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3-DIMENSIONAL IN VITRO MODELS OF MAMMALIAN TISSUES
FIELD OF THE INVENTION
The present invention relates generally to in vitro models of mammalian
tissues. More particularly, the invention relates to 3-dimensional in vitro
models simulating the behavior of a cancerous human tissue at various
stages of a solid tumour progression, to 3-dimensional in vitro model
to simulating the senescence and apoptosis of normal and cancerous human
cells, and to the use of such models for drug screening and testing. The
invention further relates to novel mathematical models for cell growth and
drug toxicity data analysis and cell-cell coupling simulation.
BACKGROUND OF THE INVENTION
Screening for bioactive chemical agents affecting in a desirable manner
functions of human organs and tissues, particularly for agents useful for
2o treating pathological conditions and disorders, is of great importance for
the
human well being and is becoming one of the fastest developing research
areas. As human testing is subject to very stringent limitations, various
models have been devised to simulate biological responses of human tissues
to such chemical agents, for example to screen for new promising drugs.
Models presently in use to simulate biological responses to chemical agents
include experimental animals, explanted tissue slices, monolayer cell
cultures,
and mathematical models based on the chemical structure of leader
compounds. None of these models offers a full compatibility with in vivo
3o human tissues and each has its advantages and shortcomings. Experimental
animals are expensive to maintain and there are ethical considerations
associated with the use of animals for such purposes. Moreover, because
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the cells and tissues of animals are different from those of humans, test
results are not always applicable to humans.
Monolayer cultures of human cells have the advantage of some biochemical
s similarity to human tissues and usually provide highly reproducible,
inexpensive, and well standardized test systems. However, 2-dimensional
(2D) cell cultures are not morphologically and physiologically similar to in
vivo
tissues and do not simulate well the state of cells and tissues in the
organism,
especially because they do not reproduce the cytoarchitecture found in the
io living organism. Furthermore, 2D cell cultures cannot reproduce the orderly
structure found in tissues formed of two or more cell phenotypes. As a
consequence, the biosynthetic activities and physiological functions
expressed by cells grown in monolayer cultures are markedly different from
those in the organism, and may frequently lead to misleading test findings.
is
The shortcomings of 2D cell cultures stimulated the development of various 3-
dimensional (3D) models of human tissues, intended to simulate more closely
the morphological and physiological characteristics of their in vivo
counterparts. Examples of such 3D models are a 3D model of brain blood
2o barrier (US 5,578,485) and a 3D tumour cell and tissue culture system (US
5,580,781 ).
3D models of human tissues known in the prior art frequently show a high
degree of sophistication and morphological resemblance to their in vivo
2s counterparts. However, such 3D structures are usually grown following
complicated protocols and even though nominally useful for screening for
agents with clinical utility, they are poorly adapted to high throughput
screening procedures typical of the modern drug development, in particular
for screening large combinatorial libraries of new compounds. There exists
3o therefore a need for 3D models of human tissues combining the advantages
of 3D tissue models and compatibility with the requirements of modern drug
testing procedures. The present invention provides such models which are
free of many prior art limitations.
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SUMMARY OF THE INVENTION
s According to one aspect, the present invention provides novel 3-dimentional
in vitro models of mammalian tissues. The models comprise 3-dimensional
cell aggregates (organoids) grown from mammalian, preferably human cells
of at least two different phenotypes in a suitable liquid growth medium.
io In a preferred embodiment, the organoids are grown from various
combinations of normal and tumour cells. These models are designed to
simulate three stages of a solid tumour progression: promotion/angiogenesis,
invasion and metastasis. In another preferred embodiment, the organoids are
grown from two sub-populations of identical cells, one of which was pre-
ss treated with a chemical agent modifying cell properties. There are two
models
in this group, designed to simulate cell senescence and cell apoptosis,
respectively. The first model uses cells pre-treated with an agent which
blocks
cell proliferation without killing the cells, whereas the second uses cell
modified with a phototoxic compound, which induces cell death upon
2o illumination of cells.
The models have a large variety of applications, particularly in drug
screening. These tests are based, for example, on measuring simultaneously
and comparing proliferation rates of at least two different cell phenotypes in
2s organoids cultured in the presence and in the absence of a candidate drug.
In
a preferred embodiment, proliferation rates are calculated from results of
flow
cytometry analysis of single cells in suspension obtained from the dispersion
of the organoids, which cells were fluorescently labeled prior to forming the
organoids.
Other advantages, objects and features of the present invention will be
readily
apparent to those skilled in the art from the following detailed description
of
preferred embodiments in conjunction with the accompanying drawings and
claims.
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BRIEF DESCRIPTION OF THE DRAWINGS
s Fig. 1A is a graph illustrating cell proliferation analysis in organoids of
the
present invention; the graph shows results of flow cytometry analysis of a
fluorescently labeled single cell suspension obtained from dispersed
organoids.
to Fig. 1 B is a graph showing the distribution of the subpopulation of cells
that
underwent 0, 1, 2, and 3 divisions, respectively, in four days old organoids.
Fig. 2A is a graph reporting the proliferation kinetics based on the
proliferation
index of Neuro2A neuroblastoma cells grown in ES/Neuro 2A cells
is aggregates containing increasing proportion of embryonic stem cells (ES).
Fig. 2B is a graph reporting the proliferation kinetics based on the
proliferation
index of the ES cells grown in the same ES/Neuro2A cell aggregates as those
in Fig. 2B. This is an example of simultaneous proliferation assessment of two
2o cell phenotypes co-cultured in the same 3D organoids.
Fig. 3 is a drawing representing schematically the design of a photoablation
experiment.
2s Fig. 4 is a gallery of graphs showing the effects of increasing the
percentage
of photoablated cells in the spheroids from 0 to 75% (top to bottom) on the
kinetics of the regeneration/proliferation rate over a 4 days period (left to
right); the analyses were performed as illustrated in Fig. 1.
3o Fig. 5 is a gallery of graphs showing gap junction intercellular
communication
in spheroids containing various proportions of ES cells and mitomycin-treated
ES cells; intercellular communication is measured by the number of cells to
which calcein was transferred (receiver cells) from the donor cells.
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Fig. 6 is a graph showing temporal changes of the proliferation index of ES
cells in cell aggregates containing increasing number of mytomycin-treated
ES cells; this is an example of simultaneous determination of the level of
s intercellular communication (Fig. 5) and cell-subpopulations flow cytometry
analyses (Fig. 1 ) in a two-phenotype organoid.
Fig. 7 is a graph showing effect of a chemical (AGA) on the maintenance of a
nonproliferative fraction (quiescent pool) in 3D embryoid bodies (EB) of
to embryonic stem cells (ES-EBs + AGA); comparison with the ES-EB control
and with a hyperplastic cell line (F9-EBs).
Fig. 8 is a graph showing effect of AGA on the development of cell coupling in
ES-EBs of two cell lines (ES and F9).
Fig. 9 is a graph showing kinetics of the cell recruitment rate from the
quiescent pool of the ES-embryoid bodies of Fig. 7.
Fig. 10 is a graph showing kinetics of the cell recruitment rate from the
2o quiescent pool of gap junction proficient and gap junction deficient
embryoid
bodies (EBs) of Fig. 8.
Fig. 11 is a graph showing simulated number of mitoses per 1000 cells in a
3D cell system over a period of 290 time units, with the heavy line showing
its
2s Fourier cosine transform.
Fig. 12 is a graph showing simulated number of cells in GO-t state per 100
cells in a 3D cell system over a period of 290 time units, with the heavy line
showing its Fourier cosine transform.
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DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
The present invention provides 3D in vitro models for simulating various
s characteristics of mammalian tissues, in particular cancerous tissues. The
models of the present invention are 3D aggregates (organoids) formed from
living cells of mammalian origin. As used herein, the term "organoid" is
intended to mean any 3-dimensional aggregation of living cells of at least two
different phenotypes, which aggregation may be grown either in an organized,
io orderly fashion, or by a random association of cells, in the presence or
absence of a solid support. Organoids may be sometimes referred to as
"aggregates", particularly when grown from cells of a single phenotype. When
characterized by an essentially spheroidal shape, organoids (aggregates)
may be also referred to as "spheroids".
Organized 3D organoid structures for in vitro models according to the present
invention may be formed, for example, by culturing cells on a solid support
either of natural or artificial origin, under conditions favoring the growth
of 3D
cellular structures, for example, by growing different cell phenotypes in
2o superposed layers on the support. Random organoid structures may be
formed, for example, by co-culturing cells of different phenotypes stepwise,
in
the absence or presence of a solid support. It is also possible to grow
suitable
organoids by various combinations of unsupported or supported, organized or
random cell culturing, according to methods well known to those skilled in the
2s art. According to a preferred embodiment, organoids according to the
present
invention are grown in suspension in rotating culture flasks (spinner flasks),
in
the absence or presence of a solid support.
Cells of different phenotypes may be either cells of different mammalian
3o tissues or cells of the same tissue having different phenotypic
characteristics.
In the latter case, the different phenotypic characteristics may be a result
of
either natural or artificial cell transformations, including but not limited
to
natural pathological cell transformations, in particular cancerous
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transformations, genetic engineering, or treatment with chemical or physical
agents. Examples of cells suitable for the growing of organoids according to
the present invention include but are not limited to, stem cells, in
particular
embryonic stem cells, endothelial, stromal, neural, liver, kidney, bladder,
s prostate, skin and heart cells. Primary cells may be isolated from normal or
pathological mammalian tissues, in particular from tumour tissues, or taken
from a large variety of commercially available cells and immortalized or
transformed cell lines, as well as various genetically engineered cells.
io The in vitro models of the present invention are intended for studying
various
characteristics of normal and cancerous mammalian tissues, in particular for
the assessment of potential anti-cancer drugs at three specific stages of
progression of a solid tumour: promotion/angiogenesis, invasion, and
metastasis. A model is also proposed for studying cell senescence and
is apoptosis and for the assessment of drugs promoting cell regeneration and
tissue repair. The latter model may also be adapted to the study of drugs
inducing cell differentiation or to preparing cells for xenotransplantation.
As
will be clear from the following, the applications of the models of the
present
invention go far beyond the assessment of anti-cancer drugs. Additional
2o applications of the models will be discussed in connection with the
preferred
embodiments of the invention.
The studies and assessment of tissue functions and characteristics using the
tissue models according to the present invention rely in most cases on
2s identifying and quantifying sub-populations of cells (cell phenotypes)
building
the model organoids under study. This usually takes place after preparing
such organoids from at least two sub-populations of cells of different
phenotypes, culturing the organoids for a predetermined period of time under
predetermined conditions, for example in the presence of a candidate drug,
3o and dispersing the cultured organoids into a suspension of individual
cells.
The identification and counting of individual cells in the suspension is
normally carried out by an automated method using appropriately marked or
labeled cells. Examples of suitable markers and labels are fluorescent,
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radioactive and immunospecific markers and labels, applied to the cells by
techniques well known to those skilled in the art. The cells may be marked
either prior to forming organoids or after dispersing cultured organoids into
a
suspension of individual cells.
According to a preferred embodiment of the invention, cells are marked or
labeled fluorescently prior to forming organoids or aggregates and are
identified and counted by flow cytometry (fluorescent analysis by cell
sorting,
or FACS) after the cultured organoids or aggregates are dispersed into a
to suspension of individual cells. The kind of the fluorescent labeling of
cell sub-
populations and specific markers or labels used depend on the property
under study. For example, using fluorescent membrane linkers, such as
PKH26 (red fluorescence) or PKH67 (green fluorescence), available from
Sigma, is preferred for measuring cell proliferation, whereas loading cells
with
is a fluorescent dye, such as calcein-AM is preferred when studying cell
communication. Simultaneous evaluation of multiple sub-populations of cells
of different phenotypes may be achieved by labeling cells of each sub-
population with markers or labels fluorescing at different wavelengths.
2o Fluorescent membrane linkers, preferably used for the measuring of cell
proliferation, allow the analysis of the distribution of cell populations
which
underwent 1, 2, 3, ..... n divisions. In this case, the proliferation of cells
in the
models according to the present invention is preferably measured by the
proliferation index (PI). The proliferation index measures the ratio of the
total
2s number of cells of a given phenotype in the analyzed sample of cells to the
calculated number of cells in the parent population at the time of formation
of
the organoid and can be expressed as:
~ Ak
PI = k=o
Ak
2k
k=0
3o where Ak represents the number of cells that underwent k divisions. The
percentage of cells that underwent k = 1, 2, ...... n division can be
estimated
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from flow cytometry measurements, based on the fact that at each generation
of daughter cells inherit only half of the fluorescence of the mother cell
generation marked with a fluorescent membrane linker.
s According to one preferred embodiment, the invention provides an in vitro
model of angiogenesis in cancer. In this model, endothelial cells of
mammalian origin labeled with a fluorescent marker are mixed in a known
proportion with tumour cells and allowed to form spheroids. The cancer cells
produce angiogenic growth factors which affect the growth rate of endothelial
to cells. The growth rate of endothelial cells can be measured and related
directly to tumour angiogenesis.
In an alternative embodiment, the fluorescently labeled endothelial cells are
allowed to adhere in a culture to microcarrier beads, such as cytodex or
is cytopore from Amersham Pharmacia Biotech Ltd. Once the beads are
covered with endothelial cells, tumour cells are added to suspension of
microbeads covered with endothelial cells and are allowed to adhere to the
latter. The tumour cells/endothelial cells spheroids are then allowed to grow
in
a suitable culture medium. In a control experiment, similar organoids
2o composed of endothelial cells and normal cells matching those of the
tumour,
e.g., normal urothelium cells in the case of bladder cancer, are prepared,
grown and analyzed.
The model of angiogenesis in cancer is particularly useful for identifying and
2s studying potential angiogenic drugs. By using organoids formed with
endothelial and other normal cells, this model can be easily adapted to
identifying and/or studying potential angiogenic agents, capable of inducing
neovascularization in the process of tissue repair.
3o According to another preferred embodiment, the invention provides an in
vitro
model of interaction between stromal and tumour cells. In this model,
fluorescently labeled normal stromal cells matching the tumour cells under
study are grown as spheroids on microcarrier beads in a stepwise manner.
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s
After the microbeads/stromal cells spheroids are formed, tumour cells
fluorescently labeled with a marker different from that used to identify the
stromal cells are added to the culture and allowed to adhere on the top of the
stromal cell layer. The spheroids are then allowed to grow in suspension.
This model is particularly useful for studying the kinetics of anticancer
drugs
blocking the invasiveness of the tumour, for measuring the growth rate of both
the stromal and tumour cells, for studying molecular interactions between
stromal and tumour cells, for measuring the expression of markers of
io proliferation and differentiation, for measuring the level of gap junction
mediated cell-cell communication, and for evaluating adhesion of tumour cells
to the stromal cells.
According to another preferred embodiment, the invention provides an in vitro
is model of metastasis. In this model, epithelial or other cells
representative of
the tissue where metastases of a given tumour are expected to develop are
grown as a monolayer (2D culture) on a lower surface of a porous solid
membrane, such as a Milipore inset. Small spheroids formed in roller flasks
from fluorescently labeled tumour cells are seeded on the upper surface of
2o the membrane. Depending on the porosity of the membrane, the tumour cells
infiltrate the membrane and establish direct contacts with the epithelial (or
other) cells. For example, human normal lung fibroblasts can be used in the
monolayer, with spheroids of appropriate fluorescently labeled tumour cells
applied to the opposite side of the membrane. Both populations of cells may
2s be fluorescently labeled if information on the growth or apoptosis of both
populations is required, for example to see if the tumour cells recruit the
quiescent cells to which they adhere and induce them to divide. This
information may be needed to check the effect of a potential anticancer drug
on the growth of a metastatic tumour and/or its intercellular communication
3o with the cells to which it adheres.
The model of metastases is particularly useful for testing potential
antimetastatic drugs, based on criteria such as the rate of growth of both
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types of cells, communication between cells, and cell adhesion, invasion
and/or differentiation.
According to another preferred embodiment, the invention provides an in vitro
s model of apoptosis, senescence, necrosis and tissue regeneration. In this
model, normal cells of any origin can be used. For monitoring the effect of
senescence, a given percentage of a single cell population is pre-treated with
a chemical agent which blocks cell proliferation without killing the cells,
such
as mitomycin. The rest of the population is fluorescently labeled with a
to membrane linker and the two cell subpopulations are mixed and allowed to
form spheroids. The regeneration of spheroids is followed by monitoring the
proliferation of cells neighboring the growth-arrested or apoptotic cells.
This model can be easily adapted to monitoring the effects of apoptosis. In
is this case, the first subpopulation of cells is loaded with a phototoxic
compound, such as chloromethyl eosine diacetate and the second one is
fluorescently labeled. The two subpopulations are mixed and allowed to form
spheroids which are then illuminated to induce cell death of cells loaded with
the phototoxic compound. The regeneration of spheroids is followed by
2o monitoring the proliferation of the surviving cells. The percentage of
cells
dying of apoptosis can be estimated by propidium iodide exposure and cell
cycle analysis by FACS, a technique well known to those skilled in the art.
For this model, original protocols were developed to measure the effect of
2s increasing the number of senescent, necrotic, or apoptotic cells on the
proliferative ability of neighboring cells. The model is particularly useful
for
testing drugs potentiating regeneration in damaged tissues, especially in
neurodegenerative diseases, for studying cell senescence and death, and to
test for apoptosis inducers. When used for the latter purpose, the model can
3o provide a positive control for testing drugs that induce apoptosis in solid
tumours. In this case, an aliquot of the tumour cells is fluorescently labeled
with a membrane linker and treated with the drug. After a predetermined
period of time, the drug-treated cells are mixed in a given proportion with
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untreated tumour cells. By comparison with the intreated subpopulation,
efficacy of the drug, possible bystander effect, number of resistant cells and
regrowth of resistant cells can be measured. When cells isolated from
biopsies are used to form the spheroids, the model can also applied in
s association with the mathematical simulation models SIMCAN to predict the
regrowth of the tumour after its irradiation (or chemotherapy) and hence the
chance for such a treatment to succeed.
Various analytical procedures have been developed to assess results of tests
io when using the tissue models of the present invention. These include the
following:
- measuring the diffusion rate of drugs, by tracking drugs labeled with a
fluorescent tag, using either imaging microscopy or sequential mild
trypsinization and flow cytometry (FACS);
is - identification of cells' phenotypes and live cell sorting using membrane
markers, in particular fluorescent membrane linkers;
- simultaneous assessment of proliferation kinetics of two cell sub-
populations, by FACS analysis of the sub-populations previously labeled
with fluorescent markers fluorescing at two different wavelengths; this
2o technique allows very fine analysis of population growth at the single cell
level; FACS data are analyzed with ModFit software (Sigma);
- measuring cell-cell communication by labeling one cell sub-population,
prior to seeding, with a red-fluorescing membrane linker (PKH26, from
Sigma) and loading the cells of the other sub-population with a green-
2s fluorescing dye (calcein), which can only transfer between the cells
through the gap junction channels;
- quantifying proliferation, differentiation and apoptotic cell markers by
immunolabeling and flow cytometry.
3o The above procedures are based on principles and techniques well known to
those skilled in the art.
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Several computer simulation and data analysis routines have been developed
in connection with the mammalian tissue models of the present invention.
These include the following:
- PROFILE - for building a profile of a subpopulation susceptible to a given
s disease using an array of various attributes of individuals;
- AD-JUST - for finding the optimal fit of a theoretical function to a set of
experimental data points;
- SIMCEL-2D - a 2D growth simulation model based on a 2D cellular
automaton that mimics monolayer cell cultures with various degrees of
io gap junction intercellular communication; for predicting of growth kinetics
and drug diffusion in monolayer cell cultures;
- SIMCEL-3D - a 3D growth simulation model based on a 3D cellular
automaton with various levels of gap junction intercellular communication
and mitotic and death rates; for predicting of growth kinetics in cell mass
is tissues, under conditions allowing for homeostatic behavior and for
homeostasis disruption.
The AD-JUST routine is applicable, for example, in analysis of the growth
related parameters of in vitro models of the present invention and in analysis
20 of drug toxicity data in both 2- and 3-dimensional tissue models.
The analysis of in vitro models of the present invention includes a routine
estimation of cell-cell coupling. Intercellular communications mediated by gap
junction channels are altered in most if not all cancers (such as breast,
2s bladder, prostate, and lung cancer) and in many other diseases
(neuropathologies, heart, lung, and kidney related illnesses, psoriasis,
etc.).
The proteins that form the channel (connexins) constitute a novel
therapeutical target. The two simulation models (SIMCEL-2D and SIMCEL-
3D) are used for the prediction of drug efficacy or gene therapies (for
3o example, the bystander effect) based on the gap junction function. The
estimation of intercellular coupling using these theoretical models, whose
validity was tested in the in vitro models of the present invention, allows
the
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prediction of dosage and temporal efficacity of drugs designed to modulate
the gap junction function.
A brief description of the above routines and the algorithms involved follows.
PROFILE
Goal
Build a profile of a sub-population susceptible to a given disease using an
to array of various attributes of individuals. Example: In a population we
know a
group of N, individuals who developed the disease. We also know K attributes
of these individuals with regard to various domains (life history, genetical,
pathological antecedents, social, professional, etc). We have the same
information for a random group of No (N,=No) individuals who did not
is developed the disease. The algorithm consists in finding out a unique
combination of attributes that best predict the occurrence and non-occurrence
of the disease. Attributes consists mostly in categorial and ordinal
information.
This model can also be applied to cell communities for which different
attributes (such as growth parameters, phenotypes, pathology, disease stage,
2o etc.) are known.
Algorithm
The overall algorithm uses a basic module based on information theory.
2s Basic module
Let f k, ;, , and f k, ;, o be the number of individuals who respectively did
and did
not develop the disease in category i of the kth attribute. Let N = N, + No be
the size of the whole sample. We derive:
n
Hk ~O~l~ - N ~ (fk,i In fk.i J k,i,O In J k.i.0 J k.i.l In fk,i,l )
i=
and
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Rk = (1n 2 - Hk (0,1))~H(k)
with
H(k) = N C N In N - ~ fk,; In fk,;J
,-
where Rk measures the efficiency of attribute k as a predictor of the
occurrence/non-occurrence of the disease. It can be tested using the
s maximum likelihood ratio.
Overall procedure
The above module is embedded in an stepwise hierarchical divisive
procedure. At each divisive step the attribute k* with the highest Rk is
selected
to as the divisive attribute and the sample of individuals subdivided
according
the categories of k*. The subsets produced are then submitted to the same
procedure
The divisive process goes on until anyone of following stopping rules applies:
is i) number of individuals too small in a subset to warrant test validity;
ii)
H(0,1)<0.3 in a subset (one of the 2 outcomes has a 0.9 probability; iii) no
attribute still available exhibits a significant Rk.
The PROFILE routine is a novel adaptation of the PEGASE routine (Phipps,
2o M. (1981 ), Entropy and Community Pattern Analysis, Journal of Theoretical
Biology 93: 253-273)
dealing with spatial pattern analysis. PROFILE, algorithm and routine, have
been designed for population targeting for medical purposes and include a
number of mathematical features which were not present in the former
2s algorithm.
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AD~J U ST
Goal
Find out the optimal fit of a theoretical function to a set of experimental
data
s points. These data may express time dependent responses, dose dependent
responses, pseudo-cyclic time series, etc. This routine includes a set of
theoretical functions: linear, exponential, reverse exponential, logistic,
reverse
logistic, polynomial functions and, in the case or time series the Fourier
cosine transform and the exponential decreasing weigh averaging (EDWA).
io For each application all functions are tested and results are given for the
3
best r~ fits.
Algorithms
Linear
is Exponential and reverse exponential
This classical growth function is usually presented as
ax+b
y= a
or its equivalent form
y = Be'~
with
B = eb
We propose an improved calculus procedure by introducing a constant c such
= c + e'~+b
Y
as
which can be linearized as
ln~ y - c~ = ax + b
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Since c is not known, the AD-JUST routine finds its value by an optimization
iterative process.
In the case of a reverse exponential, we have
y = c - e'~+b
s with
ln~y + c~ = ax + b
and the value of c is found using the same iterative procedure.
Compared with results obtained from similar commercial software available
io on the market, the goodness of fit correlation rz is considerably improved.
Logistic and reverse logistic
This classical population growth function is not commonly represented in
commercial software packages. We propose an algorithm introducing an
is overall constant c in the basic logistic equation and an original
linearization
procedure.
Let y be the basic logistic or reverse logistic equation:
1
Y - yo ~ c + e-'~+b
2o We derive the linear form using
Y= y~ v°
and
In y-c --ax+b
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Since c is not known its value is found using an optimizing iterative
procedure
and an initial value
Co= 1~~~+E
where
is the largest empirical value observed and s is an infinitesimal quantity.
Polynomial functions
Fourier cosine transform
io
Exponential decreasing weigh averaging
This model applies to time series and can be a substitute to the Fourier
cosine transform. Let x~ (: t = 1, T ) be a series of empirical values
observed
where T represents time or any other dimension like a spatial gradient. We
is want to smooth the graphic representation of this series by replacing
observed values with calculated values eliminating erratic variations.
Let v~ (: t = 1, T ) be this substituted value. It is derived from xt using:
V, = a + xf and V~ = 1 + l~(a - 1)
2o where a >1 is an integer or decimal point number that allows for more or
less
smoothing. The smaller a the smoother the data series. Optimal a values fall
in the range 1.1<a<1.5 .
Overall AD-JUST procedure
2s The overall AD-JUST procedure consists of:
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- preparing data (computation of statistical values, average, standard error,
etc. );
- testing all available functions (computing the r' goodness of fit
correlations);
s - selecting the 3 best fits;
- producing graphs, and values tables.
SIMCEL-2D and SIMCEL-3D
io
SimCel-2D and SimCel-3D are two distinct cellular automaton (CA) models
designed to simulate the dynamical behavior of 2-D and 3-D cells systems.
Given that both share a number of common functions, they will be described
under the same section heading. As the names indicate, they only differ in
is terms of their dimensionality and cell spatial arrangement.
Algorithm, structure and functions
Cell network
2o SimCel-2D is based on a 2-D regular hexagonal cell network (honey-comb
like with a 1 to 6 contact ratio).
SimCel-3D is based on a 3-D regular cell network where cell are represented
as regular dodecahedron arranged in a dense sphere packing system, with a
2s 1 to 12 contact ratio.
Accessible cell states
Individual cells have access to a set of states split in 2 subsets (resting
and
divisive):
1. Within the resting group, cells go successively through:
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- GOt state (resting with fully functional communication channels and
capacity to resume proliferation). GO-tx is the same state without
channels;
- GO-d state (differentiating resting cells with decreasing channels
s functionality and capacity to resume proliferation. GO-dx is the same state
without channels;
Dead cells.
2. Within the divisive group:
to - S state (cell synthesizing DNA);
- G2-M state (cells completing genomic replication and mitosis);
- DC daughter cells issued from mitosis;
Neighborhood
is Each cell a has a neighborhood that consists in its 6 adjacent cells in
SimCel-
2D and 12 adjacent cells in SimCel-3D.
CA transition rule
The transition from the resting group to the divisive group (i.e. GO S) or
vice
2o versa from the divisive to the resting group (i.e. DC GO) is controlled by
a
probabilistic transition rule subject to a set of probabilities pour and p,~r
that cell
a at time t adopts GO or S. Probabilities are updated at each time unit
p0u,t+1 - pout + dput and plu,t+1 - glut - dput
according to
with:
dput - YPutkut'~ut[(a0utp0ut )- (alutplut ),+ ~'~utkut
where r is an overall parameter controlling the intensity of cell-cell
communication; p~r and 8~r are age-dependent extinction factors, ao~r and a,"r
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are factors expressing neighborhood effects on the variation of both
probabilities, k~~ is a limit to this variation and E is a random Gaussian
factor.
It should be noted that GO-tx and GO-dx cells which do not develop
s communication channels, r is permanently set to 0, a mathematical condition
equivalent to the lack of channels.
Cell state change from one group to the other (i.e. resting to divisive or
vice-
versa) subject to probabilities poor and p,~~ and a pseudo random number.
1o
Within each state groups (resting or divisive) cell change their state
according
to an age dependent process and fixed time sequences.
Other functions
is SimCel-2D and -3D make provision for two additional functions: mitosis and
mortality. Six time unit after entering the divisive cycle, a cycling cell
produces
2 daughter cells and the additional new cell finds its place in the system by
a
cascade of centrifugal cell movements.
2o A cell in the resting group is assigned an age and is submitted to an age
dependent probabilistic death program subject to a probability N"~ such as:
a~~ut _1) 1n( 10.005)
hut - ~ a ' with a =
S2 - 1
where S2 is the maximum cell life span and w~~ is the age of cell a at time t.
Following a decay period, a dead cell is removed from the system and
2s replaced by a neighboring cell and a cascade of centripetal cell movements.
Initial configuration
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At the onset of a simulation run, an initial configuration is given. By random
choices, each cell is assigned a state, an age in its state subject to a
priori
frequencies, and a 0/1 (i.e. yes-or-no) capacity to communicate subject to
probabilities qo and q, that represent basic characteristics of the simulated
cell
s line.
Routine outputs
During the simulation run several dynamic graphic or numerical outputs are
displayed:
to - dynamic mapping of the cell system (for SimCel-3D an equatorial 2-D
section is shown);
dynamic analysis of state frequencies (GO-t, GO-d, S, G2M, Dead);
- dynamic graphic of cell probability;
proliferation index;
is - recruitment rate of cycling cells from the resting pool;
For the final configuration:
- assessment of the cell-to-cell intracellular diffusion (percolation) based
on
the average number of cells in the 10 longest strings of communicating
2o cells.
Previous publications on this subject
Phipps, M. Dynamical Behavior of Cellular Automata under the Constraint of
Neighborhood Coherence. Geographical Analysis 21: 197-215 (1989).
2s Phipps, M., J. Darozewski & J. Phipps. How the Neighborhood Coherence
Principle (NCP) Can Give Rise to Tissue Homeostasis. Journal of Theoretical
Biology 185: 475-487 (1997).
The present implementation extends the basic algorithm by providing
3o dynamic graphic displays, the introduction of probabilities qo and q,
determining an a priori cell capacity to communicate, and the assessment of
the percolation-diffusion capacity in the cell system.
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EXAMPLES
EXAMPLE 1
s Simultaneous measurement of proliferation and proliferation indexes in
two phenotypes grown as aggregates
Before forming aggregates, the two sub-populations (gap junction proficient
embryonic stem cells (ES cells) and gap junction deficient neuroblastoma
to cells (Neuro2A cells) were labeled with the red membrane linker PKH26 and
yellow membrane linker PKH67 (Sigma), respectively, according to modified
manufacturer instructions. The cells were then mixed in known proportions
and allowed to form aggregates that were sampled every day for monitoring
growth. For analysis, the aggregates were dissociated to single cell
is suspensions using appropriate technique suited to the cell type analyzed.
It is
known that at each division the parent cell loses half of its fluorescence
intensity. The individual cell fluorescence of no less than 10,000 cells was
measured by flow cytometry and the results analyzed using ModFit software
(Sigma). The control showing the mean fluorescence of the cell populations
2o homogeneously labeled served as the reference first peak (Fig. 1 A). The
data
illustrate one such experiment. The first peak measures the label at time 0.
The graphs obtained each day measure the distribution of the population into
subpopulations of cells having divided 1, 2, 3, ..... n times. Each successive
peak from right to left represents the percentage of cells forming each
2s subpopulation (Fig. 1 B). The proliferation index is calculated from these
data.
Fig. 2A shows that the growth (measured by the proliferation index) of one
cell phenotype is not affected by the percentage of co-cultured cells in the
aggregates. In contrast, the proliferation of the second phenotype is
3o dependent on the number of cells of the second phenotype present in the
aggregates (Fig. 2B). Cell-cell communication was also simultaneously
measured in this experiment and controls of possible growth factor secretion
by the phenotypes were included (results not shown).
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EXAMPLE 2
Photoablation
s
Photoablation was achieved using the chloromethyl eosin diacetate (CMEDA,
Molecular Probe). A suspension of individual cells was loaded with a solution
of 20 ~M of CMEDA in acetoxymethylester buffer according to modified
manufacturer's instructions, for 30 minutes at 37°C, using a 5 mM stock
to solution of CMEDA in dimethylsulphoxide (DMSO).
Cells were washed twice with the growth medium, mixed in known proportions
with untreated cells labeled with the PKH26 membrane linker and allowed to
form spheroids. The spheroids were then illuminated at a 30 cm distance with
is an incandescent lamp source for 30 min in phosphate buffered saline. The
illumination caused the death of the CMEDA loaded cells by individual
photoablation (Fig. 3).
The regeneration of the organoids following the photoablation was quantified
2o by the calculated proliferative index and the number of cell divisions. The
results are summarized in Fig. 4. The figure shows the effects of increasing
the percentage of photoablated cells from 0 to 75% (top to bottom) in the
spheroids on the kinetics of the regeneration/proliferation rate over a 4 days
period (left to right). The data show that the larger the percentage of dead
2s cells, the higher the proliferation/regeneration cell rate.
EXAMPLE 3
Senescent Model
Embryonic stem (ES) cells (D3 strain) were grown as monolayer in Dulbecco
modified Earle medium (DMEM) containing 15% ES certified fetal bovine
serum (FBS) and 1000 U/ml LIF. Actively proliferating cells were harvested by
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trypsinization and resuspended as a single cell suspension. One half of a
single cell suspension was exposed to mitomycin C for one hour (optimum
time for ES cells) using a 1 mg/ml stock solution of mytomycin. Mitomycin is a
DNA intercalating agent that irreversibly blocks the cell cycle and DNA
s synthesis at appropriate concentration. The cells which pass the restriction
point complete the cycle. Mitomycin also blocks the tyrosine kinases. The
cells remain alive but metabolically inactive and display features of
senescent
cells.
to The second half of the cell suspension (untreated) was labeled with the
membrane linker PKH26 which fluoresces in red. The cell suspensions were
then mixed at increasing ratio of mitomycin treated versus PKH26 labeled
cells (25, 50, and 75%) and allowed to form embryoid bodies (3D spheroids)
in 3.5 cm bacterial plates. Controls embryoid bodies contained 0% mitomycin
is treated cells. In an independent experiment, it was checked that mitomycin-
treated cells aggregate to form spheroids, but do not proliferate. Gap
junction
intercellular communication (GJIC) and the proliferation of the ES cells in
mixed spheroids were measured as follows.
2o GJIC measured by dye transfer
The mytomycin-treated cells are extensively communicating, as illustrated by
Fig. 5. The assay was performed as follows. A mytomycin-treated
subpopulation of ES cells was loaded with calcein, a cell-permeant dye which
fluoresces in green after de-esterification by intracellular esterases. The
2s subpopulation of calcein-loaded ES cells (donor cells) was mixed with the
untreated ES cells (receiver cells) labeled on the membrane with the red
fluorescent membrane linker PKH26 (at 75/25, 50/50 and 25/75 ratio) and
allowed to form spheroids. After being de-esterified, calcein becomes
impermeant and can only diffuse to the unloaded cells through the gap
3o junction channels. In this population of cells, receiver cells fluoresce
red,
donor cells (calcein-loaded) fluoresce green and the communicating cells
fluoresce green and red (yellow). Cells were analyzed at time 0 and 24 hrs
following disruption of spheroids to a single cell suspension. The results
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obtained from flow cytometry analysis (FAGS) are shown in graphs A through
E of Fig. 5.
The PKH26-labeled cells show in the quadrant 1 and the calcein-loaded cells
s appear in the quadrant 4. The cells that fluoresce both red and green are
found in the quadrant 2. From A to E, the graphs show the red-fluorescent
population, the green-fluorescent population (both at time 0), then the mixed
populations after 24 hrs. The ratio of untreated ES cells to mitomycin-treated
ES cells is indicated at the top of each graph.
io
The levels of dye transfer, measured in the quadrant 2, are the following:
A2 1.7 ES/ESmito: 100/0
B2 0.1 ES/ESmito: 0/100
C2 75.2 ES/ESmito: 75/25
is D2 58.6 ES/ESmito: 50/50
E2 29.7 ES/ESmito: 25/75
It can be seen from the above that:
1. A2 and B2 measure background since the cells fluoresce either red or
2o green. No dye
transfer has occurred at time 0.
2. C2 through E2 levels represent the number of cells in which the dye
transfer
occurred from mitomycin-treated ES cells to untreated ES cells in percent.
All the ES cells have received the calcein dye. The results clearly show that
mitomycin-treated cells are GJIC competent and communicate with their
untreated counterparts.
3o Proliferation analyses
The data are acquired on the PKH26-labeled, untreated ES cells following
trypsinization of spheroids to single cell suspension and FAGS analysis. The
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number of cells at 1, 2, 3..... generations is expressed as a percentage of
the
sample population analyzed by FACS (usually 10,000 cells).
At 24 hrs, all the spheroids containing mitomycin-treated cells already show a
s second peak representative of cells that underwent one division, while the
matching control does not. Therefore after 24 hrs cells in spheroids
containing
only normal (untreated) ES cells did not divide. In contrast, 3.3 to 7.15% of
the cells from spheroids containing 25 to 75%, respectively, of senescent
(mitomycin-treated) cells entered the cell cycle.
to
At 48 hrs, the second generation in controls reaches 55.92%. The spheroids
containing treated cells exhibit a third generation. The proliferative index
increases with the increasing amount of treated cells in the spheroids. The
proliferation data at 48 and 96 hrs are summarized in Tables 1 and 2.
is
Table 1. Effect of increasing percentage of senescent cells on the
regeneration rate
at 48 h rs
Generations Control 75/25 50/50 25/75
Gen. 1 43.78 40.86 21.18 20.34
Gen. 2 55.92 50.53 52.65 46.91
Gen. 3 0.00 8.12 ~ 26.15 ~ 32.62
Table 2. Effect of increasing percentages of senescent cells on the
regeneration rate
2s at 96 hrs
Generations Control 75/25 50/50 25/75
Gen. 1 19.40 20.47 13.40 5.63
Gen. 2 44.19 39.71 28.87 16.58
Gen. 3 29.23 32.41 35.47 30.00
Gen. 4 6.27 6.04 21.96 29.26
Gen. 5 0.66 1.09 ~ 0.13 ~ 18.49
In the above tables, the first number in the ratio corresponds to the
percentage of untreated ES cells. The results are summarized in Fig. 6 using
3o proliferation index data.
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It is clear from these results that the proliferation rate of the normal cells
increases in the presence of senescent (mitomycin-treated) cells when
compared to control and that the increase is directly proportional to the
s proportion of the senescent cells in the spheroids. As a whole, the results
show that:
1. cell replacement is promoted by dead or apoptotic (as in Fig. 4) or
metabolically inactive cells (as in this experiment);
2. cell proliferation is correlated with the number of dead cells;
l0 3. cell number in a delineated compartment (here organoids) is regulated;
and
4. gap junctions contribute to all processes, probably by translocating
signaling
molecules.
EXAMPLE 4
Model of angiogenesis in bladder cancer
2o Cell lines used:
Human endothelial cells from Wisent Company (Wisent, St Bruno, Canada);
RT4 cell line (ATCC HTB2), derived from well differentiated human papillary
bladder carcinoma;
J82 cell line (ATCC HTB1 ), derived from poorly differentiated invasive
bladder
Zs carcinoma;
FHs-73881 cells (ATCC HTB 160), derived from human normal urothelium.
Two single cell sub-populations, endothelial and either one of RT4 and J82 or
the control FHs-73881 cells, were mixed in a known proportion and allowed
3o to form organoids in spinner flasks in an appropriate medium. The cancer
cells produce angiogenic growth factors. The factors reflect on the growth
rate
of endothelial cells and can be directly related to tumour angiogenesis.
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The growth rate of endothelial cells were tested using a fluorescent
membrane linker, PKH26 (Sigma, USA), as described in Example 1. The
number of endothelial cells can also be monitored using the immuno-
detection of CD31 and CD34 (a membrane marker of PECAM-1 and
s microvessels, respectively). Gap Junction Intercellular Communication (GJIC)
was quantitatively estimated as described in Example 3. In addition, the CD34
immunopositive endothelial cells can be sorted out alive using FACS
technology for further microscopic studies.
to The diffusion rate of potential drugs added to the medium was measured by
adding a fluorescent tag to the drug molecule and tracking the molecule by
video-microscopy imaging. Putative positive effect of the formation of
microvasculature from the endothelial cells on the penetration of nutrients
inside the spheroids was measured using a classical fluorescent doxorubicin
is diffusion test.
Alternatively, embryonic stem cells having formed embryoid bodies that
contain microvessels were co-cultured with single cell populations of either
one of the bladder cancer cell lines to form spheroids and were analyzed as
2o described above.
In another version of the model, endothelial cells were grown on microcarriers
containing slowly released blood substitute. This extends the use of the
model from small, non-vascularized microtumours or metastases to larger
2s oxygenated tumours containing "functional" microvessels.
In this model, any cancer cell phenotype can be substituted for bladder cell
lines.
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EXAMPLE 5
Model of interaction between stromal and tumour cells.
The HCV-29 cell line of normal urothelium fibroblastic cells was used to form
s the stromal layer for the preparation of mixed spheroids. The normal stromal
cells HCV-29 were grown in spinner flasks on beads (Pierce) for three days,
after which time a single cell suspension of RT4, J82 or control FHs-73881
cells (Example 4) pre-labelled with the red fluorescent membrane linker
PKH26 was added to the bead culture and allowed to form spheroids.
io
Using this model, the following characteristics of the tumour growth were
measured, monitored, assessed, or estimated:
1. the invasiveness of the tumour bladder cells in the absence or the
presence of given
is drugs using long term membrane linker tracers;
2. the growth rate and diameter of the tumour; the growth rate was assessed
with the
technique described above using the fluorescent membrane linker PKH26
and the
2o Modfit software; the diameter of the tumour was monitored by imaging
microscopy;
3. the molecular interactions between stromal and tumour tissues;
4. the expression of proliferation markers, for example PCNA, c-myc (used as
prognosis markers in clinical usage) in the presence or absence of
2s potential anti
proliferative drugs; after spheroid dissociation, the cells were
immunolabeled and
analysed by flow cytometry;
5. the expression of the connexins in both phenotypes as well as the level of
3o cell-cell
communication; this was done by immunolabeling and dye transfer,
respectively,
followed by flow cytometry analysis;
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6. the cell adhesion; the expression of adhesion molecules was evaluated by
immunolabeling and flow cytometry analysis; the adhesion itself was
measured
using a commercially available kit.
s
In this model, any cancer cell phenotype can be substituted for bladder cell
lines.
to EXAMPLE 6
Model of metastasis.
Epithelial cells representative of the tissue where metastases are expected to
develop are grown as monolayers on the lower surface of a modified insert in
is petri dishes. Small spheroids formed from tumour cells in spinner flasks
are
then seeded on the upper surface of the insert. The insert is then placed in
its
original position in the multiwell dishes. This model was developed using
human normal lung fibroblasts CCD-37Lu (ATCC CRL 1496) as monolayers
of normal cells and human embryonal carcinoma Tera1 (ATCC HTB 105) or
2o Tera2 (ATCC HTB 106) as tumour cells. The cells forming the spheroids were
pre-labelled with a membrane linker.
The following characteristics of the model were measured:
1. growth of both cell types
2s 2. communication between cells
3. cell adhesion
4. cell invasion
5. cell differentiation
3o This model is particularly suitable for screening and testing of
antimetastatic
drugs.
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EXAMPLE 7
Model of senescence, apoptosis and tissue regeneration.
This model was developed using embryonic stem cells (D3 strain), but normal
s cells of any origin can be used, either as self-forming spheroids or as
spheroids growing on microcarrier beads. For monitoring the effect of
senescence, a given percentage of a population of single cells in suspension
was pre-treated with mitomycin at a concentration which blocks proliferation
without killing the cells. The mitomycin-treated cells become progressively
to senescent and necrosis follows. The rest of the cell population was
labelled
with the fluorescent membrane linker PKH26. Both sub-populations were then
mixed and allowed to form spheroids in appropriate medium in spinner flasks
and the regeneration of the spheroids was followed by monitoring the
proliferation of the neighbouring cells.
A similar model was designed for monitoring the effect of apoptosis. In this
model, a given percentage of a population of single cells in suspension was
loaded with a phototoxic compound (chloromethyl eosine diacetate: CMEDA).
The rest of the cell population was labelled with the fluorescent membrane
linker PKH26. Both sub-populations were then mixed and allowed to form
spheroids in appropriate medium in spinner flasks. The spheroids are
illuminated to induce cells death of CMEDA loaded cells and the regeneration
of the spheroids was followed as mentioned above by monitoring the
proliferation of the surviving cells.
This model can be customized for the study of the evolution of
xenotransplants by using co-culture of stromal cells and progenitors of the
desired phenotype grown as spheroids. Differentiation of the progenitors can
be monitored by analysing specific antigens.
Original protocols were developed for this model to measure the effect of
increasing the number of senescent, necrotic or apoptotic cells on the
proliferative ability of neighbouring cells (pancreas, kidney, liver, heart).
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This model is particularly useful for:
1. studying cell apoptosis in 3D
2. studying cell aging and necrosis in 3D
s 3. assaying drugs for tissue repair after trauma and ischemic damages.
4. testing drugs for tissue regeneration, especially for neuro-degenerative
diseases
5. monitoring the evolution of xenotransplants with and without drug treatment
6. increasing the rate of recombination events in ES engineered cells for the
to production
of transgenes.
In connection with the latter utility of the model, it was found that
induction of
apoptosis using the chloromethyl eosin diacetate (CMEDA) in a defined
is number of embryonic stem (ES) cells in spheroids (embryoid bodies)
accelerates proliferation rate of live cells and shortens their cell cycle.
The
number of recombination events is relatively rare and was shown to depend
upon the ES cell cycle duration. Therefore, the two conditions caused by the
induction of cell death in the ES population lead to an increase in the number
20 of recombinations. This model may also be used to increase the number of
spontaneous mutations.
EXAMPLE 8
2s Fitting theoretical functions to experimental data
Fig. 7 through Fig. 10 show examples of application of the AD-JUST routine
to experimental data analysis.
3o Fig. 7 shows the effect of a chemical (AGA) on the maintenance of a
nonproliferative fraction of cells in 3D embryoid bodies (EB) formed of
embryonic stem cells (curve ES-EBs + AGA) compared with the control (curve
ES-EB) and with a hyperplastic cell line (curve F9-EBs). Theoretical functions
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fitting the experimental data points are: reverse exponential (ES), negative
exponential (F9) and reverse logistic (ES+AGA).
Fig. 8 shows the effect of AGA on the development of cell coupling in ES-EBs
s of two cell lines (ES and F9). Theoretical functions fitting the
experimental
data points are logistic functions with different parameters.
Fig. 9 and Fig. 10 show the kinetics of cell recruitment rate in the three EB
types of Fig. 7 and the kinetics of cell coupling in 3D cell bodies of Fig. 8,
to respectively. The curves in Fig. 9 and 10 have been derived from the
functions fitted to data points of Fig. 7 and 8, respectively.
The following equations represent the best fit for experimental data points in
Figure 7:
0 t 0. 0 r2 0.
1s F
NPF 9 - - .066 - 02 - ,031 (gfc - 959 *)
NPFES = 1.059 - e0~037t-2.804 (g fc r2 = 0.994 * *)
1
NPFES+AGA = ~ - -0.5631+7.184 + 0.445 (gfc r2 = 0.959 * *)
1.835 + a
The following equations represent the best fit for experimental data points in
Figure 8:
1
GJICES = 99.21- 0.206t-4.652 (gfc r2 = 0.967 * *)
0.0009 + a
1
GJICF9 = 1.79 + -3.311t+5.039 (gfc r2 = 0.999 * *)
0.04 + a
1
GJICES+AGA = 101.71- 0.216t-5.984 (gfc r2 = 0989 *)
0.007 + a
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where gfc rz : goodness of fit squared correlation; correlation significant at
the
0.95 (*)
and the 0.99 (**) probability level.
s
EXAMPLE 9
Simulating behavior of 3D cell aggregates using SIMCEL
to Gap junction intercellular communication (GJIC) is involved in several
basic
biological processes such as proliferation control, cell differentiation,
organismic development, heart beat synchronization, astrocytes, neuron
coupling and cancer. They also have potential implication in medical
technologies (e.g. drug delivery through intracellular cell-to-cell diffusion
and
is bystander effect in gene therapy).
The 3D in vitro models of the present invention may be used to investigate
and test various chemical compounds (potential drugs, food additives,
environmentally harmful chemicals, etc.) with respect to their effect on these
2o biological processes. These investigations require fine kinetic analyses
that
are time consuming and expensive. Numerical experiments using simulation
software can be a useful and sensible way to complement experimental
techniques, thus saving time and money.
2s Using numerical experiments, certain features of a cell system can be
predicted based on the level of cell coupling and using previous experimental
data to set the values of parameters of the simulation model. Among these
predictable features, the cycling cell recruitment from the resting pool, the
proliferation index and the cell-to-cell intercellular diffusion are of
special
3o interest. The predictive capacity of SIMCEL-3D is illustrated in Table 3
and
Fig. 11 and 12.
Table 3 shows a comparison of simulated and experimental data for 3 cell
phases of the cell cycle and for 3 different cell lines. As the log-likelihood
test
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indicates, none of the real systems analyzed differ significantly from the
simulated system.
Table 3. Comparison of simulated and experimental data for 3 cell phases of
s the cell cycle (cell %) in 3 different cell lines.
uman neuroblastoma (IMR 32) 73.0 22.2 4.8 NS
tationarv phase L6 rat myobalsts68.1 29.9 2.0 NS
(L1 )
tationary phase L6 rat mvobalsts77.0 18.5 3.5 NS
(L2)
imulated cell system 73.9 21.9 4.2
As an example, Figures 11 and 12 display the result of an numerical
experiment performed with SIMCEL-3D. This simulation which ran over a time
io period of 290 time units (approximately 290 hours) was meant to predict the
number of mitoses as a function of the GO-t cell fraction (transient resting
cells with fully functional cell-cell communication channels. Both variables
were shown to be dynamically related.
is Although various particular embodiments of the present invention have been
described hereinbefore for the purpose of illustration, it would be apparent
to
those skilled in the art that numerous variations may be made thereto without
departing from the spirit and scope of the invention, as defined in the
appended claims.
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