ABSTRACT
This research study is about controlled drug delivery, which
is a relatively new area of mathematical modelling. In this study, there have
been two major focuses. The first is to further understand the model for drug delivery
from collagen matrices developed in [14], by solving it with a different
numerical scheme, and the second to develop a new model based on a different
geometry. Both models are based on mass conservation and Fick’s law, and are
therefore possible to compare. The two models have been discretized and
implemented, and the results compared to experimental data.
CHAPTER
1
INTRODUCTION
Disparities of collagen types can be due to variation in the
length of the helix, as well as the size of non-helical areas within it [6, 7,
15]. These areas vary from almost none (4 % for collagen I) to more than 90 %
(for collagen XII). The helix is made of three polypeptide chains, referred to
as αchains, and it is the composition
of these chains that determine the collagen type. Currently, at least 28
different types of collagen are known [15]. The predominant type of collagen is
the type 1-collagen, which amongst others is found in skin, tendon, bone and
large vessels. It consists of two identical α1(I)-chains
and one α2(I)-chain with a different
amino acid composition, or in some rare cases three α1(I)-chains [7]. Because it is the most common type, it is the
type mostly used for research, and the one used here.
Collagen is insoluble in organic solvents, and only a few
percent of the total collagen is soluble in water [6]. This is why the implants
need to be inserted into the body somewhere collagen naturally occurs. As we
have seen, this includes a lot of different tissues, which makes collagen well
suited for these kinds of implants.
1.1.1 Cross-linking
Cross-links in collagen are links between the α-chains, which makes the molecule more
stable and difficult to degrade. These occur naturally both intra- and
intermolecularly and are assembled within the non-helical areas of the
molecules. They can dwindle away by acidic reactions, but new crosslinks can be
introduced in different ways [6]. Cross-linking a collagen matrix has the
purpose of ing the degradation, so the matrix will not crumble as fast. Because
collagen-implants are used for controlled drug delivery, the ability to the process down can be useful. Changing the
molecular structure of the collagen does, however, mean that there needs to be
done new experiments, and the mathematical model needs to be altered, as this
is very new with regards to drug delivery.
There
are two ways to cross-link a collagen molecule; chemically or enzymatically.
Chemical cross-linking will be badly solvant in water, and will therefore need
to be solved in something else to react. This raises toxicological concerns,
and both the chemicals and their solvents will need to be removed from the body
[7]. Enzymatic cross-linking have very few toxicological concerns, because the
cross-linking products are activated by enzymes that will be washed away before
the implants are inserted. The cross-linked collagen used for the experiments
in this research project has been cross-linked by a
1.2 The device
The collagen implant, often called a minirod, is made by
homogenizing collagen and the drug we want to trap. Higher weight drugs, such
as proteins or polysaccharides, are commonly used [10]. This mix is made into
the minirod, formed as a cylinder, which is then inserted into the body. There
are many uses for controlled drug delivery, ranging from treatment of cancer
and diabetes to contraception and vaccination [10].
The challenges arise when we want to optimize the drug delivery.
There are many factors that influence the process, both the shape of the
minirod, whether the collagen is cross-linked or not and how much drug it
contains. From a mathematical point of view, we want to model how quickly the
collagen is degraded, and how fast the drug is deliveryd. However, due to
concurring processes, this is difficult. There are also a lot of parameters,
and although some can be determined experimentally, some will have to be
fitted. The goal is to have as few parameters as possible fitted, and we hope
that our new model will have fewer parameters that need fitting than the model
from [14].
1.3 Experiments
During my stay with the Ludwig-Maximilians University of
Munich, we did new experiments on the minirods. These were performed both with
noncross-linked collagen and cross-linked collagen. We did some short-term
measuring of the collagen degradation that was used to help determine a new set
of reaction rate constants. After I left, Madeleine Witting continued
experimenting, and in combination with the experiments we performed together
was able to give me a set of new parameters to use for my simulaions. There are
still some experimenting to do, as we did no experiments on the delivery of
drug, only the collagen degradation. However, earlier experiments suggest that
the drug delivery as well as the collagen degradation is ed down when the
collagen is cross-linked.
These systems are very promising and their
optimization is the subject of current research. As applied mathematicians we
try to contribute through mathematical modelling and numerical simulation to
the understanding and eventually the optimization of these systems.
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