Estimation of the Mean and Variance of C
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Cell Tissue Kinet. (1974) 7,201-21 1
ESTIMATION OF THE MEAN A N D VARIANCE OF
CYCLE TIMES IN CINEMICROGRAPHICALLY
RECORDED CELL POPULATIONS D U R I N G
BALANCED EXPONENTIAL GROWTH
P E T E RJ A G E R SA N D K L A SN O R R B Y
Department of Mathematics, University of Goteborg, and
II Department of Pathology, University of Linkoping, Sweden
(Received 26 April 1973; revision received 20 October 1973)
ABSTRACT
The mathematical and statistical problem of estimating the mean and variance
of cell cycle times from cinemicrographically observed durations until mitosis
(cytokinesis at late anaphase), disintegration, collision (cells superimposing each
other) or emigration (moving out of the field of vision) of randomly chosen individual cells in a population in balanced exponential growth is treated. The resulting formulae are simple and considerably reduce the average cell observation
time. They are applied to two normal human foetal cell lines and their SV40-transformed counterparts. One result is that the latter seem to have longer cycle times in
spite of their shorter doubling times. Another result is that the cell mobility seems
somewhat increased in the transformed populations. This corroborates earlier
findings, indicating that the extent of cell loss, rather than the length of cycle times,
may play a decisive role for the (net) doubling time of cultivated cell populations.
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INTRODUCTION
The cycle time, or rather the distribution of cycle times, is one of the most important and most
elusive of the factors determining the kinetics of a cell population. Direct observation in
vitro is made difficult by the length of the cycle, resulting in censored samples (Norrby,
Johannisson & Mellgren, 1967). Indirect methods like analysis of FLM (PLM)-curves are
also very involved and largely heurisitic or dependent upon narrow assumtpions. For recent
surveys (among an abundant literature) see Mendelsohn & Takahashi (1971) or Steel
(1972) and for a statistical approach MacDonald (1970).
The aim of this study is to describe a method of estimating the mean cell cycle time and the
dispersion of cell cycle times in cultivated monolayer cell populations in balanced exponential
growth. Experimentally the method is based on cinemicrographic recordings.
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Correspondence: Dr Peter Jagers,Departmentof Mathematics,Universityof Goteborg, S-402 20 Goteborg,
Sweden.
14
201
202
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Peter Jagers and Klas Norrby
Mathematically the analysis proceeds from a simple idea and the results should have a
wide range of application : as has been observed in variouscontexts,any populationin balanced
exponential growth has a stable age-distribution, related to the cycle time distribution, the
incidence of cell loss, and the distribution of the times to disintegration of disintegrating cells.
Thus, if cells are sampled at random from such a population and the time to disintegration or
mitotic division is recorded, this should provide information about the cycle time. Since
sampled cells in this procedure are usually not newly born, one would expect a considerably
shorter time of observation than by the method of following cells from their birth to mitosis
or death.
Since the cells are moving around (only seldomly cells stay essentially in one place) the
procedure is complicated by migration and collision; it is often impossible to follow cells
till mitotic division or disintegration, since they emigrate outside the field of vision or collide.
It is assumed in this paper that the migration-cum-collision rate (the mobility rate) is constant
in time and independent of the ages of cells. Nothing in our material contradicts this assumption, though apriori it can not be excluded that cells are more mobile during some stages than
during others. It is this limitation that makes it possible to disregard the loss of information
caused by migration, or rather it compensates for it. Besides that our model is the most
general one suggested for cell proliferation (Nooney, 1968; Jagers, 1970), requiring no particular form of phase time or cycle time distributions, no independence between phase durations,
and allowing cell disintegration to take place during the life of the cell according to an arbitrary law.
What we can observe is the duration of time to disintegration, division, collision or emigration of a randomly chosen cell, whichever occurs first. The statistical problem of using
such observations to estimate the mobility will be discussed. The use of the mobility rate to
find the distribution of durations to death and the main problem, of relating this to the cycle
time distribution, will also be dealt with. For the benefit of the non-mathematical reader the
outcome of these analyses are presented separately.
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MATERIAL AND METHODS
Biological
The cell lines used were two normal human foetal lines (GP 115-2 and GP 117-1) and their
SV40-transformed counterparts (GP 125-2 and GP 127-1) (for terminology seeNorrby, 1970).
The normal lines emanated from skin and lung tissue respectively. The cultivation technique
used is the one commonly employed for such normal diploid cells (Hayflick & Moorhead,
1961 ;Norrby, 1970). The transformed lines were studied in the second stage of transformation
(Girardi, Jensen & Koprowski, 1965). All cells were fed Minimum Essential Medium (Eagle,
1959)supplemented with 10 a,,: not heat inactivated newborn calf serum, penicillin (100 IU/ml)
and streptomycin (100 ,ug/ml).
For cinemicrography 175,000 k 25,000 cells were seeded in Cooper plastic Petri dishes
(Falcon Plastics, Los Angeles. U.S.A.) and subsequently maintained in humidified C0,-air
atmosphere at 37°C and pH 7-3 0.1. An inverted phase contrast microscope built into a
Zeiss Micro Cine Camera equipped with a Zeiss Microflash Unit (30 W sec) was used.
Interference and other filters transmitted less than 1 % of the G1O-j sec flash to the cells.
One frame was exposed every other minute at a magnification of about x22.
From the cinemicrographs a period of exponential growth was found in each experiment
and during this period a set of population kinetic parameters were established (Norrby et al.,
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Cell cycle times
203
1967; Jagers, 1970; Norrby, 1970; Norrby & Mellgren, 1971). Such parameters include the
effective cycle rate, the effective net growth rate, the effective death or disintegration rate, and
the probability of a cell dying without dividing. It was further concluded, among other things,
that: (i) the migration across the borders of the field of view is random, (ii) the growth fraction
is essentially unity, (iii) all cell divisions in the populations now studied yielded two daughter
cells, and (iv) distinct cellular events are determined from the cinemicrographs with an error
of a single determination less than 2 min (0.03 hr).
It is important to realize that in terms of duration of DNA synthesis and generation time
from FLM curves, as of population doubling time and ‘terminal cell density’, our normal cells
compare well with what is known about other normal human foetal fibroblast-like cells (see
Norrby & Mellgren, 1971). Transformation with SV40implies aneoplasticchange(Koprowski
et al., 1966). In the present study, in order to record durations to mitosis (to beginning
of cytokinesis at late anaphase-early telophase), to emigration (outside the field of view),
to collision (two or more cells being superimposed), and to disintegration, randomly chosen
cells at 12 hr intervals during the period of exponential growth were tracked down on the
cinemicrographs. The observation of a single cell was thus initiated a random time after the
birth of the cell. This was done from one or two experiments per cell line (Table 1, Fig. 1).
The collection of durations to mitosis was extended by similar examination of cinemicrographs from another two or three experiments per cell line (Table 2). Any duration longer than
12 hr was, of course, considered only once. A large number of cells were successfully traced
through collisions but often the tracks of two or more cells ran into each other during the
colliding event preventing further tracing of the cells involved. The number of disintegrations
recorded is possibly a minimum number since disintegrations occurring during a collision
might be camouflaged by the neighbouring and often superimposed cell(s).
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TABLE
I . Cells followed to any one of mitotic division, M, emigration out or the field of view, E, collision,
C, or disintegration, D. Columns M, E, C, D give first the mean duration (in hours) to mitosis V, etc., second
the number of cells in the cell line followed to M, E, C or D respectively. The parenthesisstates that number
as a percentage of the total number of cells observed in the line, N . The digits in parentheses after cell line
symbols indicate the number of times the lines have been subcultivated. GP 115-2 and GP 117-1 are normal
foetal human fibrcblast-like cells and GP 125-2 and GP 127-1 their SV40-transformed counterparts.
~
Cell line
M
C
E
D
N
Din
ofM
~
GP 115-2 (7)
G P 117-1 (10)
GP 125-2 (26)
GP 127-1 (16)
(19)
5.1224(51)
4.92 16 (44)
4.99 21 (49)
4.22 16 (33)
6.18 15 (30)
5.17 31 (31)
4.51 14(30)
3.61 10 (28)
4.96 15 (34)
3.53 17 (35)
3.9426 (52)
3.78 43 (43)
8.74 6(13)
10.38 8 (22)
3.54 7 (16)
5.33 3(6)
3.60 2 (6)
O(0)
47
36
43
12.5
12.5
0
3.62 16 (33)
5.93 8 (16)
4.39 24 (24)
O(0)
8.17 1 (2)
8.17 1 ( 1 )
49
50
99
0
6.7
3.2
Altogether six disintegrations were recorded (Table I ) and the complete life spanfrom birth to disintegration-could be determined in five of these cells. The life span varied
from about 2.3 to 18.1 hr indicating a wide distribution (Table 3). Since the G, cell cycle phase
in these lines probably is 3-4 hr according to the FLM technique (Norrby, 1970) it seems
possiblethat disintegrations may occur in the GI phase as wellas inlaterphasesof thecellcycle.
Table 1 may indicate that the mobility (E C ) is increased in relation to all events recorded
+
204
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Peter Jagers and Klas Norrby
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Ouration t o eniiorJtion
"
5
4t
3
L
431
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15
HOil.,
FIG. 1. Distributions of individual durations (in hours) to mitosis, collision or emigration
among ninety-eight GP 127-1 cells.
TABLE
2 . Durations to mitosis (hr) in the larger
material where only life spans ending by mitotic
division were recorded.
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Cell line
Mean
Standard
deviation
Variance
No.
G P 115-2
G P 117-1
GP125-2
GP 127-1
6.58
6.98
7.28
5.17
4.45
5.78
5.28
5.26
19.80
33.45
27.91
27.21
64
33
43
31
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205
Cell cycle times
TABLE
3. Durations to disintegration, D,
and complete life spans, L (hr) among the
six cells observed to disintegrate. In one
case L could not be determined.
~
D
L
1.23
7.20
1.57
2.10
5.10
8.17
10.47
8.15
2.31
18.07
13.23
Mean 5.23
Mean 10.44
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(M + E + C + D) in the transformed populations. Further, the number of disintegrations in
relation to the number of mitoses is decreased in the transformed cells. This is in full accordance with a decrease in cell disintegration as described by other parameters (‘death index’,
death rate, and probability of a cell dying without dividing) of transformed cells in earlier
studies (Jagers, 1970; Norrby, 1970).
Mathematical
The model is based on generalized branching processes (Crump & Mode, 1968, 1969;
Jagers, 1969) but can also be framed in the terminology of Sevast’yanov (1971). (See also
Weiner, 1966.) It is assumed that cells are independent of one another, that the cycle time has
some arbitrary unknown distribution function G, that cells which complete their cycle divide
into two new cells but that the cell cycle may have been interrupted by a disintegration. The
(known or unknown) probability of the latter event is denoted by p and the time from birth
to disintegration (in cases where it occurs) is supposed to have another completely unknown
distribution, denoted by F.
Such models are the most general ones proposed for cell proliferation. They have certain
limitations, though. First, some findings (Kubitschek, 1967; Norrby et al., 1967) indicate a
positive correlation between life-lengths of sister cells. Second, the model only concerns
cells participating in the kinetic process, leaving out of consideration, e.g. the possibility of
entering a prolonged GI or G, cell cycle phase. This should be without significance in cases like
ours with a growth fraction close to unity. The probability of disintegration is taken as less
than +,meaning that we do not consider populations bound to extinction.
In biological systems it is often difficult to judge whether an exponential population growth
actually is balanced. From the literature, however, it seems acceptable to interpret an exponential population growth as balanced, at least, as in our case, when the period of exponential
growth substantially exceeds one population doubling time (cf. Steel, 1968; Norrby &
Mellgren, 1971).
The results of mathematical analysis of such a model are as follows: Denote by T, the mean
cycle time, i.e. the expectation of the cycle time distribution G, and by V, the variance of the
cycle time. The biological reader should note that T, and V, are indeed the mean and variance
in the basic elementary meaning and not any ‘flux statistics’ (Quastler, 1963). Let t, and u,
be the corresponding entities (mean and variance) of the durations to division of not
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206
Peter Jagers arid Klas Norrby
disintegrating cells chosen at random from a population in balanced exponential growth. Let
Td stand for the doubling time of the population. Then,
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zyx
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If there is no cell loss, then p
=0
and the formulae reduce to
Tc = f ( t c Td/ln2)
v, = $(rc- t C 2+ 21, Td/ln2)- TC2
(1')
(2')
Note that the formulae are independent of the distribution F of durations to disintegration,
which is virtually not apprehensible (see Table 3, though). Also observe that even if the extent
of cell loss is not exactly known ( I ) and (2) can be used to give bounds for T, and V,, (cf. Fig.
2).
Though the relations are simple, they follow only from a relatively (in this context) advanced mathematical analysis (available from the authors on request). However, in the extremely simplified-but still popular-model where the cycle time is assumed to be constant
(=T,) and there is no cell loss (i.e. p = 0, implying that T, = Td),the relation (1 ')can be deduced
elementarily: It is known, and graphical formulations are given in almost any survey paper
(cf. Lipkin, 1970), that in this case ages in balanced exponential growth are distributed according to a decreasing exponential density over the interval from zero to T,. Indeed, the density
is 2xe-'* for 0 Q t < T, and a = (ln2)/TC.Hence the average age of cells sampled at random
from such a population in balanced exponential growth is
I te-*'dt
Tc
2r
= 2{-Tc
e-ITc + I/cc - P r c / a }
=
(TJIn2)- Tc
0
after a replacement of 2 again by (In2)/Tc.The expected time from sampling to division therefore is
t , = T, - {(T,/ln2)- Tcj = 2T, - T 4 n 2
And the right-hand side of (1') collapses into
(t,
+ Tc/ln2)/2
=
T,
as should be the case.
Statistical
Since we consider exponential growth, the doubling time Td is easily determined; if No
is the population number at time zero and N , the size t time units later, then an estimator of Td
is
f d = ( t 1n2)/ In ",IN,)
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(Recall the division by In2 in the formulae). As t o p some information was given by Jagers
( 1 970, 1973) and Norrby ( 1970).
The parameters t, and t', are estimated thus: Denote thedurations observed, according to the
procedure described above, until any one of mitosis, disintegration, collision, or emigration
occur, by xI, x2, . ,., sn,..., .I-,,,.Assume that mitosis and cell divisions occurred after times
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207
Cell cycle times
100-
80 60-
i
I8
GP 117-1
111,
12
0
,
,
I
,
I
,
,
,
GP 125-2
0
h
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6ol
2ot
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‘ 80
GP 127-1
O
°
K
16
40
0
I
I
I
I
)
J
I
J
I
I
I
005
010
015
0.20
025
P
FIG. 2. Estimates of means, fc(left), and variances, pc (right) (in hours) based on our data
but plotted for all probabilitiesof cell death,p, between 0.00 and 0.25.
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Peter Jagers and Klas Norrby
208
x,, .v2,. . . s,.Let i be the mobility rate, i.e. we assume that during a short interval the risk of a
cell emigrating or colliding is approximately itimes the length of the interval. The rate is
estimated by
m
= (no.
of collisions + emigrations observed)/ 2 x j
j=l
A basis for this is given in the appendix. Let
Then, through replacing in the relations distribution functions by cumulative empiric distributions, estimators i, and f ' t ) are obtained off, and I ( : ) = + rez, respectively:
zlc
We write
for the estimator of r,. From an intuitive point of view the blowing up of estimators by the
factor eXrjbalances the fact that due to migration and collision only the shorter ones among
durations to mitosis are observed. If i. and hence 1were null, the sums would reduce to conventional expressions, like the arithmetic mean for ic.
We arrive at the following final formulae for empiric use:
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+
+
As an illustration for cell line GP 127-1 the number of collisions emigrations was 24
43 = 67 (Fig. I ). The sum of all times observed to any one of mitosis, disintegration, collision
or emigration is
s, = 435.65 (Fig. 1 ). Hence 2 = 67/435.65 = 0.154. The estimated doubling time fdis 13.5 (Table4). Further the numbers s,,. . . x, are as exhibited in Fig. 1, durations
to mitosis. Calculation of 4 and insertion into (3) and (4) yield fcand p, for various p . For
no cell death, p = 0, Tc = 15.3 and 9, = 69 (Table 4).
TABLE
4. The last three columns give the main empiric:esults of this paper, estimates ofthe mean, standard
deviation, and variance of the cycle time distribution. Td is the estimated doubling time, h the estimated rate
of mobility, and l o o p is the percentage of cell death. For i,, etc., see the text.
Cell line
GP
GP
GP
GP
115-2
117-1
125-2
117-1
fd
161
14.3
14.7
13.5
x
0.079
0.091
0.108
0.154
i,
8-3
10.8
10.6
11.2
1 tc
5.0
6.9
5.6
6.8
tC
75.0
47.8
31.1
46.5
I OOP
Fc
3.5-6.4 14.7-15.2
11.4-18.2 12.2-13.6
0.1-0.2
15.9
15.3
0
2: 0,
fiC
1.0-2.9
8.7-9.3
6.6
8.3
1.0-8.4
76.485.8
43.343.5
69.0
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Cell cycle times
209
APPLICATION, RESULTS A N D COMMENTS
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zy
Application of (I), (2) and the estimators derived above to the four cell lines yield the results
in Table 4. The estimates Td and@of the doubling time Tdand cell disintegration probability
p are from Jagers (1970) and Norrby (1970), the latter estimate being calculated under the
assumption that mitosis has a duration independent of the cycle time. Close results have been
obtained recently without this restriction (Jagers, 1973). The rate of mobility, I, seems higher
for the transformed cells than for the normal ones. Direct cinemicrographical observations
also hint at this (Table I). A remarkable feature about Table 4 is the low cycle time variance
p, of GP 115-2 cells. We have not been able to explain this but one should note that this
culture also has the smallest ri, and by far the smallest variance of the observed durations to
mitosis (Table 2). In view of the results in Table 4, let us point out here that T, for transformed
cells may very well exceed T, for these, since variance in cycle times implies the occurrence of
individual cycles much shorter than the mean T,. Obviously they influence the doubling time.
It is noteworthy that in the transformed cells the mean cycle time, T,, appears longer despite
the fact that the effective doubling time, Tdris shorter than for the corresponding normal cells
(Table 4). This indicates that cell disintegration is significantly reduced in the transformed
lines and fits well with the actual decrease in number of disintegrations among the transformed
cells as recorded from the cinemicrographs (Table 1). We thus have further evidence that the
concept of ‘kinetic transformation’ (Norrby, 1970) is appropriate. Thus, it is not due to
increased rate of cell production that the populations of transformed cells-similarly to other
neoplastic and malignant cell populations studied by us (Norrby, 1970; Norrby & Mellgren,
1971)-expand faster than their normal counterparts but because of a reduced rate of disintegration.
More information is furnished by Fig. 2, which gives f’,and p, once the correctp is known.
Observe that it does not exhibit p, and p, as functions of p . Indeed, the cycle time, being a
property of not disintegrating cells, is independent of the frequency of cell disintegration.
Therefore negative values are conceivable in the pccolumn. They show that the corresponding
p values are impossible. We have included Fig. 2 to emphasize what has been stated before,
namely that some conclusions from formulae ( I ) and (2) are possible even in cases wherep
is unknown.
The average duration of observation of cells, as accounted for in Tables 1 and 2, was around
6 hr. This should be compared to two or three times longer average time of observation required in a method where a daughter of a dividing cell is followed till mitotic division (provided nothing interrupts the cell cycle). This shorter observation of individual cells will
probably be welcomed by those who are familiar with the tedious examination of cinemicrographic films.
However, trivially, conclusions can never say more than what was obtained in the data they
were inferred from. Therefore we wish to emphasize that our results are obtained not only from
observing cells through the narrow window provided by the interval between sampling and
division-collision-emigration or disintegration, but also from a careful exploitation of implications from the empiric fact of balanced exponential growth. Indeed, this was one of the
tasks of the mathematical analysis. Thus we feel that an advantage of our method is that it
reaches comparatively well based results using short observations of individual cells (on
comparable points the results are in conformance with what has been obtained using rather
different techniques; see Jagers, 1970; Norrby 19’70; Norrby & Mellgren, 1971). Also with
210
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Peter Jagers and Klas Norrby
the present method sampling of observed durations is made without the censoring which
operates when cycle times are recorded by following single cells from their birth to their
mitotic division (Norrby et a/., 1967). In the last end, though, only repeated application to
different populations of the procedure and estimators derived can show their reliability
and give further credit to the method and a firmer base for the biological results gained.
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APPENDIX
As mentioned, the underlying mathematical analysis is available from the authors on request.
At this place we only comment briefly upon one statistical aspect, the estimation of A. As to
other entities, the doubling time T, is easily measured, from exponential growth, and the
incidencep of cell loss has been discussed elsewhere by the authors.
The rate of mobility, ;., must be estimated from a randomly censored sample. Indeed a
chosen cell either dies (divides or disintegrates) or emigrates-collides and no information can
be available as to when a cell that died before emigration would have emigrated or collided.
Thus what we can observe is the minimum of two random variables, one of which is exponential with parameter ;. It has long been known (Deemer & Votaw, 1955) that if there is
. . ., s,)of exponential random variables with parameter I, censored at
available a sample (sI.
some fixed pointy, then with the number of x, 6 y,
13,
n
2
min(xj, y)/v,
j= 1
is a consistent maximum likelihood estimator of I/).. It is not difficult to show that the consistency and even an asymptotic normality persist in our setup wherey is replaced bya sequence
of independent and identically distributed random variables yl, . . ., yn also independent of
s,.. . ., s,:If the y j have distribution K and is the number of x j 6 y j , then
\in
n
u, =
2
min(x,, y j ) / v n
j=1
is a consistent and asymptotically normal estimator of I/;..
ACKNOWLEDGMENTS
We are grateful to A. Odin for a helpful discussion and to B. Pettersson for computational
help. Part of this study was supported by grants from the Swedish Cancer Society.
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STEEL,
G.G. (1968) Cell loss from experimental turnours. Cell Tissue Kinet. 1, 193.
STFEL,G.G. (1972) The cell cycle in tuniours: an examination of data gained by the technique of labelled
mitoses. Cell Tissue Kinet. 5,87.
H.J. (1966) Applications of the age distribution in age-dependent branching processes. J. appl. Prob.
WEINER,
3, 179.
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Cell Tissue Kinet. (1974) 7,201-21 1
ESTIMATION OF THE MEAN A N D VARIANCE OF
CYCLE TIMES IN CINEMICROGRAPHICALLY
RECORDED CELL POPULATIONS D U R I N G
BALANCED EXPONENTIAL GROWTH
P E T E RJ A G E R SA N D K L A SN O R R B Y
Department of Mathematics, University of Goteborg, and
II Department of Pathology, University of Linkoping, Sweden
(Received 26 April 1973; revision received 20 October 1973)
ABSTRACT
The mathematical and statistical problem of estimating the mean and variance
of cell cycle times from cinemicrographically observed durations until mitosis
(cytokinesis at late anaphase), disintegration, collision (cells superimposing each
other) or emigration (moving out of the field of vision) of randomly chosen individual cells in a population in balanced exponential growth is treated. The resulting formulae are simple and considerably reduce the average cell observation
time. They are applied to two normal human foetal cell lines and their SV40-transformed counterparts. One result is that the latter seem to have longer cycle times in
spite of their shorter doubling times. Another result is that the cell mobility seems
somewhat increased in the transformed populations. This corroborates earlier
findings, indicating that the extent of cell loss, rather than the length of cycle times,
may play a decisive role for the (net) doubling time of cultivated cell populations.
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INTRODUCTION
The cycle time, or rather the distribution of cycle times, is one of the most important and most
elusive of the factors determining the kinetics of a cell population. Direct observation in
vitro is made difficult by the length of the cycle, resulting in censored samples (Norrby,
Johannisson & Mellgren, 1967). Indirect methods like analysis of FLM (PLM)-curves are
also very involved and largely heurisitic or dependent upon narrow assumtpions. For recent
surveys (among an abundant literature) see Mendelsohn & Takahashi (1971) or Steel
(1972) and for a statistical approach MacDonald (1970).
The aim of this study is to describe a method of estimating the mean cell cycle time and the
dispersion of cell cycle times in cultivated monolayer cell populations in balanced exponential
growth. Experimentally the method is based on cinemicrographic recordings.
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Correspondence: Dr Peter Jagers,Departmentof Mathematics,Universityof Goteborg, S-402 20 Goteborg,
Sweden.
14
201
202
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Peter Jagers and Klas Norrby
Mathematically the analysis proceeds from a simple idea and the results should have a
wide range of application : as has been observed in variouscontexts,any populationin balanced
exponential growth has a stable age-distribution, related to the cycle time distribution, the
incidence of cell loss, and the distribution of the times to disintegration of disintegrating cells.
Thus, if cells are sampled at random from such a population and the time to disintegration or
mitotic division is recorded, this should provide information about the cycle time. Since
sampled cells in this procedure are usually not newly born, one would expect a considerably
shorter time of observation than by the method of following cells from their birth to mitosis
or death.
Since the cells are moving around (only seldomly cells stay essentially in one place) the
procedure is complicated by migration and collision; it is often impossible to follow cells
till mitotic division or disintegration, since they emigrate outside the field of vision or collide.
It is assumed in this paper that the migration-cum-collision rate (the mobility rate) is constant
in time and independent of the ages of cells. Nothing in our material contradicts this assumption, though apriori it can not be excluded that cells are more mobile during some stages than
during others. It is this limitation that makes it possible to disregard the loss of information
caused by migration, or rather it compensates for it. Besides that our model is the most
general one suggested for cell proliferation (Nooney, 1968; Jagers, 1970), requiring no particular form of phase time or cycle time distributions, no independence between phase durations,
and allowing cell disintegration to take place during the life of the cell according to an arbitrary law.
What we can observe is the duration of time to disintegration, division, collision or emigration of a randomly chosen cell, whichever occurs first. The statistical problem of using
such observations to estimate the mobility will be discussed. The use of the mobility rate to
find the distribution of durations to death and the main problem, of relating this to the cycle
time distribution, will also be dealt with. For the benefit of the non-mathematical reader the
outcome of these analyses are presented separately.
zy
MATERIAL AND METHODS
Biological
The cell lines used were two normal human foetal lines (GP 115-2 and GP 117-1) and their
SV40-transformed counterparts (GP 125-2 and GP 127-1) (for terminology seeNorrby, 1970).
The normal lines emanated from skin and lung tissue respectively. The cultivation technique
used is the one commonly employed for such normal diploid cells (Hayflick & Moorhead,
1961 ;Norrby, 1970). The transformed lines were studied in the second stage of transformation
(Girardi, Jensen & Koprowski, 1965). All cells were fed Minimum Essential Medium (Eagle,
1959)supplemented with 10 a,,: not heat inactivated newborn calf serum, penicillin (100 IU/ml)
and streptomycin (100 ,ug/ml).
For cinemicrography 175,000 k 25,000 cells were seeded in Cooper plastic Petri dishes
(Falcon Plastics, Los Angeles. U.S.A.) and subsequently maintained in humidified C0,-air
atmosphere at 37°C and pH 7-3 0.1. An inverted phase contrast microscope built into a
Zeiss Micro Cine Camera equipped with a Zeiss Microflash Unit (30 W sec) was used.
Interference and other filters transmitted less than 1 % of the G1O-j sec flash to the cells.
One frame was exposed every other minute at a magnification of about x22.
From the cinemicrographs a period of exponential growth was found in each experiment
and during this period a set of population kinetic parameters were established (Norrby et al.,
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Cell cycle times
203
1967; Jagers, 1970; Norrby, 1970; Norrby & Mellgren, 1971). Such parameters include the
effective cycle rate, the effective net growth rate, the effective death or disintegration rate, and
the probability of a cell dying without dividing. It was further concluded, among other things,
that: (i) the migration across the borders of the field of view is random, (ii) the growth fraction
is essentially unity, (iii) all cell divisions in the populations now studied yielded two daughter
cells, and (iv) distinct cellular events are determined from the cinemicrographs with an error
of a single determination less than 2 min (0.03 hr).
It is important to realize that in terms of duration of DNA synthesis and generation time
from FLM curves, as of population doubling time and ‘terminal cell density’, our normal cells
compare well with what is known about other normal human foetal fibroblast-like cells (see
Norrby & Mellgren, 1971). Transformation with SV40implies aneoplasticchange(Koprowski
et al., 1966). In the present study, in order to record durations to mitosis (to beginning
of cytokinesis at late anaphase-early telophase), to emigration (outside the field of view),
to collision (two or more cells being superimposed), and to disintegration, randomly chosen
cells at 12 hr intervals during the period of exponential growth were tracked down on the
cinemicrographs. The observation of a single cell was thus initiated a random time after the
birth of the cell. This was done from one or two experiments per cell line (Table 1, Fig. 1).
The collection of durations to mitosis was extended by similar examination of cinemicrographs from another two or three experiments per cell line (Table 2). Any duration longer than
12 hr was, of course, considered only once. A large number of cells were successfully traced
through collisions but often the tracks of two or more cells ran into each other during the
colliding event preventing further tracing of the cells involved. The number of disintegrations
recorded is possibly a minimum number since disintegrations occurring during a collision
might be camouflaged by the neighbouring and often superimposed cell(s).
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TABLE
I . Cells followed to any one of mitotic division, M, emigration out or the field of view, E, collision,
C, or disintegration, D. Columns M, E, C, D give first the mean duration (in hours) to mitosis V, etc., second
the number of cells in the cell line followed to M, E, C or D respectively. The parenthesisstates that number
as a percentage of the total number of cells observed in the line, N . The digits in parentheses after cell line
symbols indicate the number of times the lines have been subcultivated. GP 115-2 and GP 117-1 are normal
foetal human fibrcblast-like cells and GP 125-2 and GP 127-1 their SV40-transformed counterparts.
~
Cell line
M
C
E
D
N
Din
ofM
~
GP 115-2 (7)
G P 117-1 (10)
GP 125-2 (26)
GP 127-1 (16)
(19)
5.1224(51)
4.92 16 (44)
4.99 21 (49)
4.22 16 (33)
6.18 15 (30)
5.17 31 (31)
4.51 14(30)
3.61 10 (28)
4.96 15 (34)
3.53 17 (35)
3.9426 (52)
3.78 43 (43)
8.74 6(13)
10.38 8 (22)
3.54 7 (16)
5.33 3(6)
3.60 2 (6)
O(0)
47
36
43
12.5
12.5
0
3.62 16 (33)
5.93 8 (16)
4.39 24 (24)
O(0)
8.17 1 (2)
8.17 1 ( 1 )
49
50
99
0
6.7
3.2
Altogether six disintegrations were recorded (Table I ) and the complete life spanfrom birth to disintegration-could be determined in five of these cells. The life span varied
from about 2.3 to 18.1 hr indicating a wide distribution (Table 3). Since the G, cell cycle phase
in these lines probably is 3-4 hr according to the FLM technique (Norrby, 1970) it seems
possiblethat disintegrations may occur in the GI phase as wellas inlaterphasesof thecellcycle.
Table 1 may indicate that the mobility (E C ) is increased in relation to all events recorded
+
204
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Peter Jagers and Klas Norrby
zyxw
Ouration t o eniiorJtion
"
5
4t
3
L
431
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zyxwvutsrq
15
HOil.,
FIG. 1. Distributions of individual durations (in hours) to mitosis, collision or emigration
among ninety-eight GP 127-1 cells.
TABLE
2 . Durations to mitosis (hr) in the larger
material where only life spans ending by mitotic
division were recorded.
zyxwvu
Cell line
Mean
Standard
deviation
Variance
No.
G P 115-2
G P 117-1
GP125-2
GP 127-1
6.58
6.98
7.28
5.17
4.45
5.78
5.28
5.26
19.80
33.45
27.91
27.21
64
33
43
31
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205
Cell cycle times
TABLE
3. Durations to disintegration, D,
and complete life spans, L (hr) among the
six cells observed to disintegrate. In one
case L could not be determined.
~
D
L
1.23
7.20
1.57
2.10
5.10
8.17
10.47
8.15
2.31
18.07
13.23
Mean 5.23
Mean 10.44
zyxwvut
(M + E + C + D) in the transformed populations. Further, the number of disintegrations in
relation to the number of mitoses is decreased in the transformed cells. This is in full accordance with a decrease in cell disintegration as described by other parameters (‘death index’,
death rate, and probability of a cell dying without dividing) of transformed cells in earlier
studies (Jagers, 1970; Norrby, 1970).
Mathematical
The model is based on generalized branching processes (Crump & Mode, 1968, 1969;
Jagers, 1969) but can also be framed in the terminology of Sevast’yanov (1971). (See also
Weiner, 1966.) It is assumed that cells are independent of one another, that the cycle time has
some arbitrary unknown distribution function G, that cells which complete their cycle divide
into two new cells but that the cell cycle may have been interrupted by a disintegration. The
(known or unknown) probability of the latter event is denoted by p and the time from birth
to disintegration (in cases where it occurs) is supposed to have another completely unknown
distribution, denoted by F.
Such models are the most general ones proposed for cell proliferation. They have certain
limitations, though. First, some findings (Kubitschek, 1967; Norrby et al., 1967) indicate a
positive correlation between life-lengths of sister cells. Second, the model only concerns
cells participating in the kinetic process, leaving out of consideration, e.g. the possibility of
entering a prolonged GI or G, cell cycle phase. This should be without significance in cases like
ours with a growth fraction close to unity. The probability of disintegration is taken as less
than +,meaning that we do not consider populations bound to extinction.
In biological systems it is often difficult to judge whether an exponential population growth
actually is balanced. From the literature, however, it seems acceptable to interpret an exponential population growth as balanced, at least, as in our case, when the period of exponential
growth substantially exceeds one population doubling time (cf. Steel, 1968; Norrby &
Mellgren, 1971).
The results of mathematical analysis of such a model are as follows: Denote by T, the mean
cycle time, i.e. the expectation of the cycle time distribution G, and by V, the variance of the
cycle time. The biological reader should note that T, and V, are indeed the mean and variance
in the basic elementary meaning and not any ‘flux statistics’ (Quastler, 1963). Let t, and u,
be the corresponding entities (mean and variance) of the durations to division of not
zy
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206
Peter Jagers arid Klas Norrby
disintegrating cells chosen at random from a population in balanced exponential growth. Let
Td stand for the doubling time of the population. Then,
zyxwvuts
zyxwv
zy
zyx
zyxw
zyxwvu
If there is no cell loss, then p
=0
and the formulae reduce to
Tc = f ( t c Td/ln2)
v, = $(rc- t C 2+ 21, Td/ln2)- TC2
(1')
(2')
Note that the formulae are independent of the distribution F of durations to disintegration,
which is virtually not apprehensible (see Table 3, though). Also observe that even if the extent
of cell loss is not exactly known ( I ) and (2) can be used to give bounds for T, and V,, (cf. Fig.
2).
Though the relations are simple, they follow only from a relatively (in this context) advanced mathematical analysis (available from the authors on request). However, in the extremely simplified-but still popular-model where the cycle time is assumed to be constant
(=T,) and there is no cell loss (i.e. p = 0, implying that T, = Td),the relation (1 ')can be deduced
elementarily: It is known, and graphical formulations are given in almost any survey paper
(cf. Lipkin, 1970), that in this case ages in balanced exponential growth are distributed according to a decreasing exponential density over the interval from zero to T,. Indeed, the density
is 2xe-'* for 0 Q t < T, and a = (ln2)/TC.Hence the average age of cells sampled at random
from such a population in balanced exponential growth is
I te-*'dt
Tc
2r
= 2{-Tc
e-ITc + I/cc - P r c / a }
=
(TJIn2)- Tc
0
after a replacement of 2 again by (In2)/Tc.The expected time from sampling to division therefore is
t , = T, - {(T,/ln2)- Tcj = 2T, - T 4 n 2
And the right-hand side of (1') collapses into
(t,
+ Tc/ln2)/2
=
T,
as should be the case.
Statistical
Since we consider exponential growth, the doubling time Td is easily determined; if No
is the population number at time zero and N , the size t time units later, then an estimator of Td
is
f d = ( t 1n2)/ In ",IN,)
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(Recall the division by In2 in the formulae). As t o p some information was given by Jagers
( 1 970, 1973) and Norrby ( 1970).
The parameters t, and t', are estimated thus: Denote thedurations observed, according to the
procedure described above, until any one of mitosis, disintegration, collision, or emigration
occur, by xI, x2, . ,., sn,..., .I-,,,.Assume that mitosis and cell divisions occurred after times
zyxw
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zyxwvu
207
Cell cycle times
100-
80 60-
i
I8
GP 117-1
111,
12
0
,
,
I
,
I
,
,
,
GP 125-2
0
h
zyxwvutsrqp
6ol
2ot
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zyxwv
‘ 80
GP 127-1
O
°
K
16
40
0
I
I
I
I
)
J
I
J
I
I
I
005
010
015
0.20
025
P
FIG. 2. Estimates of means, fc(left), and variances, pc (right) (in hours) based on our data
but plotted for all probabilitiesof cell death,p, between 0.00 and 0.25.
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zyx
Peter Jagers and Klas Norrby
208
x,, .v2,. . . s,.Let i be the mobility rate, i.e. we assume that during a short interval the risk of a
cell emigrating or colliding is approximately itimes the length of the interval. The rate is
estimated by
m
= (no.
of collisions + emigrations observed)/ 2 x j
j=l
A basis for this is given in the appendix. Let
Then, through replacing in the relations distribution functions by cumulative empiric distributions, estimators i, and f ' t ) are obtained off, and I ( : ) = + rez, respectively:
zlc
We write
for the estimator of r,. From an intuitive point of view the blowing up of estimators by the
factor eXrjbalances the fact that due to migration and collision only the shorter ones among
durations to mitosis are observed. If i. and hence 1were null, the sums would reduce to conventional expressions, like the arithmetic mean for ic.
We arrive at the following final formulae for empiric use:
zyxwvut
+
+
As an illustration for cell line GP 127-1 the number of collisions emigrations was 24
43 = 67 (Fig. I ). The sum of all times observed to any one of mitosis, disintegration, collision
or emigration is
s, = 435.65 (Fig. 1 ). Hence 2 = 67/435.65 = 0.154. The estimated doubling time fdis 13.5 (Table4). Further the numbers s,,. . . x, are as exhibited in Fig. 1, durations
to mitosis. Calculation of 4 and insertion into (3) and (4) yield fcand p, for various p . For
no cell death, p = 0, Tc = 15.3 and 9, = 69 (Table 4).
TABLE
4. The last three columns give the main empiric:esults of this paper, estimates ofthe mean, standard
deviation, and variance of the cycle time distribution. Td is the estimated doubling time, h the estimated rate
of mobility, and l o o p is the percentage of cell death. For i,, etc., see the text.
Cell line
GP
GP
GP
GP
115-2
117-1
125-2
117-1
fd
161
14.3
14.7
13.5
x
0.079
0.091
0.108
0.154
i,
8-3
10.8
10.6
11.2
1 tc
5.0
6.9
5.6
6.8
tC
75.0
47.8
31.1
46.5
I OOP
Fc
3.5-6.4 14.7-15.2
11.4-18.2 12.2-13.6
0.1-0.2
15.9
15.3
0
2: 0,
fiC
1.0-2.9
8.7-9.3
6.6
8.3
1.0-8.4
76.485.8
43.343.5
69.0
zy
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zy
Cell cycle times
209
APPLICATION, RESULTS A N D COMMENTS
zyx
zyxwvutsrqponm
zy
Application of (I), (2) and the estimators derived above to the four cell lines yield the results
in Table 4. The estimates Td and@of the doubling time Tdand cell disintegration probability
p are from Jagers (1970) and Norrby (1970), the latter estimate being calculated under the
assumption that mitosis has a duration independent of the cycle time. Close results have been
obtained recently without this restriction (Jagers, 1973). The rate of mobility, I, seems higher
for the transformed cells than for the normal ones. Direct cinemicrographical observations
also hint at this (Table I). A remarkable feature about Table 4 is the low cycle time variance
p, of GP 115-2 cells. We have not been able to explain this but one should note that this
culture also has the smallest ri, and by far the smallest variance of the observed durations to
mitosis (Table 2). In view of the results in Table 4, let us point out here that T, for transformed
cells may very well exceed T, for these, since variance in cycle times implies the occurrence of
individual cycles much shorter than the mean T,. Obviously they influence the doubling time.
It is noteworthy that in the transformed cells the mean cycle time, T,, appears longer despite
the fact that the effective doubling time, Tdris shorter than for the corresponding normal cells
(Table 4). This indicates that cell disintegration is significantly reduced in the transformed
lines and fits well with the actual decrease in number of disintegrations among the transformed
cells as recorded from the cinemicrographs (Table 1). We thus have further evidence that the
concept of ‘kinetic transformation’ (Norrby, 1970) is appropriate. Thus, it is not due to
increased rate of cell production that the populations of transformed cells-similarly to other
neoplastic and malignant cell populations studied by us (Norrby, 1970; Norrby & Mellgren,
1971)-expand faster than their normal counterparts but because of a reduced rate of disintegration.
More information is furnished by Fig. 2, which gives f’,and p, once the correctp is known.
Observe that it does not exhibit p, and p, as functions of p . Indeed, the cycle time, being a
property of not disintegrating cells, is independent of the frequency of cell disintegration.
Therefore negative values are conceivable in the pccolumn. They show that the corresponding
p values are impossible. We have included Fig. 2 to emphasize what has been stated before,
namely that some conclusions from formulae ( I ) and (2) are possible even in cases wherep
is unknown.
The average duration of observation of cells, as accounted for in Tables 1 and 2, was around
6 hr. This should be compared to two or three times longer average time of observation required in a method where a daughter of a dividing cell is followed till mitotic division (provided nothing interrupts the cell cycle). This shorter observation of individual cells will
probably be welcomed by those who are familiar with the tedious examination of cinemicrographic films.
However, trivially, conclusions can never say more than what was obtained in the data they
were inferred from. Therefore we wish to emphasize that our results are obtained not only from
observing cells through the narrow window provided by the interval between sampling and
division-collision-emigration or disintegration, but also from a careful exploitation of implications from the empiric fact of balanced exponential growth. Indeed, this was one of the
tasks of the mathematical analysis. Thus we feel that an advantage of our method is that it
reaches comparatively well based results using short observations of individual cells (on
comparable points the results are in conformance with what has been obtained using rather
different techniques; see Jagers, 1970; Norrby 19’70; Norrby & Mellgren, 1971). Also with
210
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Peter Jagers and Klas Norrby
the present method sampling of observed durations is made without the censoring which
operates when cycle times are recorded by following single cells from their birth to their
mitotic division (Norrby et a/., 1967). In the last end, though, only repeated application to
different populations of the procedure and estimators derived can show their reliability
and give further credit to the method and a firmer base for the biological results gained.
zy
zyxwvut
zyxwvut
zyxwvu
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zyxwv
APPENDIX
As mentioned, the underlying mathematical analysis is available from the authors on request.
At this place we only comment briefly upon one statistical aspect, the estimation of A. As to
other entities, the doubling time T, is easily measured, from exponential growth, and the
incidencep of cell loss has been discussed elsewhere by the authors.
The rate of mobility, ;., must be estimated from a randomly censored sample. Indeed a
chosen cell either dies (divides or disintegrates) or emigrates-collides and no information can
be available as to when a cell that died before emigration would have emigrated or collided.
Thus what we can observe is the minimum of two random variables, one of which is exponential with parameter ;. It has long been known (Deemer & Votaw, 1955) that if there is
. . ., s,)of exponential random variables with parameter I, censored at
available a sample (sI.
some fixed pointy, then with the number of x, 6 y,
13,
n
2
min(xj, y)/v,
j= 1
is a consistent maximum likelihood estimator of I/).. It is not difficult to show that the consistency and even an asymptotic normality persist in our setup wherey is replaced bya sequence
of independent and identically distributed random variables yl, . . ., yn also independent of
s,.. . ., s,:If the y j have distribution K and is the number of x j 6 y j , then
\in
n
u, =
2
min(x,, y j ) / v n
j=1
is a consistent and asymptotically normal estimator of I/;..
ACKNOWLEDGMENTS
We are grateful to A. Odin for a helpful discussion and to B. Pettersson for computational
help. Part of this study was supported by grants from the Swedish Cancer Society.
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21 1
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P. (1973) Maximumlikelihoodestimation of thereproduction distribution in branching processes and
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F., GIRARDI,
A. & KOPROWSKA,
I. (1966) Neoplastic transformation. Cancer Res.
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G.C. (1968) Age distributions in stochastically dividing populations. J. theoret. Biol. 20, 314.
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