Asymptotic properties of the tumor growth curve estimator

Journal

of Statistical

Planning

and Inference

45

18 (1988) 45-56

North-Holland

ASYMPTOTIC PROPERTIES OF THE TUMOR GROWTH CURVE ESTIMATOR R. BARTOSZYhSKI Department Received

of Statistics, 9 November

Recommended

Abstract:

and S.M. DYNIN Ohio State University,

The estimator

Key words vergence.

USA

Puri

of growth

proposed

MSE as well as the limiting Subject

OH 43210.1247,

1986

by M.L.

its size at detection),

AMS

Columbus,

behavior

Classification: and phrases:

curve of tumors

earlier

is analyzed.

(based on a single observation The results

of the estimator

treated

concern

per tumor,

its asymptotic

as a stochastic

viz.

bias and

process.

62F12, 62FlO. Growth

curve

estimation;

bias;

MSE;

weak

convergence;

a.s.

con-

1. Introduction In this paper we shall study some large sample properties of the estimator of growth curve of tumors suggested in Atkinson et al. (1983) [see also Brown et al. (1984)]. This estimator is based on the following model. Assume that there exists a strictly increasing functionf(t) that describes the growth of cancer tumors in the following sense: the size of a tumor at time t, counting from the (unobservable) moment of its inception, is f(t/a), where a is a patient-dependent scaling factor. These scaling factors are assumed to follow a gamma distribution in the population of patients, with r and a standing for the shape and scale parameters of this distribution. Let f(0) = c denote the size of a single cell. Assume also the following mechanism of tumor detection: if a tumor remains undetected until time t, when its size isf(t/a), then the probability that it will become detected between t and t + h is bf(t/a)h + o(h). Let X denote the size of the tumor at detection, with V(x) =P[X
i?(x) = ; Some estimators 037%3758/88/$3.50

C u[l-

dV(u) V(U)]‘+“~

of the shape parameter 0

1988, Elsevier

Science

=

Lh(x), rb

r are also suggested

Publishers

(1.1)

say.

B.V. (North-Holland)

in Atkinson

et al.

R. Bartoszytiski, SM. Dynin / Tumor growth curve estimator

46

(1983); they are based on observations ferent detection constants b. These results allow for the surprising

of random possibility

variable

X in samples

of inference

about

with dif-

the function

g (hence also about the common growth curve f) using the data on X, hence the data involving only one observation for each tumor. Specifically, suppose that r is known, and let x1,x2, . . . ,xN be a sample of tumor sizes X. Let VN be the empirical cumulative distribution function of the sample. We define then

hN(X)=

d&h’) u[l-

U-2)

vN(u)]‘+“”

The aim of this paper will be to study the limiting properties of hN(x). Up to a multiplicative constant, hN(x) is an estimator of g(x), so that by taking ratios we may build schemes of inference about the growth function f = g-l.

2. Bias and MSE We shall first prove the following

theorem.

Theorem 1. Assume that l/r is an integer, say 1Ir = m. Then for every x such that V(x)< 1, we have d V(u) u[n - V(u)]m+2

Eh,,,(x) = h(x) +

-

s x

C u[l-

dV(u)

1

+0(1/N).

v(u)]“+’

(2.1)

Before starting the proof, let us remark that the assumption of the theorem is not as unrealistic as it may seem: the data in Atkinson et al. (1983) and Brown et al. (1984), as well as in Klein et al. (1986), indicate Proof.

that r is about

Let us write

I(, x)fxi)

h,,,(x) = NM f’ i=l

X;[l-

hN(X)

=

Nm f

z(c,x)(xi)

i=l Xci,(N-Ri+l)“+l

expectations

(2.2)

t;V~~j)lm+l

where IA is the indicator of the set A. Denote by X~i, the i-th order statistic from the sample, Then vN(xi) = (Ri - 1)/N, and hence

Taking

1.

we may write

and by Ri the rank of

Xi.

(2.3)

R. Bartoszytiski, S.M. Dynin / Tumor growth curve estimator

Eh,(x)

‘1x

= Nm f’ i=l

Lc I

N! I/‘-‘(u)[l-

(i-l)!

(N-i)!

Nrn

=

(N+m)!

x V”(u)[l-

dl/(u)

(N-i+l)““u

u ;=I

(N+l)...(N+m)

(i-l)!

(N-i+2)...(N-i+l+m) (N-i+l+m)!

(N-i+l)”

V(u)]‘+‘dV(u) ax

N”

=

V(U)]‘+’

47

c u[l-

(N+l)...(N+m)

N-1 c

1 V(u)]“+’

W+m)!

j=o J! (N+m-j)!

x(I+~)(l+~)...(I+~)Vj(~)[l-V(U)IN+m~idV(II)

s x

=

A(N, m)

c

’

a -v

m+l [1-;$;

(N;m)yi(l-J’)“+mPidV]

+B1+B2+-.*+B,,

(2.4)

where A(N, m) =

Nrn

(2.5)

(N+ l)...(N+m)

and

Bk=A(N,m) (2.6)

in the last sum, the summation is extended over all distinct integers al, . . . , uk satisfying 11al
AN

Nrn

m>

h(x)

(N+l)...(N+m) l-(l+;)...(l+;),, = h(x) + (1+-$.(1+;)

=WNext,

m(m+l) 2N

h(x) + 0(l/N2).

(2.7)

we have >x

I = A(N, m)

=

A(N, m)

~i c ~(l-

1

Nr

V)m+l

j=N

VN

~(l-

V)m+l

Nr j=N

(Njm)V'(~_v)N+~~-jdv

(N;m)

VjpN(l-V)N+m-jdV.

(2.8)

48

The

R. Bartoszytiski, S.M. Dynin / Tumor growth curve estimator

terms

of the

sup{l/[u(l-

last

.),+‘I:

sum

are

clulx},

bounded

N”‘m(N+

m)“‘,

so that

putting

S =

we may write 1X VNdV (/c

A(N,m)m(N+m)mS

I5

by (N+

m)”

vN+l(x)

s

s (N+l)...(N+m)

_ vN+l(c)

N+l

N+l

1

(2.9)

which is 0(1/N). As regards B,, we have x

B, = A(N,m) =

c U(l-

$1 =

x

Nrn

Mm+11 2

’ f (N;m)Vj(l-V)N+mpj;;;_+::dV V)m+’ j=o

I

1

rC u(~-V)~+*

(N+l)...(N+m+l)

(l+~)(N+~+l)VJ(l-V)N+m+l-jdV

c, +C*,

(2.10)

the two terms corresponding to multiplication through the factor Proceeding as in (2.8) and (2.9), we easily show that dV U(l-

v)m+2

1 + (m + l)/(N-j).

+0(1/N*).

(2.11)

Next, c

=

mW+112

2 I 1;1.~~~~~~~::~~51:~~~~~~~~~*1;1’ xc.

I

N”

m(m+l)*

2

max [l+E] j

(N+l)...(N+m+2)

x

1

X

m+3 ;$;

Vj(l-V)N+m+2-jdV

(,+,+‘)

.i c Nl-VI sm(m+1)2(m+3) 2

1

‘*

dV

N2 .c I u(l-V)m+s

= 0(1/N2).

(2.12)

To complete the proof, it remains to show that the sum B, + B, + ... + B, is 0(1/N2). We shall evaluate BZ; the evaluation of the remaining B,‘s follows the same lines. Thus,

R. Bartoszyriski, S.M. Dynin / Tumor growth curve estimator

49

B, = A(N,m)

Nrn = (N+l)...(N+m)(N+m+l)(N+m+2)

(2.13) Proceeding Next,

as before,

we show easily that the last expression

is 0(1/N2).

we have:

Theorem

2. Under assumptions of Theorem

Var [hN(x)] = $

X

1,

dV

i[i , c u2(1-

v)2m+2

X 1l.l +2m(m+l) 1.I c UU[ltc -mm2

1 2

dV(u) dV(o) v(u)]m+’

+0(1/N)

[l-

V(o)]“‘2

=+)+0(1/N).

(2.14)

We shall omit the proof: it is very similar to that of Theorem 1, but considerably more tedious. In Section 4, we shall show how one can derive (2.14) by a simpler method.

3. Almost

sure convergence

The results of the proceding section concerned the properties of our estimator for a fixed value of x. One can also prove the following more global result, valid for any r, integer or not. Theorem

3. If V(x) > 0 for all x, then for all a, b with CI a< b < 03, we have &(a, b) =

sup o5x5b

50

Proof.

R. Bartoszyriski, S. M. Dynin / Tumor growth curve estimator

We first bound

from above the left-hand

x dV -

1.r

+ sup acxsb

c

=A+B, Integrating

by parts,

side of (3.1) as follows:

u

say.

(3.2)

we obtain

(3.3) The first term vanishes sup

asxsb

IV,-VI

at c, and is bounded

by

l aVn(b)‘+l”’

which goes to 0 as n --t 03. The second b

sup

from above

(3.4) term can be bounded

by

du

(3.5)

K-VI

The third term may be bounded

by (3.6)

Since 6< 00 and V(x)>O, by Glivenko-Cantelli theorem, the integrals in (3.5) and (3.6) remain bounded, and hence both (3.8) and (3.6) tend to 0 as n -+ 03. Using similar argument, we show that B -+ 0, a.s., which completes the proof.

4. Central limit theorem In this section, we shall again assume that 1/r = m is an integer. 2, it was shown that EhN(X) = h(x)+O(l/N)

In Sections

1 and

(4.1)

and Var hN(x) = c(x)/N+ where c(x) is given by (2.14). Theorem

0(1/N),

(4.2)

We shall now prove:

4. As NA 00, we have for every x: fl(h,(x)

-Eh&))

= N(0, c(x)).

(4.3)

R. Bartoszyriski, S.M. Dynin / Tumor growth curve estimator

Proof.

Using

(4.1), we may write

fi(&V(X)-&V(X))

= @(Mx)-h(x))+G(l/fi),

so that we need to find the asymptotic LN

We may write,

=

fi(h’(x)

-

adding

! I

1

--~

2.4

v;+’

jf dk’,/uk”“+‘, 1 I/m+1

I

‘x d(l/,-

v) =A+B, m+l UV

+P

of (4.5)

and subtracting

.c

distribution

(4.4)

h(x)).

‘“dl/,

L/,,=fl

Next,

51

say.

(4.6)

we may write

1 +fl

‘x d(VN-

I

cc

24

V)

Vm+‘-

VJ+l

VNm+‘Vm+l

= A,+A,+A,.

(4.7)

We shall show that A, and A, tend to 0 in probability, so that the asymptotic distribution of L, is the same as that of A, + B, given in (4.7) and (4.6). Indeed, we may write

x ,s_w,I V(Y)- VN(Y)l ‘X-IdVl c

u

IWK VN)l V m-e1 m+l ’

(4.8)

vN

where D is a polynomial of V and VN of degree at most 2m. By the GlivenkoCantelli theorem, the second supremum tends to 0 in probability. The first supremum has the limiting Kolmogorov-Smirnov distribution, while the integral remains bounded in probability. Consequently, A, tends to 0. Integrating by parts, and using the same argument, we show that A, also tends to 0. Next, we may write

52

R. Bartoszyriski, SM.

=fi(V(x) -

Dynin / Tumor growth curve estimator

1‘x(m+y+y

VN(X))

c

uv

X ‘,, (m+l)dV m+2 @d(VYc c UV

+

5.1

+

‘x ‘y (m+l)dV m+2 UV 1 c rC I

V)(Y)

fld(V,(y)

- V(y))

x

=

y (m+l)du m+2 I/5Sd(V,(y) s c [.i X UV I

- V(Y)).

Consequently,

(4.9)

VI, 14flO’Z,-

It is known that fi(VS(y) denote the integrand limiting distribution as U = “S(y) ic

V,) converges to the Brownian in (4.10). We shall show that

(4.10)

bridge Z”(F’(y)). Let A, +B has the same

dZ’(F(y)).

(4.11)

We shall use the theorem of Komlos, Major, and Tusnady (1975), which asserts that there exists a probability space (S&F, P) and versions of fi(VVN) and Z”(F(y)) such that sup / $w
V,) - ZOI = 0,(N-“2

where 0, stands for boundedness We write now IU-(A,+B)j

=

log NJ

(4.12)

in probability.

IIx

S(Y)

W”(N9) - @U’-

V,)l

5 ls;Y~~zO~Y~-~~v-v,)l:I x

+

1.r c

By the quoted proof.

theorem,

WY>IZ~(~(YN

-@U’-

both terms tend to 0 in probability,

Yv)l

dy .

(4.13)

which completes

the

R. Bartoszytiski, S.M. Dynin / Tumor growth curve estimator

Finally, tions

let us use the above result to get the variance

53

given by (2.14). The calcula-

are based on the relation s(r)dZ’(F(r))]

Var which

can

be derived

EZO(F(s))ZO(F(t)) We have s(t) =

= /‘:S’(i)dF(i)-

(4.14)

[ {,$P(r)]2

by approximating integrals by sums, using the relation valid for SI t, and passing to the limit. -F(O),

=F(s)(l

“x (WI+ 1) dF( y)

1 tP+l(t)

yv”+yY)

+ If

(4.15) .

We may now write >x

‘X

I

S’(t)

dF =

L‘

+ ‘x

dF(t)

!

! [!

c t2V2”+2(t)

,C

~t

YV”+‘(Y)

‘x

+2(m+l) tl/“+‘W =A+B+C,

I

dF(t)

WY)

\ Yvm+2(Y)
say.

We shall sketch the evaluation

2

‘x (m+l)dF(y)

of the term B. Let us write (4.17)

B = ‘XG(t)2 dF(t), *\ c where ‘x (m + 1) dF( y)

G(t) =

\ *t Integrating

by parts,

YVrnf2(Y)

(4.18) .

we get ‘X

B = F(t)G2(t)j;r-2

‘xF(t)G(t)dG(t) I cc

FGzdt,

= -2

(4.19)

! rc

since G(x) = 0 = F(c). Next dG _=_ dt

(m + l)f(0

(4.20)

tv m+2(t)

so that (m + l)f(Y)

dY (m + l)f(t) tv ‘n’qt)

= 2(m + 1)2

‘* l-V(t) I C tvm+2(0

dF ‘x

dF(y)


dt

54

R. Bartoszyriski, S.M. Dynin / Tumor growth curve estimator ‘x

=2(m+1)2

dF(t)

c tv”+2

‘x (0

dF( y)

, YVMi2b9 (4.21)

Finally,

the first term,

by symmetry,

equals (4.22)

Using similar techniques, we can compute j,” S(t) dF(t) variance the expression given in Theorem 2.

5. Convergence

and

we obtain

for the

in the space D,

In this section,

we consider

QV (x) = @(h,

the sequence

(x) - EMx))

of stochastic

processes

(4.4),

3

in the space D of all functions with discontinuities of the first kind only. It is well known that D is a complete and separable metric space under Prohorov’s metric [see Prohorov (1956)]. As shown in the preceding section, we may write

DN(X)

&

=

dfl(I/,-

V)+X,dX)

where s”2b

ixNb)i

(5.2)

‘O

for every interval (a,6). In view of (5.2), all limiting finite-dimensional distributions with those of B&,(x), hence coincide with the finite-dimensional

_

‘X

‘X

!!

rC

u

dy Y2v”+l(Y)

dz°F’(y))

qf Q,,(x) coincide distributions of (5.3)

where 2’ is the Brownian bridge. Since VN is a step function, spanned by the sample xi, . . . ,x,, the function DN(x) is a function with a finite number of discontinuities of the first kind only. We shall let pN denote the probability measure on D corresponding to the N-th process. We shall prove:

R. Bartoszyhki,

S.M. Dynin / Tumor growth curve estimator

55

Theorem 5. The sequence [pN] converges weakly to a measurep such that p(C) = 1,

where CCD is the subspace of all continuous functions.

Proof. We have already shown the convergence of all finite-dimensional distributions to those of the measurep defined by (5.3). Since obviouslyp(C) = 1, it remains to show that the sequence [p,,,] is tight, that is (see, e.g., must show that for every ~1~0 one can find A such that

Billingsley

(1968)),

we

(1) pN{DN : IDN(tO)l >A} I q for all N (boundedness), and (2) for every q and E > 0, there exists a 6 with 0< 6< 1 such that

p,(D

for all N sufficiently wo(4

(5.4)

: w,(6) 2 E) I Y/

=

large.

Here (5.5)

SUP PC4 -WY)I xY lb-VI<6

is the modulus of continuity. For the first condition, it suffices to take to = c. For the second condition, let us write

P

sup

CCXCO~S

~D,(x+~)-D~(x)I>E 1

(5.6) We may take N, such that the second As for the first term.

term is bounded

above by +q for all N 2 N, .

we write

=S:i:+“+r:+“r:‘“=~+~, say.

(5.7)

We now write P(sup

IBN(x+6)-BN(~)I>~&)~P(sup

IAI>+e)+P(sup

lBI>Se).

(5.8)

R. Bartoszytiski, S.M. Dynin / Tumor growth curve estimator

56

Then

The supremum on the left is bounded from above by a random variable, which has the limiting Kolmogorov-Smirnov distribution. Choosing 6 small enough and N large enough, we can make the probability on the right-hand side of (5.9) less than $I?, say. The same reasoning

applies

to the term B, which completes

the proof.

References Atkinson,

E.N.,

function Brown, 72(l),

tumor

B.W. Brown

Math. Biosci.

B.W., E.N. Atkinson,

of human

growth

and J.R. Thompson

(1983). On estimating

the growth

67, 145-166.

R. Bartoszynski,

J.R. Thompson

rate from distribution

of tumor

and E.D. Montague

size at detection.

(1984). Estimation

L National Cancer Inst.

31-38.

Billingsley, Klein,

R. Bartoszynski,

of tumors.

P. (1963). Convergence

J.P.,

proneness Komlos,

of breast

tumors.

and sample

of Probability Measures.

and A.G.

James

(1986).

OSU Tech. Report

J., D. Major and G. Tusnady

variables Prohorov,

R. Bartoszynski

distribution

Yu.V. (1956). Convergence

Wiley,

New York.

Characteristics

of growth

#339.

(1975). On approximation functions.

of partial

sums of independent

random

A. Wahrsch. Verw. Geb. 32, 111-13 1.

of random

Teor. Veroyatnost. i Prim. 1, 157-214.

rates and metastatic

Submitted.

processes

and limit theorems

in probability

theory.