Numerical Modeling of Drug Release from Polymer Coated Arterial Stent
Abstract: Stenting is a surgical procedure in which a mesh like cage is placed within an artery/vessel when blood flow is obstructed. There are three types of arterial stents, including metallic, biodegradable and polymer coated. The polymer coated stent is the most typically used stent, such that it has a lesser rate of restenosis, a.k.a re-narrowing of a chamber. The coating of the stent is infused with a drug that releases into the artery and diffuses through the arterial wall. This diffusion can be modeled through a second order parabolic differential equation. This phenomenon can be modeled via MATLAB in order to understand the rate in which drug diffuses as it relates to time and distance traveled through the artery.
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More importantly, quantitative analysis can be applied after obtaining data from the model in order to understand the effects this can have in a postoperative patient. The first plot produced by MATLAB showed the relationship between concentration, time and distance for diffusion of Paclitaxel. The second plot showed localization of the drug at time went on for various distances. The third plot showed the amount of the drug at different distances for different sets of time.
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Every year, it is estimated that over 800,000 Americans will suffer a heart attack. A heart attack can occur as a result of occlusion through the coronary artery. The coronary artery is a vessel that transports oxygen-rich blood into the heart. When severe levels of plaque build up on the walls of coronary artery, medical intervention is often needed to allow for the restoration of blood flow to the heart and to prevent death of cardiac muscle. In order to protect against arterial obstruction, medical professional will often employ the use of a coronary stent.
A stent is a biomedical device that is used to combat the effects of atherosclerotic cardiovascular disease (figure #1). Stents are inserted into the body through a flexible catheter after an initial incision is made. For the case of the heart, the incision would typically be made near the groin and a catheter would be inserted into the inguinal canal and pushed towards the coronary artery. The catheter is tipped with a balloon that would be inflated upon arrival to the afflicted region. The balloon would press up against the walls of the the coronary artery to make it wider (angioplasty). A stent is then deployed after inflation of the vessel and is fastened to the region to keep from narrowing (restenosis) again.
There are a couple of competing stents used in the medical field. These different stents are bare metallic stents, biodegradable stents, and drug coated polymer stents. Bare metal stents are typically made from chromium cobalt alloy or stainless steel and are mesh shaped in order to allow for the flow of blood through the artery. Polymer stents are almost similar to the bare metal with the exception that they are coated with a layer of drug infused polymer layering. Biodegradable drug eluting stents are created with the intent to breakdown in the body and be resorbed after a certain period of time had elapsed [1,2,5]. Due to the fact that biodegradable stents are still somewhat of a recent phenomenon, attention will be placed on metal and polymer stenting.
Historically, metal based stents have had success in alleviating most blockages in the coronary artery. However, there are still many issues that are associated with their use in the body. One of the most prominent issues associated with metal stenting is restenosis. In the event of restenosis, the smooth muscle will begin to proliferate and thicken over the stent, causing narrowing and occlusion of blood flow. Restenosis is very common amongst the bare metal stents. In fact, studies have found restenosis to be prevalent amongst twenty-five percent of people using a metallic coronary stent . To combat restenosis, engineers have designed polymer coated stents to diffuse certain drugs (commonly paclitaxel or sirolimus) into the artery in order to keep it open. Research has shown that drug eluting stents have dropped the rate of restenosis to less than six percent by slowing down the rate of revascularization .
From data given, it can be emphasized that revascularization of the coronary artery can play a huge part as to the success or failure of a coronary stent. Being able to model the rate in which drugs are eluted from a coronary stent over a certain period of time can essential into understanding the success of a stenting procedure, and the prevention of atherosclerosis. The goal of this assignment is to understand how drug release from a polymer coated arterial stent works, and to create a numerical model for diffusion through the arterial wall, and as to what useful insight this can give towards the medical outcome of a postoperative patient.
Figure #1 (Stenting Diagram)
2.1 Arterial Environment & GeneralizedAssumptions
In order to model the way in which a polymer coated stent diffuses a drug through an artery, there needs to be an understanding of the environment that the stent is in. For simplicity reason, the artery will be assumed to have a volume of a cylindrical tube. The density of the arterial muscles is also assumed to be constant throughout the body. The stent implanted into the body is assumed to have a thin layer of drug that is uniformly distributed throughout the entirety of the stent. The diffusion coefficient of both the arterial wall and the drug is assumed to be constant throughout its lifespan. Lastly, the only chemical event that is being accounted for is diffusion through an artery, while other chemical events that could affect the way in which the drug diffuses will be neglected.
2.2 Diffusion Equation NumericalAnalysis.
The diffusion of a drug can be represented as a second order parabolic differential equation (1). The two independent variables of this PDE are the time [t] in which the drug concentration decreases and the distance [x] in which the diffusion occurs. The representation of the diffusion equation is as follows:
In this equation, Cp represents the initial concentration of the drug. The equations shows that the change in concentration over time is equal to that of the second derivative of the change in concentration as it relates to distance it diffuses over. Although this equation is useful as it relates to a one-dimensional cartesian plane, the scenario that is being dealt with relates to cylindrical geometry. Instead of x, the distance in which diffusion occurs will be represented a radius constant [r]. Next, an assumption is made that the media in which the drug will decay in will not over saturate and hinder the diffusion rate, hence an ideal sink condition is put in place (2). By assuming ideal sink condition and taking into account the cylindrical nature of this problem, a new equation (3) is used in instead of (1):
When solving for a numerical solution to this equation, it must be noted that there are three initial conditions to take into considerations. Two of these conditions are equations pertaining to the radii from the stent to the walls of the artery. The radius that extends to the inside of the arterial (r1/Ra) wall and the radius that extends to the outside of the arterial wall (r2/Ro) (figure#2). An assumption will be made that when the concentration at a particular time pertains to a distance of (r1/Ra), then the value of that concentration will be zero, since there was no change in the distance of drug delivery. This can be represented by equation (4). On the other hand, assuming the drug ends up reaching the radius (r2/Ro), then the change in the concentration can be assumed to be zero, since the drug reached its final destination. This equation is modeled by equation (5). Lastly, the last initial condition is that for time being equal to zero, it can be assumed that the concentration of the drug is zero considering no change had time to occur. This depicted in equation (6).
To find a solution that gives us a function for concentration as it relates to times and distances, then the solution will comprise of both a homogenous and a particular solution.
Figure #2 (Cross-Section of Artery-Stent Diagram)
2.3 Table #1 of givenvalues
Drug used for stent is Paclitaxel
Artery Thickness (cm)
Inner wall radius (cm)
Outer wall radius (cm)
Diffusion Constant for Artery (cm^2/s)
Diffusion Constant for Stent (cm^2/s)
Values defined in Table #1 were found in literature. The constants defined were implemented into the MATLAB program to evaluate the diffusion rate.
2.4 MATLAB CODE & FIGURES
Equation #3 was put into MATLAB in order to be modeled and to extract useful data. In order to model said equation, PDPE was implemented into the program. PDPE solves differential equations that have initial bounds in the first dimension. PDPE is very useful for evaluating parabolic differential equations including the one that is being dealt with in this problem. Code taken from MATHWORKS personal site was used in order to evaluate the PDE. A three-dimensional diagram was made of the PDE after inputting the parameters of the problem into the PDPE solver, including the distance in which the drug would travel over, a defined time frame that will be analyzed (for the purposes of this project it was set till 120 seconds) and the concentration of the drug. The surf command helped in plotting out what the diffusion model should look like (figure #4) .
Figure #4 Diffusion Model
Although figure #4 is a clean depiction of the diffusion of Paclitaxel, it’s not very helpful in giving an in depth understanding of the events going on in the artery. So two more plots were created in order to show characteristics of interest. Second plot that was created shows the amount of the drug present after time has passed for distances. T1 is the variable that represent the thickness of the stent in centimeters and T2 represents the thickness of the artery in centimeters. The following plot depicts the concentration of the drug at particular locations during release (figure #5).
Figure #5 Increasing Concentration presence at specified locations for time duration
This plot (figure #5) was created by taking values of concentration stored in a matrix plot that was formed in the PDPE solution and time. Finally, the third plot created depicts the rate at the concentration decrease for a particular distance traveled based upon the time that had elapsed during drug delivery (figure #6).
Figure #6 Decreasing Concentration rates for specified time frames are certain locations
This plot was created by taking values of concentration stored in a matrix plot that was formed in the PDPE solution and distance.
Based on the plots created in MATLAB, multiple factors can be identified. From figure #5, it can be said that at a distance closest to the arterial stent, the concentration is the highest. T1/2 would be half the distance of the arterial stent. Due to the manner in which the drug propagation occurs, this seems to be highly likely. High concentration delivery is wanted during diffusion and it is only natural that a shorter distance to the source would have the highest quantity. Subsequently, a drop in the increasing concentration rate can be seen as time goes on for farther distances. This also makes sense because drug delivery should be spanned out. If it all were to occur at once, that could potentially kill the patient, and would render the purpose of the polymer stent useless. The polymer stent should let out some drug over time to protect against revascularization.
For figure #6, the display also makes sense. When fragments of Paclitaxel are released form the stent initially (1.2 seconds) the concentration of the piece should quickly dissolve at a shorter distance. That is because the amount already in the system is so low. But, as the system gets more and more saturated, the drug will travel farther and then a decrease in the concentration of the fragmented pieces will begin to slow down.
Therefore, based on the characteristics of the plots of data obtained via MATLAB, the actions of Paclitaxel should be beneficial to the body. The reason the word should and not are is because the program is not very conclusive of real events going on in the arteries. There are other issues such as blood flow and or drug layering that can affect how diffusion occurs through the arteries. Also, the data analyzed by the graphs were only approximated by their looks and not by any statistical display. The rates of diffusion and drug concentration may be extremely abnormal. If further work was put into this project, then somehow having the ability to depict the rates of figures #5 and #6 would be worked upon. If that was done, then a definitive answer could be given as to how long this drug would last in the human body and the effect it could have on a patient.
Appendix (MATLAB CODE)
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