model for a novel cancer treatment called anti-angiogenesis.
For a long time, a positive diagnosis of cancer was synonymous with a death sentence.
We now have come a long way with all the new innovations in medicine and technology.
According to the American Cancer Society, "more than half of all people with cancer now
live at least 5 years after being diagnosed". Today, there is a wide variety of possible
treatments for cancer such as: radiotherapy, bone marrow transplants, gene therapy, surgery
or chemotherapy. This last one is by far the most common form of cancer treatment known.
Chemotherapy uses chemical agents to destroy cancerous cells. The goal of chemotherapy
drugs is to kill cancerous cells using a dose that would do the least damage to other healthy
cells in the body. Therefore the making of the drug requires the identi…cation of some
speci…c characteristics that are unique to cancerous cells in order to spare the healthy ones.
Even though chemotherapy has been successfull in numerous cases of cancer, often this
is not the case. It comes with a great deal of side e¤ects such as hair loss, weakness of
the immune system, decrease of white blood cells, and many other harmful e¤ects. Also,
because tumors cells reproduce at an abnormally high rate, they quickly develop resistance
against chemotherapy drug and some tumors are intrinsically resistant to the drugs. This
is the reason why after more than 20 years of experimenting with various cancer treatment
approaches today, anti-angiogenesis is in the spotlight as a treatment which does not develop
drug resistance and has small side e¤ects.
In this thesis we present the biological background and some scienti…c facts that sup-
port anti-angiogenesis as a form of cancer treatment therapy. We then introduce two more
responses to anti-angiogenesis formulated by a group of researchers from Harvard School of
medecine (Hahnfeldt et al [3]). This model was then analyzed as an optimal control problem
in [6] [5]. In this paper, we will formulate this problem with di¤erent objectives and analyze
how this change will a¤ect the qualitative structure of solutions. First, we will introduce the
theory behind optimal control including the general formulation for an optimal control prob-
lem, the Maximum Principle, and the nature of the resulting control. Then, the geometric
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