Optimal designs for tumor regrowth models

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Journal of Statistical Planning and Inference

Journal of Statistical Planning and Inference 141 (2011) 644–654

0378-37

doi:10.1

� Cor

E-m1 Re2 V

journal homepage: www.elsevier.com/locate/jspi

Optimal designs for tumor regrowth models

Gang Li a,�, N. Balakrishnan b,1,2

a Medicine Development Center, GlaxoSmithKline, 1250 South Collegeville Road, Collegeville, PA 19426, USAb Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1

a r t i c l e i n f o

Article history:

Received 11 January 2010

Received in revised form

16 June 2010

Accepted 13 July 2010Available online 17 July 2010

Keywords:

Local optimality

Nonlinear models

D-optimality

c-Optimality

Double-exponential

Minimally supported designs

58/$ - see front matter & 2010 Elsevier B.V. A

016/j.jspi.2010.07.009

responding author.

ail addresses: [email protected] (G. Li), bala@

search supported by the Natural Sciences an

isiting Professor at King Saud University (Riy

a b s t r a c t

Most growth curves can only be used to model the tumor growth under no intervention.

To model the growth curves for treated tumor, both the growth delay due to the

treatment and the regrowth of the tumor after the treatment need to be taken into

account. In this paper, we consider two tumor regrowth models and determine the

locally D- and c-optimal designs for these models. We then show that the locally D- and

c-optimal designs are minimally supported. We also consider two equally spaced

designs as alternative designs and evaluate their efficiencies.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Statistical modeling of tumor growth has a long history in biology and biostatistics. It is well known that the growthof an unperturbed tumor can be fitted well by the Gompertz curve (Kidwell et al., 1960; Laird, 1964; Rygaard andSpang-Thomsen, 1997)

gðtÞ ¼ aexpf�be�ktg, ð1Þ

where g(t) is the tumor volume at time t and a is the asymptotic maximum tumor volume. Gompertz function ismonotonic and the maximum growth rate ka=e is achieved at the point of reflection t¼ logðbÞ=k. However, modeling thegrowth curves for treated tumors is a more challenging problem in mathematical biology and biostatistics. After the tumoris treated by a therapy such as radiation or chemotherapy, it may shrink soon after the treatment and then grow again. Inthis situation, it is evident that monotonic functions such as Gompertz function cannot be used to characterize such non-monotonic growth curve.

Demidenko (2004) developed a regrowth curve theory by combining two existing theories: traditional growth curvetheory and survival curve theory. In this theory, traditional growth curve is used to describe the further growth of survivingcells and the survival curve is used to model the proportion of the cells killed by the treatment. Double-exponentialregrowth curve is one such regrowth curve that describes the dynamics of post-irradiated tumors based on the two-compartment model. In a subsequent work, Demidenko (2006) proposed another model, viz., LINEXP model, to describethe growth delay and regrowth for treated tumor.

ll rights reserved.

mcmaster.ca (N. Balakrishnan).

d Engineering Research Council of Canada.

adh, Saudi Arabia) and National Central University (Taiwan).

www.elsevier.com/locate/jspi

dx.doi.org/10.1016/j.jspi.2010.07.009

mailto:[email protected]

mailto:[email protected]

dx.doi.org/10.1016/j.jspi.2010.07.009

G. Li, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 644–654 645

Proper choices of experimental designs can help to improve the efficiency of the experiment as well as the quality ofstatistical inference. The theory of optimal experimental designs is an important tool for the experimenters to find the bestdesign to meet the experimental objectives. Let H denote the set of probability distributions on the Borel sets ofw¼ ½tmin,tmax�; then any x 2 H is called an approximate or continuous design (Kiefer, 1974). Let y be the vector of k

unknown model parameters and Mðx,yÞ be the Fisher information matrix induced by the approximate design measure x.Each objective of the experiment is usually expressed as a concave criterion function of the Fisher information matrix andthe optimal design is then determined by maximizing this criterion function. When the objective is to estimate the modelparameters, two commonly adopted criteria are D-optimality and c-optimality. The D-optimality criterion function isdefined as the logarithm of jMðx,yÞj, the determinant of the Fisher information matrix, if Mðx,yÞ is non-singular and �1 ifMðx,yÞ is singular (Silvey, 1980). The D-optimal design minimizes the volume of the confidence ellipsoid for y. A c-optimaldesign maximizes fcT M�ðx,yÞcg�1, where the maximum is taken over the set of all designs for which the linear combinationcTy is estimable. For some other criteria and related results, interested readers may refer to Fedorov (1972), Silvey (1980),Atkinson and Donev (1992), and Pukelsheim (1993).

For nonlinear models, the information matrix depends on the unknown parameters y. It therefore becomes quite acomplicated problem to determine optimal designs for nonlinear models. A simple and natural approach is to adopt a bestguess for the parameters, say yð0Þ, and then to consider designs which maximize the criterion function of Mðx,yÞ evaluatedat y¼ yð0Þ. The resulting design is called locally optimal design (Chernoff, 1953). Locally D- and c-optimal designs fornonlinear models have been studied extensively by many authors; see Ford et al. (1992), Sitter and Wu (1993), Hedayatet al. (1997), Han and Chaloner (2003), Dette et al. (2006), and Li and Majumdar (2008, 2009). Besides the local optimalityapproach, there are some other alternative approaches, including the Bayesian approach (Chaloner and Larntz, 1989) andstandardized maximin approach (Dette, 1997). In this paper, we consider the problem of determining locally D- and c-optimal designs for two tumor regrowth models. We first review two tumor regrowth models of interest in Section 2. InSection 3, we present some preliminary results and then apply them to determine the locally D- and c-optimal designs forthe two tumor regrowth models in Sections 4 and 5. Finally, the D-optimal designs for the regrowth models are used asbenchmarks for evaluating the merits of two alternative designs with equally spaced support points.

2. Tumor regrowth models

2.1. Double-exponential regrowth model

Demidenko (2004) developed a double-exponential regrowth model to describe the dynamics of post-irradiated tumorsbased on the two-compartment model. In this setup, tumor cells are categorized into two compartments: proliferating, P,and quiescent, Q. Under three hypotheses (i) proliferating cells divide with constant rate, n, (ii) quiescent cells die with aconstant rate, f, and (iii) a portion of proliferating cells become quiescent with rate t, the tumor growth is characterized bythe system of differential equations

dP

dt¼ nP,

dQ

dt¼ tP�fQ :

The total number of tumor cells at time t is the sum N(t)=P(t)+Q(t) with a closed-form expression given byNðtÞ ¼N0½bentþð1�bÞe�ft�, where 0obo1, and N0 is the total number of tumor cells at the starting time t=0. Assumingthe tumor volume to be proportional to the total number of cells, the natural logarithm of the tumor volume ischaracterized by

g1ðtÞ ¼ aþ ln½bentþð1�bÞe�ft�, ð2Þ

where a is the logarithm of the initial tumor volume.

2.2. LINEXP regrowth model

In a follow-up work, Demidenko (2006) proposed a LINEXP model to describe the whole range of tumor growth delayand regrowth data. Under LINEXP model, the natural logarithm of the tumor volume is

g2ðtÞ ¼ aþgtþbðe�dt�1Þ, ð3Þ

where a is the baseline logarithm of the tumor volume, g is the final growth rate, and d is the rate at which killed cells getwashed out. The LINEXP model can be applied to model both monotonic growth and non-monotonic regrowth. Note that ifg4db, the mean function is monotonic; otherwise, the mean function is non-monotonic. Consequently, this model can beused to fit the data from the untreated tumor growth and treated tumor regrowth simultaneously.

G. Li, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 644–654646

3. Preliminaries

For a general regression model

yi ¼ gðti,yÞþei ð4Þ

with independent ei �Nð0,s2Þ, the Fisher information matrix induced by the design x is Mðx,yÞ ¼Rwf ðt,yÞf T ðt,yÞdxðtÞ, where

f ðt,yÞ ¼ ðf1, . . . ,fkÞ and fi ¼ @g=@yi, i¼ 1, . . . ,k. It is easy to show that the locally optimal design problem for this model isequivalent to the optimal design problem for the linear regression model

yi ¼ yT f ðti,yð0ÞÞþei, ð5Þ

where yð0Þ is the best guess of y at the design stage.According to the Equivalence Theorem of Kiefer and Wolfowitz (1960), a design x� is locally D-optimal for the

regression model if and only if

dðx�,tÞrk 8t 2 w, ð6Þ

where dðx,tÞ ¼ f T ðt,yð0ÞÞM�1ðx,yð0ÞÞf ðt,yð0ÞÞ is the standardized variance of the predicted response at t. In addition, theequality holds at the support points of x�. For non-singular locally c-optimal designs, there also exists such an EquivalenceTheorem. Specifically, a design x� is locally c-optimal if and only if

ff ðt,yð0ÞÞM�1ðx�,yð0ÞÞcg2�cT M�1ðx�,yð0ÞÞcr0 ð7Þ

holds for all points t 2 w, with equality being attained at the support points of x�.Let H denote the set of all design measures with non-singular Fisher information matrices. For the D-optimality

criterion, the following result provides a sufficient condition to ensure that the D-optimal design for w¼ ½tmin,tmax� isminimally supported.

Lemma 3.1. If 8x 2 H, (e40 such that every function in fdðx,xÞ�kþc : 0ocoeg has at most 2k �2 roots in w¼ ½tmin,tmax�,then the D-optimal design is unique and must be supported on k points with uniform weights. In addition, both boundary points

of w (i.e., tmin and tmax) are support points of the D-optimal design.

Proof. If ~x is D-optimal with at least k+1 support points ~x1o ~x2o � � �o ~xkþ1, then there are three alternatives for thesesupport points.

Case 1:
all the support points are interior to w¼ ½tmin,tmax�; Case 2: one support point is located at one of the boundary points of w and all remaining support points are interior to w; Case 3: ~x1 ¼ tmin, ~xkþ1 ¼ tmax and ~x2, . . . , ~xk are interior to w.
For Case 1, since all the support points are interior points, then there exists some e40 such that dð ~x,xÞ�kþc has at least

2k+2 roots in w for 0ocoe. For Case 2, dð ~x,xÞ�kþc has at least 2k+1 roots in w for 0ocoe. For Case 3, dð ~x,xÞ�kþc has at

least 2k roots in w for 0ocoe. But since 8x 2 H, (e40 such that every function in fdðx,xÞ�kþc : 0ocoeg has at most 2k �

1 roots in the design region w, by contradiction, a D-optimal design is minimally supported. It follows from a standard

argument that the weights are uniform on the k support points. The uniqueness of the D-optimal design follows from the

concavity of logj � j. &

To verify the sufficient condition in Lemma 3.1 for the tumor regrowth models, the theory of Tchebycheff-system(Karlin and Studden, 1966) plays a key role in this paper. Let u0,u1, . . . ,un denote continuous real-valued functions definedon a closed finite interval I=[a,b]. These functions are called a weak Tchebycheff system over I, provided that there exists ane 2 f�1,1g such that

e � U½ðui,tiÞni ¼ 0�Z0 ð8Þ

whenever art0ot1o � � �rb, where U½ðui,tiÞni ¼ 0� is the determinant of order n+1 given by

U½ðui,tiÞni ¼ 0� ¼

u0ðt0Þ u0ðt1Þ . . . u0ðtnÞ

u1ðt0Þ u1ðt1Þ . . . u1ðtnÞ

^ ^ ^

unðt0Þ unðt1Þ . . . unðtnÞ

��

��:

If the inequality in (8) is strict, then the system fu0, � � � ,ung is called a Tchebycheff system over I, abbreviated as T-system.If the system fu0,u1, . . . ,ung is a T-system, then any nontrivial linear combination in the form of

Pni ¼ 0 aiui has at most n

distinct roots. Conversely, if any nontrivial functionPn

i ¼ 0 aiui has at most n distinct roots, then the system fu0,u1, . . . ,ung isa T-system (Karlin and Studden, 1966). Then, from Lemma 1 of Li and Majumdar (2009), we have the following result.

Lemma 3.2. If fu0,u1, . . . ,un�1,uðjÞn g is a T-system over I with e¼ 1 for j¼ 1, . . . ,m, then fu0,u1, . . . ,un�1,Pm

j ¼ 1 cjuðjÞn g is also a

T-system over I with e¼ 1 provided that cj’s are all positive constants.


The following property of T-system follows from Karlin and Studden (1966, Chapter II, Theorem 10.2). If fu0, . . . ,ung is aT-system over I, then there exists a unique function WðtÞ ¼

Pni ¼ 0 a�i uiðtÞ which satisfies the following conditions:

(i)
�1rWðtÞr1, t 2 I, (ii) these exists t�0o � � �ot�n in I, such that Wðt�i Þ ¼ ð�1Þi, i¼ 0, . . . ,n.
The function W(t) is called a Tchebycheff polynomial and the points t�0, . . . ,t�n are called Tchebycheff points. These pointsare unique if I is a bounded and closed interval and 1 2 spanfu0, . . . ,ung, a space spanned by fu0, . . . ,ung. In this case, t0* andtn* are two boundary points of I.

Let ep ¼ ð0, . . . ,0,1,0, . . . ,0ÞT denote the vector with a one in the p-th component and zeros elsewhere. Let J denote thediagonal matrix defined as J¼ diagf�1,1, . . . ,ð�1Þkg. Assume that ffiji 2 f1, . . . ,kgg is a T-system over w¼ ½tmin,tmax� andt�1o � � �ot�k are the Tchebycheff points. For the c-optimal designs in estimating the individual parameter yp, Dette et al.(2004) has the following result.

Lemma 3.3. If ffiji 2 f1, . . . ,kgg is a T-system and ffiji 2 f1, . . . ,kg\fpgg is a weak T-system over w¼ ½tmin,tmax�, then a design

supported on ft�1, . . . ,t�kg with weights

wi ¼jeT

i JF�1epjPkm ¼ 1 je

TmJF�1epj

, i¼ 1, . . . ,k

is c-optimal for estimating yp, where F ¼ ffiðt�j Þ

ki,j ¼ 1g.

In the next two sections, we will determine the locally D- and c-optimal designs for the two tumor regrowth modelsintroduced earlier in Section 2. We also consider some alternative designs and evaluate their relative efficiencies to thesedesigns.

4. Designs for double-exponential regrowth model

In this section, we consider optimal designs for the double-exponential regrowth model

yi ¼ aþ ln½bentiþð1�bÞe�fti �þei, ð9Þ

where ei �Nð0,s2Þ and y¼ ða,b,n,fÞT is the vector of parameters of interest.

4.1. D-optimal design

We have the following results for the locally D-optimal design for the double-exponential regrowth model in (9).

Theorem 4.1. (i) The D-optimal design depends on the model parameters only through b and nþf.

(ii) The D-optimal design is unique and supported on four points with equal weights. Both tmin and tmax are the support points

of the D-optimal design.

Proof. We prove Part (ii) first. Denote the gradient vector

f ðt,yÞ ¼ ðf1,f2,f3,f4ÞT¼ 1,

ent�e�ft

bentþð1�bÞe�ft,

btent

bentþð1�bÞe�ft,�ð1�bÞte�ft

bentþð1�bÞe�ft

!T

: ð10Þ

Then, the information matrix for y is Mðx,yÞ ¼Rwf ðt,yÞf T ðt,yÞdxðtÞ. Let mij denote the (i,j)-th element of M�1ðx,yÞ. The

standardized variance of the predicted response at t is dðx,tÞ ¼ hðt,yÞT M�1ðx,yÞhðt,yÞ.For any constant c, let gðtÞ ¼ ðdðx,tÞ�4þcÞðbentþð1�bÞe�ftÞ

2 and uðtÞ ¼m33b2t2e2ntþm44ð1�bÞ2t2e�2ft�2m34bð1�

bÞt2eðn�fÞt . Then, g(t) is a linear combination of

fe2nt ,te2nt ,e�2ft ,te�2ft ,eðn�fÞt ,teðn�fÞt ,uðtÞg: ð11Þ

It then follows from Karlin and Studden (1966, p. 10) that

fe2nt ,te2nt ,e�2ft ,te�2ft ,eðn�fÞt ,teðn�fÞt ,t2e2ntg,

fe2nt ,te2nt ,e�2ft ,te�2ft ,eðn�fÞt ,teðn�fÞt ,t2e�2ftg,

fe2nt ,te2nt ,e�2ft ,te�2ft ,eðn�fÞt ,teðn�fÞt ,t2eðn�fÞtg

are T-systems with e¼ 1. Since M�1ðx,yÞ is positive definite for x 2 H, we have m3340,m4440. We have shown in the

Appendix that m34o0. Thus, u(t) is a linear combination of t2e2nt , t2e�2ft and t2eðn�fÞt with all positive coefficients. By


Lemma 3.2, the system (11) is a T-system, which implies that dðt,yÞ�4þc has at most 6 distinct roots for any constant c. By

Lemma 3.1, the D-optimal design is unique and has 4 support points including two boundary points (tmin and tmax) of w.

Consider a four-point uniformly weighted design with support points {t1,t2,t3,t4}. Let ~t i ¼ ðnþfÞti, i¼ 1, . . . ,4. Then, the

determinant of the Fisher information matrix is

jMðx,yÞj ¼ðnþfÞ�4

256Q4

i ¼ 1½ðe~t i�1Þbþ1�2

½ðe~t 1þ ~t 2þe

~t 3þ ~t 4 Þð~t1�~t2Þð~t3�~t4Þþðe~t 1þ ~t 3þe

~t 2þ ~t 4 Þð~t1�~t3Þð~t4�~t2Þ

þðe~t 1þ ~t 4þe

~t 2þ ~t 3 Þð~t1�~t4Þð~t2�~t3Þ�2:

It is now straightforward to show that the locally D-optimal design depends on model parameters only through b and

nþf. &

Since we have shown that tmin and tmax are support points of the D-optimal design x�, the other two support pointscould be found numerically by maximizing the determinant of the Fisher information matrix, where t1 and t4 are replacedby tmin and tmax, respectively. Table 1 presents the support points of the D-optimal designs for selected values of b and nþfwhen the design region is w¼ ½0,10�.

4.2. c-Optimal design

As we have described earlier in Section 2, n is the rate at which proliferating cells divide and f is the rate at whichquiescent cells die. These two parameters determine the growth of the tumor to a great extent. In this section, we considerthe c-optimal designs for estimating these two individual parameters.

First of all, we prove that (10) is a T-system. This is equivalent to showing that

fbentþð1�bÞe�ft ,ent�e�ft ,btent ,ð1�bÞte�ftg ð12Þ

is a T-system since bentþð1�bÞe�ft 40. Any linear combination of (12) is also a linear combination of

fent ,e�ft ,tent ,te�ftg

which can be shown to be a T-system since any nontrivial linear combination of the system has at most 3 distinct roots.

This implies that (12) is indeed a T-system. Since 1 2 spanff1,f2,f3,f4g and w¼ ½tmin,tmax� is a bounded and closed interval, the

Tchebycheff points {t1*, t2*, t3*, t4*} are unique and t1* =tmin, t4

* =tmax. Let WðtÞ ¼P4

i ¼ 1 a�i fiðtÞ be the Tchebycheff polynomial.

Then, the coefficients of the Tchebycheff polynomial, a�i , i¼ 1, . . . ,4, and the other two Tchebycheff points, t2* and t3*, can be

determined by the system of nonlinear equations

Wðt�i Þ ¼ ð�1Þi, i¼ 1, . . . ,4,

W uðt�j Þ ¼ 0, j¼ 2,3:

We employ the Gauss–Newton algorithm to solve the system of equations and examine the numerical solution by plottingW(t) over w.

Consider the locally c-optimal design for estimating the parameter n. In order to apply Lemma 3.3, it is sufficient toshow that

fbentþð1�bÞe�ft ,ent�e�ft ,ð1�bÞte�ftg ð13Þ

is a T-system. In fact, any linear combination of (13) is also a linear combination of fent ,e�ft ,te�ftg and it has at most 2distinct roots. Thus, we have shown that (13) is a T-system. It follows from Lemma 3.3 that the locally c-optimal design forn is supported on the Tchebycheff points ftmin,t�2,t�3,tmaxg. Once t2* and t3* are determined, the corresponding weights can becomputed by the formula specified in Lemma 3.3.

Table 1Support points of the D-optimal designs for double-exponential regrowth model; w¼ ½0,10�.

b nþf Support points of the D-optimal design

0.2 0.4 {0, 2.660, 6.707, 10}

1 {0, 1.542, 4.556, 10}

0.4 0.4 {0, 2.225, 6.281, 10}

1 {0, 1.238, 4.247, 10}

0.6 0.4 {0, 1.978, 6,042, 10}

1 {0, 1.080, 4.089, 10}

0.8 0.4 {0, 1.808, 5.881, 10}

1 {0, 0.976, 3.987, 10}


We can consider in an analogous manner the locally c-optimal design for estimating the parameter f. Since we canshow that

fbentþð1�bÞe�ft ,ent�e�ft ,btentg ð14Þ

is a T-system, the locally c-optimal design for f is also supported on the Tchebycheff points {tmin,t2*, t3*,tmax} with weightscomputed correspondingly. Table 2 provides some examples of locally c-optimal designs for estimating n or f.

4.3. D-efficient design

We have shown that the locally D-optimal design for the double-exponential regrowth model is minimally supported. Itmay not be feasible to implement such a design in practical experiments since it does not allow to test the goodness of fitof the nonlinear models. However, we could use it as a benchmark to judge the efficiency of other designs that may bemore natural from a practical viewpoint. In this section, we consider two alternative designs and evaluate their relativeefficiencies. In general, the efficiency of a design x with respect to the D-optimal design x� is defined as ½jMðxÞj=jMðx�Þj�1=k,where k is the number of parameters.

When the design region is a closed interval, the support points are often chosen with an equal distance between anytwo adjacent points. Such designs are called equally spaced designs. An m-point equally spaced design on w¼ ½tmin,tmax� issupported on points fti ¼ tminþðtmax�tminÞði�1Þ=ðm�1Þ,i¼ 1, . . . ,mg. We consider two approaches to determine the weightsfor these support points. In the first approach, the weights are uniform across all support points and the resulting design iscalled equally spaced and uniformly weighted (ESUW) design. Such a design has been widely adopted in practicalexperiments due to its simplicity. In the second approach, we divide the support points into two groups, i.e., {t1,tm} andft2, . . . ,tm�1g and assign them weights w1 and w2, respectively. The values of w1 and w2 are determined by maximizing thecorresponding determinant of the Fisher information matrix under the constraint 2w1+(m�2) w2=1. We call such a designequally spaced and symmetrically weighted (ESSW) design.

Table 3 provides the efficiencies of ESUW and ESSW designs for the double-exponential model for some selectedscenarios. First of all, the m-point ESSW designs have higher efficiency than the m-point ESUW designs in all consideredscenarios when m44. When nþf¼ 0:4, the m-point ESUW and ESSW designs have high efficiencies; however, theefficiencies for such designs are much lower when nþf¼ 1, especially when m is small. This is due to the fact that thesupport points of the D-optimal designs are scattered much closer to the lower end of the design region when nþf¼ 1.This has been shown in Table 1. Since in such scenarios the equally spaced support points of ESUW and ESSW designs couldnot provide good approximations to the support points of the D-optimal design when m is small, neither ESUW nor ESSWdesigns achieve good efficiencies. However, this problem is somewhat alleviated with greater m.

Table 2

Locally c-optimal designs for estimating n and f in the double-exponential regrowth model.

w b nþf Locally c-optimal design

Parameter t�1 ot�2 ot�3 ot�4 w�1 ow�2 ow�3 ow�4

[0,1] 0.5 0.4 n {0 0.249 0.749 1} {0.161 0.327 0.339 0.173}

f {0 0.249 0.749 1} {0.172 0.338 0.328 0.162}

[0,1] 0.2 1 n {0 0.264 0.761 1} {0.159 0.325 0.341 0.175}

f {0 0.264 0.761 1} {0.188 0.352 0.312 0.148}

[0,10] 0.5 0.4 n {0 1.887 6.458 10} {0.095 0.222 0.405 0.278}

f {0 1.887 6.458 10} {0.190 0.357 0.310 0.143}

[0,10] 0.2 1 n {0 1.432 4.856 10} {0.041 0.095 0.459 0.405}

f {0 1.432 4.856 10} {0.238 0.407 0.262 0.093}

Table 3Efficiency (%) of m-point ESUW designs and ESSW designs for the double-exponential model; w¼ ½0,10�.

b nþf m-Point ESUW design m-Point ESSW design

m=4 5 6 7 8 m=4 5 6 7 8

0.2 0.4 97.2 93.4 89.2 88.9 83.2 97.2 95.0 93.3 92.3 91.6

1 62.9 79.0 82.0 80.7 78.7 62.9 79.8 84.5 85.0 84.7

0.4 0.4 92.8 92.1 88.6 85.4 82.7 92.8 93.6 92.5 91.7 91.1

1 53.6 70.6 76.4 77.3 76.4 53.6 71.3 78.7 81.4 82.2

0.6 0.4 88.9 90.4 87.7 84.8 82.2 88.9 91.8 91.6 91.0 90.5

1 49.0 65.8 72.5 74.5 74.4 49.0 66.5 74.7 78.4 80.0

0.8 0.4 85.7 88.8 86.8 84.2 81.7 85.7 90.2 90.6 90.4 90.0

1 46.1 62.5 69.7 72.3 72.7 46.1 63.2 71.8 76.0 78.2


5. Designs for LINEXP regrowth model

In this section, we consider optimal designs for the LINEXP regrowth model. By the transformation-invariance propertyof the D-optimal design (Atkinson and Donev, 1992), we re-parameterized the mean response and consider the model

yi ¼ y1þy2ey3tiþy4tiþei, ð15Þ

where ei �Nð0,s2Þ and y¼ ðy1,y2,y3,y4ÞT is the parameter vector of interest. Note that this model has the same mean

function as the return of fertilizer dressing model discussed by Atkinson and Haines (1996), where the independentvariable is the amount of the fertilizer applied. According to Atkinson and Haines (1996), the expected yield of wheatresulting from the application of an amount of fertilizer, say x, was well described by the nonlinear function, y1þy2ey3x.The economic return is obtained by adjusting the cost of fertilizer, �y4x, from the yield, leading to the same nonlinearmodel as presented in (15).

5.1. D-optimal design

We have the following result for the locally D-optimal design for the LINEXP model.

Theorem 5.1. The D-optimal design is unique and supported on four points with equal weights. Both tmin and tmax are the

support points of the D-optimal design.

Proof. Let f ðt,yÞ ¼ ð1,ey3t ,y2tey3t ,tÞT . Then, the information matrix for the design measure x is Mðx,yÞ ¼Rwf ðt,yÞf T ðt,yÞdxðtÞ.

Let mij denote the (i,j)-th element of M�1ðx,yÞ. Then,

dðx,tÞ ¼ hðt,yÞT M�1ðx,yÞhðt,yÞ¼ ½m33y

22t2þ2m23y2tþm22�e

2y3tþ½m44t2þ2m14tþm11�þ½2m34y2t2þ2ðm13y2þm24Þtþ2m12�ey3t :

If we let uðtÞ ¼m33y22t2e2y3tþ2m34y2t2ey3tþm44t2, then for any constant c, dðx,tÞ�4þc is a linear combination of

f1,t,ey3t ,tey3t ,e2y3t ,te2y3t ,uðtÞg: ð16Þ

It follows from Karlin and Studden (1966, p. 10) that

f1,t,ey3t ,tey3t ,e2y3t ,te2y3t ,t2e2y3tg,

f1,t,ey3t ,tey3t ,e2y3t ,te2y3t ,t2ey3tg,

f1,t,ey3t ,tey3t ,e2y3t ,te2y3t ,t2g

are T-systems with e¼ 1. Since M�1ðx,yÞ is positive definite for x 2 H, we have m3340,m4440. In the Appendix, we have

shown that m34y240 8x 2 H. Thus, u(t) is a linear combination of ft2e2y3t ,t2ey3t ,t2g with all positive coefficients. By Lemma

3.2, the system (16) is a T-system, which implies dðx,tÞ�4þc has at most 6 roots for any constant c. It follows from Lemma

3.1 that the D-optimal design for the design region [tmin,tmax] is unique and supported on 4 points with equal weights,

including two boundary points (tmin and tmax) of w. &

For a four-point uniformly weighted design x with support points {t1, t2, t3, t4}, the determinant of the Fisherinformation matrix is

jMðx,yÞj ¼y2

2

256½ðey3t1þy3t2þey3t3þy3t4 Þðt1�t2Þðt3�t4Þþðe

y3t1þy3t3þey3t2þy3t4 Þðt1�t3Þðt4�t2Þ

þðey3t1þy3t4þey3t2þy3t3 Þðt1�t4Þðt2�t3Þ�2:

We have the following results in this case.

Lemma 5.1. (i) The D-optimal design only depends on the parameter y3;

(ii) If x maximizes jMðx,yÞj on [a,b], then the uniformly weighted design with support points {t1+c, t2+c, t3+c, t4+c} maximizes

jMðx,yÞj on [a+c,b+c] for any constant c;

(iii) If x maximizes jMðx,yÞjy3 ¼ a on [0,b], then the uniformly weighted design with support points {t1/b, t2/b, t3/b, t4/b}

maximizes jMðx,yÞjy3 ¼ ba on [0,1] for any nonzero constant b.

If we want to find the D-optimal design x� on w¼ ½a,aþb� with the parameter vector y, using Lemma 5.1 we can first find aD-optimal design x�1 on w¼ ½0,1� with the parameter vector by, then multiply the support points of x�1 by b and add a to getthe support points of x�. So, it is sufficient to find the D-optimal design for the design region [0,1]. Since we have shownthat tmin and tmax are support points of the D-optimal design x�, the other two support points could be found numericallyby maximizing the determinant of the Fisher information matrix. We will denote the support points of the D-optimaldesign by t�1 ¼ 0ot�2ot�3ot�4 ¼ 1. Table 4 presents these support points of the D-optimal designs for selected values of y3.


5.2. c-Optimal design

Here, we consider the c-optimal designs for estimating the two individual parameters y3 and y4 in the LINEXP model.First, it can be shown that f1,ey3t ,y2tey3t ,tg is a T-system. Since 1 2 spanf1,ey3t ,y2tey3t ,tg and w¼ ½tmin,tmax� is a bounded

and closed interval, the Tchebycheff points {t1* , t2

* , t3* , t4

* } are unique and t1* =tmin, t4

* =tmax. Since f1,ey3t ,tg is a T-system, thelocally c-optimal design for estimating the parameter y3 is supported on the Tchebycheff points {tmin,t2

* , t3* ,tmax}. Once t2

*

and t3* are determined, the corresponding weights can be computed by the formula specified in Lemma 3.3. Similarly, we

can show that the locally c-optimal design for y4 is also supported on the Tchebycheff points {tmin,t2* , t3

* , tmax} with weightscomputed correspondingly. Table 5 provides some examples of locally c-optimal designs for estimating y3 or y4.

5.3. D-efficient designs

As before, we compare the efficiencies of ESUW and ESSW designs for different values of y3 in Table 6 for the LINEXPregrowth model. In all scenarios, ESSW designs achieve the same or higher efficiency than ESSW designs. For a fixed valueof m, the efficiencies of ESUW or ESSW designs decrease as y3 decreases. Here again, it is due to the fact that thedistribution of the support points of the locally D-optimal designs becomes more asymmetric in the design region when y3

decreases from �0.5 to �5.0 as shown in Table 4.

Table 4Support points of the D-optimal designs for LINEXP model; w¼ ½0,1�.

y3 t�1 ot�2 ot�3 ot�4 y3 t�1 ot�2 ot�3 o t�4

�0.5 {0 0.260 0.707 1} �1.0 {0 0.245 0.689 1}

�1.5 {0 0.230 0.671 1} �2.0 {0 0.216 0.652 1}

�2.5 {0 0.203 0.633 1} �3.0 {0 0.190 0.614 1}

�3.5 {0 0.179 0.595 1} �4.0 {0 0.168 0.576 1}

�4.5 {0 0.158 0.558 1} �5.0 {0 0.149 0.539 1}

Table 5

Locally c-optimal designs for estimating y3 and y4 in the LINEXP model.

w y3 Estimating Locally c-optimal design

t�1 ot�2 ot�3 o t�4 w�1 ow�2 ow�3 ow�4

[0,1] �0.5 y3 {0 0.235 0.734 1} {0.167 0.334 0.333 0.166}

y4 {0 0.235 0.734 1} {0.153 0.319 0.347 0.181}

[0,10] �0.5 y3 {0 1.349 5.703 10} {0.163 0.356 0.337 0.144}

y4 {0 1.349 5.703 10} {0.067 0.176 0.433 0.324}

[0,1] �2.5 y3 {0 0.182 0.664 1} {0.164 0.339 0.336 0.161}

y4 {0 0.182 0.664 1} {0.107 0.255 0.393 0.245}

[0,10] �2.5 y3 {0 0.370 2.248 10} {0.174 0.443 0.326 0.057}

y4 {0 0.370 2.248 10} {0.009 0.027 0.491 0.473}

Table 6Efficiency (%) of m-point ESUW designs and ESSW designs for the LINEXP model; w¼ ½0,1�.

y3 m-Point ESUWD design m-Point ESSWD design

m=4 5 6 7 8 m=4 5 6 7 8

�0.5 95.7 93.5 89.5 85.9 82.9 95.7 95.2 94.0 93.0 92.3

�1.0 95.1 93.3 89.3 85.8 82.8 95.1 94.9 93.8 92.8 92.2

�1.5 94.0 92.8 89.1 85.6 82.8 94.0 94.4 93.4 92.6 91.9

�2.0 92.5 92.1 88.7 85.4 82.6 92.5 93.7 93.0 92.2 91.6

�2.5 90.7 91.2 88.2 85.1 82.3 90.7 92.7 92.3 91.7 91.2

�3.0 88.5 90.1 87.6 84.7 82.1 88.5 91.6 91.6 91.2 91.0

�3.5 86.0 88.8 86.8 84.2 81.7 86.0 90.2 90.7 90.5 90.1

�4.0 83.2 87.3 86.0 83.6 81.3 83.2 88.7 89.7 89.7 89.5

�4.5 80.2 85.6 85.0 83.0 80.8 80.2 86.9 88.5 88.8 88.7

�5.0 77.0 83.8 83.9 82.2 80.2 77.0 85.0 87.3 87.8 87.9

Table 7D-efficiencies (%) under mis-specified initial parameter values for the double-exponential regrowth model; w¼ ½0,10�;

b� ¼ 0:008,d� ¼�ðn� þf�Þ ¼ �0:336.

b=b� d=d� Efficiency b=b� d=d� Efficiency b=b� d=d� Efficiency

0.5 0.3 96.6 1.0 0.3 96.5 1.5 0.3 96.5

0.5 98.3 0.5 98.4 0.5 98.3

1.0 99.9 1.0 100 1.0 99.9

1.5 99.2 1.5 99.6 1.5 99.3

2.0 98.5 2.0 96.8 2.0 94.7

2.5 93.8 2.5 89.0 2.5 85.4

3.0 85.1 3.0 78.8 3.0 74.7

2.0 0.3 96.4 2.5 0.3 96.4 3.0 0.5 96.3

0.5 98.3 0.5 98.2 0.5 98.1

1.0 99.8 1.0 99.7 1.0 99.5

1.5 98.7 1.5 97.9 1.5 97.1

2.0 92.7 2.0 90.9 2.0 89.3

2.5 82.6 2.5 80.2 2.5 78.2

3.0 71.6 3.0 69.2 3.0 67.2


6. Robustness analysis

In this paper, we have studied the D- and c-optimal designs for two tumor regrowth models. Due to the nature of thenonlinear models, the local optimality approach is adopted. It is well known that this approach depends on the choiceof the initial values for the unknown model parameters. The robustness of the optimal designs could be evaluated asfollows.

Assuming that y� are true values of the unknown model parameters, we could numerically determine D-optimaldesigns x�D and c-optimal designs x�c for estimating y and the individual parameter of interest, respectively. Let the initialguess of the parameter values be y and the corresponding optimal designs be xDðyÞ and xcðyÞ. Then the efficiencies of thesedesigns could be computed using x�D and x�c as benchmark designs. Specifically, the D-efficiency of the design xDðyÞ isfjMðxDðyÞ,y

�Þj=jMðx�D,y�Þjg1=k and the c-efficiency of the design xcðyÞ is ½cT M�ðx�c ,y�Þc�=½cT M�ðxcðyÞ,y

�Þc�.

As an illustration, we use a chemotherapy treatment example from Chapter 10 of Demidenko (2004). The data consist oflongitudinal measurements of tumor volume in one untreated group and three treated groups of mice. The data were fittedusing the double-exponential regrowth model for the three treated groups. The parameter estimates from group 3 are usedas the true parameter values in this illustration. Since a different parametrization was used in Demidenko (2004), wetransform the parameters under the parametrization setup in this paper. We get ða�,b�,n�,f�Þ ¼ ð�0:68,0:008,0:122,0:244Þand the corresponding locally D-optimal design for the design space w¼ ½0,10� is supported on {0, 3.96, 8.09, 10} withuniform weights. To explore the sensitivity of the designs with respect to the initial parameter values, we use the ratio(e.g., b=b�Þ between the initially specified value and the true value to characterize the mis-specification for each individualparameter. The value of 1 for this ratio implies there is no mis-specification. The ratio value of less than 1 indicatesparameter underestimation while the ratio value of greater than 1 indicates parameter overestimation. Table 7 presentsthe D-efficiencies for the resulted designs under a wide range of mis-specified parameter values. Since the D-optimaldesigns depend on the parameter only through b and nþf, only these two ratio indices are presented. In this example, allinvestigated designs maintain high efficiencies ð480%Þ unless the ratio is at least 3 in several scenarios. The efficiency lossseems to be slightly higher for the parameter overestimation than for the underestimation.

Similar robustness analyses could also be conducted for the locally c-optimal designs and the efficient designs weobtained in Sections 4 and 5. However, it should be pointed out that robustness analyses depend on the assumed trueparameter values and the parametrization of the model. It should be conducted on a case by case basis.

As pointed out by a referee that in practice it is very difficult to specify parameter values accurately in order todetermine the optimal designs and single-stage locally optimal designs are often criticized because of poorly selectedinitial parameter values, sequential adaptive optimal designs approach (Box and Hunter, 1965; Fedorov, 1972) is apractical approach to address this issue. The sequential optimal designs are performed in multiple stages and optimaldesigns for next stage are determined based on the parameter values estimated at the current stage.

Acknowledgements

The authors would like to thank a Referee and the Associate Editor for their valuable comments and suggestions whichimproved the paper.


Appendix A

A.1. Proof of m34o0 for DE regrowth model

For any design x based on sZ4 points t1,t2,y,ts and probabilities p1,p2,y,ps, piZ0 andPs

i ¼ 1 pi ¼ 1, let Mij be the minorcorresponding to the (i,j)-th element of Mðx,yÞ. If we let

f1ðr,s,uÞ ¼

1enr�e�fr

benrþð1�bÞe�fr

brenr


1ens�e�fs

bensþð1�bÞe�fs

bsens


1enu�e�fu

benuþð1�bÞe�fu

buenu


��

��and

f2ðr,s,uÞ ¼

1enr�e�fr


�ð1�bÞre�fr


1ens�e�fs


�ð1�bÞse�fs


1enu�e�fu


�ð1�bÞue�fu


��

��,

then it can be shown that M34 ¼P

1r io jokr tpipjpkf1ðzi,zj,zkÞf2ðzi,zj,zkÞ. In Section 4.2, we have shown that both

1,ent�e�ft

bentþð1�bÞe�ft,�ð1�bÞte�ft


( )and 1,

ent�e�ft

bentþð1�bÞe�ft,

btent


� �

are T-systems with e¼ 1. Then, M3440. Since m34 ¼ ð�1Þ3þ4ðM34=jMjÞ, m34o0 8x 2 H.

A.2. Proof of m34y240 for LINEXP regrowth model

For any design x based on sZ4 points t1,t2,y,ts and probabilities p1,p2,y,ps, piZ0 andPs

i ¼ 1 pi ¼ 1, let Mij be the minorcorresponding to the (i,j)-th element of Mðx,yÞ. We can show that M34y2 has the same sign as

~M34 ¼

1Pt

i ¼ 1 piezi

Pti ¼ 1 pizie

ziPti ¼ 1 pie

ziPt

i ¼ 1 pie2zi

Pti ¼ 1 pizie

2ziPti ¼ 1 pizi

Pti ¼ 1 pizie

ziPt

i ¼ 1 piz2i ezi

��

��,

where zi ¼ y3ti. If we let

f ðr,s,uÞ ¼

1 er rer

1 es ses

1 eu ueu

��

1 r er

1 s es

1 u eu

��,

then it can be shown that ~M34 ¼�P

1r io jokr tpipjpkf ðzi,zj,zkÞ. It follows from Karlin and Studden (1966, p. 10) that{1,ex,xex} and {1,x,ex} are T-systems. Then, f ðr,s,uÞ40 8rasau. Thus, ~M34o0 and M34y2o0 for 8x 2 H. Sincem34 ¼ ð�1Þ3þ4

ðM34=jMjÞ, m34y240 8x 2 H.

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Optimal designs for tumor regrowth models