, Optimal design of experiments for the estimation of precise hyperbolic kinetic and binding parameters, J. Theor. Biol. 90 (1981)241-263. [16] L. Endrenyi, Design ofexperimentsforestimatingenzyme and pliarmacokinetic parameters, in: Kinetic Data Analysis. Design and Analysis of Enzyme and Pharmacokinetic Experiments, ed. L. Endrenyi, pp. 137-167 (Plenum Press, New York, 1981).

166 [17] D. Verotta, P. Petrillo, A. La Regina, M. Rocchetti and A. Tavani, D-optimal design applied to binding saturation curves of an enkephalin analog in rat brain. Life Sci. (in press). [18] D.Z.D'Argenio, Optimal sampling times for pharmacokinetic experiments, J. Pharmacoldnet. Biopharm. 9 (1981) 739-756.

[19] J.J. DiStefano IIl, Design and optimization of tracer experiments in physiology and medicine, Fed. Proc. 39 (1980) 84-90. [20] R.G. Duggleby, Experimental designs for estimating the kinetic parameters for enzyme-catalysed reactions, J. Theor. Biol. 81 (1979) 671-684.