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.

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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