Transcript Cancer & Mathematical Modeling

Michael Ghoorchian
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The body is made up of hundreds of millions of living
cells.
Normal body cells grow, divide, and die in an orderly
fashion. During the early years of a person's life, normal
cells divide faster to allow the person to grow.
After the person becomes an adult, most cells divide
only to replace worn-out or dying cells or to repair
injuries.
Cancer begins when cells in a part of the body start to
grow out of control. There are many kinds of cancer, but
they all start because of out-of-control growth of
abnormal cells.
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Cancer cell growth is different from normal cell growth. Instead
of dying, cancer cells continue to grow and form new, abnormal
cells. Cancer cells can also invade (grow into) other tissues,
something that normal cells cannot do. Growing out of control and
invading other tissues are what makes a cell a cancer cell.
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Cells become cancer cells because of damage to DNA. DNA is in
every cell and directs all its actions. In a normal cell, when DNA
gets damaged the cell either repairs the damage or the cell dies. In
cancer cells, the damaged DNA is not repaired, but the cell doesn't
die like it should. Instead, this cell goes on making new cells that
the body does not need. These new cells will all have the same
damaged DNA as the first cell does.
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People can inherit damaged DNA, but most DNA damage is
caused by mistakes that happen while the normal cell is
reproducing or by something in our environment. Sometimes the
cause of the DNA damage is something obvious, like cigarette
smoking. But often no clear cause is found.
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In most cases the cancer cells form a tumor. Some cancers, like
leukemia, rarely form tumors. Instead, these cancer cells involve
the blood and blood-forming organs and circulate through other
tissues where they grow.
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Cancer cells often travel to other parts of the body, where they
begin to grow and form new tumors that replace normal tissue.
This process is called metastasis. It happens when the cancer cells
get into the bloodstream or lymph vessels of our body.
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Always named for the place where it started no matter where it may
spread. For example, breast cancer that has spread to the liver is still
called breast cancer, not liver cancer. Likewise, prostate cancer that has
spread to the bone is metastatic prostate cancer, not bone cancer.
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Different types of cancer can behave very differently. For example,
lung cancer and breast cancer are very different diseases. They grow at
different rates and respond to different treatments. That is why people
with cancer need treatment that is aimed at their particular kind of
cancer.
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Not all tumors are cancerous. Tumors that aren't cancer are called
benign. Benign tumors can cause problems -- they can grow very large
and press on healthy organs and tissues. But they cannot grow into
(invade) other tissues. Because they can't invade, they also can't spread
to other parts of the body (metastasize). These tumors are almost never
life threatening.
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Boosting the immune system
Anti-cancer vaccine — exposing some cancer cells extracted from a tumor to UV
rays for 24 hrs then injecting them back into the organism,
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Chemotherapy – Using chemicals to kill cancerous cell
Photodynamic therapy(PDT) – giving a photo sensitive drug containing
cancer killing substances which are absorbed by cancer cells then positioning a laser
beam at the tumor site, which then activates the drug that kills the cancer cells
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Radiation therapy : use of ionizing radiation as part of cancer treatment to
control malignant cells
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Targeted therapy - medication that blocks the growth of cancer cells
Gene therapy - Inserting genes into an individual's cells and tissues to treat a
disease, such as a hereditary disease in which a deleterious mutant allele is replaced with a
functional one
A generation ago, biology seemed too hard to
mathematicians to pin down with the precise laws and
calculations that work so well in physics and engineering.
 Though mathematicians did have some success in genetics
and population biology, a serious mathematical theory of
cancer seemed like a figment of the imagination.
 Now, mathematical models are popping up everywhere in
cancer research.
 Patients respond differently to treatments, so models should
be able to adjust on different cases.
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Renee Fister , PhD ;
Mathematician at Murray State University in Kentucky, studies
optimal-control models that promise to provide physicians
the best time tables for drug treatments.
Carl Panetta, PhD ; St. Jude Children’s Research
Hospital in Tennessee, uses systems of elementary
differential equations—typically 30 or 40 at a time—to
predict a patient’s response to a given drug regimen.
The goal is to answer to these questions :
“how much of the drug is getting into the patient’s cells and how fast it is
getting cleared.”
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To answer these questions, they use a giant system of linear differential
equations that model cell-to-cell and drug-to-cell interactions.
Because they are linear, the systems can be solved exactly.
Solving them is only half the job. The other half is parameter-fitting since every
patient reacts a little bit differently to a drug regimen.
The differences show up in the constant coefficients—the parameters—in the
differential equations.
The right parameters for each patient have to be teased out statistically from the
data.
Fister and Panetta started with one simple differential equation:
N : tumor volume,
r : growth rate,
Θ : maximum size for the tumor.
G(N, u(t)), : interaction between the physician-prescribed treatment u(t) and the tumor volume N.
There are 3 scenarios :
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you kill the most cancer cells when the tumor is largest .
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The drug is most effective when the tumor is small but has the highest
growth rate.
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Saturation effect - drug becoming less potent as fewer proteins are
available for binding.
The answer depends on the goal of therapy.
• Is it to minimize tumor size?
• Is it to minimize the expense to the patient, which might be
expressed as an integral of u(t), the amount of drug given?
• Is it to maximize the patient’s quality of life, which might be
represented by a functional that penalizes high doses, possibly an
integral of u(t)2 ?
• Perhaps the therapy should aim for a combination of all the above.
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Panetta and Fister found that each of these considerations has an
important influence on the design of the optimal therapy.
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The tumor-size hypothesis implies that it is beneficial to delay
treatment until the tumor is large, while the other two lead to
recommendations for early treatment.
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The linear patient-cost model implies that “bang-bang controls”
are best, in which the physician gives the patient either a
maximum dose or no drugs at all.
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Based on the nonlinear patient-welfare model, the physician
would phase the drugs in and out more gradually.
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Although optimal-control theory has a long history in engineering,
its application to cancer treatment is still very new.
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Mathematical Modeling and Cancer ; By Dana Mackenzie ; SIAM News, Volume
37, Number 1, January/February 2004 ; By Dana Mackenzie
http://www.cancer.org
SIAM Journal on Applied Mathematics (Vol. 63, No. 6). 2003
http://stjude.org/cancercenter
http://en.wikipedia.org/wiki/Cancer Treatment
http://en.wikipedia.org/wiki/Cancer_research
http://en.wikipedia.org/wiki/Photodynamic_therapy