AGE DESTREBUTIONS FOR
MULTEPHASE BRANCHLNG PROCESSES
AND THEIR APPLICATIONS

Thesis for the Degree of Ph. D.
MECHiCM STATE UHWERSITY‘
WEN~HOU KUO '
{973

,4.
/‘ .......
........

 

LIB R A R Y
Michigan Stat:
University

‘—

This is to certify that the

thesis entitled
Age Distributions for I-Zultiphase Branching

Processes and Their Applications

presented by

hen-Hon Kuo

has been accepted towards fulfillment

of the requirements for
Ph.D. Statistics

 

degree in

 

(/ Major professor

Date Z/’ 9/73

07639

 
  
   

   
 

glNDING BY
‘ HUM; & sons
1 800K wow me.
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ABSTRACT
AGE DISTRIBUTIONS FOR MULTIPHASE BRANCHINC
PROCESSES AND THEIR APPLICATIONS
By

Wen-Hon Kuo

A supercritical age-dependent branching process is considered
in which the lifespan of each individual is composed of four phases

whose durations have joint probability density f(t1,t2,t3,t4).

Starting with a single individual of age zero at thme zero we con-

sider the asymptotic behavior as t a a of the random variable

2(4)(a0,...,an,t) defined as the number of individuals in phase 4

at time t for which the elapsed phase durations Y01,...,Y04,

..., Y11,...,Y'14,...,Yn4 of the individual itself and its first n

ancestors satisfy the inequalities Yij s aij’ i

j = 1,...,4. We also state an analogous result which defines (con-

= 0,...,n,

ditional on eventual non-extinction of the population) the asymptotic
joint distribution of the phase and elapsed phase durations of an
individual drawn at random from the population and the phase durations
of its ancestors.

The correlations between the lifespans of parent and daughters
are considered in a more general model in which we treat the popula-
tion as a multiphase, multitype branching process. Moreover, the

possibility of random cell removal is also taken into account.

Wenéflou Kuo

The application of the results to the analysis of cell
labelling experiments is described. Finally we include some

numerical results.

AGE DISTRIBUTIONS FOR MULTIPHASE BRANCHING
PROCESSES AND THEIR APPLICATIONS

By

Wen-Hon Kuo

A THESIS
Submi tted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

1973

TO MY PARENTS

ACKNOWLEDGEMENTS

I wish to express my thanks to Professor P.J. Brockwell for
suggesting the problem and for his patient guidance before and during
the preparation of this thesis. His comments and suggestions led
the way to theorems and simplified proofs.

I am also grateful to the Department of Statistics and
Probability, Michigan State University for financial support during

my graduate studies.

iii

Chapter
I

II

III

TABLE OF CONTENTS

INTRODmTION 0..0.0.0....0.0.0.0000...0.000.000.0000

AGE DISTRIBUTIONS AND LABELLING FUNCTIONS FOR A
SINGLE'TYPE MULTIPHASE BRANCHING PROCESS 00000000000

§l
§2
§3
§4

§s

§6
§7
§8

Notation .00....0..0.00.0000...OOOOOOOOOOOOO.COC0

(4)

Asymptotic Behaviour of M (ao,...,an,t) .....

Mean Square Convergence of 2(4)(a0,...,an,t)e-Ct

The Almost Sure Convergence of

-ct

~0,...,gn,t)e ..........................
Random Selection of a Cell From the Population
as a Whole .....................................
Generalization to Include Random Cell Removal ..
The Case of an Arbitrary Initial Population ....
Application of the Asymptotic n-fold Age
Distribution to the Analysis of Labelling
Experiments ....................................

AGE DISTRIBUTIONS AND LABELLING FUNCTIONS FOR MULTI-
TYPE MULTIPHASE BRANCHING PROCESS ..................

§1
§2
§3

§4
§5

§6

§7
§s

§9

Introduction ...........................s.......
Notation .......................................
Summary of the Known Results in Multitype Single-
Phase Branching Process ........................

, 4
Asymptotic Behaviour of M: )(go,iO,---.gn,1n:t)

Mean Square Convergence of

(4) t
Zi (

‘C
80’10’...’gn’in’t)e oooooooooooooooooooo

~

The Almost Sure Convergence of

4 -ct
2:)(8 ,io’...,gn’in’t)e .QOOOIOOOCOOCOOOOOCO
Random Selection of a Cell from the Population
asaWhOIC OOOOOOOOOOOOOOOOOOOOOCOOOOOOOCCOOCCCC
Application of the Asymptotic n-fold Age Dis-
tribution to the Analysis of Labelling
Experiments ....................................
Generalization to Include Random Cell Removal ..

iv

Page

12

19
21
24

26

29

29
31

34
4O

42

45

47

49
53

Chapter Page
IV A MODEL FOR CELLS OF THE CORNEAL EPITHELIUM ........ 58

§1 IntrOduction OOOOOOOOOOOOOOOOO0.00.00.00.00.00.0 58
§2 FLU Function and Some Numerical Results ........ 60

BIBLImRaAPI'IY 00......00000000000000.0.0....000000000000.0.0.0... 7o
APPENDICES

A INTEGRAL EQUATIONS FOR THE MEAN FUNCTIONS M(t\i,a)
MD U(t‘1,a) OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 72

B CLU FUNCTIm OOCIOOIOOOIIOOCOOOOOI0.000000000000000. 75

c ASYMPTOTIC BEHAVIOR OF THE FUNCTIONS M(t\i,a) AND

U(t\i,a) ........................................... 79

Table

LIST OF TABLES

vi

Page

66

65

LIST OF FIGURES

Figure Page
1 67
2 68

vii

CHAPTER I

INTRODUCTION

The life-cycle of many biological cells can be divided into four
phases numbered 1,2,3,4. During the second phase the cell is synthesizing
DNA and during the fourth phase the cell is in the process of dividing
(cells in phase 4 are said to be mitotic and can be distinguished from
cells in phases 1,2 and 3). The phases l,2,3,4 are often referred to as
61’ S, 03 and M respectively. In this paper we shall consider a
pOpulation of cells each having a 4-phase life-cycle as described above.
It will be assumed that the joint distribution of the four phase durations

T1,T2,T3 and T4 for any new-born cell is specified by a probability

density

f(t1,t2,t3,t4) = f(£) .
The distribution of total lifetime for any cell thus has the probability
density

g(u) = F it I f(t1,t2,t3,u-tl-tz-t3)dt1dt2dt3 .
t1 2 3Su

We shall suppose moreover that the process of cell multiplication is a
supercritical age-dependent branching process (see Harris [7], Chapter 6)

in which the lifetime distribution of any individual is specified by the

probability density 3 and the distribution of the number of offspring
per parent has probability generating function,

a

h(2) = E szk .

k=0
where the factorial moments h'(l) and h"(l) are assumed to satisfy
1 < h'(l) < m and h"(l) < m respectively. We shall assume also that
the density g satisfies the condition f[g(y)]pdy < m for some p
such that p > 1. (The latter condition is not used until the proof
of Theorem 4.2 of Chapter II.

The analysis of the model we have just described can easily be
modified to allow for the removal of cells from the population if the
removal process is as follows: for any cell, independently of its age
or phase and independently of other cells, the probability of removal
in any small time interval (t, t+5t) is x5t'+ 0(5t). The generaliza-
tion to this case of the results derived in Chapter II are indicated in

Section 6.

A powerful tool in the experimental study of the cell cycle is the
class of so-called "labelling" experiments, in which a radioactive substance,
e.g. tritiated thymidine, is injected into the population. If the in-
jection is a pulse administered at epoch t then the effect is to label

0

all cells which at epoch t are in phase 2 and no others. Subsequently

0
when the labelled cells divide they pass the label on to their offspring.
After injection of the pulse at t0 we observe the cells in phase 4

and count the fraction at epoch t0 +'t which are labelled. The resulting
function Pto(t), t 2 O, is known as the FLM- (fraction-labelled-mitoses-)

function. If instead of using a pulse at epoch to we inject continuously

from epoch t0 onwards and again count the fraction of labelled cells in

phase 4 at time t0 +'t which are labelled, we obtain a function C (t),
to
t 2 0, known as the continuous-labelling-function. For large to the

function Pt has been studied in the case of independent phase durations
O
by Barrett [1], Trucco and Brockwell [17], and Takahashi [16] and for a

more general model by Macdonald [9],‘who also considered the function

Ct . These analyses all assume that the pOpulation is large, that a
0

limiting age distribution has been attained by epoch t0 and that the
random variables involved may be replaced by their expected values. Note
that in practice we do not observe the fraction of 211 cells in mitosis
at time t0 + t which are labelled but only a sample of N mitotic

cells. The number of labelled cells observed in such a sample will be

(under our model) binomially distributed with parameters N and Pt (t)

. O
or Ct (t).
0
In Chapter II we show how the asymptotic forms P and C of the
functions P and C as t a m can be derived for the model
to to 0

defined in the first paragraph. The functions P and C. are obtained
from the asymptotic joint distributions of the phase durations of a cell
selected at random in phase 4 and its first n ancestors. (The cell
itself is said to be its own zeroth ancestor, its parent is its lSt
ancestor, etc.) For a population which starts at epoch zero with one
member of age zero we establish the (almost sure) existence of these
asymptotic joint distributions given that the pOpulation does not die out
and determine them explicitly. They generalize the so-called "carrier-
distributions" of E.O. Powell E4] (see also Brockwell and Trucco [4],

p. 172 for a heuristic derivation of the correSponding results for the

model with independent phase durations).

The purpose of deriving such theoretical expressions for P and
C is to use them to estimate parameters of the model from experimental
data. We do not consider this question here but refer the reader to
Barrett [2'], Brockwell, Trucco and Fry [5], Macdonald [9], Steel and
Hanes [15] and Mendelsohn and Takahashi DO] who deal with the estimation
problem for a variety of particular models.

There is experimental evidence to suggest that in many populations
correlations exist between the lifespans of parent and daughter cells.
In order to take this into account we consider in Chapter III a more
general model in which we treat the population as a multiphase, multi-
type branching process. We assume that there are m types of in-
dividual, the joint phase distribution for the ith type being Specified
by a joint density fi(t1,t2,t3,t4). The expected number of type j
offspring from a type i parent is m and the second moment of the

11

total number of offspring per parent is assumed to be finite.

CHAPTER II

AGE DISTRIBUTIONS AND LABELLING FUNCTIONS FOR
A SINGLE-TYPE MULTIPHASE BRANCHING PROCESS

§l. Notation

The model to be considered here (a single-type multiphase
supercritical branching process with correlated phase durations) was
defined in Chapter I. The following notation will be required in

addition to that already introduced.

Consider the population at time t descended from a single ancestor

(4)
k

in phase 4 at time t which are kth generation descendants of the initial

of age zero at epoch zero. Define Z (ao,t) to be the number of cells

cell and for which the times Y01,Y02,Y03,Y04 Spent (prior to epoch t)

in phases l,2,3,4 satisfy

(1.1) Toi saOi , i =1,...,4.

(We shall consistently use the vector notation Y s a as an abbreviation

for the set of inequalities Y0i S aOi between corresponding components

of Y and a Note also that the 0th generation descendant of a cell

0 ~0°
means the cell itself, a first generation descendant is a daughter cell,

(4)
k

number of cells in phase 4 at time t which are kth generation descendants

etc.) More generally we define Z (ao,a1,...,an,t), k 2 n, to be the

of the initial cell and whose 1th ancestor, i = 0,1,...,n has spent

times Yi1""’Yi4 in phases l,...,4 satisfying Yi 5 a1. The quantities

(4)

2k (a0,...,an,t) are random variables defined on a probability space
of family trees (whose construction is essentially the same as that in
Harris [6], Chapter 6). We shall also make use of the random variables

2 <4)

Z(t), the total pOpulation size at time t, (t), the number of cells

in phase 4 at time t, and

(4) = °° (4)
Z (30"°"3n’t) kin Zk (30,...,an,t) .

Our primary interest is in the ratio 2(4)(a ,...,an,t)/Z(4)(t),
particularly in its asymptotic behaviour as t _. on (given that Z(t) /-o O)
which leads to the asymptotic joint distributions of the phase durations
of a cell selected at random from those in mitosis and its first n

ancestors. (Clearly the analysis which follows could easily be modified

to treat the case of a cell selected at random from those in any particular
phase but bearing in mind the application to labelling experiments we
shall consider only cells selected from phase 4. A corresponding result
for sampling from an arbitrary phase is given in Section 5.)

We shall use M and S with apprOpriate subscripts and superscripts

to denote first and second moments of the corresponding random variables, e.g.

(4) _ (4)
Mk (30900'agnst) " E zk (fos°°°ssnst)a

4 4 4
SRL)(3 ,...,an,t,T) = E[Z: )(ao,...,3n,t)2: )

(20’...,Sn’t+T)],
M(t) = E Z(t), S(t,T) = E[Z(t)z(t+'r)],

“(4) (t) = E 2(4)“), 8(4)(t,'r) = E[z(4) (t)Z(4)(t+T)], etc.

These moments are known to be finite for all finite t, T because of

the finiteness of M(t) and S(t,T) (see Harris [7]).

(4)

§2. Agymptotichghaviour of M (a0,... an,t)

3
~

The following decomposition,

(2.1) M(4)(ao,...,an,t) = Mi4)(ao,...,an,t) +' =§+1 M£§)(ao,...,an,t),

~

(4)

1

leads at once to a renewal equation for M

t
(2.2) M(4)(EO,...,an,t) = M514) (30,...,an,t) + A]; M(4)(S ,...,an,t-u)g(u)du,

~ ~

where A = h'(l) > 1, g is the probability density of the total lifetime

of a newborn cell, and Mia) is easily shown (by induction) to be
(4) _.
(2.3) Mn (EO’°°°’En’t) —
n
j... A f<y )...£<y >1 (t-y -y - ..-y )
yisai, ii1,...,n, ”0 ~n [0,y04A804) 03 02 “1

$8 = ... 3
ij 0:], j 13 s
dyO ... dy .

Applying the standard asymptotic theory of the renewal equation to equa-

tion (2.2) (see Feller [6], p. 468 or Harris [7], p. 161) we find that as

t - co
(2 4 M(4)( K<4) cc “ (a )
. ) 30"°"En’t) ~ e 120 Y1 Ni
where
(2.5) Viki) = f ¢i(z)dy/f¢i(y)dy .
~ 2‘31 ~ ~
‘C(y1+...+y4) so
(206) ¢0(Z) = e I f(y1,y2,y3,U)du,
3'4
-c(y1+...+y4)
(2.7) ii(Z> = e r<z). 1 = 1.2....

8

-C(y1+. . .+ya) on
I3 [I f(Y1sY23Y3 a“) duldz

y
(2.8) R“) = _c 4
A [Y e yg(y)dy

and c is the unique positive root of the equation
(2.9) AJ'e'Cugmmu = 1 .

It is easy to show, using precisely the same argument (applied to

M(4)(ao,t) with all aOj = m), that the expected number of cells in

phase 4 at time t is given asymptotically, as t a m, by

(2.10) it“) (c) ~ 154)..“

with K(4) and c defined as above. Equations (2.9) and (2.10)
immediately suggest the following. If a cell is selected at random from
those in phase 4 at time t then the joint distribution of the times
Yij’ j = l,...,4, i = O,...,n spent by the ith ancestor of the cell in
phase j prior to time t has the following asymptotic form as t a m:
n
(2°11) Fn(ZO’°"’Zn) = PEZO S ZO"°°’Zn S Zn] = .fl Yi(Zi)
1=0
where the distribution functions Yi(yi) are defined by equations (2.5)-

(2.7). (Note that the functions Y i = 1,2,..., are identical.)

1’
We shall refer to the function Fn defined by (2.11) as the asymptotic

n-fold age distribution of cells in phase 4. It is reasonable to suppose

that conditional upon non-extinction of the pOpulation; the actual fraction

of cells in phase 4 satisfying I S f "'°’Zn 5 2n converges to Fn

for each n. The purpose of the following sections is to show that this

is indeed true in the following sense: First we recall (Harris [7],

Chapter 6) that under the conditions specified in Chapter I the total

9

population size Z(t) has the property that
(2.12) Z(t)e-cc « KW both in mean square (m.s.) and almost surely (a.s.),
where

A — 1

K = 2 _c
C A [Y e yg(y)d>'

and W is a random variable with expectation

EW = 1 ,
and variance
2
UW>02

moreover the events {Z(t) a O} and [W = O} are a.s. equal. We shall

show that analogous results hold for the random variables 2(4)(t) and

2(4)(a0,...,am,t); more precisely we shall show that

8.8.
-ct m.s.
_.

(4)

2(4) w ,

(t)e K

8.8.

) -ct mgs. K(4)

2(4) (a ,t e

~0,ooo,fn Fn(g ’...,fn)w

for all a. 2 0, i = 0,1,...,n and for all n = 0,1,2,..., and in fact

(30,...,an,t)e Fn(ao,...,an)w

for all a 2 O, i = 0,1,...,n and for all n B 0,1,2,...}a.s.

i

The methods used are similar to those in Harris [7], Chapter 6, but the
arguments are somewhat complicated by (a) the multiphase structure of the
process and (b) the simultaneous consideration of the elapsed phase dura-

tions of cells and their ancestors.

10
(4)(a

-Ct

§3. Mean Square Convergence of Z ..,a ,t)e

~0" ~n

4
It is convenient to decompose the second moment S( )(a0,...,an,t,T) =

(4) (4) . - .
E[Z (30,...,an,t)z (30,...,an,t+w)] In the following manner.

~

(4)(30,...,3n,t,7) = Q(t.r) + 2 2 8(4)

(3.1) S ,
iafiljafil U

(8 ’°",an)t’T)’

where 8:?) was defined in Section 2 and

(3.2) O s Q(t,T) s E[Zn(ao,...,an,t)2(t+7) + Z(t)Zn(ao,...,an,t+T)] .

. th .
(Here Zn(ao,...,an,t) 18 the total number of n generation descendants

of the initial ancestor in the population at time t.)

From equation (3.1) we obtain the renewal equation

(3.3) s“) (go.---.gm.t.r) = cum)

(4)

t
+ h"(1)]}4(4) (30’ - - - .gm.t-u)M (go. - - - .fm.t+r-u)s(U)du

t
+ A 33(4)(a0,...,Em,t-u,T)g(u)du .

Writing R(t,T) for the sum of the first two terms on the right side of

(3.3) it is not difficult to show that R(t,'1-)e"2m:-CT a

u (4) 2° -2cy
h (1)[K Fm(ao,--.,am)] Se g(y)dy uniformly for T 2 O as t a m.
(This follows from equation (2.4) and Schwarz's inequality applied to the

right-hand side of (2.2), noting that Zn(a0,...,an,t) s
n 4 ~ "
Zn(a0,...,an, 2 z a, ) and that E[Z(t)2]e-2cc «.M < m (Harris
" “‘ i=0 j=l 3
[7], p. 144).)

Writing §(4) for 8(4)e-2Ct-CT

we obtain from (3.3)

'§'(4)(a -2ct-cT

~0,...,am,t,'r) = R(t,T)e

t
+ g 3(4)(ao,...,am,t-U,T)A e-Zcug(u)du

11

whence, by Lemma 4 of Harris [7], p. 163, we obtain

h"(1)[K(4)Fm(g .---.gm)]2]e'2°ys(y)dy

-(4>
(3.4) lim 3 (30"°°’Sm’t’T) =

t-sco

 

m 2c
1 - [A e yg(y)dy

uniformly in T 2 0.
Starting from a renewal equation similar to (3.3) for
(4) g .
E[Z (30,...,am,t)Z(t+w)] C(ao,...,am,t,r) we can proceed in the

same way as above to obtain

h"(i)[R(4)Fm(ao,...,am)]R £8-2cyg(y)dy

 

-2 t-
(3.5) lim C(SO,...,3m,t,T)e ° CT =

t—oco

” -2c
1 - ]A e yg(y)dy

uniformly for T 2 0.

Equations (3.4) and (3.5) imply that as t ~ m,

(4) 2 -2ct

(4)
(3.6) E[(KZ (a ,...,3m,t) - K Fm(30,...,gm)2(t)) e ] a O ,

from which it follows that

(4)

-Ct “1.8.
c.

(3.7) 2(4) (30,...,am,t)e K Fm(30’...’fm)w 9

where W is the random variable appearing in (2.12).

12

(4)(fo.~-o.gn.t)e-Ct

We shall need the following decomposition of the random variables

Z(4) (a

§4. The Almost Sure Convergence of Z

0,...,an ,t):

(4) = e I ..
(4,1) Z (EO,...,an ,t) 2E (a. ’81,...,En’t) ‘ Z "(EO’EI’°°"En’t 804)

U
X(go’fl’ooo’gn,t,804) ,
I
where 30 denotes a three component vector (a03,802,a01) and

ZE(36,...,an ,t), Ze (a' ..,an ,t) are respectively the numbers of cells

~O’.
entering phase 4 in [0,t], [0,t) with ancestral phase durations

Yn1 satisfying Y' s a' Y s a Y s a

Y03,Y02,...,Y n3,Yn2, 0 0, ~1W1,..., ~n.

Similarly X(a

0,...,an,t,a) is the number of cells entering and leaving
phase 4 in [t-a,t] with Y' s fO’°'°’Zn s an.

Derivation of the asymptotic behaviour of the first and second moments

ME(36,...,an ,t) = E[zE (a' .,fn,c)] and 33(36,...,3n,t.w) =

~0,OO
E[ZE(a0',...,an,t)ZE (a0,...,an,t+T)] follows the pattern of Sections 2
and 3 so we shall simply write down the relevant renewal equations and

state the final results without proofs.

t
(4.2) ME(36..--.fn.t) = ME(35.....gn.c> + jME(35....,gn.c-u)g<u>du .

where
(4'3) ”g(35""’fn’t) = £ "' [ Apf‘Zo)'°'f(Zn)I[o,m)(t'yo3‘yoz'
yf ai,i= .,n
~0 ~0
...-yn1)dyo...dyn ,
and

(4.4) sE<a'.....a .c.w> = QE(t.T)
+h"(1)]NE(aO,. ...,an ,t- -u)MF(ao, ...,an,t+T-u)g(u)du

+ AISE (a',...,a ,t-u,T)g(u)du
~0 ~n

13

where

(4.5) O SQE(C,T) sE[ZE(aé,---.an .t)ZB (t+T) + 2B (02: (8' .an:t+‘r)]

0,...

and ZB(t) is the total number of births in [0,t], and z:(36,...,an,t)
is the contribution to ZE (a0,... ,an,t) of nth generation descendants of
the initial cell.

Lemma 4.1. Under the conditions specified in Section 1, as t a a,

 

 

(4.6) ME(36,...,3n,c)e'°t = pE(56,...,fn)[1 + 0(1)] ,
E 2 -2cu
_ _ h"(1)p (é'n-ufi) e g(u)du
(4.7) SE(a6,...,an,t,T)e ZCt CT = 0 _2C3 I [1 + 0(1)]:
" " 1 - A.Ie g(u)du
and
E - t . . E
“-8) 7- (36’--~3.2t>e ° ”*8 p saw-.29“ »
-C(y *7 “'7 )
y'Jsa' "0 "O n
where pE(a',...,a ) - "0 "O n Y1(a )
~O ~n -cy i
CAJY e g(y)dy i=1

Remark 4.1. If we apply Lemmas 3, 4 and 5 of Harris [7], p. 162, to
equations (4.2) and (4.4) we obtain the stronger result that the terms
0(1) in (4.6) and (4.7) can be replaced by o(e-et) where g is in-
dependent of t and T and g > 0. (It is here that we use the con-
dition j(g(t))pdt < m for some p > 1.) This fact is crucial in the
proof of almost sure convergence which follows. The argument follows

that of Harris [7], Theorem 6.21.1.

 

-ct a.s. E
Theorem 4.1. As ti» o, ZE (ao,...,an,t)e a p (36,...,an)w for all
a' 2 0, a 2 0,1 = 1,...,n and for all n - 0,1,2,...

0 1
Proof For fixed a' a let Y = ZE(a' a t)e"Ct and
_. ~0,O O O ’ ~n t ~o, O O I ,~n’

Y I p E(a' .,an)W. Then from Lemma 4~1 and Remark.4.1 it follows that

0’ '

l4

°° 2
SEWt - Y) dt < m and hence

(4.9) g(Yt - Y)2dt < m a.s.

Since ZE(a6,...,an,t) is non-decreasing in t it follows that,
(4.10) Y 2 e Y

Denote by (0,3,P) the probability space of family trees on which the
random variables are defined and let w denote a point of n. Then if

IT; Yt(w) > Y(m) it follows from (4.10) that

t-coo

(4.11) farm) - Y(u)))2dt = a. .

Similarly (4.11) holds if llE.Yt(W)‘< Y(w). Hence by (4.9) the set
tdm
[m E n : lim Yt(w) B lim'Yt(m) - Y(w)} has probability 1.
-Ct 8.8.

e E
Remark 4.2. 2 (a6,...,3n,t)e a p (3',...,an)W for all 26 2 9,

a. 2 0, i = l,...,n and for all n = 0,1,2,..., the proof of this

~

~

assertion being identical to that of Theorem 4.1.
We now turn to the somewhat more complicated problem of establishing
the a.s. convergence of X(a',...,an,t,x)e-Ct (see equation (4.1)). Let

m be the function defined on [0,m) as follows:

1 , u s B

(4.12) (p(u) =

e-o(U'B), u >‘B

where o,8 > 0. Consider now the random variable X¢(a6,...,an,t)

defined (a.s.) as the Lebesgue-Stieltjes integral,

(4.13) x‘i’(36,...,3n,t) = j¢(u)x(56,...,an,c,du)

~

3 aye-aux(369 ° ° ° :En’t ,LH’B)dU,

15

(the second equality can be established by integrating by parts). Notice

also that if 0 s B‘< x,

(4.14) x°"(55.....gn.t) s X(g(;...-.gn.t.x) + e'“(x'8)x(g(;.....gn.t.w)

. . Q g = ¢ I T '
Defining M (50,...,an,t) E[X (50""’fn’t)] and S (20,...,an,t,r)

= E[X¢(a6,...,an,t)x¢(a' ...,an,t+T)] we can write down the renewal

~O’
equations
t
(4.15) M‘P(36,...,gn,c) =Mff(36,...,3n,c) + [[NCP(36,...,gn,t-u)g(u)du ,
where
T I 3 n _ _ -
(4.16) Mn(ao,...,an,t) A f(yn)--of(zo)q>(t 3'03 3'02

' '.... =

Zo‘fo’Zi‘fi’i 1""’“'
+0..

y04+Y03 +"..2+"nl‘t

,,,-yn1)dzoo..dzn ,
and

(4.17) s¢(gg,....gn.t.w> = Q‘Pun)

t
+ h"(l)gM¢(36,...,an,t-u)M¢(26,...,gn,t+T-U)8(U)du

t
+ A.g8¢(aé,...,an,t-u,T)g(u)du ,

where

(P E [i B B E S
0 sQ (t,1') s 1?.[2n 30""’fn’t)z (t+1-) + z (t)zn(go,...,fn,t+r)] .

The upper bound here is the same as in equation (4.5). Although it could
easily be improved it is all we needto establish the following analogue
of Lemma 4.1.

Lemma 4.2. Under the conditions specified in Section 1, the quantities

‘Mw, S¢ and Xm satisfy the relations (4.6), (4.7) and (448) respectively

16

provided p E(a0,... ,an) is replaced by
-c(y0 4+3. .+y

' I
x0‘50

I e 01 [SN ¢(“+yoa)d“]f(Zo)dZo

 

(4.18) p¢(a',...,a ) = _
° “ A jy e Cyg(y)dy

n
Ilium.)-
i=1 1 "1

As in Lemma 4.1 the terms 0(1) may also be replaced by o(e-et) where

e > O and s does not depend on t or T.

Proof. Exactly the same as that of Lemma 4.1 and Remark 451.

 

(.p -Ct 8:.80 (p ' |
Theorem 4.2. X (a0,...,an,t)e p (50,...,an)w for all 30
a1 2 0, i = l,...,n and for all n = 0,1,2,... .

2 0,

Proof. The proof of Theorem 4.1 carries over without change once we have

established an inequality to play the role of (4.10). If we write

m -ct = pm
Vt= X (a0,. . .,an,t)e and V p(ao,...,an)w then the required
inequality is
-(c+a)T
(4.19) V£+T 2 e Vt

which follows from (4.13) and the fact that

X¢(a6,...,an,t+¢) 2 e"CT X¢(a',...,an,t) for all T 2 0 .

N

With (4.19) replacing (4.10) the proof is now identical to that of Theorem

4.1.

I
- t ass. 0,”

Corollary. X(a6,...,an,t,m)e ..,an)w where

pIEO ’m)(a0,...,an ) is obtained from (4.18) on replacing ¢ by the

indicator function I

[09”).
Proof. Take 3 I'm» in the definition (4.12) of m, in which case

XCP(a0 ,...,an ,t) becomes X(ao,..

Theorem 4.2 all remain valid.

Theorem 4-.3. X(a0,...,an ,t,x) ”4 pIEO’x](a %)w where

~0’
pI[0,x](ao ,...,an ) is obtained from (4-.18) on replacing m by the

.,an,t,m) and the arguments leading to

17

indicator function I

ro.x1'
Proof. Observe that for 0 s g.< x, equation (4.13) implies that

(4.20) e-CtX¢(26,...,fn,t) - e-a(x-8)e-CtX(g',...,fn,t,m)

~ct -ct — ’
Se X(g',...,an,t,x)5e xcp(36,...,an,t)

-a(U-X)

where '$(u) = 1 if u s x, e if u > x.

Letting t 1 w in equation (4.20) we find that a.s.

I
p¢( ..,fn)W - e-a(x-B)p [O’Q)(E6,...,an)w s lim_e-CtX(a'

~0,...,an,t,x)

a6,.

s lim e-CtX(3',...,an,t,x) S p¢(a' .. a )w .

3' !
tfiw ~O 2n

Now letting a t m then a 1 x we obtain the assertion of the Theorem.

(4) -ct , (4)
Theorem 4.4. P[lim Z (20"°°’En’t)e K Fn(ao,...,gn)w for all

.1i 2 o, i = 0,1,...,n and for all n =o,1,2,...] = 1, where K

Fn were defined by equations (4.4) and (4.11).

and

Proof. The fact that for each fixed ao,...,a ,

~

2(4)(a

~

(4)

'CC 8.8 o
..q

oa°°':gnst)e K Fn(so,...,a )W

~n
as t a m follows at once from Theorems 4.1 and 4.3 and Remark 4.2.
The stronger assertion made by Theorem 4.4 is a consequence of the a.s.

(4)

monotonicity properties of Z (a0,...,an,t) for each t, the continuity
of Fn and the denseness of the rationals in the real line.
Corollary. Given that Z(t) L’O as t a m, the ratio

2(4)(go,...,a t)

a ’

 

converges to the joint distribution function

2(4)(t)

F (a ,...,a ) for all a 2 0,...,a 2 O and for all n = 0,1,2,...,
n ~0 ~n ~ ~ ~n N

O
with probability 1.

18

Proof. We note first that Z(A)(t)e-Ct a.s. K(4)W. (The proof of this

fact is similar to but much easier than the proofs leading to Theorem 4.4
above.) Moreover it is shown by Harris|:7], p. 147 that the events
[Z(t) a 0} and {W - 0} are a.s. equal. The assertion of the corollary

is an immediate consequence of these facts.

19

§5. Random Selection of a Cell From the POpulation as a Whole.

The corollary of Theorem 4.4 gives the asymptotic joint distribu-
tion of YO,...,Yn for a cell selected randomly from those in phase
4 at time t. Analogous results could clearly be derived for sampling
from any particular phase. We now indicate the corresponding result

for sampling from the entire cell population.

Let Z(i)(a(i)a a1,. . .,an,t) be the number of cells in phase i at
time t for which the elapsed phase durations Y01,...,Y01,..., ,Y n4"°"Ynl
i
of the cell itself and its first n ancestors satisfy Y01S aéi),. ..,
8(1)
0 .0. S I
Y01 5 a01 ’ ’Yn4 ‘ an4’ ’Ynl anl

Then by the same arguments as we have used for the case i 8 4 it

can be shown that, conditional on Z(t).Fv 0,

z<i>(géi>

 

’31""’5n’t) d x(1) F§1)(8(i)

 

 

(5.1) Z(t) K ,...,a“) for all
~éi)2 20,...,an 2 0 for all n = 0,1,2.... almost surely,
where
'C()’ +. . .+y ) co
[e 1 1 [£ Ef(Z)]yi=udu]dy
(5.2) g(i) = _ i
A [y e Cyg(y)dy
and
’C(y1+- ”+7 1) on
(5.3) I (1)8 If [15(2)]yi duldz
(y 9 - ° ° #1)“ n
(1) (i) a 1 51
Fn (~0 9 o . o ’3“) _c(y1+. . .+yi ) a kill Yk(ak)

fa

[j [f(Z)] duldz

yl

20

(The function Y was defined in Section 2.)

k
Equation (5.1) defines in considerable detail the asymptotic
age—phase structure of the pOpulation as t a m, conditional upon non-

. (i) (i) - . .
extinction. We note that Fn (a0 ,...,a“) defined in (5.3) gives

the asymptotic jOint distribution of YOi"°"Y01’Zl"°°’Zn for a

cell selected randomly from those in phase i at time t.

21

§6. Generalization to Include Random Cell Removal.

We now consider the following modification of the process con-
sidered in Sections 1-5. Suppose that in addition to elimination of
cells by death at the end of the cycle there is "removal" of cells in
the following manner: for any cell, independently of its age or phase
and independently of other cells, the probability of removal in any
small time interval (t, t+5t) is th + 0(5t); the probability that
more than one cell is removed in (t, t+6t) is o(5t). This model
describes the growth of the population of proliferating cells in a
population where cells may differentiate and become non-proliferative
during the cycle. As before, c stands for the unique positive root
of equation (2.9). We consider the following three cases.

Case 1. “L;§_g. It is not difficult to check that if we again start
with a single cell with age zero at epoch zero but now define Z(t)

to be

)‘t

20:) = e .zm

where Z(t) is the number of cells in the population at time t, and

(4) kt

«a A
likewise Z (a ,a ,t), Z (t), etc. each to be e times their

~0’°°' ~n
definitions given in Section 1, then the whole of Sections 2-5 carries
over verbatim to the new process. We end this case by remarking that
the condition 1 < c is used at the place where we establish the m.s.
convergence.

Case 2. L_:_g. This is the case corresponding to the so called
critical branching process. The procedures used in Section 3 do not

hold in this case. In order to discuss the asymptotic behavior of

the random variable Z(t) we define Ik to be the number of cells

22

in the kth generation. Observe that the probability that a cell is removed
from the cycle before multiplication is 1 - f*(x). If we regard a

cell removed from the cycle as a death (giving no birth), then the

number of daughter cells per parent is a random variable with proba-
bility generating function H(s) = 1 - f*(x) + f*(x)h(s). Proceeding

the same arguments used in Theorem 5.1 and Theorem 5.2 of Harris [7],

pp. 127-128 with minor changes we conclude that the random variable

1k are a Galton-Watson branching process with generating function

H(s) and two events {Z(t) « 0 as t a m} and {IR is equal to

zero for some k] are a.s. equal. The above results, together with

the fact H'(l) = 1, imply that

P(Z(t) a O as t a w) = l .

Case 3. A > c. Since the loss rate A is greater than the growth
rate c, an immediate question to ask is "Will the population die out
eventually?" In order to answer this question we make use of the

following decomposition,
E D
(6.1) Z(t) =z (t) - Z (t),

where ZE(t) is defined to be the total number of cells entering the
cycle during the time interval [0,t] and ZD(t) is defined to be
the total number of cells leaving the cycle during the time interval
[0,t]. Again, using the method suggested in case 1, we obtain
immediately the following results.

Theorem 6.1. If we denote Mg(t) = EZE(t) and MD(t) = EZD(t), then

(i) M(t) -0 0 88 t -0 m

23

(ii) ME(t) an and MD(t) «B as t-aao,
where B = [l-AE e-xtg(t)dt]-1,
(iii) zE(c)-+wE and 2130;) aw” as taco,
where WE and WD are random variables,
(iv) EWE=EWD=B and P(VP<oo) =P(W‘D<co) =1
and
(v) P(Z(t) —oO as t ..oo) = l .
2593:. (i) and (ii) are the immediate consequence of applying the
standard asymptotic theory of the renewal equation to the equations
obtained by using the method suggested in case 1. (iii) follows from
the monoticity of the random variables ZE(t) and ZD(t). (iv)
follows from (iii) and the fact that B is a finite constant. It
remains to show (v). Equation (6.1) and (iii) together imply that

Z(t) converges to a random variable W, W =.QF - WD, as t approaches

infinity. Since Z(t) is a nonnegative function of t we have

w 2 0. This, together with the fact that EW = 0, establishes (v).

24

§7. The Case of an Arbitrary Initial Papulation.

We have assumed from Sections 2-6 that the process starts with
a single cell of age zero at epoch zero. In the following we indicate
that the corollary to Theorem 4.4 remains valid regardless of the
phase and elapsed phase durations of the initial cell and indeed of
the number of initial cells.
Case 1. One initial cell. Suppose at epoch zero the population con-

sists of a single cell with age a in phase i (age in phase i

0

being measured from entry into phase i). Assuming random cell

removal as defined in Section 6 and letting Z(t) denote the product

ext - Z(t) (where A = 0 if there is no cell removed) we have

* = M - -
(7.1) Z(t) e I (t U)I(_m’0)(t L)

(“330)

N
(L -U) ZZ i(t -U),

‘+ exUI (t-U)I
) i=1

(0 ,6) (O! m)

where U, L and N are independent random variables (U is the epoch
at which the initial cell completes inacycle, L is the epoch at which
the initial cell leaves the system before completing its cycle and N
is the number of daughter cells which it produces) while

{21,22 ,23,...} is a sequence of independent copies of the process

starting with a single ancestor of age zero. U,L,N,Z are all

1,22,...
independent and can be defined on a single product Space endowed with
the appropriate product measures. From equation (7.1) it follows

that, for l< c,

N
(7.2) e CtZ(t);{os 8' Redo.”U E W ,

25

where K was defined in equation (2.12) and Wi is the a.s. (and

m.s.) limit of e-ct Zi(t) as t a m, and, for A > c,

N

(HI) 2 zi<t-U>

(7.3) Z(t) = I(.m,o)(t”u)1(«m,0)(t‘L) + I(O:¢)(t'u)1(o,m) 1.1

which implies that
8. .
(7.4) Z(t)-.3 o as ts...

We observe that the two events {Z(t) « O} and {W1 = W2 =...= W = 0}
are a.s. equal.

The same reasoning shows that Theorem 4.4 and the results of

Section 5 remain valid provided the random variable Z(h)(go,...,an,t)

is interpreted as ext times the definition given in Section 1 and
N

the random variable W is replaced by e-(c-)‘)U 2 W1. The corollary
i=1

of Theorem 4.4 remains valid as it stands.

Case 2. K initial cells. If A < c and at epoch zero there are K
cells with given phases and ages (measured from entry into the given
phase) then Theorem 2.4.4 and its corollary and the results of Section

5 remain valid provided the random variable W is replaced by

N
K -(c-).)Uj j
e 2 W1
j=l i=1
where Uj is the epoch at which the jth initial cell divides (j=l,...,K)
and N1 is the number of daughter cells it produces. (The proba-

bility space on which the W 's are defined is now a K-fold product of

i

Spaces like that considered in case 1 of this section.) Similarly,

for the case 1 > c, Theorem 6.1 remains valid.

26

§8. Application of the Asymptotic n-fold Age Distribution to the

Analysis of Labelling Experiments.

 

To make the results derived in this section more readable, we

use the following notation: We denote by f the

1,2,3,4(ti’t2’t3't4)
joint probability density of the four phase durations T1,T2,T3,TA

for a new born cell. Marginal probability densities will be denoted

in the usual way, e.g. f2 3 for the density of T2 and T3, and the
9

joint densities of sums of phase durations will be indicated by dele-
tion of apprOpriate commas in the subscripts. Thus for example, the

trivariate joint density of T1. T2+T3: T4 “111 be denoted f1’23,4

and the bivariate denSity of T1+T2, T3+T4 will be denoted £12,34'

We will still use g to denote the probability density of T1+T2+T3+T4.

Laplace transforms will be denoted by a superscript *. Thus for

example

_* Q G
<8» I31,23,4(01’Bn0‘g g.-au-ev-w.

G—fia

1’23,4(u,v,w)dudvdw .

The Laplace transform of a function of several variables will some-
times be taken with respect to a subset of the variables. The
particular subset involved will be indicated by underlining the

appropriate subscripts, e.g.

-cu-(c+s)vf (u,v,w)dudv .

* new
(8.2) flJ2234(c,c+s,w) = g &e 1,23’4

Suppose first that the label is administered continuously to a cell
population (growing according to the model we have described) from time
t onwards. All unlabelled cells entering phase 2 after time t0 will

0
then become labelled and all offspring of labelled cells will be labelled

27

from birth. Let Ct (t) be the fraction of cells in phase 4 at time
0

to +’t which are labelled. Assuming that t0 is large and that the
observed pOpulation is a realization of the process which does not die
out, then for each fixed t as to a m, Cto(t)' is just the asymptotic
probability that a cell selected at random from those in phase 4 entered

phase 3 at time t or less prior to selection, i.e.

£- Fo(da )
804 80351:

(8.3) lim c (t)
t
todw O

 

t _ Y
i C * £9 CYiEf:2 3(c,u)-f:2’34(c.u)]dudy,
£123 (C) -8 (C) .., —

where F0 is the distribution function defined by equation (2.11).

Similarly for pulse labelling (administered at time to) the fraction

of labelled cells in phase 4 at time t0 +-t satisfies (under the assump-

tions of the previous paragraph)

(0

(8.4) lim Pt (t) = Z P (t),

t do 0 v=0 V

0
where

= ,...,d
(8.5) Ple) OL-a -a - -a -a A Fv(d§ ,dfl Ev)
04 03 "° v4 v3 v2

The probabilities Pv(t)’ v = 0,1,2,... can be calculated by carrying
out the straightforward integration. If we take Laplace transforms in
(8.5) we find that

* * -1
C[f123(C)-f (6)] * *
c(s+c) [£12,3(c’9+°)‘fi,23

 

(8.6) P;(s) = (c.s+c)

* *
-f12,34(c,s+c) + £1,234(c,s+c)]

and

28

(8. 7) p* (s) = ““8 (“‘91 [f:23(s+c)-g*(s+c)][f:2 34(mm)
c<s+c)[f123<c) -g (c)] '

*
- f

1’234(c,s+c)], v = 1,2,...

Summing over v = 0,1,2,... we obtain

* * -1
CEf (C)-s (C)
(8. 8) ge 3‘ lim P (t) = 123 * ] {[l-Ag*(s+c)][f:2 3(c,s+c)
t0 s(s+c)[l-Ag (s+c)] ’

 

*
- f1’23(c,s+c)]-[l-Af:23(c+s)][f:2,34(c,c+s)-f:,234(c,c+s)]} .

It is easy to check that the result (8.8) reduces to the result of Trucco
and Brockwell F17] when the phase durations are independent and that both
(8.3) and (8.8) reduce to the results of Macdonald [9] when the duration'

of phase 4 is independent of the durations of phases 1, 2 and 3. (Macdonald
considers pulse-labelling with a pulse of non-zero duration 5. The
labelling function for such a pulse administered over the interval

[t0,t0+5] is obtained from (8.5) on replacing an2 in the inequalities

defining the range of integration by an2 +'6-)

CHAPTER III

AGE DISTRIBUTIONS AND LABELLING FUNCTIONS
FOR MULTI-TYPE MULTIPHASE BRANCHING PROCESS

§1. Introduction

The model considered in the previous chapter assumes that every
cell has the same phase-duration distribution. In this chapter we shall
remove this assumption to make the model more general. Consider a
population consisting of m (m > 1) types. We assume that the life-
cycle of a cell of the ith type, i = l,...,m, is composed of four
phases whose durations are random variables with joint probability

density function

The distribution of total lifetime for any cell of type i thus has

the probability density function
(1.2) gim) = i i Ifi(t1,t2,t3,u-t1-t2-t3)dtldt2dt3 .
t1 t2 t3$u

At the end of its life a cell of ith type gives rise to a random
vector N = (N1,...,Nm9 of numbers Nj of cells of type j,
j = l,...,m. Each new-born cell then goes through the cycle appropriate

to its type. We denote the probability distribution of the offspring

vector N of a cell of type i by

29

30

(1.3) Pi(n) = Pi(N = n), where n = (n1,. . m j —

and j = l,..,,m,

and the generating function by

n
(1-4) hi(S) = E Pi(g)(§)~ ,
n
n " m n,
where s = (s ,...,s ), (s)~ = H s.1 and s. are complex numbers
~ 1 m N i=1 i 1

such that ‘Si‘ 5 l, i = l,...,m.

The daughter cells of any parent are assumed to behave independently
of each other. Starting with a new-born cell of ith type, it has
been shown that

. ‘Ct
P{lim Ei(t)e = (3} -.= 1

t-«n

where Zi(t) = (Zi(1,t),...,Zi(m,t)), Zi(j,t) is defined to be the
number of cells of type j at time t and W is a random vector.
We will obtain the analogous results which will determine the
asymptotic type-phase-age distribution of the population for large
values of time t. This distribution will play a very important role

in determining the FLM function which will be discussed in Section 8.

31

§2. Notation

Consider the population at time t descended from a single

ancestor of type i having age zero at epoch zero. Define

2:2)(a0,io,t) to be the number of cells of type i0 in phase 4 at
time t which are kth generation descendants of the initial cell and
1Lo 1.Lo 10 1o
for which the time 201, Y02’ Y03’ YO4 spent (prior to epoch t) in
phase l,2,3,4 satisfy
10 10
(2.1) YOi ‘ aOi , i = l,2,3,4 .

i
We shall again use the vector notation YO0 3 a0 as an abbreviation
i ~ ~
for the set of inequalities YOi s aOi between corresponding components
i
(4)

O .
of IO and 20' More generally we define Zki (ao,io,...,an,in,t),

k 2 n, to be the number of cells of type i in phase 4 at time t

O
which are kth generation descendants of the initial cell and whose

jth ancestor was of type i
i 1 3 1

le,...,ng in phases l,...,4 satisfying Y j s a
(4)

zki

space of family trees (whose construction is essentially the same as

, j 8 0,1,...,n and spent times
1' The quantities

(a0,io,...,an,in,t) are random variables defined on a probability

that in Mode [13]). We shall also make use of the random variables
Zi(j,t), the number of cells of type j at time t, 2:4)(j,t), the

number of cells of type j in phase 4 at time t, and

(4) . ° (4) .
(2.2) 21 (20,10’...,fn’in,t) 8 kgnzki (209109.0’93n’1n’t) '

We shall also use the following vector random variables.

(4) . g (4) . ,
(2.3) 21 (go’31’11,000’2n,1n,t) (Zi (50,1,21’11’000’5n’1n,t),
(4) .
000,21 (go’m’gl’il,ooo’En,1n’t)),

32

(2.4) 29%;) = (zg )(1 c).....z ( )(m m ,
and
(2.5) 31m = (zi<l,t),....zi(m,t>> .

where m is the total number of different types considered in this
model. Our main interest is to find the asymptotic joint distribution
of the types and the phase durations of a cell selected at random from
those in mitosis and its first n ancestors. A corresponding result
for sampling from an arbitrary phase is given in Section 7.

We shall use M and S with appropriate subscripts and super-
scripts to denote first and second moments of the corresponding random

variables, e.g.

Méi)(go’il’...,gn,in,t) =Ez 2(4)(go,i ’10,...,an,1n ,t),
(4) . g (4)

Mi (30,10....,gn.1n,t) E2 21 (50.10,...,3n,in,t),
31:2)1(80 10,...gan,1n:t :T) =EEZé:)(ao,io,...,gn,in,t)

Z(4)
2&1 (ao,i 0,...,

an’in ’tl'l)]’
(a) ___ (4) -
Mi (.1 at) E 21 (j at) a Mi(jst) "' E Zl(j at) s

8:28....» = Biz 2(4)“ c>z“"<k Hm.

Si(j,k,t,'r) = E[Zi(j,t)Zi(k,t+'r)], etc.
Si(j,t,T) ‘2 Si(j:j3taT),

Those moments are known to be finite for all finite t, 7 because of

the finiteness of Mi(j,t) and Si(j,t,T), provided “M“ and

33

Max “ijk (to be defined later) are finite (see Mode [11]).
ifij,k *
We shall also denote M 3 (mij) and H(x) 8 (gi(x)mij), two
*
m X m matrices with ijth element m1:] and gi(),)mij respectively,

where the superscript * stands for Laplace transform and

m. . = Djhi(§)\

1] s=(1,...,1)

with Djhi(s) standing for the first derivative of the function

hi(s) with respect to the jth element in the vector 8. We set

b1 (i) = (-1) ”hum, where A1

1 j(x) is the complementary minor of
the ijth element in I - H(x) and I is an identity matrix, A<l> =
\I - H(x)\, the determinant of I - H(x), and [130)]ij = hji(1), the
adjoint of the matrix [I - H(x)].

We shall use the matrix norm “M“ defined by
(2.6) us“ a Max 2 ‘mijl
i J

2 2
jkhi(g)‘§=(l,...,l)’ where Djk stands for the

second-order partial derivative of the function h (s) with respect

and denote ”ijk - D

to the elements 31 and 3k in the vector 3 = (31,...,sm).

34

§3. Summary of the Known Results in Multitype Single-Phase Branching

Process

We summarize some of the known facts in the following.
Lemma 3.1. (a) If M is a matrix of positive elements (every element
is positive) and the Perron-Frobenius root (unique positive largest
eigenvalue) of M is greater than 1, then there exists a positive
number c such that A(c) = O. The number c has the following
properties.
(b) The Perron-Frobenius root of the matrix B(c) is l and correspond-

ing to this root there are positive left and right eigenvectors n and

N

E such that B = flfl(c), “(C)E.= E. and flli = l.

(c) c is the root of the determinantal equation 4(1) = O with largest
real part and has multiplicity 1.

(d) H(c)B(c) = B(c) and B(c)H(c) = B(c).

(e) B(c) = “i3 = Kfliinj) for some positive constant R and the
positive numbers pi and “j are the ith and jth elements in the
vectors E and n.
grggf; See Theorem (3.1) of Mode [11].

Remark 3.1. The condition that every element in M is positive can
be relaxed to some extent. All we need is to assume that M is a
matrix of nonnegative elements with at least one positive element and
that there exists a positive integer n such that Mn is a matrix of
positive elements (see Karlin [8], pp. 469-484).

The above lemma leads to the following result.

Lemma 3.2. If M is a matrix of positive elements such that the

Perron-Frobenius root of M is greater than 1 and if gL E Lp for

35

some p > 1 for all i = l,...,m, then

(3.1) Mi(j,t) .-. cijeCtu + 0(e'et)], t _. on for some 6 > o,
where

b‘i(c) m -c
(3.2) cij = mg e ya - cj(y))dy .

3
defined in Lemma 3.1 and A'(c) is the derivative of A(x)

at c.
Proof: See Theorem 3.2 of Mode [11].

It follows from (e) of Lemma 3.1 that

. °° *cy ..
(3.3) cij chinjie (1 Gj(y))dy

k
A'(C)

 

where d =

Now we let dij be the ijth element in the matrix
and also use the following notation;

m

- d2 * 2
(3.4) d, - 81‘ c)L’E‘1uupLuk .
m
(3.5) c. = Z d. d ,
1 M u. I.
Q -cy
(3.6) vj = 81 g e (1 - cj<y>)dy.
(3.7) wins) = 210,06“,

and

(3.8) 51m = <w1(1.c).....wi(m.t))

y
C (y) a £3j(t)dt, bij(c) and A(c) were defined in Section 2, c was

evaluated

(I-u<2c))'1

36

Lemma 3.3. If M is a matrix of positive elements such that the

Perron-Frobenius root of M, is greater than 1, 3i E‘Lp for some

p > 1 for all i = l,...,m and Max u < a, then
ijk
2 + 1:1:k
. Ct CT
(1) Si(j:kataT) ~ CiVije 9 t “ w.

(ii) As t a m, E[Wi(j,t+n) - Wi(j,t)]2 » 0 uniformly in T, T 2 0.
(iii) Wi(j,t) converges in mean square to a random variable ”1(1)

as t d'm.

(iv) E w1(j) - cij.
2 2 2
(v) Var Wi(j) = (C1 - d ui)vj

(vi) Var Wi(j) > O.

Eroof: The proof of (i)-(v) can be found in Theorem 3.2, Mode [12].
It remains to show (vi). It has been shown that Mi(j,t) and

Si(j,k,t,T) satisfy the following integral equations

t m
(3.9) M,(J.c> = 611(1 - ci(c)> +' z miva<j.c-u>gi(u)du
v-l
and
t m
(3°10) Si(j9k9t9T) = fljk(t’T) +' 2 mivsv(j’k’t-U’T)gi(u)du
v=l
where
(3.11) lim f (t ¢)e'(2°t+“T) = E u c c 8*(20)
tax 11k , v v'=l ivv' Vj V'k 1

Applying (iii) to equation (3.9) and (3.10) and using the fact that
* * 2
81(2c) > (gi(c)) we obtain

2 m
(3-12) E[Wi(1)] > 8.
v,v =
va'

[* 2+
$1.1»ngijch 81(C)]

m

2 * 2
Vglcuivv + miv)E[wv(j)] [g1(c)]

37

and
3 2 m * 2 m [at 2
(3-1 ) [E "1(j)] ’ [Vilmivcvj8i(c)] a V Eg'lmivmiv'cvjcv'j 8i(c)]
v¥v'
+ gmz EEW (1)121: [(6)]2 -
v=l iv V 31

By the assumption that the daughter cells of any parent behave in-

” '. o 2
dependently we have uivv' mivmiv' for v i v Inequality (3 l )
2
. - - 2 ,
and (3 13), together with the fact that uivv miv miv O

establish (vi). This completes the proof of the lemma.

The result (vi) in Lemma 3.3 is used to show that {21(t) ~ 0]
and {Wi = 0} are a.s. equal, where Wi is a vector random variable
with components Wi(j).

(n) th
I be the number of cells of j type belonging to

J
(n) ISO) ;

the nth generation. It can be shown that {1(n) 8 (I1 ,...,

Now let

n = 1,2,...} forms a multi-dimensional Galton-Watson process with

generating function

(3.14) B = (1.1,. . . ,hm)

and that the following result.

Lemma 3.4. Let A be the event In = g for some n 2 1, given that
at time zero there is a new born cell of type i and let B be the
event that Ei(t) is equal to zero for some t > 0. If “M“ < m,
then P(A) -P(B) for i = l,...,n.

2399:; See Theorem 4.1 of Mode [13].

Corollary. P{Ei(t) 4'9, as t ~»m} is the smallest root q of the

~

equation

(3.15) r = h(r) 0

Proof: See Theorem 7.1, p. 41, of Harris [7].

Consider the moment generating function of W (t) and Wi

-sW 1T“) . -sW'Ii:“'i
given by Qi(s,t) = E e "~ and Qi(s) = E e "" , Re(s) 2 0.

Let Q(s) = (Q1(s),...,Qm(s)). The first jump equation for
Qi(s,t) is obtained by considering the time at which the initial cell
first leaves the cycle. Thus

ct
'3 e

a Q t .-
(3.16> Qi(§.t) = e 1 {giflfldu +j'hi(Q(§e °“,c-u>>gi(u>du .
0

The m.s. convergence of Wi(t) to Wi implies that Q1(s,t) -.Qi(s)

as t a Q. So we obtain from (3.16) that
(3.17) Qicg) - £h1(Q(ge-cu))g1(u)du .

Note that equation (3.17) can be found in Theorem 3.2 of Mode [12].
It is obvious that Zi(t) a 0 implies that W1 8 0. We want to show

that these two events are a.s. equal. This can be done by showing that
(3.18) P(Wi = 9) = q1
where qi is the probability of extinction of Zi(t), i.e.

qi =P(Ei(t) 49 as t -+ on).
If we let P(Wi = 9) = fii, then, by letting a a m in (3.17), we obtain

(3.19) a. = him) .

where a = (81,...,&m).

39

It follows from (vi) of Lemma 3.3 that W1 is non-degenerate,
so we cannot have ii = l for all i, and hence from Corollary of
Lemma 3.4 we must have ai = q1 for all i. This completes the proof
of the following lemma.

Lemma 3.5. If the conditions of Lemma 3.3 are satisfied, then the
two events {Zi(t) a 9 as t a m} and {W1 = O} are a.s. equal.

This lemma will be used to obtain the asymptotic joint type-
phase durations distribution. Furthermore it can be shown from
equation (3.17) (see Theorem 3.2, Mode [12]) that
Lemma 3.6. If the conditions of Lemma 3.3 are satisfied, then there
exists a scalar random variable W1 and a positive constant K1 such

that

(3.20) wig) = KlWivj for all j = l,...,m .

Now we state the last lemma of this section which has been
shown in Theorem 4.1 of Mode [12].
Lemma 3.7. If the conditions of Lemma 3.3 are satisfied, then

W1(t) « W1, with probability one, as t a m.

40

§4. Asy_ptotic Behavior of ME4 )(a,i a ,in,t)

...’01°0" ".,n

The equation (2.2) leads at once to an integral equation for

Ms) ,
l

(4) . . __ (4) .
(4.1) Mi (30.10....,gn,1n.c> 'Mni (30.10.-u.gn.in,t) +

t m

(4) . _
2 “‘1va (8: 91'0" ’gn’ln’t u)gi(u)dua
v=l
(4) . .
where MM is easily shown (by induction) to be
M(4) _
(4.2) Mni (a0,i 0,. ”’2n’ ”in t) -

5. m ...m . ......... f (y )...f (y )x
lln inin-l 1110 I I in .,n i0 ...0

Z) sgj,j=l,...,n.

Yojsaoj ’jal $293

I (t‘yoa‘yoz '°°°' yn1)d2'.o °°°dZn
[0'Y0443043

If we use exactly the same arguments as Mode [11], pp. 11-14, we

obtain that as t -1 co

 

(4.3) M§4)(go.i 10.....an,1 .c)~ ( )<go.1o,....gn.1n)e°‘ .
where
C(4) s (4)
(4,4) (80,10,...,8n ,1“) K1 F “9(20 103°°°afn31n)9
“:nm j"nin-l.” mi110 n
(4.5) Fn(30’1o’°'°’5n’in) - kgb Yik.k(3k) .
1." 11‘6ka 0(y)dy

k=l

41

(4.6) YL’k(gk) = y£a *L,k(z)dZ .
-C(y1+. . .4‘y4) on
(4.7) $11,0(Z) = e j‘ fL(yl,Y2,y3,U)du 9
y4
-c(y1+. . .474) .
(4.8) ¢L,k(Z) = e £L(Z)’ k = 1,2,...,
(4) _ m .
(4.9) Ki — K U1 1: nk f 1k 0(y)dy/A (c),
k=l ' ” "

and c is the unique number determined in Lemma 3.1.
It is easy to show, using precisely the same argument, that

as t—+oo

Kw)
(4.10) Mi“)<j.c> ~ —.*—1 $1”qu e“t .

Hence we obtain the asymptotic behavior of the expected number of cells

in phase 4 as t a m

(4)

1 (t) ~ K?) e“ .

(4.11) M

Equations (4.3) and (4.11) immediately Suggest the following.
If a cell is selected at random from those in phase 4 at time t, then
I
the joint distribution of the type IR and the times Ykkj’ j
9

t
is an integer between 1 and m, spent by the k

= l,...,4,

h
k = 0,...,n, Ik

ancestor of the cell of type I in phase j prior to time t has

k
the following asymptotic form as t a m:
i i

. . = = . o = -
(4.12) Fn(20310’°°"zn’1n) P(IO 10’ 30 ‘ 20,00031‘1 1“, .Y-n 0'“

We shall again refer to the function Fn defined by equation (4.4) as
the asymptotic n-fold age distribution of cells in phase 4. This dis-
tribution is essential in deriving the FLM function which will be dis-

cussed in Section 8.

42

(4 )

§5. Mean Square Convergence of Zi (801 io:---:an,in,t)e-Ct

It is convenient to decompose the second moment

(4) . (4) (4) . .
Si (ao,io,...,gn,in,t,T)= --E[Zi (a0,i 0,. ..,an,in,t)Zi (20’10’°°°’gn’1n’t+T)]

in the following manner:

(5.1) 8:4)(3 .10.....gn.1n.t,1> =Qi(t,'r)

m m

+ 2 E S

(4)
(a,
k=n+l.L=n+l k4

~09109-009gn9in9t97)9

where

(4)
skm

was defined in Section 2 and

(5.2) 0 s(21(t,T) s E[Zni(a ,i

~O 990°93n91n9t)zi(io9t+7)

o
21(10,t)zni(50,10,...,gn,1n,t+w)]

From equation (5.1) we obtain the following integral equation

(5.3) SE4)(30:10,...,En,in,t,T) =(21(t,T)

" m M4() (4)
+ £ 2' ”ivv'M v (a0”10""’an’in’t-U)Mv' (a0, ,i0,...,an ,in ,t+¢-u)
v,v =1
31(U)du
t m (4 )
+ Vilmivsv (a0, i0,...,an,1n,t-u,T)gi(u)du .

Writing Ri(t,T) for the sum of the first two terms on the right side

of equation (5.3) and letting R;(t,T) = Ri(t,T)e-ZCt-CT it is not
difficult to show that
— “‘ C4() (4)
(5.4) lim R.(t,T) = 2 n (so, 0,...,an ,in )Cv' (a ,i ,...,a ,i n)
1 I ivv CV ~0 0 ~n
tam v,v =1

*
81(20)-

The convergence being uniformly for T 2 0.

43

Now letting 5:4)(a0,i0,...,an,in,t,w) =
(4) . -2ct-cT ,
i (a ,i0,...,an,in,t,T)e and applying Theorem 2.2 of Mode

[12] to equation (5.3) we obtain under the conditions of Lemma 3.3 that

. . ' -l—
(5.5) lim S(4 )(ao,10,...,an,1n,t,T) = (I - H(2c)) R(t,7),
t N N
where g(4) and fil are column vectors with ith component 3(4 )

and E; respectively. If we let

(4) n
(506) (80,10 ,...,a n’i”) nim . i 000 mi i. [1 Vi ’k(ak),
n 1n n-l 1 O k=0 k
where Y1 R was defined in equations (4.6)-(4.8), then it follows
k’

from equation (4.4) that

(5.7) 0:4)(30,10,...,3n,1n)=du1(v4‘)(ao,io,...,31,1n)

and from (5.4) that

di[v(4)

. -' _ . . 2
(5.8) lim Ri(t,T) — (30’10’°°°’3n’1n)]

tam

where di was defined in equation (3.4).

So we see from (5.8) that the ith element in the column
. (4) . 2
vector in (5.5) is ci[v (30’10’°°°’3n’in)] , where ci was de-

fined in equation (3.5). Thus we have

_ (4) 2
900°:an91n3tfl') - CiEV (20,i0’...’gn’i )1 °

(5.9) lim 3(4 )(ao,i n

i 10

t-ocn
The convergence is uniform in T, T 2 0.
It follows from (5.9) that

Theorem 5.1. 'If M is a matrix of positive elements such that the

 

Perron-Frobenius root of M is greater than 1, 81(t) E Lp for some

44
p > 1 for all i = l,...,m and Max pi k < m, and if we define
i j, k
w(4)

w].

. _ (4) , -ct
(a O’i0""’gn’1n’t) — Zi (a ,i0,...,an,1n,t)e , then

(i) As c _. co, EDI?) (a

”0:10,. 0 ° ainsinat'i'T) '

 

Wia )(30,i 10,. °°’En’in’t)]2 « O uniformly in T 2 0,
(ii) W(4)(go,i 10,...,an,in ,t) converges in mean square to a random
variable W44 )(a, 10,...,an,in) as t a m, and
(4)
K
... (4) . . i . . .
w ’ ’00. g = ’ ’00., , o o
(111) 1 (30 10 ’3n 1n) ci 1 Fn430 1o in 1n)wi(10) a s ’
0

where wi(io) was defined in Lemma 3.3.

Proof: (i) and (ii) follow from (5.9) and completeness of Lp space.

(iii) can be established by writing down the integral equation for

(4)(

E[Zi a ,in ,t)Zi(i0,t)] and using the same arguments as

(2’0 ia‘o’...’n
in this section to deduce from these that

E[Cii wi(io,t)-K§4)

O

; 2
F n(ao,io,...,an,in)wi(ao,io,...,an,1n,t)] a O as t a a.

Since the procedures are quite similar and lengthy, the proof is omitted.

Remark 5.1. By using (3.20) we can rewrite (iii) of Theorem 5.1 as

follows.
4
(5 10) W(4)(a i a i ) = KIK: ) F (a i a i )Q a s
o i NO, 0,ooo,~n, n dJ‘i n~0, 0,.oo,~n,.n i o o

4)

. - ‘t
§6. The Almost Sure Convergenc1. of Z( )(a0,i0,...,an,1n,t)c (

 

4
As in Section 4, Chapter 2 we decompose z: )(a0,io,...,an,in,t)

as follows:

(6.1) 2:4)(a ...,En,in,t) = Z§(a'

0,io,al,i1,...,an,in,t) -

e . . .
i ~0’1031’11’°°"i‘n ’ln ’ $04) X (a0,10a1, 1’°”’En"n’t'ao4)’

I
where 30 denotes a three component vector (a01,a02,a03) and
2E i(a' i a in ,t) Ze(a' i a i t) are res ectivel the
~0’ 0,...,4.“ ’ 3 ' o’o"°”n’n 3 p y

number of cells entering phase 4 in [0, t], [0, t) with ancestral phase

i
1 i0 in ini .150 Y1

. n
durations Y03’Y02’°“’Yn3’Yn2’Yn1 satisfying Yo $30, ~1 s

i
fl"°°’Znn s gn' Similarly X(a',...,an,in,t,a) is the number of
.0

cells entering and leaving phase 4 in [t-a,t] with Y0 S 80,. ,

Derivation of the asymptotic behavior of the first and second
moments of these three random variables follows the pattern of Sections
4 and 5 (alSo see Section 4, Chapter 2) so we shall omit the pro-
cedures and simply state the following result.

Theorem 6.1. If the conditions of Theorem 5.1 are satisfied, then
(4)
K K

(4 ) —c 1 1 . .
P{li.mZi (a0,io,...,an, in,t)e d“ F n(ao, ,i0,...,an,1n)Wi for
t—vao i
all a1 2 0, i = 0,1,...,n and for all n = 0,1,2,...} = l, where
K1 and Fn were defined in (3.20) and Fu was defined in (4.5).

Similarly, under the conditions of Theorem 5.1,

t=KKui

(6.2) lim.Z§4)(j,t)e-C X1?2751"“ji*j 0(y)dy Wi a.s. (see

t-m

equation (4.10)).

If we let 2(4 )(t) = 2 m2(4)(j,t), then it follows from (6. 2) that
j- =1

K K“)
(6.3) lim 2(4)(t)e-Ct = —L-1—— fi. a.s.
t-m 1 (14,1 1

Hence we obtain the following result.

Corollary. Given that Zi(t) A 0 as t a m, the ratio

4) . .
2: (80,10,uoo,an,1n,t)
~ (4) " converges to the joint distribution
Z
l

 

(t)

Fn(fo,io,...,gn,in) for all a 2 0,...,a 2 0, 1n = l,...,m and

0 ...n

for all n = 0,1,2,... with probability one.

Egggf: It follows from (6.3), Theorem 6.1, Lemma 3.5 and Lemma 3.6.
Remark 6.1. We have assumed throughout that the process starts with a
single cell of type i with age zero at epoch zero. It is not dif-
ficult to see that the preceeding Corollary remains valid regardless

of the type, the phase and elapsed phase durations of the initial cell

and indeed of the number of initial cells (see Section 7, Chapter 2).

47

§7. Random Selection of a Cell from the Population as a Whole

The Corollary of Theorem 6.1 gives the asymptotic joint dis-
1 i

g . 0 g n -
0 10, 30 ,...,In in, In for a cell selected

randomly from those in phase 4 at time t. Analogous results could

tribution of I

clearly be derived for sampling from any particular phase. We now
indicate the corresponding result for sampling from the entire cell

population.

Let zij)(g(j) 10 81:111-oo13n11n1t) be the number of cells

of type 10 in phase j at time t for which the elapsed phase
i i i i
Yog,...,Y02,Yll,...,Ynn of the cell itself and its first
" " i i
. O 0
n1 ancestors oi type 1n satisfy Yojs aoj,...,Y01 S 801’
l n

Y1 s a1....,Yn s an. Then by the same arguments as we have used ,

durations

for the case j = 4 it can be shown that (using (3.20)), conditional

on Zi(t) F 9,

Z(J)(2(J) 1

 

1 ’ o""’° ’in t) “(1) (j) (1)
(7.11 z (1) 1.11 -iL—Fn (1,10,...,.M,1)
i
z c
1=1 L‘
(1)

for all 80 2 0,...,8n 2 0, in = l,...,m and for all n = 0,1,2,...

N

with probability 1, where

 

(j) m -c(yl+°' +Y.) 0: '

(7.2) Ki = K1, 2 1, Je 3 [j [£<y)1y =udU]dy/A (c),
k=1 yj " j "

(7.3) Féj)(a(j),io,...,an,in) =
. -c()'1+ oooflj>m [X [f( 'ij)] and 1d

1]. m ...m y u y

1n inin-l i110~(y1,...,y ).<a(j)e yj "

m -c(yi+...+yj)m

z flk J9 [f [13(3)]yj WdUde

k=1 yj ~ n

X HY' (a):
k=1 lk’k ”k

48

m
and Z.(t) = X Z.(j,t). (The function Y, was defined in
1 1 1 ,k
k=1 k
Section 4.)

Equation (7.1) defines in considerable detail the asymptotic
age-type-phase structure of the population as, t a m, conditional

upon non-extinction. We note that

(j) (j)
Fn (30

,...,fn,in) defined

"0
in (7.3) gives the asymptotic joint distribution of IO = i0,
i i i i
0 o = . 1 g . n
Y0j""’Y01’ 11 11, 31 ""’In 1“, Zn for a cell selected

randomly from those in phase i at time t.

49

§8. Application of the Asymptotic n-fold Age Distribution to the

Analysis of Labelling Experiments

For a better representation we shall use the following notation
in this section. The joint probability density of the four phase
durations for a new born cell of type i will be denoted
fi’1’2’3’4(t1,t2,t3,t4). Marginal probability densities will be de-
noted in the usual way and the joint densities of sums of phase dura-
tions (type i is fixed) will be indicated by deletion of appropriate
commas in the subscripts (see Section 8, Chapter 2). Laplace trans-

*
forms will be denoted by a superscript , e.g.

e'au‘Bv-me ,w)dpdvdm .

* o: oo
f1,1,23,4(°”B’Y) = [E (E 1,1,23,4(u’v

d—w

The Laplace transform of a function of several variables will some-
times be taken with respect to a subset of the variables. The particular
subset involved will be indicated by underlining the appropriate '
subScripts (see Section 8, Chapter 2).

Suppose first that the label is administered continuously to
a cell population (growing according to the model we have described)
from time t0 onwards. Let Ct (t) be the fraction of cells in phase

0

4 at time tO + t which are labelled. Assuming that t0 is large

and that the observed population is a realization of the process which

does not die out, then for each fixed t as to a m, Ct (t) is just
0

the asymptotic probability that a cell selected at random from those

in phase 4 entered phase 3 at time t or less prior to selection, i.e.

50

 

m
(8.1) lim Ct (t) = 2 I F (i, da )
t fl” t0 i=1 a +a 0 "O
O 04 03St
m cni t cyy
= 121m gee [f: 12 3(c, u) -f: L2 34(c,u)]dudy,
2 nktf: 123(c) -g:(c>]
k=l
where F0 is the distribution function defined by equation (4.5).
Similarly for pulse labelling (administered at time to) the
fraction of labelled cells in phase 4 at time t0 +'t satisfies (under

the assumptions of the previous paragraph)

(8.2) lim Pt (t)= 2 P n(t) ,
to-*no 0 n=0
where
m m
P (t) = z ... 2 I'................... I F (dao,1,...,da ,i ) .
n - . = _ _ _ _ _ i0 ...n n
10 1 1n 1 OSt a04 a03 ... an4 an3san2v

The probabilities Pn(t), n = 0,1,2,... can be calculated by carrying
out the straightforward integration. If we take LaplaCe transforms

in (8.2) we find that
3 3 9* —

(c.c+s>-£T

° 3 n [f* 3,12,34

'k *
-1 J J 1z3 (C’°+S)‘fj11.23(°’°+S)+fj.11234(°'°+S’1

 

m * *
8(s+C)kS “REfk 123(C) - gk(C)]
=1 ’

‘k
(8.4) P1(s) =
m m * * * *
c121 1ElnimijEf1,12,34(c’°+s)41,1,234(°’°+8)][fj,123(c+8)'gj(°+3)J

m * *
8(S+C)k81nk[fk’123(c) ' gk(c)]

51

and, for n 2 2,

*
(8.5) Pn(s) =

m m

m m * *
C 2 2 ~-- 2 8 n m.. ---m. g. (C+S)---8 - (0+8)
i=1 11=1 1n_1=1 j=l i 111 1n-lj 11 1n-1

m * *
8(s+C) E nk[fk 123(C) - gk(6)]
k=1 ’

*

L J

'k * 'k
fi,12,34(°’°+s) ‘ f1,1,23z.(°’°+8)][fj 423““) ' g (“’1'

According to a remark of Bellman and Cooke [3], p. 259, the Patron-
Frobenius root of the matrix H(c+s), s > 0, is strictly less than
the corresponding root of the matrix B(c). But the Perron-Frobenius
root of the matrix B(c) is l, which means that the largest positive
root of the matrix H(c+s), s > O, is less than 1, which implies that

Hn(c+s) a 0, the m X m zero matrix, as n a m, which hmplies that

(8.6) [I - I-I(c+s)]'-1 = I + H(c+s) + H2(c+s) +3.., 8 > 0.’

(n)
hij

then (8.4) and (8.5) can be rewritten in the form

If we denote (c+s) the ijth element of the matrix Hn(c+s),

(8.7) P:(s) =

m m (n) * -1 * * * *
c E zlnihij (c+S)[gi(c+S)] Lfi 12 34(c1c+S)-fi 1 234(c1c+8)][fj 123(c+S)-gj(c+8)]
i=1 J: a a 3 a a

m * *
8(s+C) 2 flk[fk 123(C) - sk(C)]

k=1 ’

Summing over n = 0,1,2,..., and using the fact that [I - H(c+s)]-1 8

fig- we obtain

52

m * * -
C( 2 nk[fk 123(C) - gk(C)])

k=1 ’

A(s+c)s(s+c)

1

 

m
(3-8) re-Stlim Pt (t)dt =
b to-‘co 0

m

rm b + *+ ..1 ‘k *‘ * + *+
tigl jflfli ji(S C)[81(S c)] [£1,12,34(C,s+c)-f1’1,234(c,s+c)]ij’123(s c)-gj(s c)]

m m * -1 * * * *
-A(S+C) 2: r. niigi<s+cn [fi 123(c,s+c)-fi 1 234(c.s+c)][fj 123<s+c>~gj<s+c>1
1:1 1:1 ) : 9 9

*
(c,c+s)-f

m * *
-A(s+C) Z nJEfj’123(c1c+S)-fj j 123

*
j=1 (c,c+s)+fj,1,234(c,c+s)] } .

,12,34

Example. Consider the case where M = (mij) is a m X m matrix such

. 1, . _ =
that mij ' m > m f°r every 1’1 ' 1""’“’ °“d f1.1.2.3,4 f1.2.3.4

for each i = l,...,m. Let A = a u: > 1. Then for every m-dimensional

non-negative vector x = (x1,...,xm),

~

 

m m
Emijxj EZXJ
def 1:: a
904) = Min 1 3 ’11 smm=A .
x _ x. Max x
N i-l,...,m 1 _ i
1-1,ooo,m
m,
But, for x 3 x =...= x we have 2 x / Max x, = m . Thus the
~ 1 m j= j i=1 m 1
,...,

Perron-Frobenius root of M, p(M), is

m
Emu":
p(M)=Max Min L=fnm=A>l.
x i=1,...,m xi

~

It is easy to check that 4(k) = \I - H(x)\ = 1 - Agr(x). Thus the
unique positive root c of A(x) = 0 is the unique positive root

c 9 Ag*(c) - 1. Now we see that P;(s) in (8.3) is the same as

P;(s) in (8.6), Chapter 2 and P:(s), n = 1,2,..., in (8.4) and (8.5)
are the same as P:(s), n I 1,2,..., in (8.7), Chapter 2. This is not

a surprising result but provides a useful check on the computations.

53

§9. Generalization to Include Random Cell Removal

We now consider the following modification of the process con-
sidered in Sections 1-7. Suppose that in addition to elimination of
cells by death at the end of the cycle there is "removal" of cells in
the following manner: for any cell of type i, independently of its
age or phase and independently of other cells, the probability of removal
is any small thme interval (t, t+6t) is list +-o(5t); the probability
that more than one cell is removed in (t, t+6t) is o(5t). Suppose
at epoch zero the population consists of a new born cell of type i.

We consider the following two cases.
Case 1. *1 are all equal, Let k = ‘1’ i = l,...,m. It is not dif-
ficult to check that if we assume that the conditions of Theorem 5.1

are satisfied and l < c and if we define 21(j,t) to be

. _ )\t
21(Jat) - e zi(j9t):

where c is the number determined in Lemma 3.1, Zi(j,t) is the number
of cells of type j in the population at time t, and likewise
2:4)(ao,io,...,an,in,t), etc. each to be eKt times their definitions
given in Section 2, then the whole of Sections 3-8 carries over
verbatim to the new process. We remark that the condition x1< c is
used at the place where we establish the m.s. convergence. Similarly,
if k > c, then the population will eventually die out (see Section 6,
Chapter 2). For the critical case A = c the proof given in Section

3 and Sections 5-7 do not hold.

Case 2. *1 are not all equal. The equation (2.2) again leads at
(4)

once to an integral equation for M).- ,

54

(4) M<4)

(9.1) Mi (30,10,...,an,1n,t) = ni (20,10,...,an,1n,t)‘+
t m ‘1 u
(4) . _ i
g 2 miV'Mv (30,10....,an,in,t u)e gi(u)du ,
v=1
where
(9 2) M(4)(a i a i t)=5 m m ° IL (x l, (x y
‘ ni ~0’ 0’°°"~n’ n’ ii i 1 "° 1 1 J‘°° i n’yn)°°' 1 o’~o
n n n-l ’ 0 A n 0
I (t'yoz'yoz‘°°°‘yn1’dxndZn °"dxodZo ’
[o’yoaAao4]

A = [y, S a., j = l,...,n, yoj S an’ j = 1,2,3,

~J ~J
4 n 4
x > 2 y , j = l,...,n, x > t - 2 2 y. 11
j k=1 j’k 0 j=1 k=1 3

'1.x
and Li(x,y) = x1e 1 11(2), 1 = l,...,m.

. _ ’xit / * * - * “ - “
Let gi(t) _ e gi(t) gi(xi), mij - 31(xi)mij. M - (mij)
).

A* A
and 9(3) = (gi(s)mij

Remark 9.1. It is easy to check that, for some positive integer n,

A

Mn is a positive matrix if and only if Mn is a positive matrix.

.+.
Again we SEC Bi (8) = ('1)1 JDij(s), where Di (3) is the complementary

j 3
minor of the ijth element in I - 8(3), D(s) = \I - fi(s)|, the

determinant of I - 3(3), and 8(3) = (B (3)), the adjoint of the matrix

(4)
1

ji
1 - fi(3). As before the asymptotic behavior of M will be associated
with the roots of the determinantal equation D(s) = 0.

Assumption 9.1. M is a positive matrix such that the Perron-Frobenius

root of g is greater than 1, gi(t) 6 LD for some p > 1 and for all i,and

)

55

Max p

. < m.
1,1,11 15"

If Assumption 9.1 is satisfied, then there exists a positive
number a such that D(a) = 0 (see Lemma 3.1). By using exactly

the same arguments as Mode [11], pp. 11-14, we obtain that as t 41m

 

(4) . . ~ (4)~ . . at
(9.3) Mi (30,10,...,an,1n,t) Ci Fn(ao,10,...,fn,1n)e ,
where
* hinminin-ludni’io n
(9'4) Fn(30’10’°."2n’1n) = n A . kEOQ k,k(gk)’
8 flkjpk 0(y)dy
k=1 ’ ~ ~
(9.5) Q (a ) = ' ¢ (y)dy .
Lsk ~k £48k {wk ~ "'
(9 6) C(4) = k “ 2 fl ' ( )d /D'( )
° 1 “1 Fl kj¢k,0 Z Z a ’

and ¢L,k(y) is the same as ¢L’k(y) defined in equations (4.7) and
(4.8) with c replaced by 0&8; where a is the unique number determined
under Assumption 9.1 and 31 and fij are determined by the relation
g(a) = k(fiifij) (see Lemma 3.1).

We note that Assumption 9.1 can be achieved in many ways. One
of those is

Lemma 9.1. The conditions stated in Theorem 5.1 and c > Max *1
i=1 ... m
imply that Assumption 9.1 is satisfied, where c is the uniqde ndmber

determined in Lemma 3.1.

A

Proof: We only need to show that the Perron-Frobenius root of M

is greater than 1. The condition c > Max Xi implies that
i=l,...,m

(M)ij > (H(c)) for every (i,j). Using the fact that the Perron-

ij

S6

Frobenius root of B(c) is l and applying the theory of positive
matrices (Bellman and Cooke [3], p. 259) we complete the proof of
the lemma.

Using the above method to repeat the whole of Sections 5 and 6
we obtain the following result.
Theorem 9.1. If Assumption 9.1 is satisfied, then, given that

Zi(t) A 0 as t -+ co, the ratio

 

4 . .
Z: )(80’31’... ,an,1n,t)
~ " converges to the joint distribution
(4)
21 (c)
Fn(ao,10,...,an,1n) for all so 2 0,...,3n 2 0, 1n = l,...,m and for

all n = 0,1,2,... with probability one.

Using this fin to repeat the whole of Section 8 we find that

(9.7) lim Ct (t) =
t-m o
0

‘hi t -(or+>.. )yy ,1,
e g[f1 12 3(od’li 9“) '

“MB

1

1 m A * * ‘1
k=1 ’

 

*
£12113 “(cm ,u)]dudy

and
(9'8) lim Pt (t) = :1)“ (t),

t “m t0 n=0

0
where
9 9 11*
( . ) O(8)

1 m "L. [f * J.+ -j£* +3
2: n 151: 4(ar+>1k)

Sk=l ’

‘k
- fj 1 13(jo ..oH'l Jj+s)+f (M’s-i .MljflH,

,1, 234

57

*
(9.10) P1(s) =

 

1 m m a * *
m ‘ * 131 jflnimiij:4(a+xj+s)[fi,12,34(d+xi’d+xi+s)
8 13311115ka (MM)

* .
' f1,1,234(””1’°’+"i+s)] ’

for n 2 2,

*
(9.11) Pn(s) =

n-l

 

m m a . *
,n * . 21.. 2 1111T1mi i 8i (a+li +3)]
k=1 ’

* -1 * u *
[8i (U+Xi‘+s)] Li ’4(G+Xi +S)[fi ’12,34(G+Ai afl+Xi +3)
n n n n 0 O O

*
' fio,l,234(°’fl‘io’°'fi‘io+s)]

and
* * * -1
(9.12) Lk,4(8) = [fk 123(s) - gk(s)]s .

We end this section by making the following remark. If *1
are all equal to x and A < c, then it is not difficult to see that

c a 01+) and in defined in (9.4) is the same as Fn defined in (4.5).

CHAPTER IV

A MODEL FOR CELLS OF THE CORNEAL EBITHELIUM

§1. Introduction

In the corneal epithelium of mice the cell population can be
divided into two subpopulations, a proliferative population occupying
the inner or basal layer, and a non-proliferative population occupying
the outer or super layer. When a cell in the basal layer multiplies
(giving birth to two new cells) one of the daughter cells remains in
the basal layer while the other moves up into the super layer with
probability (l-p) and remains in the basal layer with probability p.
Cells in the super layer do not reproduce, but cells in the basal
layer do reproduce and we shall suppose that the reproductive cycle is
as described in Chapter II with independent phase durations, i.e. the

four phase durations T T ,T3 and T for any new-born cell are

1’ 2 4
independently distributed with probability density function f1,f2,f3
and f re3pectively, and mean number of proliferative daughter cells

4
per parent, A = l +'p. The distribution of total lifetime for any cell

thus has the probability density
a e * *
3(0) f1 f2 f3 13401) 1

where * stands for the convolution.
The departure of cells from the basal layer into the super layer

appears to be a mixture of departures of daughter cells at birth and

58

59

departures occurring randomly throughout the proliferative cycle (see
Section 6, Chapter II). The departure mechanism is such that the popula-
tion of cells in the basil layer is approximately constant. Thus the
rate 1 of random loss from the basal layer must be related to p
in such a way as to maintain the expected number of cells in the basal
layer approximately constant. In fact we shall assume that the basal
layer population is a very slightly supercritical branching process
and use the results derived in Chapter II to analyze an experiment used
by R.J.M. Fry [5] to estimate the value of p.

A radioactive substance such as tritiated thymidine will be in-

jected into the population at epoch t to label all cells which are

0

in phase 2 and in the basal layer at epoch to and no others. Sub-
sequently when the labelled cells divide they pass the label on to their
two offspring. The cells which have migrated from the basal layer do
not pick up the label, but they retain any label they had before leaving
the basal layer. We are interested in the FLU functiOn defined to be
the ratio of the number of labelled cells in the super layer to the
total number of labelled cells (in the basal and super layer).

The purpose of studying FLU functions is to try to esthmate
the parameter p. The means and variances of the phase durations for
the proliferative cells in the corneal epithelium of the CF1 AnL mouse
have been estimated previously using FLM data by Brockwell, Trucco and
Fry LS]. However, the FLM function is extremely insensitive to change

in the value of pl so that additional information, such as provided by

the FLU function, is necessary for the estimation of p.

60

§2. FLU Function and Some Numerical Results

Again at time zero a pulse of tritiated thymidine is administered
into a cell populatiOn to label the cells which are in phase 2. The
experimental FLU(t) is the fraction of labelled cells in the super layer
at time t. In this section we shall first derive a theoretical FLU
function which is defined to be the ratio of the expected labelled
cells in the super layer to the total expected number of labelled cells
at time t. Then we use this function to fit the experimental data

for the corneal epithelium of the CF AM, mice which was prepared by

1
R.J.M. Fry. If we let LB(t) and LU(t) be the expected number of
labelled cells in the basal and super layer at time t respectively

and let LT(t) =‘LB(t) + LU(t), then

(2.1) FLU(t) = :‘fi = 1 - $8- .

We shall also define M(t‘i,a ) and U(t\i,a ) to be the expected
number of cells in the basal layer and in the super layer at time t,
re3pectively, given that at time zero there is a single cell (i,a )
in phase 1 with age a in the basal layer. If the initial joint
phase-age distribution g(i,a), i = l,2,3,4 and a 2 0, is known,

then it is clear that

(2.2) LB(t) = A M(t|2,a)§(2,da)
and
(2.3) LU(t) =£ U(c\2,a)g(2,da) .

The mean functions M(t‘i,a) and U(t\i,a) are determined in Appendix

A.

61

Since the initial phase-age distribution cannot be measured
it is reasonable to take as initial distribution the limiting distribu-
tion as t a m if suCh a limiting distribution exists. When xt>(:and p > 0
the limiting phase-age distribution g(i,a) 'exists (see Chapter II)

and takes the form

11‘ 1)

 

. . (i) (i)
(2.4) §(1,a .) = 1 ——-—F (a ,...,a)
01 a 1m,k:T,...,n K n ~0 ~n
aojtao,j=1,...,i-1
C 800i '0 m
= 4 * £ e yf£i(u)dudy ,
(A-l) n fm(c) 7"

m=i

where K(1) and Fél) were defined in equations (5.2)-(5.3), Chapter

II.

We recall here that c is the unique positive root of the
equation Ag*(c) = l and superscript * stands for Laplace transform.

What we are going to do is to obtain FLU(t) under the assumption
that l is less than c by a very small positive number and that the
initial population of cells in the basal layer has joint phase-age
distribution g(i,a) given by (2.4). Then we take lim FLU(t) as the
theoretical FLU function to analyze the experimental dzza. We proceed
as follows.

If we denote FLU(t) = lim FLU(t), LB(t) = lim'LB(t),‘LU(t) =

... ... 11$. 11C
lim‘LU(t) and LT(t) = LB(t) +-LU(t), then it follows from (2.1) and

110
the monotonicity of M(t‘2,a) and U(t‘2,a) in A that

(2.5) Effie) = 1 - Iii-’49- .
LT(t)

where

62

(2.6) i§(c) = r lim.M(tl2,a)§(2,da)
U xtc
and
(2.7) LU(t) = g lim U(t\2,a)g(2,da)
11c

We note that the functions M(t‘2,a) and U(t‘2,a) actually depend
on 1 although not shown explicitly. It follows from equations (A.6),

(A.ll) and (A.12) of Appendix A and a straightforward calculation that

*
—_.* * 1 - f2(c) *
_ .______._. .11.
(2.8) LB (s) — N1f1(c)[ C(s+c) + s+c H (3)] .
2 __* _ * 1 - f:(c) s+c-ps *
( .9) LU (3) -N1f1(c)[ -;?;:;y—-'+ s(s+c) H (8)] 9
1 f*< )

" C
(2.10) 13*(11) = le:(c) {—53-— + i H*(s)] ,

* *

f2(c) - f2(s+c) 1

 

 

(2.11) H*(s) = [f:(s+c)f:(s+c) . s > 0.

8 *
1 - Af (3+c)

and N1 is a fixed constant depending on the size of the population

in the basal layer at time zero.

 

If we let Tkj’ j = 1,2,3,4; k = 0,1,2,3,... be a sequence
of independent random variables with probability densities gk j
defined by
t

e-Ct g f4(u)du

m t , if k = 0, j = 4,

Se'ct gf4(u)dudt
(2.12) gkj(t) =

e-ctf1(t)

, otherwise,

 

£e-thj(t)dt

63

and let Ev(t)’ v 0,1,2,... be a sequence of mutually exclusive

events defined by

(2.13) E0(t) = {104 + 103 _<. c and 104+ 103 + 102 > t} ,

while for v 2 l,

v- -1 4
2.14 E = + + d
( ) (t) {(jgozk=1Tjk) T 4 T113 3 t an
v-1 4
+ 1' +1 + T > t ,
(12031.59 04 v3 v2 }

-— -—-* -—— ——*
then the inverse functions LB(t) of LB (3) and LT(t) of LT (s)

can be written as follows;

(2.15) ITB-(t) = N1[f:(c)f:(c)e’°t + gags)- 2 Pv(t)] .
v=0
t a
-—— e * 1
(2.16) LT(t) = N1 [f1(c)f2(c)+ CRT—+1») )3 P (t) +-—— z; .0,Pv(u)du]

1+1)

where
(2.17) Pv(t) is the probability of occurrence of the event Ev(t)'

Now we include some numerical results for this FLU function.
Example 1. fj(t)’ j = l,...,4 are gamma type which are characterized

by their means m. and coefficients of variation 6 , j 8 l,2,3,4.

J 1

Assume that m1 = 87.3 hr., m = 10.9 hr., m3 = 3.5 hr., m4

= 0.14, c c4 = 0.00. These figures are

2 B 0.5 hr.,

= 0.37, c = 0.3, and

cl 2 3
taken from [5]. A fairly complicated Fortran program produces the

following numerical results for varied values of p.

64

 

TABLE 1
p p=0.05 p=0.2 p=0.4 p=0.5 p=0.60337

TIMEhr.) EEE’ ELE' Eifi' Eifi' EEE'
1.0 0.00048 0.00180 0.00334 0.00404 0.00472
2.0 0.00108 0.00370 0.00676 0.00814 0.00947
3.0 0.00641 0.00956 0.01339 0.01489 0.01630
4.0 0.03498 0.03484 0.03441 0.03347 0.03235
5.0 0.09025 0.08200 0.07066 0.06443 0.05790
6.0 0.15272 0.13471 0.11053 0.09824 0.08551
7.0 0.20987 0.18296 0.14721 0.12936 0.11095
8.0 0.25964 0.22513 0.17950 0.15683 0.13348
9.0 0.30320 0.26214 0.20797 0.18111 0.15347
10.0 0.34159 0.29482 0.23321 0.20269 0.17131
11.0 0.37563 0.32383 0.25569 0.22197 0.18731
12.0 0.40563 0.34943 0.27559 0.23908 0.20159
13.0 0.43092 0.37109 0.29253 0.25374 0.21394
14.0 0.45032 0.38786 0.30592 0.26552 0.22409
15.0 0.46344 0.39950 0.31567 0.27438 0.23206
16.0 0.47237 0.40679 0.32235 0.28080' 0.23822
17.0 0.47544 0.41105 0.32689 0.28550 0.24312
18.0 0.47745 0.41353 0.33016 0.28919 0.24727
19.0 0.47837 0.41513 0.33279 0.29237 0.25104
20.0 0.47882 0.4634 0.33514 0.29532 0.25464
24.0 0.47979 0.42052 0.34399 0.30665 0.26860
48.0 0.48619 0.44549 0.39493 0.37104 0.34713
72.0 0.50854 0.48187 0.44879 0.43312 0.41735
73.0 0.51042 0.48411 0.45131 0.43572 0.41999
101.0 0.57598 0.55024 0.51678 0.50042 0.48367

65

We remark that FLU(t) is approximately linear between

t = 20 and t = 101. We plot this table for some values of p in

Figure 1. The circles represent the experimental data.
Example 2. Four phase durations are deterministic with (11 = 87.3 hr.,
3 = 3.5 hr. and d4

produces the following numerical results for varied values of p

d2 = 10.9 hr., d = 0.5 hr. A quite simple program

(Table 2).

We plot this table for some values of p in Figure 2.
Both Table 1 and Table 2 indicate that the FLU functions

obtained from gamma phase durations and detenninistic phase durations

are about the same for each fixed p, p > 0. This is a very interesting

and surprising result. Due to this fact we can, in fitting the case

with gamma distributions,assume that the four phase durations are

8 87.3 hr., d = 10.9 hr., d = 3.5 hr. and

3

to obtain the best agreement be-

determinisitic with (11 2

d4 = 0.5 hr. and vary the value of p
tweentiustheoretical and experimental FLU curves in the sense that the sum
A non-linear minimization

of aquares of the residuals is minimized.

procedure program produces the following results.

 

Assumed parameters: d1 = 87.3, d2 = 10.9, d3 = 3.5, d4 = 0.5
Fitted parameter: p = 0.60337
TABLE 3
Experimental Fitting

TIME (hr.) @191, FLU Residual

5.0 0.025 0.05790 0.0229

17.5 0.185 0.2453 0.0603

24.0 0.262 0.2686 0.0066

48.0 0.418 0.3471 -0.0709

72.8 0.499 0.419 ~0.08

For details see Table l and Table 2.

66

 

TABLE 2
p=0.05 p=0.2 p=0.4 p=0.5 p=0.60337 p=0.7

“1;“; 1:51? 3'1; 31:; 11:17 11:13 51-11
1.0 0.00048 0.00178 0.00329 0.00396 0.00461 0.00518
2.0 0.00095 0.00356 0.00656 0.00790 0.00920 0.01033
3.0 0.00143 0.00534 0.00983 0.01183 0.01376 0.01546
4.0 0.00191 0.00711 0.01308 0.01574 0.01831 0.02055
5.0 0.08169 0.07436 0.06405 0.05872 0.05312 0.04781
6.0 0.14915 0.13135 0.10740 0.09538 0.08292 0.07126
7.0 0.20694 0.18027 0.14476 0.12704 0.10875 0.09169
8.0 0.25700 0.22273 0.17729 0.15468 0.13139 0.10969
9.0 0.30078 0.25992 0.20589 0.17905 0.15142 0.12570
10.0 0.33940 0.29279 0.23124 0.20070 0.16929 0.14006
11.0 0.37372 0.32203 0.25388 0.22009 0.18535 0.15304
12.0 0.40441 0.34823 0.27422 0.23756 0.19988 0.16485
13.0 0.43203 0.37184 0.29261 0.25339 0.21311 0.17566
14.0 0.45701 0.39322 0.30933 0.26783 0.22521 0.18561
15.0 0.47757 0.41091 0.32335 0.28006 0.23562 0.19435
16.0 0.47782 0.41196 0.32557 0.28291 0.23914 0.19852
17.0 0.47807 0.41301 0.32779 0.28575 0.24265 0.20267
18.0 0.47832 0.41406 0.33000 0.28858 0.24614 0.20680
19.0 0.47857 0.41510 0.33220 0.29139 0.24961 0.21091
20.0 0.47882 0.41615 0.33439 0.29420 0.25307 0.21499
24.0 0.47981 0.42030 0.34310 0.30531 0.26675 0.23113
48.0 0.48574 0.44460 0.39301 0.36841 0.34370 0.32121
72.0 0.49159 0.46787 0.43913 0.42577 0.41257 0.40073
73.0 0.49184 0.46882 0.44097 0.42804 0.41528 0.40384
101.0 0.49858 0.49470 0.49020 0.48818 0.48622 0.48450

67

 

 

 

 

 

H MMDUH m

 

F IGURE 2

68

 

 

p .1 0.05

 

 

 

 

70

50

40

69

We summarize the results obtained in this section as follows:
1. The FLU functions obtained by using deterministic phase
durations and by using gamma phase durations are almost
identical over the range of parameters relevant to our prob-
lem. The deterministic FLU function (which is much easier
to compute) was therefore used to simplify the fitting problem.
2. The FLU function is very sensitive with respect to the value
of p.
3. The fit is not as good as could be hoped but it seems to be
compatible with experimental errors and indicates that p
is not either 0 or 1 as has been assumed by various authors.
For the experiment in which the label is administered con-
tinuously to a cell population, we are also able to calculate the
CLU(t) defined to be the ratio of the expected number of labelled
cells in the super layer to the total expected number of labelled cells
at time t. This CLU function is presented in Appendix B. We devote
Appendix C to the discussion of the asymptotic behavior of M(tli,ao)

and the random variable 23(t‘i,ao).

B IBLIOGRAPHY

10.

11.

12.

BIBLIOGRAPHY

Barrett, J.C. A mathematical model of the mitotic cycle and its
application to the interpretation of percentage labelled mitoses
data. J. Nat'l. Cancer Inst., 37, 443-450, 1966.

Barrett, J.C. Optimized parameters for the mitotic cycle. Cell
and Tissue Kinetics 3, 349-353, 1970.

Bellman, R.E. and Cooke, K.L. "Differential-difference equations",
Princeton Univ. Press, Princeton, New Jersey, 1963.

Brockwell, P.J. and Trucco, E. On the decomposition by generations
of the PLM-function. J. Theoret. Biol., 26, 149-179, 1970.

Brockwell, P.J., Trucco, E. and Fry, R.J.M. The determination of
cell-cycle parameters from measurements of the fraction of labelled
mitoses. Bull. Math. Biophys., 34, 1-12, 1972.

Feller, W. "An Introduction to Probability Theory and its
Applications", 2nd ed., Vol. II, John Wiley and Sons, Inc., New
York, 1971.

Harris, T.E. "The Theory of Branching Processes", Springer Verlag,
Berlin, 1963.

Karlin, S. "A First Course in Stochastic Processes", Academic
Press, New York and London, 1968.

Macdonald, P.D.M. Statistical inference from the fraction labelled
mitoses curve. Biometrika, 57, 489-503, 1970.

Mendelsohn, M.L. and Takahashi, M. A critical evaluation of the
fraction of labelled mitoses method as applied to the analysis of
tumor and other cycles. Chapter III of The Cell Cycle and Cancer
(R. Baserga, ed.), Marcel Dekker, Inc., New York, 1972.

Mode, C.J. A multidimensional age-dependent branching process
with applications to natural selection I. Math. Biosci. 3, 1-18,
1968.

Mode, C.J. A multidimensional age-dependent branching process
with applications to natural selection II. Math. Biosci. 3,
231-247, 1968.

70

13.

14.

15.

16.

17.

71

Mode, C.J. A generalized multidimensional age-dependent branching
process. ORNL-433l, Oak Ridge National Laboratory.

Powell, E.P. Growth rate and generation time of bacteria, with

special reference to continuous culture. J. Gen. Microbiol., 15,
492-511, 1955.

Steel, 0.6. and Hanes, S. The technique of labelled mitoses:
analysis by automatic curve fittings. Cell Tissue Kinet., 4,
93-105, 1970.

Takahashi, M. Theoretical basis for cell cycle analysis II. J.
Theoret. Biol., 18, 195-209, 1968.

Trucco, E. and Brockwell, P.J. Percentage labelled mitoses curves
in exponentially growing cell populations. J. Theoret. Biol.,
20, 321-337, 1968.

APPENDICES

APPENDIX A

INTEGRAL EQUATIONS FOR THE MEAN FUNCTIONS M(c\1,a) AND U(t‘i,a)

Define Z(tli,a) to be the number of cells in the basal layer
at time t, given that at time zero there is a single cell (i,a) in
phase i with age a in the basal layer. Let Zv(t‘i,a) be the
number of 0th generation of the initial cell (i,a) in the basal
layer at time t. We shall denote M(t|i,a) = E Z(t‘i,a) and
Mv(t\ 1,8) = E Zv(t\i,a).

Consideration of the time at which the initial cell first

leaves the cycle leads to the following integral equations for the

function Mv(t\i,a).

(A.1) Mo(t\i,a) = P(the initial cell (i,a) is itself in the basal
layer at time t)

e-xt l y(i,a ,u)du

and, for v 2 1,

t
(4.2) Mv(t\i.a> = Agnwlu-uh.0)e'”‘y(i.a,u>du .
where
(A.3) y(i.a.u) = r(i.a.-) * fi+1 *---* f4(u)1
(A.4) r(i,a,t) = £i(s+c)/j fi(u)du, t 2 0,

a

and A and fi’ 1 = l,2,3,4 are defined in Section 1, Chapter IV.

72

73

Taking Laplace transform on both sides of equations (A.l) and
(A.2) and using the equality
a)
(A.5) M(t‘i, a) = 2 M v(t‘i, a)
v=0

we obtain

(A.6) M*(s\i,a)=1 '1 *(ixa 5m +L£J§JLW ago 038* (8.11))"

3+1 3+1

 

8*(8‘1'1) v‘l
* 4 *
where superscript * stands for Laplace transform and g (s) - n f1(s).
i=1

Now define Zs(t\i,a) to be the number of cells in the super
layer at time t, given that at time zero there is a single cell (i,a)
in phase i with age a in the basal layer. Let z:(t\i,a) be
the number of vth generation of the initial cell (i,a) in the super
layer at time t. We shall also denote U(t\i,a) = E 28(t\i,a) and
Uv(t\i,a) = E 2:(t\i,a). Consideration of the time at which the initial
cell first leaves the basal layer leads to the following integral equa-

tions.

(A.7) U0(t\i,a ) P(the initial cell (i,a ) is itself in the super
layer at time t)

1: co
gxe-xudu J. Y(isa' 99)d9 :
U

1'. t
(1-p)£e-xuy(i,a ,u)du +’Ag UO(t-u\l,0)e-xuy(i,a ,u)du,

(A.8) U1(t\i,a )

and, for v 2 2,
t -),u
(A.9) uv(t1i,a ) = A; UV_1(t-u‘l,0)e y(i,a ,u)du .
Again taking Laplace transform on both sides of equations

(A.7) - (A.9) and using the relation

74
(A.10) U(t\i,a) = E U (t\i,a)
v=O V
we obtain
* . 1- * * .
(A.1l) U(s\1,a) = 3:51.? [32+ A U (s\1,0) - my (i,a.s+1)

and

(11.12) U*(s\1,0) = (183’- 8*(s+>1) + m- [1 - 8*(s+>.)]) z [Ag*(s+>.)]v.
v=0

APPENDIX B

CLU FUNCTION

Suppose that the label is administered continuously to a cell
population. Let CLU(t) be the ratio of the expected number of labelled
cells in the super layer to the total expected number of labelled cells
at time t. As in Section 2, Chapter IV we shall assume that 1 is
less than c by a very small number and that the initial population
of cells in the basal layer has joint phase-age distribution g(i,a)
given by equation (2.4), Chapter IV. If we define LB(t‘i,a) and
LU(tli,a) to be the expected number of labelled cells in the basal
layer and in the super layer at time t respectively, given that at
time zero there is a single cell (i,a) in phase i with age a in

the basal layer, then it is clear that

(3.1) CLU(t) = 1% = 1 - %% ,
where
4 m
(B.2) LB(t) = 2 gLB(t\i,a)§(i,a)
=1
and
4 e.
(3.3) LU(t) = 2 A'LU(t\i,a)§(i,a) .
i=1

Again we are interested in the function CLU(t) = lim CLU(t).

Me
It is clear that

75

76

-- _ LB(t)
(B.4) CLU(t) - l - LT(t) ,
where
__ 4 m
(B15) LB(t) = E g lim.LB(t\i,a)§(i,da)
i=1 11c
and
__ 4 w
(8.6) LU(t) = E g 11m,LU(t\i.a)§(i.da)
i=1 1tc

It remains to determine LU(t‘i,a) and LB(t‘i,a). It is obvious

that

(3.7) L8(t\2,a) = M(t‘2,a)
and

(B.8) LU(t‘2,a) = U(t12,a),

where M and U were defined in Chapter IV (also in Appendix A).
Consideration of the time at which the initial cell first
leaves the phase 1 leads to the following integral equations.

t

(3.9) LB(t\1.a> = 1 M(t-u\2.o>e"“r(1.a.u)du.
t -

(8.10) LB(t13,1) ==£ LB(t-u\4,0)e A“r(3,a,u)du,
t -

(8.11) LB(t‘4,a) = A& LB(t-u\1,0)e x”1(4,a,u)du,
t

(8.12) LU(t\1,a) =

g U(t-u\2,0)e-xur(1,a,u)du,

t
(8.13) LU(t‘3,a) = J LU(t-u‘4,0)e-xur(3,a,u)du

77

and

t
(B.14) LU(t\4,a) = AJ; LU(t-u\1,0)e A"t(4,a,11)dn,
where r(i,a,u) was defined in (A.4), Appendix A.
Taking Laplace transform on both sides of equations (B.7),

(B.9), (8.10) and (B.11) we obtain

(B.15) L8*(s11,a) = m*(s\2,0)t*(1,a,s+1) ,

(8.16) L8*(s\2,a) = M*(s‘2,a),

(B.17) L8*(s\3,a) = A M*(s12,0)f:(s+mf:(s+),)r*(3,a,s+1)
and

(8.18) L8*(s\4,a) = A M*(s\2,0)£:(s+1)t*(4,a,s+i) .

It follows from equation (A.6), Appendix A and a straight-

forward calculation that

f* 1 13* 1*
1(e)( - 2(e)) (C)

I - * * * *
C(S+C) + S(3+C) [1 Af1(s+c)f2(c)f3(S+C)f4(S+C)

 

(B.19) N2f8*(s) =
+ f* f* *
P 2(c) 3(s+c)f4(s+c)],

where N2 is a fixed constant depending on the initial population
size.

Similarly, equations (3.8), (B.12), (B.13), (B.14) and equations
(A.11) and (A.12) of Appendix A lead to the explicit form of LB(t)

in terms of Laplace transform.

fI(c)(1'f:(°)) f*(°) * * *
s(s+c) + 8 (8+6) {C'ACf1(3+C)f3(S+C)f4(s+c)

 

(8.20) N21'F(s) =

+ [(s+c>-ps]f:(c)f§(s+c)f:(s+e)1

78

Thus we have

* 'k *
f <c><1-£ <c>> f (c)
1 cs 2 + JC— [13139118138£3<s+e>£Z<s+c>

 

(8.21) 11217211) =

* * '*
+ f2(c)f3(s+c)f4(s+c)] .

-——* -—a*
We note that it is very easy to invert LB (3) and LT (s) into

LB(t) and LT(t) respectively.

APPENDIX C

ASYMPTOTIC BEHAVIOR OF THE FUNCTIONS M(t\1,a) AND U(t\i,a)

The notation to be used in this Appendix is the same as those
in Chapter IV and in Appendix A. Observe that the number of cells
in the basal layer in the time interval [0,t] is non-decreasing in

t and finite for every finite t. Thus Sup M(T‘i,a) < m for

OSTSt
every finite t. This leads to the following obvious lemma.

Lemma C.1. If M(t‘i,a) “’Cl asd t a m and C1 is a positive

constant then

on

(i) g M(t\i,a)dt = m

(ii) 3 C2 independent of t 9 M(t‘i,a) 3 C2 .

Now we are ready to state the following result.
Theorem 0,1. A necessary and sufficient condition for the function

M(t‘i,a) converge to a constant K(i,a) as t a m is that A = c.

Then K(i,a) > 0 and

4
c-1(A-1)r*(i,a,c) H f*(C)

(0.1) K(i,a) = ”33:1: .

m -c
A~gye yg(y)dy
Proof: (i) It follows fromequations (A.l), (A.2) and (A.5), Appendix
A that
on t

(0.2) M(t\i,a) = e“): £y(i,a,u)du +-A.£M(t-u\l,0)e-xuy(i,a,u)du .

79

80

Setting i = l and a = 0 in equation (C.2), multiplying
both sides of equation (C.2) by ext and letting M(t‘l,0) =‘M(t\l,0)ext
(this is the method suggested in Section 6, Chapter II) we obtain a
renewal equation for the function M(t‘1,0). By applying the
asymptotic theory of the renewal equation we establish the result
that 1 = c is a sufficient condition for M(tli,a) converging to
K(i,a) as t a m. (ii) Suppose that lim.M(t|i,a) = K(i,a). If

t-Doo

I > c, then it follows from (A.6), Appendix A that

*
m*<s\i.a0> 33° %[1-7*(i.a.1)] + £35in y*<i.ao.i) Tit-8- < .. .
-Ag 3

This contradicts (i) of Lemma C.1 (see Feller [6], pp. 418—420). Thus
A must be less than or equal to C. If p = 0, i.e. A = l and c = 0,
then we have completed the proof of the theorem. If p > 0, i.e.
c > 0, it remains to rule out the possibility I > c. Suppose I > c.
We can find a positive 30 such that k+so < C. For such a 80 we
have that ; [Ag*(so+1)]v = m, i.e. M*(so\i,a0) = m. But for any
v=0

s > 0,

T m
M*(s\ i,ao) = ge'StMu‘ i,ao)dt + £8.5th i,a0)dt

1 s
< ge-StM(t\i,ao)dt + (K(i,a0) + .1) ie'Stdt < ..,

where a. T are chosen properly from the assumption that

lim.M(t\i,a0) = K(i,a So we cannot have I < c. This completes

).
tam 0
the proof of the theorem.

Theorem 0.2. If I = c, then

* * *
D(t‘isa) ~ KIA)’ (i,a,c)[(1-p)g (C) + (1'3 (C))]t9 t "‘ m °

81

Proof: It follows from equations (A.7) - (A.lO), Appendix A that

t on t
(C.3) U(t\i,a) = gxe kudu §y(i,a,e)de + (I-p)ge—xuy(i,a,u)du
u
t _)\u .
+ A U(t-u\1,0)e y(i,a,u)du .
Taking Laplace transform on both sides of equation (C.3) we
obtain

* - =.___1.. 1:2. * _.__l___ *
(0.4) u (1111,11) mm +[ s + AU <s\1.0) mm}: (i,a.sm.

Setting i = l and a = 0 in equation (0.4) and solving for

*
U (311,0) we obtain

[1-g*(s+i)1) l

* = 1:2 *
(0.5) U (1111.0) ( s g (3”) + s(s+>.) 1-Ag (11+).)

, 3 > 0.
By letting s a 0 in (C.5) we find that
2 * * *
s U (8\1.0) -° K[(1-p)8 (C) + (1-8 (e))]
which implies

2 * * * *
(C.6) s U (s‘i,a) « K1Ay (i,a,c)[(1-p)g (c) + (l-g (c))] as 3 4 0

and

-k
U (Ts‘i,a)
(0.7) * -
U (m‘i,a )

 

'l as T a 0 ,
s
m -0t -1
where K1 8 [Agte g(t)dt] .
It follows from standard Tauberian theorem (e.g. Feller [6],
pp. 418-420) that

t
«2.8) lgum i,amy ~ §KlAy*(i.a.c>[<1-p>g*<c>+<1-g*(c>>1t2. :2 - .. .

82

Since U(t\i,a) is non-decreasing in t, the Lemma on p. 422, Feller
[6] can be applied to complete the proof of the theorem.
We also state the following two obvious theorems.

Theorem C.3. If 1 > c, then

* . *
1 a -

(i) U(t\i,a) s 1 *
1'A80.)

as t—ioo

(ii) Zs(t\i,a) a W8(i,a) as t - m, where W8 is a random variable.

(iii) E w"(i.a) - lim 0(t\i,a)

t-oco
(iv) P(Ws(i,a) < m) = 1.
Proof: Proceed exactly the same arguments used in Section 6, Chapter II.

Theorem 0.4. If p > 0, c > 1 and f E Lq for some q > 1, then

(i) U(t|i,a) ~ C(i,a)e(c-X)t, t a m where

*
cu.a) -- MJ- 131—19— + U- g*(c))A y*(1,a,e)
c-x c c-x
(ii) 23(t\i,a)/c(i,a)e(c->‘)t converges to a random variable
Ws(i,a) with probability one as t a m
(iii) E Ws(i,a) = 1

(iv) Var Ws(i,a) > 0 .

Y R

"'WfiifimflfilflmfiulfiflilfifilfiiiuuflimmsF

1291