Modeling the Propagation of Aortic Dissection

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Modeling the Propagation of Aortic Dissection

SELDA S H E R I F O V A

Master of Science Thesis in Medical Engineering Stockholm 2015

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This master thesis project was performed in collaboration with Institute of Biomechanics, TU Graz Supervisor at TU Graz: Prof. Gerhard A. Holzapfel Gerhard Sommer

Modeling the Propagation of Aortic Dissection Modellering av Fortplantningen av

Aortadissektion

S E L D A S H E R I F O V A

Master of Science Thesis in Medical Engineering Advanced level (second cycle), 30 credits Supervisor at KTH: Svein Kleiven Examiner: Dmitry Grishenkov School of Technology and Health TRITA-STH. EX 2015:003

Royal Institute of Technology KTH STH SE-141 86 Flemingsberg, Sweden http://www.kth.se/sth

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Abstract

Aortic dissection is a diseased condition of the aorta in which there is an initial tear to the intimal layer propagating in the radial direction initially then causing delamination of the arterial layers creating a false lumen. It is estimated to effect 30 individuals in one million per year. The dissection is fatal when it ruptures, and 90 % of the patients die within three months if not diagnosed.

This study presents the first steps towards modeling the propagation of aortic dissection.

The aorta was treated as a three-layered fiber-reinforced composite structure, and the Tsai- Wu failure criterion was employed to obtain the 3-D failure surface for the healthy and dissected human aortic media. To be able to obtain Tsai-Wu coefficients, uniaxial tensile tests in the axial, circumferential, and radial direction, and additionally in-plane (axial- circumferential plane) and out-of-plane shear tests in different orientations were performed on human aortic medias. To our knowledge the combination of applied tests and performing of out-of-plane shear tests on aortic tissues is novel.

The results showed that the aortic media was the weakest in radial direction under tensile loading. Furthermore, the media was much stronger under out-of-plane shear loading than under in-plane shear loading. In order to consider influences of stress coupling between axial and circumferential directions, an optimal specimen geometry was designed for biaxial tensile testing by the help of finite element analyses. A cruciform geometry with a reduced cross-section in the biaxially loaded zone was found to fit our purposes the best. The preparation protocol to achieve this geometry is currently under investigation.

For aortic tissues, all compressive strengths and some biaxial tensile strengths needed to be assumed since they are yet not possible to obtain from mechanical tests. Finally, failure surfaces described by the Tsai-Wu criterion were plotted in 2-D using the analyzed experimental data, with different assumptions in compressive and biaxial tensile strengths.

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Sammanfattning

Aortadissektion är ett sjukdomstillst˚and av aortan där det finns en antydan till en reva p˚a det innersta skiktet som utbreder sig i den radiella riktningen initialt, vilket därefter or- sakar att de olika arteriella skikten lossnar fr˚an varandra och skapar ett falskt h˚alrum.

Det uppskattas att det ˚aterfinns hos 30 personer p˚a en miljon individer per ˚ar. Dissektio- nen är dödlig när den spricker, och 90% av patienterna dör inom tre m˚anader om den inte diagnostiseras.

Denna studie presenterar de första stegen för att modellera spridningen av aortadissektion. Aortan behandlades som en treskiktad fiberförstärkt kompositstruktur, och Tsai-Wus brottkriterium användes för att erh˚alla en tredimensionell brottyta för de friska och dissek- erade mänskliga mellanlagren av aortan. För att kunna f˚a Tsai-Wus koefficienter, utfördes enaxlad dragh˚allfasthetsprovning i axiell, periferi och radiell riktning, och dessutom i planet (axial-periferiplanet) ur-planet i olika orienteringar p˚a mänskliga aortamellan- lager. Enligt v˚ar vetskap s˚a är kombinationen av tillämpade tester och genomföring av belastningstester av krafter p˚a aortavävnader ny.

Resultaten visade att aortalager var svagast i radiell riktning under dragbelastning. Dessu- tom var denna effekten mycket starkare i ur-planet belastningstester än i i-planet belastningstester. För att kunna undersöka kopplingen av belastningen mellan axiell- och periferi-riktning, s˚a var en optimal provgeometri designad för tv˚aaxlig dragprovning med hjälp av finita element analyser. En korsformad geometri med ett reducerat tvärsnitt i den tv˚aaxlade p˚afrestade zonen passade v˚ara syften bäst. Beredningsprotokollet för att n˚a denna geometri är för närvarande under utredning. För aortavävnader behöver alla tryckh˚allfasthetskrafter och n˚agra tv˚aaxiala dragh˚allfasthetskrafter antas eftersom de

ännu inte är möjliga att f˚a fr˚an mekaniska tester. Slutligen ritades brottytor beskrivna av Tsai-Wus kriterium i 2D med hjälp av analyserade experimentella data med olika antagan- den för tryck och den tv˚aaxiala dragh˚allfastheten.

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Acknowledgment

I would like to express my gratitude to my supervisors Prof. Gerhard A. Holzapfel, Dr.

Gerhard Sommer, and Prof. Svein Kleiven for their help and support, including their as- sistance for writing the reports during the time I have been working on this project. I would like to thank my colleagues from the Institute of Biomechanics of TU Graz, Justyna Niestrawska, Osman Gultekin, Andreas J. Schriefl, Kewei Li and others for the good dis- cussions we have had, their valuable inputs and their support as friends. I would also like to thank Franz Seiringer for performing the experiments needed for this study, and David Walk for the laboratory trainings. It has been a great pleasure to work with all of them.

Furthermore, I appreciate the help of Niklas Roos on the translation of the abstract into Swedish.

The greatest deal of thanks I owe to my family for the support they provided me at every stage of my life. I would also like to thank all of my friends that have been there for every- thing, bringing many laughters and tears into my life.

Finally, it should be recognized here that it would not be possible to conduct this work without the fundings from NIH (National Institutes of Health, USA) and the samples received from Graz University Hospital and School of Medicine at New York University.

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List of Tables

4.1. Material Parameters used in simulations . . . 24 5.1. Ultimate strengths obtained from uniaxial tension (σ_1t^u, σ^u_2t), direct tension

(σ_3tû), and shear tests (σ₄û, σ₅û, and σ₆û) . . . 47 A.1. A list of MMPs, TIMPs and their target structures . . . 69

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List of Figures

2.1. Structure of elastic arteries . . . 4

2.2. Elastic fiber assembly . . . 4

2.3. An illustration of the propagation of aortic dissection . . . 5

2.4. Classification of aortic dissection . . . 6

3.1. Principal directions of the lamina . . . 15

3.2. Change of failure surface with F₁₂for different P values . . . 17

4.1. Sketch of the uniaxial and direct tension test specimens . . . 20

4.2. Case I – Change of failure surface with different compressive strengths . . 21

4.3. Case IIA – Change of failure surface with different compressive strengths 22 4.4. Case IIB – Change of failure surface with different compressive strengths 22 4.5. Case III – Change of failure surface with different compressive strengths . 23 4.6. Dimensions of the main geometry, Geometry A . . . 25

4.7. Results of simulation set 1, σ₁ . . . 27

4.8. Results of simulation set 1, σ₂ . . . 28

4.9. Results of simulation set 2, σ1 . . . 29

4.10. Results of simulation set 2, σ2 . . . 30

4.11. Results of simulation set 1,2, and 3 for geometry E, σ₁and σ₂ . . . 31

4.12. Shear properties of aorta . . . 32

4.13. Schematic of the direct tension test and uniaxial tension test apparatus . . 34

4.14. Fixing of direct tension specimens . . . 35

4.15. Sketch of the in-plane shear test specimens . . . 36

4.16. Sketch of the different out-of-plane shear test specimens . . . 36

4.17. Sketch of the out-of-plane shear test specimens . . . 37

4.18. A sketch of the CNC mounting table . . . 40

4.19. Drawings of the specimen holders . . . 41

4.20. Sketch of the mounted specimen on the CNC table . . . 42

5.1. Results of uniaxial tensile rupture tests . . . 44

5.2. Results of direct tension tests . . . 44

5.3. Results of in-plane shear tests . . . 45

5.4. Results of out-of-plane shear tests . . . 46

5.5. Comparison of in-plane and out-of-plane shear test behavior . . . 46

5.6. Failure surface in 12- plane with different σ_c^uand F12 = 0 . . . 48

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XVI LIST OF FIGURES

5.9. Failure surface in 12- plane with different σ^u_c and F12 from the Tsai-Hahn equation . . . 49

5.10. Failure surface in 13- plane with different σ^u_c and F₁₃ from the Tsai-Hahn equation . . . 50

5.11. Failure surface in 23- plane with different σ^u_c and F23 from the Tsai-Hahn equation . . . 50

5.12. Failure surface in 12- plane with σ_c^u= 0.0001 kPa and F₁₂calculated from biaxial tensile tests . . . 52

5.13. Failure surface in 13- plane with σ_c^u= 0.0001 kPa and F₁₃calculated from biaxial tensile tests . . . 52

5.14. Failure surface in 23- plane with σ_c^u= 0.0001 kPa and F₂₃calculated from biaxial tensile tests . . . 53

5.15. Failure surface in 12- plane with σ_c^u = 1000 kPa and F₁₂ calculated from biaxial tests . . . 54

5.16. Failure surface in 13- plane with σ_c^u = 1000 kPa and F₁₃ calculated from biaxial tests . . . 54

5.17. Failure surface in 23- plane with σ_c^u = 1000 kPa and F₂₃ calculated from biaxial tests . . . 55

B.1. Dimensions of geometry A. . . 74

B.2. Simulation results from set 1 . . . 75

B.4. Dimensions of geometry B. . . 77

B.7. Dimensions of geometry C. . . 80

B.10. Dimensions of geometry D. . . 83

B.13. Dimensions of geometry E. . . 86

B.17. Dimensions of geometry F. . . 90

C.1. Failure surface in 13- plane with σ_c^u= 0.0001 kPa and F13calculated from biaxial tensile tests . . . 93

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LIST OF FIGURES XVII

C.2. Failure surface in 23- plane with σ_c^u= 0.0001 kPa and F23calculated from biaxial tensile tests . . . 94 C.3. Failure surface in 13- plane with σ_c^u = 1000 kPa and F₁₃ calculated from

biaxial tests . . . 95 C.4. Failure surface in 23- plane with σ_c^u = 1000 kPa and F₂₃ calculated from

biaxial tests . . . 96

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1. Introduction

Aortic dissection is a disease of the aorta with high mortality rates if undiagnosed. The condition starts with an initial tear or damage to the innermost layer of the aorta, propagating in the radial direction through the medial layer, then due to the blood flow, followed by delamination of the aortic layers and creating a false lumen. In other words, the failure starts with a crack propagation and it is followed by delamination of the plies. The blood flowing into the false lumen causes a dilatation of the aorta, which is followed by the rupture in most severe cases. Considering the structure of the elastic arteries, one can see similarities between aortic tissue and conventional engineering materials, i.e. a mix- ture of polymer–matrix composites and laminar composites. Hence, the main task in this thesis project was to determine which failure criterion should be employed and how. The strength, crack initiation and crack propagation of such composites are highly dependent on the fiber length, fiber density, orientation of fibers within the ply and the orientation of plies (Callister, 2007). The aorta can be viewed as a pipe, consisting of three plies of which the two are reinforced with symmetrically arranged collagen fibers.

Imaging with X-ray radiation, CT angiography and MR angiography are among the di- agnostic methods for aortic dissection in the medical field, of which similarities can be found in application to conventional engineering materials. This field of analysis and mea- surement is known as nondestructive evaluation (NDE) (Department of Applied Science, College of William and Mary). Radiographic testing, magnetic testing, eddy current testing, infrared testing, and ultrasonic testing are among the examples of NDE which are also used in medicine. For example, ultrasound is used for inspection of rails, whereas eddy current testing is used to investigate the location and depth of surface breaking cracks in pipes (NDT Resource Center, 2001; Department of Applied Science, College of William and Mary).

To provide a background to the project and better understanding of the condition, the following Chapter 2 describes the physiology of the aorta, and gives an overview of aortic dissection. Next chapter, Ch. 3, deals with the material properties of the aorta followed by review of the failure criteria available for composite structures. Methods to employ the criterion are presented in Chapter 4. Experimental results and failure surfaces are discussed in Chapter 5. Finally, concluding remarks and future perspective are given in Chapter 6.

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2. Medical Background

2.1. Physiology of the Aorta

Arterial structures consist mainly of collagen, elastin and smooth muscle cells. These biological structures play an important role in the mechanical response of the artery due to their structural properties, interconnections and remodeling which differ in the healthy and the diseased states. For example, He and Roach (1994) showed that the composition of the arteries in collagen, smooth muscle cell, and elastin differ significantly between the abdominal aortic aneurysm subjects and healthy subjects.

Elastic arteries like the aorta consist of three main layers: tunica intima, tunica media and tunica adventitia, as shown in Fig. 2.1. The intima, the innermost layer, consists of a single layer of endothelial cells and has no significant contribution to the mechanical response of the artery in a young healthy artery. However, with increased age or with disease one can observe thickening and stiffening of the intima, which may play a significant role in the mechanical response, for example, in the case of atherosclerosis. Nevertheless, the endothelial cells have the function to respond to the blood flow, i.e. mechanotransduction due to shear loading, and send stimuli to the surrounding cells in the connective tissue and to the smooth muscle cells allowing the arteries to change their composition and diameter accordingly (Alberts et al., 2007). The media itself consists of several layers which are rich in collagen and elastin. It is composed by a complex network of smooth muscle cells, elastin, and collagen. The adventitia, rich in collagen, consists of cells producing collagen and elastin, called fibroblasts (Holzapfel et al., 2000a). Collagen is a protein which is found in all multicellular animals and it is mainly responsible for the nonlinear mechanical response of the tissue (Alberts et al., 2007; Holzapfel et al., 2000a). So far 28 different collagen proteins which are all encoded by different genes have been identified in the human body (Fratzl, 2008). In the arteries mainly fibrillar collagen type I and type III are found, which are seen in Fig. 2.1 as two symmetric families of fibers. Nonfibrillar, basement membrane associated collagen type IV is found in the surrounding of the smooth muscle and endothelial cells (Wu et al., 2013). Collagen type IV molecules assemble to a sheet network, while type I and type III are arranged in a helical cable structure. Collagen molecules are secreted by fibroblasts in the form of propeptides and then moved outside the cell. Here they assemble in the form of collagen fibrils which are covered by proteoglycans and then these fibrils arrange themselves into cable-like helical fibers with cross-links. If the cross linking does not happen for a reason, the tensile strength of the collagen fibers are reduced significantly resulting in deficiencies. Collagen fibers are attached to smooth muscle cells by the attachment protein fibronectin, which has connections to myosin and

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4 CHAPTER 2. MEDICAL BACKGROUND

Figure 2.1.: Structure of elastic arteries (Holzapfel et al., 2000a).

actin through integrin dimers (Alberts et al., 2007).

Elastin is the main protein which forms elastic fibers. It is responsible for the elastic part of the mechanical response of the arteries (Holzapfel et al., 2000a). It is secreted in the form of tropoelastin into the extracellular matrix by fibroblasts or smooth muscle cells, where it is assembled to elastic fibers. Elastin core is covered by microfibril glycoproteins, for example, elastin attachment proteins fibrillin and fibulin, as seen in Fig. 2.2.

Figure 2.2.: Elastic fiber assembly: elastin, fibrillin, and fibulin molecules, adopted from Muiznieks and Keeley (2013).

FBN1 and FBN2 are the genes responsible for the production of the fibrillin, whereas fibulin genes are responsible for fibulin production and attachment. The FBN2 gene is active during the embryotic growth. It is taken over by the FBN1 gene later in the growth process. Mutations in the FBN1 gene, which effect the elastic fiber assembly through

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2.2. Aortic Dissection 5

fibrillin proteins, were found to result in Marfan’s Syndrome, a connective tissue disorder which makes arteries prone to dissection (Dietz et al., 1991).

TGF-β is a family of growth factors which bind to appropriate matrix proteoglycans to stimulate or inhibit cell proliferation. They also effect the protein synthesis, promote or inhibit depending on the concentration in the environment. It can up-regulate or down- regulate the production of matrix metalloproteases (MMPs). These MMPs are the enzymes which are responsible for degradation of extracellular matrix proteins, i.e. elastin and collagen. Tissue inhibitors of matrix metalloproteinases, TIMPs, also play an important role in the regulation process. One type of MMP can be activated to start the degradation process of one protein, but it effects many. To stabilize the concentration of the other target proteins, several other MMPs and TIMPs come into play. A list of MMPs and their corresponding substrates are provided in the Appendix A, taken from Visse and Nagase (2003).

2.2. Aortic Dissection

Aortic dissection is a condition in which there is an initial tear to the intimal layer propagating in the radial direction through the media and then axially due to the blood flow, causing delamination of the aortic layers and creating a false lumen which is filled with blood. The point where the tear starts to propagate axially is either between the media and the adventitia, or very close to the adventitia but still within the media, see Fig. 2.3.

Figure 2.3.: An illustration of the propagation of aortic dissection

The false lumen may propagate distally or proximally to the aortic valve. It narrows the true lumen decreasing the blood flow to the tissues, and increases the diameter of the aorta at the area of the dissection. The most severe cases occur when the remaining intact layers rupture through the adventitia. This condition is said to effect no less than 30 individuals in a million per year (Criado, 2011).

The risk groups for aortic dissection include patients with connective tissue disorders such as Marfan’s or Ehlers-Danlos syndrome, high blood pressure combined with advanced age, cocaine abuse, pregnancy, aortic dilatation, congenital bicuspid valve, and so forth.

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Aortic dissection can also be induced in surgery during catheterization for stent placement, valve surgery, bypass, etc. Furthermore, there are reported cases of car accidents causing a trauma to the chest, and from thereon aortic dissection. Patients may present a very sharp,

‘ripping’ chest pain or back pain (Hagan et al., 2000; Pape et al., 2007; Thalmann et al., 2011; Criado, 2011).

2.2.1. Classification

There are different classifications of aortic dissection in the literature. Based on the anatomical location, two widely used ones are DeBakey and Stanford classifications. Stan- ford Type A dissection includes any dissection which effects the ascending aorta, while Type B includes only the descending aorta. According to DeBakey classification, there are three types. DeBakey I and II effects the ascending aorta, type I initiated with a tear in the ascending aorta followed by a dissection in the entire aorta while type II includes only ascending aorta. DeBakey III which effects the descending aorta is further divided into IIIa and IIIb. IIIa goes until the visceral segment, while IIIb extends to involve abdominal aorta and iliac arteries (Golledge and Eagle, 2008; Criado, 2011; Ayala and Chen, 2012). Figure 2.4 summarizes these two classifications and shows their difference.

Figure 2.4.: Classification of aortic dissection: DeBakey type I starts at the ascending aorta and effects the entire aorta, while type II involves only the ascending aorta. Both DeBakey types I and II are classified as Stanford Type A (any dissection involving the ascending aorta). DeBakey type IIIa effects the descending aorta until the visceral segment while IIIb extends to involve iliac arteries. IIIa and IIIb together correspond to Type B in the Stanford classification.

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Aortic dissection may occur if aortic ulcer, intramural hematoma, or intramural hemorrhage is present (Erbel et al., 2001). Hence, a new classification was proposed by Svensson et al. (1999). In class I, there is an intimal flap between the true and false lumen. In class II, media is damaged with intramural hematoma or hemorrhage. In class III, there is a bulge at the tear site without hematoma due to ulceration of aortic plaque after rupture. In class IV, there is plaque rupture which leads to ulceration with surrounding hematoma and, finally, in class V there is trauma to the aorta due to surgery or impact (i.e. car accident).

This classification is said to serve as a subdivision to Stanford or DeBakey classifications (Erbel et al., 2001).

Another classification is based on the time course of the disease. The first two weeks is considered as an acute dissection. However, the first 48 hours are the most critical, so some sources consider 2-14 days as sub-acute. After 14 days, the dissection is classified as chronic. These classifications are based on the mortality rates, which show that untreated aortic dissections have a mortality rate of 50% within the first 48 hours, 75% within 14 days, and 90% within 3 months without being diagnosed. What is more interesting is the mortality rates of Stanford type A aortic dissection. It has been reported that 50% of the patients die within the first 2 days, and 80% within 2 weeks (Criado, 2011).

2.2.2. Diagnosis and Treatment

Diagnosis includes a review of the patient history to decide whether he is in any of the risk groups (i.e. drug abuse, high blood pressure, old age, connective tissue disorders, recent surgery, etc.). Cardiovascular or pulmonary pathology, musculoskeletal disorders (related to connective tissue disorders) will help the physician to come up with a diagnosis. Common imaging methods include chest X-ray, CT angiography, MR angiography and Transesophagel echocardiography (TEE). Among these methods, most reliable and accepted ones are MR angiography and CT angiography with contrast agents. MR is preferred over CT because it is the most accurate one. However, it is not always readily available which leads to the choice of CT. TEE is the fastest imaging tool used in the diagnosis of aortic dissection and commonly preferred for the acute cases, although it is the least reliable procedure (Mukherjee and Eagle, 2005; Haase et al., 2010).

The decision for medical treatment or surgery is based on which class of aortic dissection the patient belongs to. The aim of the surgical treatment is to prevent the dissection from rupture. Patients with acute type A dissections are immediately treated surgically.

Depending on the extension of the dissection, surgical procedure can involve aortic valve replacement. Most Stanford type B patients are treated non-surgically with medication.

The follow ups after the medical and surgical treatments showed that the survival rates after surgery – mortality rate 31 % – were found too high compared to non-surgical treatment – mortality rate 10.5% (Haase et al., 2010; O’Gara and DeSanctis, 1995). In treatment of both types, surgical intervention is suggested if the diameter of the aorta is found to be more than 5.5 cm – revised later to 5 cm, still ineffective (Pape et al., 2007). In surgical treatment of Stanford Type B dissection, other criteria for surgery include severely effected

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blood flow to an organ, persistent difficulty in breathing, persistent pain, and small amount of bleeding (Khan and Nair, 2002).

The performed surgery is endograft replacement which includes the removal of the dissected part, and placement of a graft, which is glued to the remaining ends of the artery and then sutured. The cases in which the dissection involve the aortic valve, valve replacement is necessary and is performed together with the endograft replacement (Akin et al., 2010;

Criado, 2011).

2.2.3. Pathology

As introduced in Section 2.1 and motivated in 3.1, collagen and elastin play an important role in the mechanical properties of the aorta. It is important to know how these compo- nents are effected in the aortic dissection. There are several studies in the literature, which focus on the structural changes and the underlying mechanisms of aortic dissections. Sar- iola and colleagues (Sariola et al., 1986) observed an increase in the amount of collagen I and III, as well as type IV collagen and fibronectin around the smooth muscle cells in the dissected media. Ishii and Asuwa (2000) have found increased levels of MMP1, MMP2, and MMP9, and also their inhibitors TIMP1 and 2. They investigated the enzymes in the media and the intima and compared the levels at the entry site and remote site of the dissection. There was a decrease at the levels of type IV collagen and elastin, but no information was provided on type I and III collagen. They observed a significant difference in the levels of the remote and of the entry sites, and concluded that remote sites have closer levels to the healthy control subjects. Furthermore, they observed a phenotypic change of the smooth muscle cells from contractile to synthetic. Lesauskaite et al. (2001) observed an increase in MMP1, MMP2, MMP9, TIMP1, TIMP2 together with the phenotype change from contractile to synthetic but no smooth muscle cell apoptosis. Wang et al. (2005) observed smooth muscle cell apoptosis together with an increase in the levels of type I and III collagen, and a decrease in fibulin-5 and elastin levels. Furthermore, they had higher dif- ferences in the levels of these proteins between the healthy tissue and the aortic dissection, than between the healthy tissue and other diseases such as atherosclerosis. In another study by Wang et al. (2006) similar results were reported, but instead of studying fibulin-5 they looked at levels of TGF-β, which was found to increase. Zhang et al. (2009), in a review article, observed an increase in the levels of MMP1, MMP2, MMP9, and their inhibitor TIMP2. There was also an increase in the smooth muscle cell apoptosis, while elastin and type IV collagen levels decreased.

Cheuk and Cheng (2011) observed a decreased activation of fibulin-1 gene, which in turn resulted in a decrease in TIMP1, TIMP2, type IV collagen, and elastin levels. The deactivation triggered the increase in the levels of MMP9, MMP11, type I and III collagen, and smooth muscle cell death. Wang et al. (2012) looked at the SMC phenotypes and their relation with the extracellular matrix disorders. They found out that MMP1 levels stayed the same while the smooth muscle cell the phenotype changed from contractile to synthetic.

MMP2, type I and III collagen levels increased while type IV collagen and elastin levels

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decreased.

Song et al. (2013) investigated the effect of the MMP12 levels, which was found to be increased, effecting type IV collagen and elastin levels adversely. A recent review article by Wu et al. (2013) summarized the changes. Turning off the Fibrillin 1 (FBN1) gene in mice resulted in an excessed amounts of TGF-β, decreased attachment of smooth muscle cells to elastin which triggered a smooth muscle cell phenotype change, and overexpression of MMPs which may all play a role in the progression of aortic dissection. Turning off the Fibrillin 2 (FBN2) gene did not have any significant effect, since it is naturally active during fetal development, and turned off afterwards. Mice with fibulin 5 deficiency had disorientation of the elastic fibers, while fibulin 4 deficiency resulted in increased TGF-β levels. Deficiency of fibulin 1 showed no effect on elastic wall integrity. The changes in these genes resulted in increased levels of MMPs and TIMPs, which in turn resulted in increased levels of type I and III collagen and decreased levels of type IV collagen and elastin.

In summary, the presented studies are in a good agreement that elastin and type IV collagen are decreased, while collagen type I and III increased during the time course of the aortic dissection. Since these elements are known to provide the characteristic mechanical response of the arteries, significant difference in the mechanical behavior of dissected and healthy arteries is expected.

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3. Mechanical Background

3.1. Material Properties of Arteries

Proximal arteries show elastic material behavior, like aorta, and they are modeled as hyperelastic incompressible materials. The incompressibility of a material means that it does not go under volume change and it has been shown by Lawton (1954). Muscular type distal arteries show viscoelastic behavior (Holzapfel et al., 2002). The mechanical response of elastic arteries is highly nonlinear with the ability to recover from large defor- mations. Furthermore, they show exponential stiffening at higher stress values. Hysteresis, energy dissipation, can be observed in the loading-unloading curves of arteries (Holzapfel et al., 2000a). This behavior with different loading and unloading response is known as pseudoelasticity(Fung, New York (1993). Roach and Burton (1957) showed that collagen is mainly responsible for the nonlinear response and elastin is responsible for the linear response in the arteries. Weisbecker et al. (2013) treated the human thoracic aorta partially and fully with collagenase and showed that elastin is responsible for the linear response.

Another important characteristic of arterial walls is the existence of circumferential and axial residual stresses, which is attributed to growth and remodeling. It has been shown that the residual stresses are responsible for homogeneous stress distribution in the arteries in vivo. It is important to note that these residual stresses are layer and location specific.

For detailed references and further discussion see Holzapfel et al. (2007).

In their work Holzapfel et al. (2000a) used a linear and a nonlinear part in the constitutive equation to characterize the mechanical response of elastic arteries i.e.

Ψ = µ(I₁− 3) + X

i=4,6

k₁

k₂(exp[k₂(I_i− 1)²] − 1) , (3.1)

where I1, I₄, I₆ are the invariants of C = F^TF and the structural tensor, C is the right Cauchy Green tensor and F is the deformation gradient. The linear isotropic part in this equation is attributed to the ground substance together with elastin, and the anisotropic nonlinear part is attributed to collagen; I4 stands for the square of the stretch in the first fiber family direction while I₆stands for the square of the stretch in the second fiber family direction. Schriefl et al. (2011) identified two symmetric families of collagen fibers in the intima, media, and adventitia in which the layers differed from each other with the preferred direction. Material parameters µ, k₁, k₂ are determined experimentally.

Furthermore, it has been shown that arteries show different characteristics in the axial and circumferential direction. Lanir and Fung (1974) tested rabbit carotid arteries, and

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12 CHAPTER 3. MECHANICAL BACKGROUND

showed that the response in two orthogonal directions are different from each other. The anisotropy in the axial and circumferential directions and the difference in the mechanical behavior of layers are attributed to the collagen orientation and content (Holzapfel et al., 2000b).

3.2. Failure Criteria

A failure criterion for biological soft tissues has not yet been developed to the student’s and supervisors’ knowledge. The decision was to investigate the Tsai-Wu criterion further, and analyze the application while reviewing the failure criteria for composite materials available in the literature, since the arterial tissue can be considered as a fiber reinforced composite material.

A wide range study called World Wide Failure Exercise (WWFE) was performed under the leadership of Kaddour and Hinton in three stages, which aims to compare the effec- tiveness of the 12 different composite failure criteria by evaluating the results of given test cases. Each stage had Part A and Part B. Part A covered the predictions of the theories for a given set of data, while Part B dealt with the comparison of the predictions with experimental results. WWFE-I was performed to identify the gaps with the composite failure theories in the case of in-plane loading, whereas the aim of WWFE-II was to assess different criteria to predict the mechanical response of materials under 3-D loading. WWFE-III addressed the damage evolution under in-plane stress conditions with the effects of ther- mal loading and stress concentrations (Kaddour and Hinton, 2012a,b, 2013; Kaddour et al., 2013).

In WWFE-II the criteria of Bogetti, Carrere, Tsai-Ha, Hansen, Huang, Pinho, Wolfe, Hashin, Christiensen and Puck were tested by different groups. Studying Table 5 of WWFE-II Part A, the Tsai-Ha failure criterion was the only one which was able to predict the initial and the final failure of all cases, including polymer, unidirectional lamina, and multi-directional laminates. The failure of multi-directional laminates was also predicted by Carrere, Huang, Hansen, Rotem, and Wolfe. Among these, Wolfe’s theory (maximum strain-energy method) which is originally developed by Sandhu (1976), and Tsai-Wu criterion, more general version of Tsai-Ha, were selected to be investigated further.

After the investigations, the Tsai-Wu criterion was chosen to be adopted in this project.

3.2.1. Maximum Strain-Energy Method

The maximum strain-energy method assumes that the strain energies for nonlinear orthotropic materials are independent of each other in the longitudinal, transverse and the shear modes in 2-D (Wolfe and Butalia, 1998; Zand et al., 2012). Function is expressed as follows:

X

i=1,2,6

1 K_i

Z

i

σ_id_i

mi

= 1 , (3.2)

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3.2. Failure Criteria 13

where mi are the energy shape factors and K_i =

"

Z

^u_i

σ_id_i

#mi

. (3.3)

The explicit form reads R

1σ₁d₁ R

^u₁ σ₁d₁

!m1

+ R

2σ₂d₂ R

^u₂ σ₂d₂

!m2

+ R

6σ₆d₆ R

^u₆ σ₆d₆

!m6

= 1 . (3.4)

The theory uses the incremental constitutive law for plane stress, with piecewise, cubic- spline interpolation functions. The algorithm starts with approximating d^◦_n+1as

d^o_n+1= [A]⁻¹_n+1dNn+1 (3.5)

where [A]⁻¹_n+1 represents average compliance properties of the laminate and dN_n+1 is the load increment at n+1 th step. However, [A]⁻¹_n+1is not known at the n+1 th load increment.

Hence, the following is used:

d^o_n+1 = [A]⁻¹_n dN_n+1 . (3.6)

Strain increments are calculated from Eq. (3.6). It is assumed that the stress increments are the same for all plies and laminate d^o_n+1 = d_k, where k is the ply(lamina) number.

The strain increment then, is substituted into

dσ_k=C_kd_k . (3.7)

Eq. (3.7) in the global laminate axes. C_krepresents the stiffness matrix of the lamina with respect to laminate axes. It is calculated from:

C_k = [T_σ]⁻¹_k [C]_k[T]_k . (3.8) [C]_k is the stiffness matrix of the lamina with respect to lamina material axes, [C]_k = [S]_k, [S]_k being the compliance matrix of the lamina. Next, stress and strain increments are calculated in lamina directions from:

dσ_k = [T_σ]_kdσ_k , d_k = [T]_kd_k (3.9) The lamina strains are then used to calculate the equivalent strain increments, d₁|_eqv and d₂|_eqv, as

d₁|_eqv = d₁

1 − ν₁₂B , d₂|_eqv = d₂

1 − ν₂₁B (3.10)

where B = dσ₂/dσ₁. The equivalent strains are used to calculate the total stress and strains, which are used to calculate new [A]⁻¹from:

[A]⁻¹=

n

X

k=1

t_kC_k . (3.11)

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Next, again using Eq. (3.6) new d^o_n+1is calculated. This procedure is repeated for each ply (lamina) until the criterion in Eq. (3.2) is satisfied (Wolfe and Butalia, 1998).

Even though the theory was modified for 3-D state of loading for WWFE-II (Zand et al., 2012), the problem of adopting the maximum strain energy method for soft biological tissues, arises from the use of the compliance matrix and several others such as knowing the maximum strain energy. Tangent moduli E₁ and E₂, Poisson’s ratio ν₁₂, and shear modulus G12 need to be known for the plies, since they are used to calculate entities of [S]_k. Furthermore, the number of experiments needed do determine ultimate strain energy and material constants, m₁, m₂ , and m₆, is not clear compared to employing the Tsai-Wu criterion.

3.2.2. Tsai-Wu Criterion

Tsai and Wu (1972) failure criterion states that the failure occurs for a unidirectional lamina when

F_iσ_i+ F_ijσ_iσ_j > 1 , i, j = 1, . . . , 6 , (3.12) with the stability condition for the failure surface to be an ellipse (in 2-D) or an ellipsoid (in 3-D), i.e.

F_iiF_jj − F_ij² ≥ 0 . (3.13)

Subscript 1 corresponds to the fiber direction (longitudinal), 2 to the transverse direction and 3 to the out-of-plane axis, as shown in Fig. 3.1. F_iand F_ijare experimentally identified parameters, describing the difference between tensile and compressive failures and the stress interactions, respectively. Fij is a symmetric matrix, with positive diagonal entities.

This becomes clear following the experimental procedures. An explicit version of the Eq.

(3.12) is as follows:

F₁σ₁+ F₂σ₂+ F₃σ₃+ F₄σ₄+ F₅σ₅+ F₆σ₆ +F₁₁σ₁²+ F₂₂σ²₂+ F₃₃σ²₃+ F₄₄σ₄²+ F₅₅σ₅²+ F₆₆σ₆² +2F₁₂σ₁σ₂+ 2F₁₃σ₁σ₃+ 2F₁₄σ₁σ₄+ 2F₁₅σ₁σ₅ + 2F₁₆σ₁σ₆

+2F₂₃σ₂σ₃+ 2F₂₄σ₂σ₄+ 2F₂₅σ₂σ₅+ 2F₂₆σ₂σ₆ +2F34σ3σ4+ 2F35σ3σ5+ 2F36σ3σ6

+2F₄₅σ₄σ₅+ 2F₄₆σ₄σ₆+ 2F₅₆σ₅σ₆ > 1 .

(3.14)

Determining the elements Fi and Fij

Proposed by Tsai and Wu, the first three entries of Fiand the first three diagonal entities of F_ij are identified from uniaxial tension and compression tests. For example, along the axis 1, when the failure stresses are σ1cand σ1tand all the other stress entities are zero, Eq.

(3.12) reduces to:

F₁σ_1t+ F₁₁σ²_1t = 1 , F₁σ_1c+ F₁₁σ_1c² = 1 . (3.15)

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Figure 3.1.: Principal directions of the lamina: 1 fiber direction, 2 cross-fiber direction, 3 out-of- plane direction.

Solving the system of equations for F₁and F₁₁we obtain F₁ = 1

σ_1t − 1

σ_1c , F₁₁ = 1

σ_1tσ_1c . (3.16)

Repeating the same procedure for the second and the third directions:

F₂ = 1 σ_2t − 1

σ_2c , F₂₂ = 1

σ_2tσ_2c , (3.17)

F₃ = 1 σ_3t − 1

σ_3c , F₃₃ = 1

σ_3tσ_3c . (3.18)

The rest of the terms of F_i and the other diagonal elements of F_ij are determined from simple shear tests. For simple shear in the 1-2 plane, σ6p and σ6n being the positive and negative failure stresses, Eq. (3.12) will reduce to:

F₆σ_6p+ F₆₆σ_6p² = 1 , (3.19)

F₆σ_6n+ F₆₆σ_6n² = 1 . (3.20)

Imagine that shear towards the right is applied to the upper edge of a square in 2-D until it fails, and now imagine that this time it is applied towards left until it fails. The right and left (positive and negative) failure stresses will be equal to each other. Hence, for the orientation in Fig. 3.1, negative and positive shear tests, σ_6nand σ_6p, have the same value σ6, which leaves us with:

F₆ = 0 , F₆₆ = 1

σ₆² . (3.21)

Performing simple shear tests in the 1-3 and 2-3 planes will give:

F5 = F4 = 0 , F55 = 1

σ₅² , F44= 1

σ₄² . (3.22)

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For the orientation in Fig. 3.1, the theory states that there are no interactions between normal stresses and shear stresses. Hence, the following relation is obtained for the off- diagonal terms of F_ij:

F_ij = 0 , i = 1, 2, 3 , j = 4, 5, 6 . (3.23) In the same orientation mentioned above, there are also no interactions between shear stresses which means F45 = F₄₆ = F₅₆ = 0. After these assumptions and tests we have the following:

F_i =









 F₁ F₂ F₃ 0 0 0











, F_ij =







F₁₁ F₁₂ F₁₃ 0 0 0 F₁₂ F₂₂ F₂₃ 0 0 0 F₁₃ F₂₃ F₃₃ 0 0 0

F₄₄ 0 0

F₅₅ 0 F₆₆







. (3.24)

To determine the normal stress interaction terms F12, F13and F23a number of different experiments can be performed according to Tsai and Wu (1972), such as biaxial tension or compression tests, off-axis shear tests, off-axis tension or compression tests. Pipes and Cole (1973) experimented with boron epoxy composites and showed that off-axis tensile test was not appropriate for determining F₁₂. Suhling et al. (1984) concluded that none of the off-axis tests were in good agreement with the experimental results nor are suitable to determine the interaction term. They also concluded that F12 = 0 was a reasonable suggestion in their case. Theocaris and Philippidis (1991) argues zeroing the term F₁₂ without a justification, ensures a closed failure surface and might contradict the existing experimental results.

Tsai and Hahn (1980) suggest to calculate the interaction term as:

F12 = −

√F₁₁F₂₂

2 . (3.25)

The term F12determines the rotation of the ellipse in the 1-2 plane, F13in the 1-3 plane and F₂₃ in the 2-3 plane. It is not necessary to perform an equibiaxial test to determine the interaction terms. However, as an example, let us say that equibiaxial tension test is performed to determine the value of the interaction term in the 1-2 plane, and failure is observed at a stress value of P . Eq. (3.12) becomes:

F₁P + F₂P + F₁₁P²+ F₂₂P²+ 2F₁₂P² = 1 . (3.26) To be able to better understand the effect of F12 on the failure surface, MATLAB (The MathWorks Inc., Natick, USA) scripts were prepared. The term had an important effect on the failure surface, as shown in Fig. 3.2. For arbitrarily chosen strength properties all coefficients were the same except F12. This condition was satisfied by choosing different P values and the same strength values for each curve [100 100 100 100 0 0 0 0 0 0 50 50].

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The elements of the given array in 2-D correspond to stresses in 3-D, as shown below:

[σ_1u_tσ_1u_cσ_2u_tσ_2u_cσ_3u_tσ_3u_cσ_4u_pσ_4u_nσ_5u_pσ_5u_nσ_6u_pσ_6u_n] . (3.27)

σ₁

σ 2

−200 −150 −100 −50 0 50 100 150 200

−200

−150

−100

−50 0 50 100 150 200

3.8889e−005 2.0408e−006

−2.1875e−005

−3.8272e−005

−5e−005

Figure 3.2.: Change of failure surface with F12for differentP values: F 12 values can be seen on the label. As it is getting bigger in absolute value, the amount of rotation is increased.

As it changes from positive to negative, the direction of rotation is changing.

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4. Methods

On the basis of what is explained in Chapter 3, methods to adopt Tsai-Wu failure criterion for aortic tissues are explained in detail together with the necessary assumptions. This chapter starts with introducing the necessary experiments and assumptions, together with the biaxial specimen design in Section 4.1 and continues with explaining the experiments considering the specimen preparation and the experimental procedure in Section 4.2.

4.1. Adopting The Tsai-Wu Criterion for Aortic Dissections

The Tsai-Wu failure criterion was adopted by Cowin (Cowin, 1985, 1986; Keaveny, 2011) trabecular bone. He used the density of the bone and fabric tensor to be able to characterize the Tsai-Wu coefficients related to the structure of the bone, i.e. porosity.

This strategy makes it possible to apply it for different anatomical locations, however, this would require a lot of data for to statistically confirm the 26 unknowns introduced with the use of the density dependent fabric tensor (Keaveny, 2011). A profound difference between the behavior of bone and aorta, even though they can both assumed to be orthotropic, is that bone shows a linear behavior at small strains and it can be modeled using Hooke’s law.

Important difference in terms of adopting the failure criterion is that bones can be subjected to compressive testing until failure to decide Tsai-Wu coefficients, whereas this cannot be done for soft biological tissues. Under compression the trabecular bone is the main load carrier and the behavior is linear, but once the strains get larger it fails and nonlinearity is observed due to the porous structure. Bones are adapted to carry compressive loads, while the aorta needs to resist and assist pulsating blood flow, and the blood pressure and these loading conditions effect the collagen orientation and composition of both tissues, hence the mechanical properties.

Using the symmetry of the fibers in the aorta, as shown by Schriefl et al. (2011), the layers of the aorta are assumed to have orthotropic symmetry. With these assumptions, directions 1,2, and 3 correspond to θ, z and r respectively, for the intima, the media and the adventitia. The assumption of orthotropy leaves us with the coefficients for all layers as shown in Eq. (3.24). To be able to identify these coefficients for a dissected aorta, mechanical tests should be performed, as explained above. There are several limitations to these tests since the material is a soft biological tissue. The following sections describe the limitations on experiments together with the necessary assumptions, and which tests should be performed on the arterial layers to obtain the Tsai-Wu coefficients.

19

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20 CHAPTER 4. METHODS

4.1.1. Uniaxial Tests

Uniaxial tension tests in the axial, 2, and the circumferential, 1, directions with bone shaped specimens and direct tension test, with circular specimens in the radial direction, 3, were decided to be performed with the specimens shown in Fig. 4.1 to obtain tensile strengths σ_it^u where i = 1, 2, 3. Compressive strengths needed to be assumed since it was not possible to observe failure of the soft biological tissues under compression like it can be done for bone, metals or concrete. Using the same MATLAB scripts, which were used above to understand the effect of the interaction term, F₁₂, the influence of different compressive strength values were analyzed in 2-D.

In Case I (see Fig. 4.2), the interaction term F12is calculated by an analytical expression, Tsai-Hahn, Eq. (3.25), and the compressive strengths are chosen in the 1 and 2 directions, while holding the other strength values constant. Figure 4.2 shows the results with very low and very high compressive strength values. 60 is the value of P , which does not have any effect on the results since it is not used to make any calculations. However, the assumption of the compressive strength has an important effect on the interaction term F₁₂. On the third quadrant, another effect of different compressive strengths can be seen by looking at the axes interception points.

(a) (b)

(c)

Figure 4.1.: Sketch of the uniaxial and direct tension test specimens: (a) bone shaped specimen in circumferential direction, (b) bone shaped specimen in axial direction, and (c) direct tension specimen, a circular punch-out through the thickness of the artery with an incision around the circumference to induce failure.

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4.1. Adopting The Tsai-Wu Criterion for Aortic Dissections 21

σ₁

σ 2

60

−150 −100 −50 0 50 100 150 200 250

−300

−200

−100 0 100 200 300

[100 10 100 10 0 0 0 0 0 0 50 50]

[100 100 100 100 0 0 0 0 0 0 50 50]

[100 200 100 200 0 0 0 0 0 0 50 50]

[100 500 100 500 0 0 0 0 0 0 50 50]

[100 1000 100 1000 0 0 0 0 0 0 50 50]

Figure 4.2.: Case I – Change of failure surface with different compressive strength values when F₁₂is calculated with Tsai-Hahn equation: As the compressive strength is increased the elliptic failure surfaces shift towards the third quadrant and get larger in diameter.

They are getting closer to each other in the first quadrant.

In Case II, the interaction term is calculated from an assumed ultimate stress value for a biaxial test, P and the compressive strengths are chosen in the 1 and 2 directions while holding the other strength values constant. For a better understanding of Case II, surfaces with two different P values – Case IIA and Case IIB– are plotted in Figs. 4.3 and 4.4. From Case IIA (see Fig. 4.3), it can be seen that the curves are overlapping in the first quadrant, but getting apart in the third quadrant due to the axes interception points. In Case IIB (see Fig. 4.4), the curves are in an acceptable agreement until very high values of compressive strength. However, when the compressive strength reaches 500 and 1000 it can be noticed that the failure surface is no longer an ellipse but a hyperbola. This means that the stability condition in Eq. (3.13) is not satisfied. It can be seen that the hyperbolas are getting steeper.

It means that the material will not fail even under very large compressive stress states.

In Case III (see Fig. 4.5), the interaction term is omitted and the compressive strengths are the same as in Case I and II. The curves are further apart from each other in the first quadrant compared to Case II, and the axes of the ellipses are parallel to the horizontal and vertical axes.

The compressive strength assumption needs special attention, since it also effects the interaction term and it can change the failure surface dramatically. It has been reported by Holzapfel et al. (2000a) that axial and circumferential compressive stresses are not observed, while very small amounts of compressive radial stresses are encountered in the human aorta. Taking that as a basis for the assumption, small or very large compressive strengths can be assumed for circumferential, 1, and axial, 2, directions. Assumption in the radial direction, 3, needs careful examination since the assumptions for interaction terms F₁₂, F13and F23will also be effected. These cases should be analyzed simultaneously.

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22 CHAPTER 4. METHODS

σ₁

σ 2

60

−150 −100 −50 0 50 100 150 200 250

−300

−200

−100 0 100 200 300

[100 10 100 10 0 0 0 0 0 0 50 50]

[100 100 100 100 0 0 0 0 0 0 50 50]

[100 200 100 200 0 0 0 0 0 0 50 50]

[100 500 100 500 0 0 0 0 0 0 50 50]

[100 1000 100 1000 0 0 0 0 0 0 50 50]

Figure 4.3.: Case IIA – Change of failure surface with different compressive strength values when F12 is calculated using the failure stress, P = 60, obtained from biaxial tests: The elliptic failure surfaces are overlapping perfectly in the first quadrant as the compressive strength is increased. Due to the change of axes interception points in the third quadrant, the ellipses are getting larger in diameter.

σ₁

σ 2

100

−150 −100 −50 0 50 100 150 200 250

−300

−200

−100 0 100 200 300

[100 10 100 10 0 0 0 0 0 0 50 50]

[100 100 100 100 0 0 0 0 0 0 50 50]

[100 200 100 200 0 0 0 0 0 0 50 50]

[100 500 100 500 0 0 0 0 0 0 50 50]

[100 1000 100 1000 0 0 0 0 0 0 50 50]

Figure 4.4.: Case IIB – Change of failure surface with different compressive strength values when F12is calculated using the failure stress, P = 100, obtained from biaxial tests: The elliptic failure surfaces are close to each other in the first quadrant as the compressive strength is increased. However, when it is increased to 500 and 1000, the stability criterion is not satisfied anymore and open failure surfaces observed in the shape of hyperbolas. Due to the change of axes interception points in the third quadrant, the ellipses are getting larger in diameter.

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4.1. Adopting The Tsai-Wu Criterion for Aortic Dissections 23

σ₁

σ 2

60

−150 −100 −50 0 50 100 150 200 250

−300

−200

−100 0 100 200 300

[100 10 100 10 0 0 0 0 0 0 50 50]

[100 100 100 100 0 0 0 0 0 0 50 50]

[100 200 100 200 0 0 0 0 0 0 50 50]

[100 500 100 500 0 0 0 0 0 0 50 50]

[100 1000 100 1000 0 0 0 0 0 0 50 50]

Figure 4.5.: Case III – Change of failure surface with different compressive strength values when F₁₂is assumed to be0: the elliptic failure surfaces are in a worse agreement in comparison to Case IIA, and Case IIB, but better than Case I. Again, as the compressive strengths are increased the failure surfaces are getting closer in the first quadrant and getting apart in the third quadrant due to the change of axes interception points.

4.1.2. Biaxial Tests

Biaxial tests were designed to be performed only with the medial layer in the 1-2 plane.

It can be clearly seen that it is not possible to perform biaxial tests in the 1-3 and 2-3 planes to identify F₁₃ and F₂₃, respectively, due to the limitations of the specimen size.

The data from the biaxial tests in the 1-2 plane can be used to choose an analytical expression, such as Tsai-Hahn term, as in Eq. (3.25), to be able to assign a value to these coefficients. The media is treated as one ply in the formulation of the Tsai-Wu coefficients, even though it was mentioned in the Section 2.1 that the media itself consists of several layers. Furthermore, by considering the thickness of the intima and the adventitia and their mechanical properties, it did not seem possible to perform the discussed procedures to specimen preparations on these layers. As a result, an analytical expression should be used to obtain interaction terms for the intima and the adventitia.

The biaxial tests on soft biological tissues are performed mainly to be used for constitutive modeling. The aim is to have a homogeneous stress distribution in the biaxially loaded zone. Biaxial testing of soft biological tissues is performed in several different ways in the literature. Lanir and Fung (1974) performed experiments on rabbit skin, using square specimens which were attached to the force distributors by small staples and silk thread.

Sommer et al. (2013a) performed biaxial tests on human abdominal adipose tissue with a different design to improve preoperative simulations. They used an adhesive to glue the specimen ends from the top and bottom with acrylic glass plates and mounted the plates

Modeling the Propagation of Aortic Dissection

Modeling the Propagation of Aortic Dissection

Modeling the Propagation of Aortic Dissection Modellering av Fortplantningen av

Aortadissektion

Abstract

Sammanfattning

Acknowledgment

Contents

List of Tables

List of Figures

1. Introduction

2. Medical Background

2.1. Physiology of the Aorta

2.2. Aortic Dissection

2.2.1. Classification

2.2.2. Diagnosis and Treatment

2.2.3. Pathology

3. Mechanical Background

3.1. Material Properties of Arteries

3.2. Failure Criteria

3.2.1. Maximum Strain-Energy Method

3.2.2. Tsai-Wu Criterion

4. Methods

4.1. Adopting The Tsai-Wu Criterion for Aortic Dissections

4.1.1. Uniaxial Tests

4.1.2. Biaxial Tests