Steady State Computer Model of Human Vascular for Analyzing Effects of Stenosis

Transcription

1 Steady State Computer Model of Human Vascular for Analyzing Effects of Stenosis University of Illinois at Chicago Professor Andreas Linninger Michael Naskrent December 2, 2013

2 Table of Contents Abstract Introduction Motivation Background Locations of atherosclerosis Goal Methods Introducing the Model Assumptions Equations Building the Model Initial Conditions Solving the Equations Creating the Stenosis Validation Results Carotid Artery Disease Pressure Analysis Flow Analysis Coronary Artery Disease Pressure Analysis Flow Analysis Renal Artery Disease Pressure Analysis Flow Analysis Peripheral Artery Disease Pressure Analysis Flow Analysis Venous vs. Arterial Flows Discussion Appendix References... 14

3 Naskrent 1 STEADY STATE COMPUTER MODEL OF HUMAN VASCULAR FOR ANALYZING EFFECTS OF STENOSIS Michael Naskrent mnaskr2@uic.edu Abstract A steady state computer model for simulating the pressures and flow rate of the circulatory systems was generated. The model incorporated the main arteries and veins of the system, while simplifying the capillary structures. The model was designed to simulate the effects of a stenosis by constricting portions of the internal carotid artery, proximal coronary artery, renal artery, and deep femoral artery, which mimics carotid artery disease, coronary artery disease, renal artery disease, and peripheral artery disease respectively. This was used to determine how the blood was redistributed throughout the body, and how the added pressures effects the system. The model showed how the circulation in the body was altered when a major arterial pathway was incrementally obstructed, which provided insight into the effects of a stenosis. The model revealed that the four stenosed arteries had similar diminishing blood flow and pressure pattern, which started around 60% stenosis and rapidly approached a nominal flow at 90% stenosis. The flow rate of the arteries up stream of the stenosis essentially decreased by the same degree as the stenosed artery, while the pressures increase was minimal. The branches above the stenosis only had a slight increase in flow rate and pressure. This study showed that an artery can be significantly impaired before any effects on the body become dangerous or life threatening. 1. Introduction 1.1 Motivation Cardiovascular disease which is caused by atherosclerosis is the leading cause of death and disability in the world, with 17.3 million deaths in Most of these conditions are caused by atherosclerosis in the form of heart disease, myocardial infarction, and stroke [1]. The four simulations will improve the understanding of some of these most common complications of atherosclerosis. Strokes which can be cause by carotid artery disease is the second leading cause of death in the world. It accounts for 4.4 million or 9 percent of all deaths each year [2]. With this prevalence, it is beneficial to study and understand the effects that carotid artery disease can have on the body. Carotid artery disease is the reduction of blood to the brain tissue as plaque builds up in the carotid artery, which deprives the brain of vital nutrients and oxygen and can cause a stroke [3]. Coronary artery disease accounts for 1.7 million or 3.5 percent of all deaths each year [4]. It is when the major arteries that supply blood to the heart become blocked. The most common cause of this is due to a buildup of plaque [5]. In the U.S. alone, 400,000 patients with renal artery disease are being treated with dialysis, and 170,000 patients are on the kidney transplant list [6]. Renal artery disease is the stenosis of arteries that transport blood to the kidneys. This reduces the oxygen levels, and the kidneys will not be able to properly filter waste products and remove excess fluids [7]. It is approximated that 8 million people in the U.S. have peripheral artery disease. This includes 12-20% of individuals over 60 [8]. Peripheral artery disease is the stenosis of arteries in your extremities, most commonly in legs [9]. [10] Figure 1 [10]. The progression of atherosclerosis from initial lesion to complete thrombosis with timeline.

4 Naskrent Background Atherosclerosis is the buildup of fatty materials, cholesterol, cellular waste, calcium, and fibrin on the arteries wall. This coagulation of materials that impedes the flow of blood through the artery is called plaque [11]. Figure 1 demonstrates at plaque will build up over decades, which slowly blocks off the artery. There are two outcomes of this process: a piece of the plaque may break off, flow downstream, and lodge itself in a smaller artery or a blood clot (thrombus) forms on the plaque s surface. In both cases, the artery will be completely blocked, which causes a heart attack or stroke [12]. There are three stages of plaque formation, which is called atherogenesis: Initiation, plaque progression, and plaque disruption. During the initiation phase a fatty streak forms on the lumen of the artery due to endothelial dysfunction, which can be caused by various agents, such as physical stress and chemical irritants. It is observed as a yellow discoloration that is slightly elevated. There are no symptoms associated with initiation, and it can also subside over time. The plaque progression phase consists of the continuation of material build up in the artery. There are two layers. First, the core of plaque, which consists of low density lipoprotein (LDL). Second, the fibrous cap covers the internal lipid layer, which is comprised of dead foam cells, macrophages, smooth muscle cells, lymphocytes, and extracellular matrix. In the third stage, plaque disruption, the body attempts to remove the growing mass to restore blood flow by extracellular matrix metabolism. It consists of classes of photolytic enzymes and inflammatory cytokines, which activates apoptosis in the smooth muscle and foam cells. The cellular debris will build up and add to the size of the plaque. At this point, the plaque can continue to stenos the artery or become unstable, causing the fibrous cap to rupture. The necrotic core will release its content including tissue factor, which when in contact will blood will start the formation of a thrombus. Depending on the composition of the plaque and quantity of lipids in the blood stream, the thrombus can dissipate or form a clot. The clot can occlude the artery or break off and block arteries and capillaries downstream [1]. 1.3 Locations of atherosclerosis There are many factors that determine the complications of atherosclerosis, but in many cases the outcome can be fatal. The most common locations of atherosclerosis are the dorsal section of the abdominal aorta, proximal coronary arteries, popliteal arteries, descending thoracic aorta, internal carotid arteries, and renal arteries. Figure 2 shows some of the locations of atherosclerosis along with the corresponding diseases and physiological reasons for the conditions [1]. 1.4 Goal The goal of the computer model is to effectively simulate the conditions of a stenosis and record the change in bulk flow and pressure of the surrounding veins and arteries. Four arteries will be incrementally constricted to simulate stenosis: right internal carotid (39), proximal coronary (179), right renal (24), and right deep femoral (15). These four stenosis locations will effectively simulate Carotid Artery Disease, coronary artery disease, renal artery disease, and peripheral artery disease respectively. For each of the four experiments, graphs of the flow versus the percent blockage and the pressures versus the percent blockage of the surrounding arteries and veins will be created. This will determine where the flow is diverted throughout alternate pathways during a stenosis, how the pressure is redistributed, and when the percent blockage effects the body s ability to maintain sufficient oxygenation of tissues. Figure 2 [1]. A picture of the human body with the locations of different complications of atherosclerosis are shown. The reason for the conditions are also given.

5 Naskrent 3 2. Methods 2.1 Introducing the Model To better understand the effects of atherosclerosis throughout the body, a one dimensional computer simulation of the human circulatory system was created. The model would be able to find the pressure and flow rate of any main artery and vein using a cylindrical tube based system. To model a stenosis, the size of a tube was decreased, and the change in the bulk movement of blood and the pressure throughout the system was tracked. This provided insight into the inner working of a circulatory system when effected by atherosclerosis. 2.2 Assumptions To create the model, some assumptions for the one dimensional cylindrical system was used. The blood in the system was viscous, and it behaved as a Newtonian fluid. It had a uniform density, which means that it could not be compressed. The flow of blood was smooth, which allowed the flow to be laminar. This was crucial in creating a system with no change in shear stress, pressure, and velocity with respect to time. The flow of blood was fully developed, which created a systems where the maximum velocity was at the center of the blood vessel. Also, there were no edge effects [13]. These assumptions allowed for the simplification of the system, while retaining the level of detail required to effectively model the stenosis. 2.3 Equations The flow network was governed by two crucial equations, the conservations of momentum and the Hagen-Poiseuille equation. First, the conservation of momentum states (Accumulation) = (In Flow) (Out Flow) ± (Generation) (3) Since, the blood cannot accumulate and there is no generation, the equation simplifies to n n i fi, in = i fi, out. (4) This was very simple but powerful when analyzing the branching points of the system. With the equations governing the flow network in place, the network was built. Second, the Hagen-Poiseuille equation solves for the drop in pressure, P, in a cylindrical pipe given a flow rate, f, and a resistance, α. The formula is P = fα (1) with α = 128μL, (2) πd 4 where μ is the dynamic viscosity, L is the length of the pipe, and D is the diameter of the pipe [14]. Figure 3. The visual representation of the human arterial network is used to model blood flow through the body with labels for each of the arteries in the body. The names and information corresponding to the points can be found in table 1 of the appendix. Figure 4. The visual representation of the whole body network was created off of the arterial network, which was used to simulate the circulatory system. This model consists of arteries (red and in foreground), capillaries (purple and on the transverse axis), and veins (blue and in the background).

6 Naskrent Building the Model Figure 3 shows the basic model of the human arterial network, which was used as a starting point for creating the whole body network [14]. The names and information of each artery is given in the human arterial network information chart, which is in table 1 of the appendix. It also provides the lengths and diameters of each artery. To simulate the blood flow through the capillaries and organs, a resistance was added to each terminal node. This was used to find the pressure drops at each terminal node, thereby satisfying the boundary conditions. These resistances can be found in table 2 of the appendix, and were used to simulate the capillaries. The network was then expanded by adding the venous network. This was accomplished by replicating the length, width, and branching pattern of the arteries. They were then connected at the terminal points of the network, which created one inflow and one outflow for the entire system. The completed whole body network can be seen in figure Initial Conditions When just modeling the arteries, there were 27 boundary conditions, but when the capillary and venous structures were added, only 2 boundary conditions remained, the pressure flowing in and out of the heart. The average systolic pressure for the ascending aorta is 120 mmhg, and the pressure in the vena cava is 5 mmhg [15, 16]. These boundary conditions simulate the pressures leaving and entering the heart. Since, the entire human body circulatory system consisting of the arteries, capillaries, and veins, was completed, MATLAB was used to solve the system. 2.6 Solving the Equations Since the system consists of 197 conservation of momentum equations and 152 Hagen-Poiseuille, it was represented in matrix form. An example of the two types of equations are F a F b F c = 0 (5) 1 F 1 P 2 + P 3 = 0. (6) Solving the 331 linear equations was accomplished by using the matrix form Ax = b. (7) The A matrix contained the left hand side of the conservation of momentum and Hagen-Poiseuille equations. They were set equal to zero so the left hand sides of the equations, b vector, were all 0 except for the boundary conditions. x is the list of unknown pressures and flow rates. The 331 unknowns were solved by using MATLAB functions. The solutions for the unknowns represents the flow and pressures throughout the body. 2.7 Creating the Stenosis The four locations of stenosis had to be created by simulating a reduction in size of a 0.5 cm segment of the artery. This subpart of the artery simulates the stenosis with additional accuracy, since the average stenosis that is surgically removed using percutaneous transluminal coronary angioplasty is 0.5 cm in length [17]. This was achieved by using a modified resistance formula, α = 128μL s πd s μL r πd o 4, (8) where L s is the length of the stenosis, L r is the length of the remaining segment, D 0 is the original diameter, and D s is the diameter of the stenosis. For the experiment, each of the four sub-segments were reduced in size from 100% to 1%. The flow through the segment, as well as adjacent segments, was graphed to illustrate its effect on the network. 2.8 Validation Before proceeding, it was necessary to verify that the program was operating correctly. This was accomplished by verifying the solutions for flows and pressures of a small network, when solved by hand. The subsequent step was to verify that the pressure results of the human vasculature were accurate. All of the pressures were positive and between the end point pressures of 120 mmhg and 4 mmhg. Also, the flow rate in and out of the system were equivalent. This furthermore lends credibility for the validity of the program. It was also essential to verify the system behaved in a similar fashion to real human vasculature. First, all of the flows were going in the proper direction. The flow through the heart (end points of the network) was 4.5 L/min, which is comparable to the literature value of 5.25 L/min [18]. It can also be noted that the kidneys, G.I. tract, spleen, and liver were within 10% of the given values, but the flow to the brain was 70% lower than expected. This could be why the model had an overall lower flow rate than the literature values.

7 Naskrent 5 To correct this issues, the resistance of the veins and arteries entering and leaving the brain were optimized to reflect the literature values of 0.75 L/min. This not only fixed the flow to the brain, but it also corrected the literature value of the blood flowing through the heart to 5.1 L/min, which is only a 1.4% difference. This can be found in table 3. The blood through the heart is the single most important factor in verifying the proper flow through the human vasculature model. With this being very close to actual values, it was reasonable to assume that there is a high confidence in this model. Human Blood Flow Comparison Organ Flow (L/min) Literature Value [18] Results 1 (L/min) % Difference Before Optimizing Brain Flow Results 2 (L/min) % Difference After Optimizing Brain Flow Heart Kidneys G.I. Tract Spleen Liver Brain Table 3. The literature values of blood flowing through organs in the body was compared against the proposed model. The percent before optimization of blood flowing to the brain gave the percent of blood flowing through the organs before any modification were made on the network. The percent difference after optimizing blood flow to the brain gave the percent of blood flowing through the organs after the flow rate to the brain was modified to match literature values. The small differences in blood flow lends credibility to the model. 3. Results 3.1 Carotid Artery Disease Pressure Analysis A. B. C. Figure 5. These figures model the arterial pressure effects of a Carotid Artery Disease by stenosising the right internal carotid artery. A. This graph shows the change in pressures for points in the arteries that are in close proximity to the stenosis, which is indicated by the blue dot. B. The arterial network pressure figure shows the points that were being observed during the stensosis. C. This is the same graph as A without the artery point directly preceding the stenosis, which gives added detail to the other three points. This system models the conditions that are observed in carotid artery disease. The pressures of the arteries and veins in close proximity to the stenosis were analyzed to understand the system. Figure 5 A shows the pressure in the right internal carotid artery as the stenosis increased in size, which is indicated by the blue dot in figure 5 B. It also shows the pressures in the r. innominate, r. subclavian, and aortic arch A, which are up stream of the stenosis. Figure 5 C is the same graph, but focused on the r. innominate, r. subclavian, and aortic arch A to have a clear visualization of the increased pressure. Figure 6 shows the pressures in the veins that are equidistant from the point of the stenosis. In figure 5 A, the r. internal carotid artery pressure was measured after the stenosis. It started to decrease significantly around 65% and tapers off at 5%. It initially had a pressure of 118 mmhg, and it decreased to 4 mmhg. The pressure was reduced by 50% at 85% stenosis. The three other arterial points had a slight increase in pressure due to the stenosis, which can be seen in figure 5 C. In figure 6, the venous points behaved in a similar fashion, but the r. internal carotid started at

8 Naskrent mmhg and then decreased to 4.3 mmhg along with the r. internal carotid vein. This is because the r. internal carotid vein and the innominate vein are adjacent to each other. The r. subclavain vein and the superior vena cava only decreased slightly since they are farther away from the stenosis. Figure 6. This figure models the venous pressure effects of carotid artery disease by stenosising the right internal carotid artery. It shows the change in pressures for points in the veins that are in close proximity to the stenosis, which is indicated by the blue dot in figure 5 B. Figure 7. This figure models the venous flow effect of a stroke by stenosising the right internal carotid artery. It shows the change in flow for the veins that are in close proximity to the stenosis, which is in the right internal carotid artery Flow Analysis The flow rate of the stenosed artery can be seen in figure 7. The right internal carotid artery, initially flowed at 0.38 L/min and decreased to 0 L/min at 95% stenosis. The flow rate started to substantially decrease around 67% stenosis, and it had a 50% reduction of flow at 85% stenosis. The points preceding the branch leading to the stenosis, right innominate and aortic arch A, have a very similar decreasing flow curve as the stenosed artery, however they are higher on the flow rate axis. The flow rate of the right subclavian artery increased slightly as the stenosis increased and reached the same flow rate as the right innominate as they approached 100% stenosis. This was due to the two arteries becoming one when the flow through the right internal carotid artery was 0 L/min. 3.2 Coronary Artery Disease Pressure Analysis A. B. C. Figure 8. These figures model the arterial pressure effects of coronary artery disease by stenosising the coronary artery. A. This graph shows the change in pressures for points in the arteries that are in close proximity to the stenosis, which is indicated by the blue dot. B. The arterial network pressure figure shows the points that were being observed during the stensosis. C. This is the same graph as A without the artery point directly preceding the stenosis, which gives added detail to the other three points.

9 Naskrent 7 This system models the conditions that are observed in coronary artery disease. Figure 8 A shows the pressure in the right internal carotid artery as the stenosis increased in size, which are indicated by the blue dot in figure 8 B. It also shows the pressure in the thoracic aorta 2, the thoracic aorta 3, and the ascending aorta, which were up stream of the stenosis. Figure 8 C is the same graph, but focused on the thoracic aorta 2, the thoracic aorta 3, and the ascending aorta. Figure 9 shows the pressures in the veins that are equidistant from the point of the stenosis. In the model, the coronary artery was stenosed. The pressures of the arteries and veins in close proximity to the stenosis were analyzed to understand the system. The pressure in the coronary artery starts to increase significantly around 50% and tapers off at 95% as shown in figure 8 A. It initially had a pressure of 108 mmhg, and it increased to 119 mmhg, which is very close to the highest pressure in the system, 120 mmhg. The pressure was increased by 50% at 75% stenosis. The three other arterial points have a slight increase in pressure due to the stenosis, which can be seen in figure 7 C. In figure 9, the venous points all decrease. The coronary vein started out at 16 mmhg and then decreased to 4.4 mmhg along with the coronary sinus 2 vein. This was because the coronary vein and coronary sinus vein are adjacent to each other in the model. The coronary sinus 3 vein and the superior vena cava vein only decreased slightly since they are farther away from the stenosis. Figure 9. This figure models the venous pressure effects of coronary artery disease by stenosising the coronary artery. It shows the change in pressures for points in the veins that are in close proximity to the stenosis, which is indicated by the blue dot in figure 8 B. Figure 10. This figure models the venous flow effect of coronary artery disease by stenosising the coronary artery. It shows the change in flow for the veins that are in close proximity to the stenosis, which is in the coronary artery Flow Analysis The flow rate of the stenosed artery can be seen in figure 10. The coronary artery initially flowed at 0.7 L/min and decreased to 0 L/min at 90% stenosis. The flow rate started to substantially decrease around 55% stenosis, and it had a 50% reduction of flow at 69% stenosis. The points preceding the branch leading to the stenosis, ascending aorta and thoracic aorta A 2, have a very similar decreasing flow curve as the stenosed artery, however they are higher on the flow rate axis. The flow rate of the thoracic aorta A 3 artery increased slightly as the stenosis increased and reached the same flow rate as the right innominate as they approached 100% stenosis. This was due to the two arteries becoming one when the flow through the coronary artery was 0 L/min. 3.3 Renal Artery Disease Pressure Analysis This system models the conditions that are observed in renal artery disease. Figure 11 A shows the pressure in the right renal artery as the stenosis increased in size, which is indicated by the blue dot in figure 11 B. It also shows the abdominal aorta 2 artery, the left renal artery, and the abdominal aorta artery, which are up stream of the stenosis. Figure 11 C is the same graph, but focused on the abdominal aorta 2 artery, the left renal artery, and the abdominal aorta artery. Figure 11 shows the pressures in the veins that are equidistant from the point of the stenosis.

10 Naskrent 8 A. B. C. Figure 11. These figures model the arterial pressure effects of renal artery disease by stenosising the right renal artery. A. This graph shows the change in pressures for points in the arteries that were in close proximity to the stenosis, which is indicated by the blue dot. B. The arterial network pressure figure shows the points that are being observed during the stensosis. C. This is the same graph as A without the artery point directly preceding the stenosis, which gives added detail to the other three points. In the model, the right renal artery is stenosed to simulate a renal artery disease. The pressures of the arteries and veins in close proximity to the stenosis were analyzed to understand the system. The pressure in the right renal artery starts to decrease significantly around 50% and tapers off at 90% as shown in figure 11 A. It initially had a pressure of 110 mmhg, and it decreased to 10 mmhg. The pressure was reduced by 50% at 72% stenosis. The three other arterial point have a slight increase in pressure due to the stenosis, which can be seen in figure 11 C. In figure 12, the venous pressures behaved in a similar fashion, but the pressure in the right renal vein started at 25 mmhg and then decreased to 10 mmhg along with the right abdominal aorta 2 vein. This was because the right renal vein and the abdominal aorta 2 vein are adjacent to each other. The left renal vein and the abdominal aorta vein only decreased slightly since they are farther away from the stenosis. Figure 12. This figure models the venous pressure effects of renal artery disease by stenosising the right renal artery. It shows the change in pressures for points in the veins that are in close proximity to the stenosis, which is indicated by the blue dot in figure 11 B. Figure 13 This figure models the venous flow effect of renal artery disease by stenosising the right renal artery. It shows the change in flow for the veins that are in close proximity to the stenosis, which is in the right renal artery Flow Analysis The flow rate of the stenosed artery can be seen in figure 13. The coronary artery, initially flowed at 0.7 L/min and decreased to 0 L/min at 90% stenosis. The flow rate started to substantially decrease around 55% stenosis, and it had a 50% reduction of flow at 69% stenosis. The arteries preceding the branch leading to the stenosis, ascending aorta and thoracic aorta A 2,

11 Naskrent 9 have a very similar decreasing flow curve as the stenosed artery, however they are higher on the flow rate axis. The flow rate of the thoracic aorta A 3 artery increased slightly as the stenosis increased and reached the same flow rate as the right innominate as they approached 100% stenosis. This was due to the two arteries becoming one when the flow through the coronary artery was 0 L/min. 3.4 Peripheral Artery Disease Pressure Analysis A. B. C. Figure 14. These figures model the arterial pressure effects of peripheral artery disease by stenosising the right deep femoral artery. A. This graph shows the change in pressures for points in the arteries that are in close proximity to the stenosis, which is indicated by the blue dot. B. The arterial network pressure figure shows the points that were being observed during the stensosis. C. This is the same graph as A without the artery point directly preceding the stenosis, which gives added detail to the other three points. This system models the conditions that are observed in peripheral artery disease. Figure 14 A shows the pressure in the right deep femoral artery as the stenosis increased in size, which is indicated by the blue dot in figure 15 B. It also shows the right external iliac artery, the right femoral artery, and the right internal iliac artery, which are up stream of the stenosis. Figure 14 C is the same graph, but focused on the right external iliac artery, the right femoral artery, and the right internal iliac artery to have a clear visualization of the increased pressure. Figure 15 shows the pressures in the veins that are equidistant from the point of the stenosis. In the model, the right deep femoral artery was stenosed to simulate a peripheral artery disease. The pressures of the arteries and veins in close proximity to the stenosis were analyzed to understand the system. The pressure in the right deep femoral artery starts to decrease significantly around 58% and tapers off at 88% as shown in figure 14 A. It initially had a pressure of 101 mmhg, and it decreased to 17 mmhg. The pressure was reduced by 50% at 75% stenosis. The three other arterial points have a slight increase in pressure due to the stenosis, which can be seen in figure 14 C. In figure 15, the venous points behaved in a similar fashion, but the right deep femoral vein started at 38 mmhg and then decreased to 17 mmhg along with the femoral vein and the internal iliac vein. This is because the right deep femoral vein, the right femoral vein, and the external iliac vein are adjacent to each other. Only the internal iliac vein decreased slightly since it is farther away from the stenosis Flow Analysis The flow rate of the stenosed artery can be seen in figure 15. The right deep femoral artery initially flowed at 0.1 L/min and decreased to 0 L/min at 94% stenosis. The flow rate started to substantially decrease around 55% stenosis, and it had a 50% reduction of flow at 79% stenosis. The right external iliac is upstream of the stenosis, and it had a very similar decreasing flow curve as the stenosed artery, however it was higher on the flow rate axis. The flow rate of the right femoral artery increased slightly as the stenosis increased and reached the same flow rate as the right deep femoral approached 100% stenosis. This was due to the two arteries becoming one when the flow through the right internal carotid artery was 0 L/min.

12 Naskrent 10 Figure 15. This figure models the venous pressure effects of peripheral artery disease by stenosising the right deep femoral artery. It shows the change in pressures for points in the veins that are in close proximity to the stenosis, which is indicated by the blue dot in figure 14 B. Figure 16. A. This figure models the venous flow effect of peripheral artery disease by stenosising the right deep femoral artery. It shows the change in flow for the veins that are in close proximity to the stenosis, which is the right deep femoral artery. 3.5 Venous vs. Arterial Flows A. B. C. D. Figure 17. These figures show the venous and arterial flows rates for the four models. The arterial flows are in the same locations as the arteries, but they are down stream of the stenosis. A. It simulates the conditions of carotid artery disease by stenosing the right internal carotid artery. B. It simulates the conditions of coronary artery disease by stenosing the coronary artery. C. It simulates the conditions of renal artery disease by stenosing the right renal artery. D. It simulates the conditions of peripheral arterial disease by stenosing the right deep femoral artery.

13 Naskrent 11 When the pressure of the four systems were analyzed, the arteries and veins were found to have different pressures at the parallel locations of the network. This was not the case for the flows. Figure 17 shows the eight flows that correspond to the arterial and venous flows for each of the four systems. It can be seen that for each graph only four lines are readily apparent. This was because the four flows of the arterial system mimic the venous flows exactly. Since the limiting factor of the system was the capillaries, the flow through the arteries must be the same as the corresponding veins, because the venous system was created by copying the arterial system. This is presented in sections 3.4 and proven by the conservation of momentum in equation 4. This means that there was no reason to study the flow of the venous system, because it will give the same results as the arterial system. If the human vasculature was refined to more accurately model the true venous system, some difference may have been seen, but this level of detail was not useful for the purpose of understanding the stenosis. 4. Discussion By creating a model to simulate the effects of a stenosis in the internal carotid, proximal coronary, renal, and deep femoral arteries, it gave insight into the effects of a carotid artery disease, coronary artery disease, renal artery disease, and peripheral artery disease. The model was able to predict how the circulation in the body was changed when one of these arterial pathways were blocked. In the carotid artery disease, the pressure begins to drop significantly at 65% and the flow rate begins to drop significantly at 67%. In the coronary artery disease, the pressure begins to drop significantly at 50% and the flow rate begins to drop significantly at 55%. In the renal artery disease, the pressure begins to drop significantly at 50% and the flow rate begins to drop significantly at 55%. In the peripheral artery disease, the pressure begins to drop significantly at 55% and the flow rate begins to drop significantly at 58%. In the four systems, the pressure after the stenosed artery started to decrease substantially between 50% and 65% stenosis, and the flow rate was reduced by 50% between 72% and 85% stenosis. The pressure directly before the occluded arteries equaled the pressures after. The points in close proximity but not directly next to the stenosis did not have a significant pressure change. The flow rates of the four systems behaved in a similar manner. They started to decrease considerably between 50% and 67% stenosis, and they were reduced by 50% between 68% and 85% stenosis. The flow rate of the arteries up stream of the stenosis almost decreased as much as the stenosed artery, but the branches above the stenosis only had a slight increase in their flows. Lastly, the flow rate of the arteries and the veins were the same due to mirroring effects. Doctors will normally surgically remove a stenosis when the size of the artery is reduced by 50% or more [19, 20]. It was readily apparent why it is not necessary to surgically remove the debris until it has reached a 50% reduction in size, after analyzing the model. This study showed that an artery can be significantly impaired before any effects on the body become dangerous or life threatening.

14 Naskrent Appendix Human Arterial Network Information Flow # Flow Name Proximal Radius (cm) [21] distal radius (cm) [21] Avg. Diameter (m) Initial End point Resistance (Ns/m^5) [21] 1 R. anterior tibia E+09 2 R. posterior tibia E+09 3 L. posterior tibia E+09 4 L. anterior tibia E+09 5 L. posterior tibia L. femoral L. femoral L. femoral L. deep femoral E L. external iliac L. internal iliac E R. femoral R. femoral R. deep femoral R. deep femoral E R. external iliac R. internal iliac E R. common iliac L. common iliac Abdominal aorta E Inferior mesenteric E Abdominal aorta D Abdominal aorta C R. renal E superior mesenteric superior mesenteric E L. renal E Abdominal aorta B Gastric E abdominal aorta A Thoracic aorta B Celiac A Hepatic E Celiac B E Splenic E L.. Innominate R. Innominate R. internal carotid R. internal carotid E R. subclavian A R. external carotid R. external carotid R. external carotid R. external carotid E L. external carotid L. external carotid L. external carotid E L. subclavian A L. internal carotid E L. subclavian B L. subclavian B L. subclavian B L. subclavian B L. ulnar A L. ulnar B E+09

15 Naskrent L. interosseous L. interosseous E L. Radial L. Radial E R. subclavian B R. subclavian B R. subclavian B R. ulnar A R. Radial R. interosseous E R. ulnar B E R. Radial E Thoracic aorta B Thoracic aorta B Ascending aorta (start) Aortic Arch A Thoracic aorta A Thoracic aorta A Thoracic aorta A Thoracic aorta A Thoracic aorta A Table 1. Each artery in the model is documented in the human arterial network information table. It provides the name of the arteries and the corresponding number. It also has the proximal and distal radius of the blood vessels, which is used to calculate the average diameter. The diameter is used in the model to find the resistance through each artery. The end points of the arterial network are given along with their resistances, which simulate the capillaries of the system. Flow # Flow Location Capillaries Information End Point Artery's End point Veins Capillary Resistance (mmhg*min/l) [2] Diameter (mm) R. Foot R. Foot L. Foot L. Foot L. Thigh L. Pelvis R. Thigh R. Pelvis Large Intestine R. Kidneys Intestine L. Kidneys Stomach Liver Abdominal Cavity Spleen R. Brain R. Head L. Head L. Brain R. Hand R. Hand R. Hand L. Hand L. Hand L. Hand Intercostal Table 2. The table gives the specific information regarding the added capillaries, which connect the arteries and veins. The flow location, end points, diameter, and resistance are essential in understanding the model.

16 Naskrent References [1] R. Delewi, H. Yang and J. Kastelein, "Textbook of Cardiology," in Atherosclerosis, [2] University Hospital, "Stroke Statistics," [Online]. Available: [Accessed 12 November 2013]. [3] Mayo Clinic Staff, "Carotid artery disease," Mayo Clinic, 13 July [Online]. Available: [Accessed 29 November 2013]. [4] T. Gaziano, A. Bitton, S. Anand, S. Abrahams-Gessel and A. Murphy, "Growing Epidemic of Coronary Heart Disease in Low- and Middle-Income Countries," Curr Probl Cardiol, vol. 35, no. 2, pp , [5] Mayo Clinic Staff, "Coronary artery disease," Mayo Clinic, [Online]. Available: [Accessed 13 November 2013]. [6] National Kidney and Urologic Diseases, "Kidney Disease Statistics for the United States," National Institutes of Health, vol. 12, no. 3895, [7] Mayo Clinic Staff, "Renal artery stenosis," Mayo Clinic, [Online]. Available: [Accessed 13 November 2013]. [8] A. MA, H. E and D. JO, "Ethnic-specific prevalence of peripheral arterial disease in the United States," American Journal of Preventive Medicine, vol. 32, pp , [9] Mayo Clinic Staff, "Peripheral artery disease (PAD)," Mayo Clinic, [Online]. Available: [Accessed 13 November 2013]. [10] Wikipedia, "Atherosclerosis," 4 November [Online]. Available: [Accessed 4 November 2013]. [11] A. Maton and C. W. M. S. J. M. Q. W. D. L. J. D. W. Roshan L. Jean Hopkins, Human Biology and Health, Englewood Cliffs: Prentice Hall, [12] American Heart Association, "Atherosclerosis," 1 May [Online]. Available: 64_Article.jsp. [Accessed 28 October 2013]. [13] G. Truskey, F. Yuan and D. Katz, Transport Phenomena in Biological Systems, New Jersey: Pearson Education, [14] A. Linninger, "BioE Biological Systems Analysis," in Hunam Arterial Tree, Chicago, [15] M. E. Klingensmith, L. E. Chen and S. C. Glasgow, The Washington manual of surgery, Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins, [16] G. Pesola, H. Pesola, M. Nelson and R. Westfal, "The normal difference in bilateral indirect blood pressure recordings in normotensive individuals," The American Journal of Emergency Medicine, vol. 19, no. 1, pp , [17] M. Bernhard, A. Gruentzig, J. Hollman, T. Ischinger and J. Bradford, Does Length or Eccentricity of Coronary Stenoses Influence the Outcome of Transluminal Dilatation, Dallas: American Heart Association, 1983.

17 [18] D. Elad and E. Shmuel, STANDARD HANDBOOK OF BIOMEDICAL ENGINEERING AND DESIGN, McGraw-Hill, [19] Hational Hear, Lung, and Blood Institute, "How Is Carotid Artery Disease Treated?," National Institutes of Health, 1 November [Online]. Available: [Accessed 14 November 2013]. Naskrent 15 [20] Johns Hopkins Medicine, "Carotid Artery Disease," Johns Hopkins Medicine, [Online]. Available: 248/. [Accessed 14 November 2013]. [21] N. Stergiopulos, D. F. Yound and T. R. Rogge, "Computer Simulation of Arterial Flow With Applications to Arterial and Aortic Stenoses," Biomechanics, vol. 25, no. 12, pp , 1992.