[CANCER RESEARCH 43, 556-560, February 1983]
0008-5472/83 / 0043-0000502.00
Growth of Mammalian Multicellular Tumor Spheroids
Alan D. C o n g e r 1 and Marvin C. Z i s k i n 2
Radiation Biology [A. D. C.] and Diagnostic Radiology Research Laboratory [M. C. Z.], Temple University School of Medicine, Philadelphia, Pennsylvania 19140
ABSTRACT
MATERIALS AND METHODS
The in vitro growth of small (0.05 to 3 mm diameter) avascular
multicellular tumor spheroids from six rodent and two human
tumor lines has been analyzed. Surprisingly, the radial increase
of multicellular tumor spheroids is linear with time after a brief
initial period of geometric growth. These multicellular tumor
spheroids are shown to have a constant thickness of proliferative outer crust and of middle nonproliferative but viable
mantle. An analytical model for their growth is developed which
explains the growth pattern. This constant crust thickness
model leads to a progressively diminishing growth fraction as
radius increases and should be applicable to such early growth
of micrometastases in vivo. The model also provides a procedure for determining cell cycle time.
Several rodent and human tumor lines of different origin were studied
and are listed in Table 1. Cell lines and MTS are grown in Eagle's
complete medium with added 10% fetal calf serum and antibiotics
(Grand Island Biological Co., Grand Island, N. Y.) in a 37 ~ incubator
with humidified 5% CO2:95% air-gas.
To grow MTS, a trypsinized cell suspension in liquid complete
medium taken from nearly confluent monolayer flasks is inoculated
over an agar-solidified base layer of complete medium in Petri dishes
(14). The cells cannot adhere to the agar surface and so cohere to
each other. In one to four days, the initial aggregation of cells and cell
clumps has formed small MTS that commence solid tumor growth as a
unit. From such production dishes, MTS of uniform size are selected
under a dissecting microscope with a calibrated ocular micrometer
("sized") and placed individually in agar-based wells of 24-well plates.
Medium in the wells is changed twice per week, and the MTS is
transferred into fresh agar-based wells once per week; more frequent
feeding does not enhance growth rate. The dishes are periodically
agitated. Subsequently, diameter of the MTS in their growth wells is
measured at x 1 2 5 [to ___2#m] under an inverted microscope, and an
average diameter and error limits are determined for the 12 or sometimes 24 MTS at increasing times, yielding growth curves. Accuracy is
such that, within any sized lot of MTS, the coefficient of variation in
diameter is --<10% and is usually -<5%. MTS thus grown are virtually
spherical; extreme difference between greatest and least diameter is
<5%.
INTRODUCTION
The in vivo growth rate of solid tumors is difficult to measure
accurately, although it can be done, but at small tumor volumes
measurement is virtually impossible. The development of an in
vitro method by Sutherland e t al. (12) to grow multicellular
tumor spheroids and its later simplification by Yuhas et al. (14)
permitted detailed and accurate measurement of growth rate
in these small MTS, 3 which serve as an in vitro model system
for solid tumor " m i c r o m e t a s t a s e s " growing in vivo.
An early and unexpected finding with these MTS was that
the radius of the spheroids increased linearly with time in the
period after initial single-cell aggregation to form the MTS and
before the spheroids became so large that a plateau phase of
s e n e s c e n c e set in at about 2 to 4 mm diameter. In oncological
terms, this linear-with-time increase in diameter applies to early
state " m i c r o m e t a s t a s e s " in the period before the development
of intratumoral circulation; when that occurs, diffusion of metabolites and catabolites is no longer solely through the external
surface of the solid spheroid, and different conditions apply (3,
4).
Since the observed MTS radial growth is apparently linear
and not curvilinear as would be expected in geometric growth
processes, some other model for the kinetics of spherical solidtumor growth must apply.
We will present data on the in vitro growth of MTS for several
tumor lines and on the characteristics of the growth fraction
within them and from this development a growth model for
tumors that is compatible with the observations.
Recipient of National Cancer Institute Grant CA 24658.
2 Recipient of National Institute of General Medical Sciences Grant GM 25477
to whom requests for reprints should be addressed at Diagnostic Radiology
Research Laboratory, Temple University School of Medicine, 3400 North Broad
Street, Philadelphia, Pa. 19140.
3 The abbreviations used are: MTS, multicellular tumor spheroids; GF, growth
fraction (cycling cells per total cells).
Received February 15, 1982; accepted November 5, 1982.
556
RESULTS
Fitted growth curves with their 9 5 % confidence limits for the
8 lines described in Table 1 are given in Chart 1. In such
experiments, MTS of uniform diameter of about 2 5 0 /~m are
sized from the production plates and placed in individual wells
of 24-well plates ( " D a y 0 " ) ; at this size, the initial aggregation
process and geometric growth phase are over. At larger sizes
and longer growth times, these growth curves will start to bend
over and finally flatten out. Repeated feedings with fresh medium in fresh agar-based wells will not prevent this. Volume
becomes too large for surface area, and diffusion exchange
becomes insufficient in these nonvascularized solid tumors.
With some tumor lines at this stage, the MTS will " m e t a s t a s i z e "
by splitting up into several smaller spheroids, and these will
resume growth.
The data of Chart 1 have been fitted to the linear relation,
D(#m) = a + bt(d~y~)
and it can be seen that over this size range, 2 0 0 to 1500/~m
(a range in volumes of 4 2 2 to 1), the diameter increases
apparently linearly with time; the 9 5 % confidence limits and
correlation coefficients of 0 . 9 9 2 minimum to 0 . 9 9 9 maximum
confirm that the linear fit is a reasonable one. The curvilinear
geometric increase that would apply if all cells divided [Rn =
Ro(2) t~
or if a constant fraction divided [Rn = Ro(1 +
GF)t~
where R is the radius of MTS and Tc is the cell cycle
time, can be excluded.
CANCER RESEARCH VOL. 43
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Research.
Solid Tumor Growth
Table 1
Tumor lines studied
Tumor line
Species
498 ("Line 1 ")
MCa-11
QTA-31
KA-31
3M2N
RLN-13-K2
MDA-361
NB-1 O0
Type
Organ
Lung
Breast
Muscle
Muscle
Breast
Kidney
Breast
Nerve
Mouse (BALB/c)
Mouse (BALB/c)
Mouse (BALB/c)
Mouse (BALB/c)
Rat
Rat
Human
Human
Growth rate of diameter (~m/
day)~
Remarks
Spontaneous origin
Radiation induced
Derived from 3T3
Derived from 3T3
Type II carcinoma, alveolar
Mammary epithelium
Fibroblast, myoepithelium
Fibroblast, myoepithelium
Squamous cell carcinoma
Embryonal nephroblastoma
Epithelial carcinoma
Neuroblastoma
99.6
28.5
42.8
31.2
33.4
66.5
18.3
42.4
Mixed cell types
Insulin-dependent
Spontaneous
a Values obtained from growth curves in Fig. 1.
0
7OO
600
.....
I
5
10
I
I
-
-50
/{/,
40
~--
500LINE NB 100
400
1500
500
,~/
-
/{/
80
.,-..
- 4 0 0 -~
-10
<~ 3 0 0
_
1
_
'
I
,/, '
I
'
I
_
co
6
X
~
I-LU,,~a-Z m
1 0 -~&"/1.e
1000
KT-2
,
, ~ / /
wz_
--5
LL
oo ~<
9
m
0.5 m
z
0
1:if i--~
LINEM:a-ll t1:~
0 1
5
10
:D,
rr
UJ
b--
uJ 5 0 0
<_ 4 0 0
3OO
200
'~" ~"~
Z / * :: .~-"~ .S"Y " ,~"-Y;~".~
-'><{.-".-~/'"
800
15
GROWTH TIME (DAYS)
Chart 1. In vitro growth of MTS for 8 tumor lines (described in Table 1). All
growth curves start at about the same diameter, 250 ffm, but have been displaced
vertically on the graph to avoid overlap. The highest (498) and lowest (MDA361) MTS lines have their actual diameter indexed on the ordinates. ~ ,
regression fits, with 95% confidence limits to:
D(#,m) = a 4- bt(days)
where D is the diameter [correlation coefficients, r, from 0.991 (QTA) to 0.999
(3M2N)]; | A, [~, x,diameter. Bars, S.E. (when no error bars are shown, S.E.
limits are smaller than the size of the symbol); numbers with arrows, diameters
(mm) of MTS. Points, average of 12 or 24 MTS.
Even though growth at these larger sizes is linear with time,
as seen in Chart 1, this is not the case for very small sizes.
When the MTS are very small, the diffusion distances to their
centers are short, and all of the cells are proliferating. Thus,
initial growth is geometric with time.
From 2 tumor lines, we selected MTS as small as could be
sized and handled properly and plated them into individual
wells, with the results given in Chart 2. The initial part of these
curves is largely but not entirely the geometric growth that
occurs at still smaller sizes. Actually, the initial part of these
curves occupies an intermediate phase between the early
wholly geometric growth and the later linear growth phases.
The initial parts of the curves of Chart 2 are curvilinear and do
not conform, statistically, to the latter growth pattern these
same MTS lines assume (Chart 1).
FEBRUARY
",d"
~'- ~ ;,/~,d~'"
~ .~-" 0.5. ~~. . ~-/ s "
"---. -
500
.'$A .+=" . ~ - - "
I
I
I
1
0
10
GROWTH
20
TIME (DAYS)
30
-
200
Chart 2. Initial curvilinear (mostly geometric) and later linear growth of MTS
for a mouse and a human line. Data are fit to:
where D is the diameter, for the initial growth, and to:
D~
=
a + bt(day=~
for the later growth . . . . . . . .
95% confidence limits (for the later linear growth,
95% confidence limits fall within the width of fitted line); | z~, diameter. Bars,
S.E. (when no error bars are shown, S.E. limits are smaller than symbol).
DISCUSSION
The growth of these MTS proceeds through successive
phases, as seen in Chart 3. There is a first stage of cell and
cell cluml2 aggregation to initiate the MTS, followed by a period
of geometric growth when diffusion is not limiting and all cells
are dividing. As the MTS increase in size still more, diffusion
1983
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Research.
557
A. D. Conger and M. C. Ziskin
VIABLE RIM t CRUST (prolif.)
tMANTLE (non-pr o l i ~ c o n s t a n t
CORE ( . . . . . tic)t:!~ ( -
~-" lit
)
..~.~.~3
i
GROWTH TIME
Chart 3. Growth phases of MTS. Initial aggregation, growth, and development
of MTS 3-layered heterogeneity are correlated with the growth curve. Note that
crust and mantle thicknesses are constant (once those thicknesses are attained).
profiL, proliferative; non-prolif., nonproliferative.
gradients of nutrients and catabolites set in, and it becomes a
2-layered solid with an outer well-nourished and proliferative
" c r u s t " and an inner deprived and nonproliferative " m a n t l e . "
Still later and at a size which is line characteristic, the MTS
develop a central necrotic " c o r e " which continues to increase
proportionally as overall size increases. Eventually, the MTS,
like a solid tumor in vivo, are cellularly heterogeneous solids
with 3 layers: the outer proliferative crust; a middle viable but
nonproliferative mantle; and a central necrotic core.
At still later times of plateau and senescence (outside the
scope of the model presented here), cell shedding of the MTS
and cell packing in the interior may become appreciable. In
general, however, their magnitude is less, and they do not
significantly affect the growth rate during the initial and linear
growth phases. If cell packing or shedding were appreciable
during these phases, the growth rate would be less than linear,
but this is not observed (Chart 1 ).
Yuhas and Li (13) have shown experimentally for 7 different
tumor lines that the outer proliferative crust does indeed have
a constant thickness, characteristic for each line (range, 18 to
110/~m thick for the lines that they studied). Tumor lines with
thick proliferative crusts should grow faster than other lines
with thin crusts as long as cell cycle times do not differ much;
we show ( " A p p e n d i x , " Table A1) that both these conditions
are fulfilled for 4 tumor lines. Yuhas and Li (13) found the
correlation of growth rate with crust thickness for their 7 tumor
lines was 0.97. We will show also how this constant proliferative
crust thickness results in the linear-with-time increase in MTS
diameter (Chart 1) that is observed after the initial geometric
growth (Chart 2).
We have shown experimentally 4 that the mantle also maintains a constant thickness; hence, the entire "viable rim" has
a constant thickness as diameter increases over the full range
of linear growth. This constant viable rim thickness is line
characteristic and ranges from 35 to 206 /~m for the 6 lines
thus analyzed by us. 4
Since nutrient and oxygen concentration of the medium at
the MTS surface is constant, and then if metabolic rate is
constant with time, it follows that a gradient of nutrients in and
4 Unpublished observations.
558
catabolites out will be established within the MTS. Also, if there
is a nutrition or catabolite threshold level at which proliferation
ceases, then it is easy to see why the proliferative crust
thickness is constant. Unless metabolic rates, proliferation
thresholds, and gradients are identical for all tumor lines, we
should expect that the crust thickness would be line specific
and different for different lines; this expectation agrees with the
findings of Yuhas and Li (13).
The idea that the proliferative outer crust maintains a constant thickness as the MTS grow is counter to the most usual
assumption that a solid tumor has a more or less constant GF.
The 2 schemes lead to quite different consequences as growth
proceeds.
A difference of growth pattern results from the 2 cases: A, a
constant fraction of cells divide (GF _< 1); and B, it is the
thickness of the proliferative crust that remains constant. The
size for these spheroidal tumors as they grow is given as
follows:
A. Constant GF model:
Vn + 1 = Vn + (GF)Vn
Rn + 1 = R~(1 + GF) 1/3
where R,, is the radius of MTS at end of nth cell cycle; R,, § 1 is
the radius, one cell cycle later; V,, and Vn +1 are volumes of
MTS; and n is the number of cell cycles.
B. Constant crust thickness model:
R, + ~ = R,[1 + 3 (k/R~) - 3 (k/R~) 2 + (k/Rn)3] ~/3
where k is crust thickness, and, in this latter case, the GF
continually decreases as radius increases with time. The above
equations are given because they are simple and readily understood. They assume, for simplicity, synchronous division.
However, division is a continuous asynchronous process, and
the exact solution for continuous division is given in
" A p p e n d i x , " Equation A3. It has been shown that MTS cell
cycle time for a tumor line is constant and is independent of
MTS age or size (2). With our constant crust thickness model,
given only the growth rate [ ~ R / d a y , #m (Chart 1)] and crust
thickness, k (Table A1, Column 2), it is apparent that an
estimate can be made of cell cycle time, Tc, within the crust of
the MTS. This point is developed more fully in the " A p p e n d i x , "
and estimates of Tc are given in Table A I .
Chart 4 illustrates the difference in MTS growth that results
from these 2 different proliferative models, GF constant, or GF
continually decreasing with time as it does when proliferative
crust thickness is constant. The curves show that, for the case
of constant crust thickness, growth quickly approaches approximate linearity with time and would be difficult to discriminate from linearity. But curvilinearity persists with a constant
GF.
The fact that the GF actually decreases with growth has been
verified by us in the 4 tumor lines thus analyzed. We found that
the GF decreases by .-40% in only 5 cell cycles and by ~-60%
in 10 cell cycles following initial measurements.
The key element for the present model is based on the
existence of a constant thickness of proliferative crust. This
leads directly to the approximately linear-with-time increase in
radial dimension. This linear-with-time increase shown for MTS
has been found also for growth of fungal spherical pellets
growing submerged in shake cultures; they appear to depend
on a similar constant thickness of surficial hyphal growth ex-
CANCER RESEARCH VOL. 43
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Research.
CANCER
1
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VOL.
43
radius of the entire MTS will be less than the eventual constant value of crust
thickness, k; hence, the entire MTS will be proliferative. Therefore, the growth
prior to the time when the MTS radius equals k will be of the usual logarithmic
pattern. The constant thickness model becomes applicable at the moment the
MTS radius equals k and continues to be applicable until senescence sets in.
For the purposes of this model, we make the following assumptions: Assumption 1, MTS consist of 2 distinct regions, an outer crust in which all cells are
proliferating and an inner region in which no cells are proliferating (there is no
need in this derivation to distinguish between the mantle and the necrotic core,
both of which are nonproliferative); Assumption 2, growth occurs in such a
manner that the inner region increases in diameter, but the thickness of the
proliferative crust remains constant and is line specific; Assumption 3, all daughter cells remain attached to the MTS (i.e., no shedding); Assumption 4, the
volume of any region is proportional to the number of cells contained within that
region, and the proportionality is constant throughout the MTS; Assumption 5, at
any given moment, the rate at which new cells are formed in the MTS is
proportional to the number of cells in the crust at that time. The constant of
proportionality equals (In 2)/Tc, and where Tc is the cell cycle time and does not
vary over time. Assumption 5 is expressed mathematically as:
|
/
/
II
I
RESEARCH
NUMBER OF CELL C Y C L E S , N
Chart 4. Relative increase in MTS radius with time (in cell cycles), according
to 2 models for MTS growth. - . . . . , constant GF for GFs of 1.0, 0.75, 0.5, and
0.25:
~Vtotal
R. + ~ = R~[1 + 3 (kiRk) - 3 (k/R~) 2 + (k/Rn)3] 1/3
(A1)
where dN/dt is the rate at which the total number of cells increases, N .... t is the
number of cells in the crust, and e is a constant of proportionality (rate constant).
It is related to the effective MTS cell cycle time by 4x equals (In 2)/tc; tc is the
average cycle time for the cells within the crust and is constant over time.
Assumption 4 permits conversion of Equation A1 into:
R~ + 1 = R~(1 + GF) ~/3
where n is number of cell cycles, and Ro and R~ are the radius, initially and at nth
cell cycle, respectively; - - ,
constant crust thickness, k, for initial k/Ro ratios of
1.0, 0.75, 0.5, and 0.25:
aN .... t
-
dt
= ~V .... t
(A2)
This equation is the essence of this model. It deviates from the usual logarithmic
growth in that the variable on the right side of the equation (V .... ,) is not the same
as that on the left (Vtota~). Expressing this in terms of R, we get an expression for
the rate of growth of the MTS radius:
An exact solution for this model is given in the "Appendix."
dt
tension,
although
analytically
the model used to describe
different
After completing
this work
different
be mathematically
(7) was based
vivo.
later (1978)
but had considered
basis
constant
empirical
to this same
knowledge
for
effect
thickness
or
an
model
analytical
shows
develops,
of the deviation of the growth
The
model
for
analytically
radial increase
for estimating
----=_tan-
(A4)
in
+--+1/2Ink
1 +3
+3
+~tan-
-~
.~
crust
that GF
The first 3 terms on the right side of the above equation equal a constant and can
be simplified if we measure time from the point at which the radius equals k.
Then Ro equals k, to equals 0, and:
(2) but did not give
linear-with-time
up to senescence.
+ 3
prior to
of constant
titative measure
method
of tumors
model
this as an assumption.
size and age increase
this
e ( t - t o ) = _R.._s V21n 1 - 3
by Mayneord
in t h e p a s t h a v e m a d e t h e o b s e r v a t i o n
crust
a
although
we were able to prove to
as MTS
for any stage
we discovered
a model which,
linear growth
that he came
thickness
Some studies
model,
to ours. This paper
only on the observed
the much
the
on our
in i t s e x p r e s s i o n ,
It i s r e m a r k a b l e
decreases
In orde r to obtain an e x p r e s s i o n for the radius, p e r s e , Equation A3 must be
integrated to obtain:
that presented
equivalent
3
was
from ours (6).
very early paper (1932)
somewhat
their growth
and provides
model
it. T h e
how
+In
1 -3
+
(A5)
+ ~/3~1tan-1
provides
a
cell cycle time within the MTS.
ACKNOWLEDGMENTS
[.2(k/R)-3~
L
a quan-
rate from linearity
also
t=-
the
,/~
_ 1.3023}
J
Equation A5 provides the relationship between radius and time of growth.
Although the radius cannot be expressed as an explicit function of time, time can
be expressed as an explicit function of the radius; i.e., the required time for
growth can be determined directly for any given radial size, provided that we
know the values of e and k.
It is convenient at this point to replace ~xwith (In 2)/Tc and rearrange Equation
A5 into:
We wish to thank Dr. John Yuhas for the cultures of tumor lines that he kindly
furnished us,
Tc-
In2
+ In
1 - 3
+
(A6)
+ ~tt a n
APPENDIX
Constant Crust Thickness Model.
The purpose of this appendix is to present the derivation of the equation for
the size of a MTS as it grows. It will also be shown that this model predicts a
nearly linear-with-time radial growth rate and that the slope of the empirical
growth curve of any MTS line can be used to determine its cell cycle time.
At this point, it should be noted that in the very early stages of growth, the
FEBRUARY
-1
[2(k/R)-3],/_3 J
- 1.3023
}
In Equation A6, the MTS radius, R, appears only in association with k. R/k
represents a nondimensional measurement of size which is normalized for the
differing values of k across cell lines. Similarly, t/Tc is a nondimensional measurement of time equaling the number of cell cycles of MTS growth. A graph of
MTS growth, as predicted by Equation A6, is shown in Chart A1. The advantage
of using the nondimensional ratios R/k and t/Tc is that all cell lines are
represented by this one curve. The apparently differing empirically observed
1983
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1983 American Association for Cancer
Research.
559
A. D. Conger and M. C. Ziskin
Table A1
9
i
;
/
LOGARITHMIC
GROWTH PATTERN //
(GF-1.0)
i/
8
"------~ I
/
7
6
M T S cell cycle times
/
Tumor line
~
/
/ ~ t = ,//."_ j / z ~
CONSTANT CRUST
THICKNESS MODEL
Es
n-
43
1
0
J,e--
- ~
- --I
-5
..~
- 3
+
= ~xk - - -
+-3R 2
As R becomes large, relative to k, the second and third terms on the right
side of the equation become progressively smaller. Thus, for large R, (i.e., small
k / R ) we can drop the last 2 terms to obtain the asymptotic solution:
=c~k(forlargeR/k)
(A7)
asymptotic
The right hand side of Equation A7 is a constant, and linearity is fulfilled when
the first derivative equals a constant. Therefore, this model predicts that MTS
growth is linear for sufficiently large R relative to k.
By expressing e in terms of cell cycle time and rearranging Equation AT, we
obtain the formula for determining the MTS cell cycle time:
560
49.8
21.4
15.6
14.3
31.1
41.2
40.5
32.6
18.6
13.1
21.0
15.9
kin2
(A8)
I
15
R
(dR)
93
53
38
28
Tc -
growth curves in the previously given Charts 1, 2, and 4 are, in fact, merely
differing intervals on this curve. Where any individual tumor line would begin on
this generalized curve is dependent on its crust thickness, k, and its radius, R,
when growth measurements commence.
One advantage of this model is that it shows how MTS growth pattern is
completely determined for any cell line if we know the cell cycle time within the
MTS and the crust thickness. The crust thickness, k, can be determined empirically as stated previously, but the cell cycle time within an MTS remains to be
computed. Direct experimental measurement of this cell time is difficult and to
date has not been accomplished. An important feature of the current model is
that it provides a formula for determining average cell cycle time within MTS. It
is derived as follows.
Recall that Equation A3 provides an expression for the rate of radial growth.
Rewriting Equation A3, we obtain:
3
Monolayer
Tc (hr) a
QTA 31
3M2NJ
I
I
0
5
10
NUMBER OF CELL CYCLES (t/tcy)
eR
3
498 ("Line 1 ")
QTA-31
KA-31
MCa-11
/
_ ~ . ~ 498
~..-Iq=.,~MCa 1 1
Chart A I . MTS growth. - - . ,
radial growth curve as determined by the
constant crust thickness model (Equation A6). For comparison, growth curve
resulting from a truly logarithmic process is indicated ( . . . . . ). Region in lower
left, time in which the radius is smaller than k and where logarithmic growth does
occur. For each of several tumor lines, the smallest size ( R / k ) actually measured
(Charts 1 and 2; Table A1) is indicated on the curve. Notice how rapidly the
constant crust thickness model growth curve tends to straighten out and approach linearity.
dR_
dt
Av MTS Tc
(hr) c
a Empirically determined values taken from Table 1, the report of Yuhas and
Li (13).
Taken from Chart 1.
c Calculated from Equation A8.
/
,,~1---/~ KA 31
2
k (Ftm)a
MTS radial
growth rate
(/~m/day) b
/
where (dR~tit)asymptotic is the maximum rate of change of radius and is measured
readily as the slope of the linear phase of the radial growth curve (or is equal to
one-half of the slope of the diameter growth curve). Table A1 lists the average
cell cycle times computed for 4 tumor lines as determined by Equation A8. For
comparison, the cell cycle times for monolayer growth are also shown; in this
case, all cells have immediate and equal access to the nutrient medium.
REFERENCES
1. Carlsson, J., Stalnacke, C.-J., Acker, H., Haji-Karim, M., Nilsson, S., and
Larsson, B. The influence of oxygen on viability and proliferation in cellular
spheroids. Int. J. Radiat. Oncol. Biol. Phys., 5 : 2 0 1 1 - 2 0 2 9 , 1 979.
2. Durand, R. E. Cell cycle kinetics in an in vitro tumor model. Cell Tissue
Kinet., 9 : 4 0 3 - 4 1 2, 1976.
3. Folkman, J. Tumor angiogenesis: therapeutic implications. N. Engl. J. Med.,
285: 1 1 8 2 - 1 1 8 6 , 1971.
4. Folkman, J., and Hochberg, M. Self-regulation of growth in three dimensions.
J. Exp. Med., 138: 7 4 5 - 7 5 3 , 1973.
5. Greenspan, H. P. Models for the growth of a solid tumor by diffusion. Studies
Appl. Math., 5 2 : 7 1 7 - 7 4 0 , 1 972.
6. Koch, A. L. The kinetics of mycelial growth. J. Gen. Microbiol., 89: 2 0 9 216, 1975.
7. Mayneord, W. V. On a law of growth of Jensens rat sarcoma. Am. J. Cancer,
1 6 : 8 4 1 - 8 4 6 , 1932.
8. McEIwain, D. L. S., and Morris, L. E. Apoptosis as a volume loss mechanism
in mathematical models of solid tumor growth. Math. Biosci., 3 9 : 1 4 7 - 1 5 7 ,
1978.
9. McEIwain, D. L. S., and Ponzo, P. J. A model for the growth of a solid tumor
with non-uniform oxygen concentration. Math. Biosci., 34: 2 6 7 - 2 7 9 , 1977.
10. Shymko, R. M., and Glass, L. Cellular and geometric control of tissue growth
and mitotic instability. J. Theor. Biol., 63: 3 5 5 - 3 7 4 , 1976.
11. Sutherland, R. M., and Durand, R. E. Hypoxic cells in an in vitro tumor
model. Int. J. Radiat. Biol. Relat. Stud. Phys. Chem. Med., 23: 2 3 5 - 2 4 6 ,
1973.
12. Sutherland, R. M., McCredie, J. A., and Inch, W. R. Growth of multicell
spheroids in tissue culture as a model of nodular carcinoma. J. Natl. Cancer
Inst., 46: 1 1 3 - 1 2 0 , 1971.
13. Yuhas, J. M., and Li, A. P. Growth fraction as a major determinant of
multicellular tumor spheroid growth rates. Cancer Res., 38: 1 5 2 8 - 1 5 3 2 ,
1 978.
14. Yuhas, J. M., Li, A. P., Martinez, A. O., and Ladman, A. J. A simplified
method for production and growth of multicellular tumor spheroids (MTS).
Cancer Res., 37: 3 6 3 9 - 3 6 4 3 , 1977.
CANCER
RESEARCH
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VOL.
43
Growth of Mammalian Multicellular Tumor Spheroids
Alan D. Conger and Marvin C. Ziskin
Cancer Res 1983;43:556-560.
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