IN DEGREE PROJECT TECHNOLOGY,FIRST CYCLE, 15 CREDITS

, STOCKHOLM SWEDEN 2018

A Study of Vascular Plaque in the Carotid

KRISTIN SALMI

VICTOR THUN

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

INOM EXAMENSARBETE TEKNIK,GRUNDNIVÅ, 15 HP

, STOCKHOLM SVERIGE 2018

A Study of Vascular Plaque in the Carotid

KRISTIN SALMI

VICTOR THUN

KTHSKOLAN FÖR TEKNIKVETENSKAP

Abstract

In 2015, cardiovascular diseases caused the death of 17.7 million people - 31 % of all deaths

globally - making it deadlier than any other cause.

Cardiovascular diseases often result from Arteriosclerosis, a disease where plaque builds up and

clogs arteries. As this occurs in the carotid arteries and the plaque material is subject to forces

caused by blood pressure and flow, the stress that occurs within could lead to rupture of the

material. The ruptured particles could then travel to the brain and cause a stroke.

To identify and prevent these strokes, computed tomography and ultrasound scans are used

today. With the help of such examination, it’s possible to see how much of the artery that is

clogged and decide whether or not to operate on the patient. Another way of analyzing the

plaques vulnerability of rupture is with the help of biomechanics. Based on the geometry and

material composition of the plaque, and using a finite element analysis, the stresses and risk of

rupture can be estimated.

This thesis invetigates how different plaque geometries, with focus on plaque length and cross-

section area coverage, affects the resulting stress in the material. The study was carried out in

a total of 12 different plaque models. The models were based on two different lengths, 3 mm

and 10 mm and four different cross-sectional coverages.

With computed tomography as reference, models of the carotids and plaque were built using

a computer-aided design program. The models were imported to the finite element program

COMSOL and analyzed using fluid-mechanical and solid-mechanical simulations. The simula-

tions were executed on non-linear solid-state simulations and linear-elastic tissue models and

the Newtonian fluid assumption.

based on the investigated plaque functionalities, this study found that the stress in the plaque

tissue shows a peak at 50-60% plaque coverage of the cross-sectional area, such that the risk of

rupture seems to be the highest at this area coverage.

i

Sammanfattning

2015 var hjärt- och kärlsjukdomar orsaken till 17,7 miljoner människors död - 31 % av alla

dödsfall världen över - vilket gör det till världens vanligaste dödsorsak.

Hjärt- och kärlsjukdomar resulterar inte sällan från Ateroskleros, en sjukdom där plack byggs

upp längs blodkärlsväggar och är än mer farlig om det sker i halspulsådern. När plackmaterialet

utsätts för krafter från blostrycket och flödet uppkommer spänningar som kan leda till bristning

i materialet. Partiklar färdas då till hjärnan vilket kan orsaka stora skador.

I dagsläget används datortomografi för att identifiera och bedöma denna sjukdom. Det är hur

stor del av artären som är blockerad av plack som avgör huruvida patienten bör opereras eller

inte. Ett annat sätt att avgöra detta är med hjälp av biomekanik. Då står plackets geometri

och materialkomposition till grund för analysen och de uppkomna spänningarna och risken för

bristning kan beräknas med ett finita elementprogram.

Denna uppsats behandlar olika plackgeometriers påverkan på.de resulterande spänningarna

och därmed också.risken för bristning. Studien fokuserar på.skillnader i plackens längd och

tvärsnittsarea där totalt 12 stycken fall studeras. Modellerna är gjorda i två olika längder, 3

mm och 10 mm, med varierande grad av plackens tvärsnittsarea.

Med hjälp av datortomografi som referens modellerads halspulsådern och placken i ett design-

och konstruktionsprogram. Dessa modeller importerades sedan till ett finita elementprogram

och analyseras med hjälp av fluid- och struktursimuleringar. Simuleringarna gjordes med linjärt

elastiska material och en ickelinjär stationär studie.

Slutsatsen av denna studie är att vid 50-60 % täckning av tvärsnittsarean nås en maximipunkt

då.spänningen är som störst, vilket också.innebär högst risk för att placken brister.

ii

Innehåll

Abstract i

1 Introduction 1

1.1 Arteriosclerosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Carotid Artery Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Risk assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Method 4

2.1 Study and Modeling of the Carotid . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Modeling of the Plaque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Finite element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Result and Conclusion 12

3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Source of Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

iii

3.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Appendices 18

A.1 Coordinates of the centerline of the carotid . . . . . . . . . . . . . . . . . . . . . 19

A.2 Different models of plaque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

A.3 Results of COMSOL-studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

A.4 Mesh of vessel and plaque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

iv

Kapitel 1

Introduction

1.1 Arteriosclerosis

Arteriosclerosis is a progressive cardiovascular disease where plaque builds up inside the arteries

over time. The plaque consists of fat, cholesterol and calcium. As the disease progresses it leads

to narrow blood vessels with thick and stiff vessel walls. The narrow blood vessel restricts the

flow of blood to organs and other parts of the body. See Figure 1.1

There isn’t an exact known cause for the disease. One accepted theory is that the build-up

of plaque begins when the arteries are damaged. There are some risk factors that are known

to cause damage to the arteries and these are, to mention some, high blood pressure, high

cholesterol and smoking. [1]

Arteriosclerosis is the main reason behind heart attacks and strokes. It is consequently the

leading cause of deaths and disabilities in the developed world. Preventing and treating cardi-

ovascular diseases requires the highest share of health care costs in the western world.

1

2 Chapter 1. Introduction

Figur 1.1: Cartoid vessel with plaque in yellow that restricts the blood flow through the anas-tomosis. [2]

1.2 Carotid Artery Disease

Arteriosclerosis that affects the carotid arteries is called Carotid Artery Disease, and might

have fatal consequences for the patient as it could lead to a stroke. The great risk lies in when

the plaque material ruptures, which causes parts of the plaque tissue to travel into the brain.

This has possibly fatal consequences if the tissue would get stuck, blocking the blood flow and

thus causing a stroke, as illustrated in Figure 1.2.

1.3 Risk assessment

Today, there is no good way in identifying plaque about to rupture. This might lead to un-

necessary operations that are costly and a risk for the patients. In order to decide when to

operate or not the plaque is analyzed. If the plaque covers more than 50% of the cross-sectional

area it is highly recommended that an operation is done. This is a method that excludes many

1.4. Objectives 3

Figur 1.2: This shows how the brain and carotid artery relates to one another, it also depictsa plaque rupture with the following broken-free piece of plaque that travels to the brain andleads to blocked blood vessels, in other words, a stroke. [3]

variables such as the plaque material composition and the geometry of the vessel. With many

unknown variables it is nearly impossible to individualize a risk assessment. This might lead to

unnecessary operations of patient not in the risk of having a stroke.

1.4 Objectives

The objective of this thesis was to create a biomechanical model of the Carotid artery and the

built-up plaque in order to achieve a better understanding for which plaque build-up is at risk

of rupture. To do this, multiple plaque models were studied in order to explore what plaque

shapes lead to high stresses.

Kapitel 2

Method

2.1 Study and Modeling of the Carotid

In order to conduct a study on the Carotid, a model was created based on Computed Tomograp-

hy scans (CT-scans) received from Karolinska Institutet. CT is a common imaging method used

in a variety of diagnostic purposes. The images produced by this method are cross-sectional,

slices", of the body [4]. In total, three sets of CT-images from male patients were received and

studied.

The pictures were analyzed in a medical image viewer [5] showing cross sectional images of the

head. Figure 2.1 shows one of these images. As reference, the teeth are visible in the top of the

picture and the throat is seen as the black area in the middle. The Carotids are in the same

picture marked with red arrows, one on each side. It was put into a coordinate system, after

which the coordinates of the center line could be read from different points on the z-axis.

As seen in Figure 2.1, the centerline was pointed out by its z-, y-, and x-coordinates. The same

was done for several scans around the branching in order to plot a approximation of the Carotid

in 3D. All coordinates can be found in appendix A.1. The results were plotted in MATLAB

(Figure 2.2), with the red line representing the smaller branch. Note that the measures are

given in millimeters and the coordinate system is given according to the CT-scans.

4

2.1. Study and Modeling of the Carotid 5

Figur 2.1: A CT-scan showing a cross-sectional image of the head. In the picture, the teethcan be seen in the top and the throat in the middle as a black area. The carotids, one on eachside, are marked with arrows. The figure illustrates the measuring method used for pointingout the z-, x-, and y-coordinates of one of the carotid’s centerline. The same is done for severalcross-sectional images in order to create a 3D-image of the center line.

Figur 2.2: With all centerline coordinates collected from CT-scans, a plot for each patientexamined was created in MATLAB. The units are in millimeters and based on the CT-scansfrom where they were taken. The red line represents the smaller branch, after bifurcation. Thesecenterlines will later be used to create a 3D-model of the carotid.

6 Chapter 2. Method

Figur 2.3: The pictures shows the lumen i.e. the domain where the blood flows. The 3D-models ofthe carotid were created using the CAD software Solid Edge. For this, the centerlines measuredfrom CT-scans were used, and in addition diameters of the inlet and outlets, measured fromCT-scans as well. The lines that are visible in the models represents the centerlines

The next step was to turn the centerline into a 3D model using the computed aided design

(CAD) software Solid Edge [6]. For this, the centerlines from Figure 2.2 were used, as well as

the diameters of the inlet and the two outlets. The inlet and outlets were measured at the first

and last cross-sectional view that were used to plot the centerline. The resulting 3D-models

represent the lumen and are shown in Figure 2.3. In addition, a vessel wall was created outside

of the lumen. The vessel wall was assumed to be solid and have a constant thickness of 1.00

mm [7].

The CT-images from case No 1 (Male, 1937) had the best images for reconstruction, hence the

resulting model from this case was used in further analysis.

2.2 Modeling of the Plaque

As CT is insensitive to soft tissue, the plaque was approximated. It was created with CAD by

sweeping the area between a circle and the vessel wall in the cross-sectional view. The resulting

plaque model uses the same centerline as the corresponding artery (No 1, 1937). The plaque

model was made based on percentage coverage of the lumen’s cross-sectional area. The cross-

sectional area of the lumen was calculated as Atot = πr2tot with measured radius rtot = 3.8 mm,

2.3. Finite element analysis 7

Figur 2.4: 3D-models made in Solid-Edge of plaque covering 50 % of cross-sectional lumen areaand with the respective lengths of 3 mm and 10 mm

Atot = 45.36 mm2. The area the plaque covered in the cross-sectional view was calculated with

Aplaque = Atotα, where α corresponds to the percentage of plaque in the cross-sectional view.

Different geometries made with corresponding α that was to cover 25 % (α = 0.25), 50 %

(α = 0.50), 75 % (α = 0.75) and 90 % (α = 0.90). Aplaque was then used to get the radius of

the circle rplaque used for sweeping the plaque model rplaque =√

Aplaque

π. In order to explore the

impact of the plaque length on the stresses, the models were made in two different sizes, 3 mm

and 10 mm long, as seen in Figure 2.4. In total eight resulting plaque models were made and

can be found in appendix A.2.

After receiving results from the FEA, it was concluded that the highest stresses were obtained

for the blood vessels with 50 % plaque coverage. To analyze this further, two more cases of 40%

and 60 % stenosis cross-sectional area coverage of the lumen were examined.

2.3 Finite element analysis

The objective of the finite element analysis (FEA) was to explore how different plaque geo-

metries affected the stresses in the plaque material. Especially interesting are the peak stress

levels that would indicate when the plaque is most likely to rupture.

8 Chapter 2. Method

The CAD 3D-models were imported to the FEA software COMSOL Multiphysics [8] and stu-

died. First, a CFD (Computational Fluid Dynamics) was conducted to explore how the blood

pressure develops in the vessel. After this, structure studies were done to find the stresses in

the plaque material.

Fluid simulation

The objective of the CFD simulation was to find the pressure in the vessel. The Reynolds

number Re was calculated as Re = ρbloodubloodDµblood

≈ 1500, where the parameters ρblood, ublood and

µblood are given in Table 2.1. The carotid’s mean diameter was measured to 7.6 mm from CT-

scans as described in section 2.1. As flow can be assumed laminar for Re < 2000 [9], the CFD

was made with laminar flow.

The CFD simulation was done on a rigid wall CFD Model with a no slip condition. First, the

model of the carotid (No 1, Male (1937)) was imported to COMSOL from Solid Edge using

the function Livelink. In order to minimize points of singularity, the function Remove Details

was used on the geometry. The material blood was added with the density ρblood and dynamic

viscosity µblood. Laminar flow physics was added with the boundary conditions of inlet, outlet

and wall as shown in Figure 2.5. Inlet blood pressure as well as outlet blood velocity is defined

in table 2.1. The simulation was done with a rigid wall CFD Model

Meshing was done with COMSOL’s inbuilt physics-controlled mesh, using normal element size.

This tool generates a linear tetrahedral mesh.

A stationary study was carried out and the distribution of the hydrostatic pressure can be seen

in Figure 3.1 in the results section. Figure 3.1 shows that because of the pressure distribution

the pressure P = 20 kPa can be assumed to be constant throughout the vessel.

2.3. Finite element analysis 9

Structural Analysis

Next, using the pressures received from the CFD study, a solid mechanics simulation was done

calculating the stresses in the plaque. The results from the CFD show that the assumption of

a constant pressure P = 20 kPa throughout the vessel can be used in further studies. Because

of the complex geometry that our model represents and our limited resources with regards to

computing power and time, the study on the plaque in the solid mechanics simulation was done

only taking into account the vessel geometry and plaque that occurs proximal to the bifurcation.

This simplification saves a lot of resources because most of the complex geometry exists distal

of the bifurcation.

The specific model that was to be studied was imported from Solid Edge using Livelink. The

command Form Composite Faces was then used on the geometry in order to remove all the

points of singularity that our complex geometry created. Linear elastic material was used for

both the vessel wall and the plaque; parameters are defined in table 2.1.

The solid mechanics study was done by simulating the constant pressure acting in the lumen,

this is done by inserting a constant boundary load on all the relevant faces, as seen in Figure

2.6a. The model is then fixated with Fixed constraints at the top and bottom faces of the vessel

as seen in Figure 2.6b.

The vessel and plaque was then meshed in a linear tetrahedral mesh predefined in COMSOL

as "extra fine", as shown in Appendix A.4.

A nonlinear stationary study was then carried out plotting the maximal von Mises stresses with

results as can be seen in Appendix A.3. The solution time for the 10 mm long plaque model

with 50% lumen coverage was 75 seconds.

10 Chapter 2. Method

Figur 2.5: The 3D-model of carotid (No 1, Male (1937)) imported into COMSOL with boundaryconditions set for CFD simulation. Inlet marked in purple in the left figure and outlet markedin purple in the right figure. The wall condition is set for all the remaining surfaces.

(a) Boundary loads (b) Fixed constraints

Figur 2.6: The 3D-model imported to the software COMSOL Multiphysics and set up foranalysis. The pictures show a model of 10 mm plaque covering 50 % of the cross-section areaand its vessel wall. In (a), the boundary loads i.e. pressure from the blood flow, are applied ondomains in blue. In (b), the blue domains show fixed constraints. These are introduced to makeup for the rest of the artery which prevents it from moving in the z-direction.

2.3. Finite element analysis 11

Tabell 2.1: Defined parameters for study in COMSOL Multiphysics. [10]

Density of blood ρblood 1060 kg/m3

Dynamic viscosity of blood µblood 0.005 Ns/m2

Inlet blood pressure 150 mmHgOutlet blood velocity 0.5 m/s

Young’s modulus plaque Eplaque 500 kPaPoisson’s ratio plaque νplaque 0.4

Density plaque ρplaque 1090 kg/m3

Young’s modulus vessel wall Ewall 1000 kPaPoisson’s ratio vessel wall νwall 0.4

Density vessel wall ρwall 960 kg/m3

Kapitel 3

Result and Conclusion

3.1 Results

This study has been carried out by first creating a 3D-model of a section of the blood vessel

with plaque of different geometries. The vessel was then analyzed in a CFD Simulation and

then in a quasi-static structure analysis.

Result of fluid simulation

The CFD that was used in order to find the pressure along the vessel from the blood flow,

resulted in pressures between 19.9 and 20.0 kPa. The resulting von Mises stress distribution

can be seen in Figure 3.1. As there’s a small diametrical change through the vessel the near

constant pressure is not surprising. The resulting flow velocity through the vessel can be seen in

Figure 3.2. The resulting velocity of zero at the vessel wall is an effect from the no-slip condition

used in the study.

12

3.1. Results 13

Figur 3.1: Units in kPa. The result of fluid simulation in order to find the pressure distributionin the plaque caused by the blood flow. As the pressure varies very little through the partexamined, the pressure can be assumed to be constant. This was used in the later studies ofthe blood pressure’s impact on the structure.

Figur 3.2: Units in m/s. The resulting flow velocity from the CFD plotted in 15 vertical slices.The resulting velocity of zero at the vessel wall is an effect from the no-slip condition used inthe study.

14 Chapter 3. Result and Conclusion

Tabell 3.1: The results of the FEM-analysis. The table shows the retrieved von Mises stressesdepending on the geometry.

Plaque length [mm] 3 10Plaque coverage [%] 25 40 50 60 75 90 25 40 50 60 75 90

Maximum stress [kPa] 208 185 182 239 153 138 246 287 369 217 216 306

Figur 3.3: The von Mises stress received from the FEA plotted as a function of the plaquecoverage. The green and red line show the results for 3 mm respectively 10 mm long plaque.The blue line shows the percentage for when surgery is carried out today. The results show thatstresses can be expected to be higher for plaque covering a longer section. They also show anincrease of stress at 50 - 60 % for both lengths.

Result of the structure analysis

The resulting peak von Mises stresses from the structure analyses on surfaces in the plaque can

be seen in table 3.1 as well as plotted in Figure 3.3. The results are presented in appendix A.3.

The stress peak occurs in the 10 mm long plaque model that is covering 50% of the lumen

diameter. In general, the 10 mm long plaque measured higher stresses than the 3 mm plaque.

As can be seen in Figure 3.3 both plaque geometries seem to reach a maximum stress value at

50 % respective 60 % lumen coverage.

3.2. Conclusions 15

3.2 Conclusions

The results show that the highest stresses in the plaque material are found in plaque with 50-

60% lumen cross-sectional area coverage. This founding is quite surprising, since we expected

the stresses to increase with increased plaque coverage. This result might be caused by the

geometry of the models studied and in particular the bending of the plaque.

Figure 3.4 illustrates two types of plaque geometries, one thin and one thicker, viewed in a

longitudinal section. The black marks show where the highest stresses were found. A thin

plaque tissue is easily bent which means that it doesn’t take up as much stress as a thicker one

would. A very thick plaque on the other hand is too stiff to bend and won’t cause the same

bending stress. At around 50 % plaque area coverage there is a peak stress, which means that

the plaque is thick enough to take up stress but not too rigid that it doesn’t bend.

The results from our study in Figure 3.3 doesn’t just only show that peak stresses occurs in

plaque with 50-60% lumen coverage but also that after the peak, the more plaque the less the

stress, this is especially evident in the 3 mm long plaque (green line). This suggests that surgery

done today at 50% coverage (blue line) is done when the risk of rupture is at the highest and

an operation is the most crucial in order to prevent a stroke.

3.3 Source of Error

The study that has been carried out in this thesis has shortcomings that needs to be taken into

consideration when reviewing the results.

• The blood model used in the study is a Newtonian viscosity model, when in fact blood is

non-Newtonian.

• In the CFD simulation the blood flow was considered steady state flow. In fact, blood

flow is pulsating.

16 Chapter 3. Result and Conclusion

Figur 3.4: An explanation of the tissue stresses in the plaque. The Figure shows an exampleof a blood vessel with little plaque (left) and one with a lot (right). The vessel is viewed in alongitudinal section with the red and yellow area representing the vessel wall respectively theplaque. The black marks show where the highest von Mises stresses were found.[11]

• The Plaque and wall models used in the study are linear models, when in fact they’re

both non-linear.

• The composition of the plaque material is approximated to be one homogeneous material,

when in fact it is a complex inhomogeneous material made up by cholesterol and calcium.

• The CT-scans provided was hard to decipher and the resulting geometry of the vessel is

a severe simplification.

• The geometry of the plaque is an approximation as in reality it is not evenly covering the

sides of the cross-sectional view of the vessel. The plaque usually builds up from one side

and the coverage is irregular with respect to the cross-sectional view.

• The boundary conditions in the study is a simplification that doesn’t represent reality.

In conclusion our model is a simplification and does only contain average properties of plaque

according to previous research. Our results should therefore be used as a motivation for further

research in the field.

3.4. Future Work 17

3.4 Future Work

Suggestions for future work based on this report is to do the analysis on different geometries

with more relevance to patients, since this seems to have significant impact on the stresses. The

plaque modeled in this work is evenly spread out on the walls, which is an approximation that

might be misleading as plaque rarely builds up in an even manner.

Further work would also be modeling of the plaque material more accurately. There are many

types of plaque material, such as soft and hard tissue as well as different matrix compositions,

with great variations from person to person. These variations all play a big role in the stress

levels and the risk of rupture.

From this study the progress of future work could go two ways according to us:

1. To individualize the data in order to create more patient specific models.

2. To make a more general model where an approximation of the geometry and composition

of the materials is made from patients.

Appendices

18

***** Coordinates of the Centerline no 1*****

Anonymous male 1937

-----------------------

z-led -70.052

y-led 54.7

x-led 82.5

-----------------------

z-led -68.804

y-led 54.7

x-led 82.5

-----------------------

z-led -66.932

y-led 55.2

x-led 83.0

-----------------------

z-led -64.748

y-led 55.7

x-led 83.5

-----------------------

z-led -62.252

y-led 56.6

x-led 82.0

-----------------------

z-led -57.26

y-led 58.1

x-led 77.1

-----------------------

z-led -54.14

y-led 59.1

x-led 75.7

-----------------------

z-led -52.258

y-led 60.1

x-led 75.2

-----------------------

*** Branching ***

-------------------------------------

z-led -51.02

y-led 60.1 y-led 60.1

x-led 75.7 x-led 75.7

-------------------------------------

z-led -50.084

y-led 60.1 y-led 59.6

x-led 75.2 x-led 79.1

-------------------------------------

z-led -49.148

y-led 60.5 y-led 59.6

x-led 75.2 x-led 79.6

-------------------------------------

z-led -47.588

y-led 61.0 y-led 60.1

x-led 75.2 x-led 79.6

-------------------------------------

z-led -45.716

y-led 60.5 y-led 60.1

x-led 73.7 x-led 79.1

A.1. Coordinates of the centerline of the carotid 19

A.1 Coordinates of the centerline of the carotid

***** Coordinates of the Centerline no 2*****

Anonymous male 1940

-----------------------

z-led 6.18

y-led 35.6

x-led 48.8

-----------------------

z-led 8.052

y-led 34.7

x-led 47.9

-----------------------

z-led 10.236

y-led 33.7

x-led 47.9

-----------------------

z-led 12.108

y-led 34.2

x-led 47.9

-----------------------

z-led 14.292

y-led 33.7

x-led 45.9

-----------------------

z-led 16.164

y-led 33.7

x-led 45.4

-----------------------

*** Branching ***

-------------------------------------

z-led 17.412

y-led 32.2 y-led 37.1

x-led 44.4 x-led 49.3

-------------------------------------

z-led 19.284

y-led 34.2 y-led 37.6

x-led 43.5 x-led 48.8

-------------------------------------

z-led 19.908

y-led 33.2 y-led 37.6

x-led 43.0 x-led 49.3

-------------------------------------

20

***** Coordinates of the Centerline no 3*****

Anonymous male 1950

-----------------------

z-led -284

y-led 49.6

x-led 61.7

-----------------------

z-led -281

y-led 50.5

x-led 61.7

-----------------------

z-led -279

y-led 51.0

x-led 60.4

-----------------------

z-led -276

y-led 50.5

x-led 59.9

-----------------------

z-led -273

y-led 51.9

x-led 59.9

-----------------------

z-led -270

y-led 52.8

x-led 59.0

-----------------------

z-led -268

y-led 53.7

x-led 59.9

-----------------------

z-led -267

y-led 52.8

x-led 59.0

-----------------------

z-led -266.5

y-led 53.2

x-led 59.0

-----------------------

*** Branching ***

-------------------------------------

z-led -264

y-led 53.2 y-led 55.9

x-led 58.1 x-led 63.1

-------------------------------------

z-led -262

y-led 52.3 y-led 57.7

x-led 56.8 x-led 63.1

-------------------------------------

z-led -259

y-led 52.8 y-led 58.1

x-led 56.8 x-led 64.9

-------------------------------------

z-led -256

y-led 53.2 y-led 59.5

x-led 56.4 x-led 65.3

-------------------------------------

z-led -254

y-led 53.2 y-led 59.5

x-led 57.3 x-led 57.3

-------------------------------------

A.1. Coordinates of the centerline of the carotid 21

25 %, 3 mm 50 %, 3 mm 75 %, 3 mm 90 %, 3 mm

25 %, 10 mm 50 %, 10 mm 75 %, 10 mm 90 %, 10 mm

22

A.2 Different models of plaque

25 %, 3 mm

25 %, 10 mm

40 %, 3 mm

40 %, 10 mm

50 %, 3 mm

50 %, 10 mm

60 %, 3 mm

60 %, 10 mm

75 %, 3 mm

75 %, 10 mm

90 %, 3 mm

90 %, 10 mm

A.3. Results of COMSOL-studies 23

A.3 Results of COMSOL-studies

24

A.4 Mesh of vessel and plaque

Artery and plaque proximal the bifurcation meshed in COMSOL. This figure shows a 5 mm

plaque with 50% lumen coverage. The mesh is predefined by COMSOL as "extra fine".

Litteraturförteckning

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https://www.nhlbi.nih.gov/health-topics/atherosclerosis#Diagnosis

[2] Sterling Care. Carotid Artery Problems. Picture retrieved 2018-07-13 from

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[PubMed]

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25

26 LITTERATURFÖRTECKNING

[10] Marieb, Elaine N.; Hoehn, Katja. The Cardiovascular System:Blood Vessels". Human ana-

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[11] Matt Penfold. Atherosclerosis. Picture retrieved 2018-07-13 from

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www.kth.se