A series of mathematical models for in vitro multicellular tumor spheroids response
to chemotherapeutic treatment vs. plate cultures Women in Science Program
(WISP)
Michelle Chen, Paula Chen, Ann Dunham, Milan Huynh, and Evan Rheingold
Department of Mathematics, Dartmouth College, Hanover, NH 03755 USA
Background
Abstract
Cancer tumor growth may be modeled as a monolayer culture
or as a spheroid culture. However, various authors state that
spheroids are a more accurate model of in vivo tumors than a single
cell line. In vivo tumors do not consistently grow. Initially at a small
size, a tumor rapidly proliferates, similar to a cell culture. As the
tumor approaches an intermediate stage, only the outermost cells
perform active cell division, mitosis. This is referred to the tumor’s “shell” and the tumor’s “inner core” consists of living cells not engaged in mitosis, quiescent cells. At the tumor’s late stages, the tumor’s very inner core is necrotic tissue, quiescent cells comprise the middle layer, and proliferating cells remain on the outmost layer.
Often medical doctors administer chemotherapy treatments to
kill the proliferating cells. However, since a tumor is not fully of
proliferating cells implies that certain chemotherapy doses are not
optimally administered. Furthermore, prior research suggests that
the quiescent cell layer may cause the tumor to acquire resistance to
chemotherapy, possibly by supplying nutrients to the proliferating
layer.
Professor Dorothy Wallace and Xinyue Guo in January 2013
provided a potential spheroid growth model that accounted for 4
factors: 1. Proliferating cells, 2. Quiescent cells 3. Necrotic cells 4.
Total spheroid size. Thus, previous mathematical models describe
the disparity between cell plate cultures and spheroid cultures.
The research topic is a mathematical model of tumor growth in response
to chemotherapy treatments. Interest in mathematical models arises from
the behavior of spheroid tumors which differ from observed cell plate
culture growth. Thus, approaching tumor growth by mathematical
modeling spheroids produces more accurate growth models. This allows
a better understanding of the effects of chemotherapy drugs and
optimizing the administration of treatments. The method to approach first
begins with modeling a cell line’s plate culture growth without any
chemotherapy treatment for 4 cell lines: A-549 (lung cancer), SK-N-SH
(neuroblastoma), SK-N-MC (neuroblastoma), and MCF-7 (breast
cancer). Then, using literature data of treatment effects on monolayer
cultures allows the analysis of a treatment’s effectiveness. From the cell line model, the calculated values of a cell line’s growth and response to chemotherapy are then applied to a spheroid model. These models were
generated using MATLAB, a multi-paradigm numerical computing
environment.
1) Introduction: The cell plate culture model
G1
c1 G1
S
cs S
S
𝐛𝐆𝟏
Fig 4. An example of literature data used. This is
the cell-cycle analysis of SK-N-SH cells treated
with various doses of 15-deoxy-PGJ2. a, control;
b, 2 µM; c, 4 µM; d, 8 µM; e, 12 µM; and f, 16
µM 15-deoxy-PGJ2 treatment for 24 h. (Kim
2003) Literature tends to combine the G2 and M
cell cycle stages to G2.
Expressing the cell-cycles as a vector:
𝐺
𝑋= 𝑆
𝐺
G1
Q
𝐞𝐐
N
Fig 6. An example of determined
constants for a particular
treatment (15-deoxy-PGJ2) and
a cell-line (SK-N-SH).
𝐟 𝐍𝐆𝟐
𝐜𝐐
𝐦𝐍
The rate of P, Q, N growth is modeled as
𝑑𝑃
= 𝐺 𝑃 − 𝑏 , 𝑃 + 𝑐 , 𝑄 − 𝐹 𝑃, 𝑄, 𝑁 − 𝑑𝑃
𝑑𝑡
𝑑𝑄
= 𝑏 , 𝑃 − 𝑐 , 𝑄 − 𝑐 , 𝑄 + 𝐻 𝑃, 𝑄, 𝑁
𝑑𝑡
𝑑𝑁
= 𝑐 , 𝑄 − 𝑚𝑁
𝑑𝑡
Which has the nonzero equilibrium points of
𝑏
𝑄=
𝑃
𝑐+𝑒
𝑒
𝑁= 𝑄
𝑚
0 = 𝑤𝑋 − 𝑣𝑋 − 𝑢
Assumptions:
1) 𝑠𝑝ℎ𝑒𝑟𝑜𝑖𝑑 𝑔𝑟𝑜𝑤𝑡ℎ 𝑖𝑠 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓
𝑡ℎ𝑒 𝑠𝑝ℎ𝑒𝑟𝑜𝑖𝑑. 𝑇ℎ𝑢𝑠, 𝐺 𝑃 = 𝑎 𝑃 + 𝑄 + 𝑁 / ,
𝑑𝑃
= 𝑎 𝑃 + 𝑄 + 𝑁 − 𝑏 , 𝑃 + 𝑐 , 𝑄 − 𝑓𝑁𝑃 − 𝑑𝑃
𝑑𝑡
2) 𝑄 𝑖𝑠 𝑚𝑎𝑑𝑒 𝑎𝑡 𝑡ℎ𝑒 𝐺 𝑠𝑡𝑎𝑔𝑒 𝑎𝑠 𝑎𝑛 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔 𝑆.
𝑇ℎ𝑢𝑠, 𝑡ℎ𝑒 𝑐𝑒𝑙𝑙 𝑝𝑙𝑎𝑡𝑒 𝑐𝑢𝑙𝑡𝑢𝑟𝑒 𝑠 𝐺 = 𝑠𝑝ℎ𝑒𝑟𝑜𝑖𝑑 𝑚𝑜𝑑𝑒𝑙 𝑠 𝑄
3) 𝑡ℎ𝑒 𝑒𝑓𝑓𝑒𝑐𝑡 𝑜𝑓 𝑛𝑒𝑐𝑟𝑜𝑠𝑖𝑠 𝑜𝑛 𝑝𝑟𝑜𝑙𝑖𝑓𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑐𝑒𝑙𝑙𝑠 𝑜𝑐𝑐𝑢𝑟𝑠 𝑎𝑡
𝑡ℎ𝑒 𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑠𝑡𝑎𝑔𝑒. 𝑇ℎ𝑢𝑠, 𝑓𝑁𝑃 = 𝑓𝑁𝐺
4) 𝑝𝑟𝑜𝑙𝑖𝑓𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑐𝑒𝑙𝑙𝑠 𝑎𝑟𝑒 𝑜𝑛𝑙𝑦 𝑎𝑡 𝑡ℎ𝑒 𝑠𝑝ℎ𝑒𝑟𝑜𝑖𝑑 𝑠 𝑠𝑢𝑟𝑓𝑎𝑐𝑒
𝐿𝑒𝑡 𝑇 𝑏𝑒 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑜𝑓 𝑎𝑙𝑙 𝑐𝑜𝑚𝑝𝑎𝑟𝑡𝑚𝑒𝑛𝑡𝑠
𝐿𝑒𝑡 𝑃 = 𝐺 + 𝑆 + 𝐺
Let 𝑎 be the scaling constant for volume to surface area
∴ 𝐺 = 2𝑐 𝐺 𝑃 𝑇 − 𝑐 𝐺 − 𝑏𝐺 + 𝑐𝑄
𝑆 =𝑐 𝐺 −𝑐 𝑆
da G2
A model of the basic cell cycle (for a cell line)
𝐺 = 2𝑐 𝐺 − 𝑐 𝐺
𝑆 =𝑐 𝐺 −𝑐 𝑆
𝐺 =𝑐 𝑆−𝑐 𝐺 −𝑑 𝐺
𝐜𝐬 𝐒
𝐝𝐚 𝐆𝟐
G2
𝐺 = 𝑐 𝑆 − 𝑐 𝐺 𝑃 𝑇 − 𝑑 𝐺 − 𝑓𝑁𝐺
𝑑𝑄
= 𝑏𝐺 − 𝑐𝑄 − 𝑒𝑄
𝑑𝑡
𝑑𝑁
= 𝑒𝑄 − 𝑚𝑁
𝑑𝑡
B
Then, 𝑋 = 𝐴𝑋
−𝑐
𝑐
0
0
−𝑐
𝑐
2𝑐
0
−𝑐 − 𝑑
𝑋
Fig 7. Time vs. The total number of cells for SK-N-SH (A)
and SK-N-MC (B) in response to various doses of 15deoxy-PGJ2. These graphs are generated using MatLab
with parameters to the plate culture mathematical model
which best correlate to literature data.
𝑔∗
𝑋=𝑒
𝑠∗
𝑔∗
where 1) λ is a constant that controls doubling time and
2) 𝑔∗ , 𝑠 ∗ , 𝑔∗ are proportions of cell in each stage in the limit (𝑔∗
+ 𝑠 ∗ + 𝑔∗ = 1 )
Works Cited
1) Carlsson, J., et al. "Formation and growth of multicellular spheroids of human origin." International Journal of Cancer 31.5 (1983): 523-533.
2) Cheng, Yeung-Leung, et al. "Acetone extract of< i> Angelica sinensis</i> inhibits proliferation of human cancer cells via inducing cell cycle arrest and apoptosis." Life
sciences 75.13 (2004): 1579-1594.
3) Kim, Eun Joung, et al. "Peroxisome proliferator-activated receptor-γ activator 15-deoxy-Δ12, 14-prostaglandin J2 inhibits neuroblastoma cell growth through induction of
apoptosis: association with extracellular signal-regulated kinase signal pathway." Journal of Pharmacology and Experimental Therapeutics 307.2 (2003): 505-517.
4) Michael, R. King. "Gather Round: In vitro Tumor Spheroids as Improved Models of In Vivo Tumors." Journal of Bioengineering & Biomedical Science (2012).
5) Kyrk, John. "Cell Structure." Cell Index. N.p., 12 Nov. 2013. Web. 22 May 2014. <http://www.johnkyrk.com/CellIndex.html>.
6) Wallace, Dorothy I., and Xinyue Guo. "Properties of tumor spheroid growth exhibited by simple mathematical models." Frontiers in oncology 3 (2013).
𝐜𝟏 𝐆𝟏
G1
A
A solution looks like:
Fig 2. The cell cycle stages (Kyrk 2013)
Fig. 8
Fig. 3
𝑋 =
Fig 1. Pathophysiological gradients in a late-stage
tumor spheroid (King 2012)
Using the Wallace and Guo 2013 paper’s spheroid model 3E:
Let P, Q, and N be the number of proliferating, quiescent, and
necrotic cells respectively.
𝑔∗
and X ∗ = 𝑠 ∗ is an eigenvector of 𝐴 with eigenvalue 𝜆, 𝐴𝑋 ∗
𝑔∗
= 𝜆𝑋 ∗
Let D be the doubling time:
ln 2
𝑒 =2→ 𝜆=
𝐷
Let 𝑑 , the natural apoptosis rate,
= 0 since cell death is accounted for in the doubling time. Now,
−𝑐 𝑔∗ + 2𝑐 𝑔∗ = 𝜆𝑔∗
𝑐 𝑔∗ − 𝑐 𝑠 ∗ = 𝜆𝑠 ∗
𝑐 𝑠 ∗ − 𝑐 + 𝑑 𝑔∗ = 𝜆𝑔∗
𝜆 + 𝑑 𝑔∗
∴𝑐 =
𝑔∗
2𝑐 𝑔∗ − 𝜆𝑔∗
𝑐 =
𝑔∗
𝑐 𝑔∗ − 𝜆𝑠 ∗
𝑐 =
𝑠∗
Fig. 9. A plot of time vs. tumor spheroid’s total number of cells for the literature data (red-line) and for the mathematical model. The
cell line shown here is SK-N-MC. Literature data used to compare
the spheroid model is from the Carlsson 1983 paper.
Fig 5. An example of a summary of cell plate cycle
parameters for plate culture (default parameters)
Step 2) Treatments on Plate
Cultures
Literature usually states the particular cell cycle
stage(s) that a chemotherapy drug targets in a
cancer cell. For treatments on a cell plate culture,
using the part of the cell cycle affected and/or the
resulting observed death rates creates a series of
models and computer runs to estimate the new
death rate constants for each type of treatment/cell
lines. This step determines a mathematical
constants for a cell plate culture’s response to a treatment and these constants will be applied to
spheroid models.
Total Number of Cells
The current research prominently relies upon five articles to
provide a basis of a mathematical spheroid tumor model,
empirical cell plate culture data without treatment, empirical cell
plate culture data in response to treatment, and empirical tumor
spheroid data without treatment, and empirical tumor spheroid
data in response to treatment.
Step 2) Cont.
Step 3) Spheroid Models
𝐺 𝑜 = 0.682 ∗ 4189000 = 2.86 × 10 ;
𝑆 𝑜 = 0.227 ∗ 4189000 = 9.51 × 10 ;
𝐺 𝑜 = 0.091 ∗ 4189000 = 3.81 × 10 ;
𝑄 = 0;
𝑁 = 0;
𝑏 = 0.65;
𝑐 = 0.00000001 = 1 × 10 ;
Time (hours)
Discussion of Results
The research project accurately mathematically modeled the growth of
monolayer cancer plate cultures in response to certain treatments for four cell
lines: A-549, SK-N-SH, SK-N-MC, and MCF-7. The values determined for 15deoxy-PGJ2 effect for SK-N-MC was accurately mathematically modeled to the
Carlsson 1983 paper. The research project is still in the process of accurately
modeling spheroid growth to for A549 and breast cancer. Current results are
promising and exemplify the potential of mathematical modeling of cancer tumor
growth. The approach described in the research project provides a general
method to mathematically model any cell line that produces tumor spheroids
using cell plate culture data to identity basic growth and treatment parameters.