Computational Modeling and Simulation Examples in Bioengineering. Группа авторов

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СКАЧАТЬ authors used the generalized string model as the structure of blood flow in compliant vessels and arteries [50–56]. Causin et al. [57] described this string model as a structural model derived from the theory of linear elasticity for a cylindrical tube with small thickness. Nobile and Vergara [56] emphasized that the generalized string model neglects bending as well. Čanić et al. [58–61] claimed that there are no analytical results which are able to prove the well posedness of FSI problems without assuming the structure model that includes the higher‐order derivative terms, capturing the viscoelastic behavior. For blood flow, there is a strong added mass effect issue in which the fluid and structure have comparable densities.

      All the above studies demonstrate the importance of biomechanical modeling of AAA by using FEM approach with and without FSI and nonlinear wall deformation. This mechanical approach provides an additional understanding of potential indicators of rupture risk.

1.4 Experimental Testing to Determine Material Properties

      Experimental tests are used to determine the mechanical properties of AAA. In‐vivo measurements are based on the imaging modality. For in‐vivo measurements, the main difficulty is to accurately determine the true force and the displacement distribution for the aorta wall. For isolating samples often unknown changes of their behavior affecting the results of such tests. MacSweeney et al. [62] found that elastic modulus was higher in aneurysmal abdominal aorta compared with controls. More recently, van't Veer et al. [63] estimated the compliance and distensibility of AAA by means of simultaneous instantaneous pressure and volume measurements obtained with the magnetic resonance imaging (MRI). Ganten et al. [64] by using time resolved electrocardiography (ECG)‐gated CT imaging data from 67 patients, found that the compliance of AAA did not differ between small and large lesions. Molacek et al. [65] did not find any correlation between aneurysm diameter and distensibility of AAA wall and of normal aorta. Uniaxial extension testing is the simplest and most common of ex‐vivo testing methods. The recorded force‐extension data are converted to stress/strain. Di Puccio et al. [66] provided a recent review of the incompressibility assumption on soft biological tissue. Biaxial test was used as an initial square thin sheet of material which is stress normally to both edges. Even this test is not sufficient to fully characterize anisotropic materials [67, 68], although it can capture additional information regarding the mechanical behavior of the specimens with respect to uniaxial one.

      One of the most complete data for biaxial mechanical behavior of aorta and AAAs is described in Vande Geest et al. [69, 70]. They reported on biaxial mechanical data for AAA (26 samples) and normal human AA as a function of age: less than 30, between 30 and 60, and over 60 years of age. They found that the aortic tissue becomes less compliant with age and that AAA tissue is significantly stiffer than normal abdominal aortic tissue.

At Clinical Center in Belgrade, Serbia, we developed an experimental procedure for bubble inflation test. Intraoperatively, specimens of anterior wall of AAA are harvested, stored in saline at 4° C immediately, and placed in the laboratory setup that simulates natural forces by exposing aortic tissue specimen to inflation with pressurized solution [71] (Figure 1.1). The model consists of the mechanical pump with Crebs–Ringer solution, heater and heat exchanger, tissue container, pressure sensor, and transducer with camera. Camera and pressure transducer system were connected by USB connection with the laptop serving as a control unit and data collector. After heating the Crebs–Ringer solution in the system to 37 °C, the pump was turned on and the pressure was gradually increased exposing aortic tissue to the maximum pressure until the moment of tissue rupture that was recorded by a webcam placed above the tissue. Pressure value in the moment of rupture was known due to the dedicated software defining tissue (failure) strength.

      Ex‐vivo testing is mostly based on uniaxial or biaxial stretching of the intraoperatively harvested tissue specimens. These tests have the ability to characterize intrinsic properties of the tissue itself with independent physical meanings, stiffness, failure stress, and strain. Tissue sample is exposed to extension along its length at a constant displacement rate, while the force is recorded during extension and until failure of the tissue. Uniaxial extension testing is the simplest and most common. Van de Geest et al. [72] reported uniaxial extension testing of 69 AAA specimens, from 21 patients. A novel mathematical model to estimate physically meaningful measures such as stiffness varied from 21.2 to 19.3 N/cm², mean 80.5 N/cm².

      Improvement of uniaxial testing was achieved by performing biaxial testing, when specimen is exposed to determined forces between the two orthogonal directions. Using biaxial testing, the same authors compared the tissue of AAA and normal aorta and compared their behavior in longitudinal and circumferential direction. They found that aneurysmal degeneration of the abdominal aorta is associated with an increase in mechanical anisotropy, with preferential stiffening in the circumferential direction. Information related to stiffness and strain of aortic tissue was gained. Thubrikar et al. [73] were testing different regions of aneurysm in terms of yield stress, yield strains, and other mechanical properties, and found that the anterior wall of AAA is the weakest. In the circumferential direction, the yield stress of the lateral region was greater than that of the anterior or posterior region (73 + 22 N/cm² versus 52 + 20 N/cm² or 45 + 14 N/cm², respectively).

      Our results showed rupture of the tissue at the inflation pressure of 9.8 N/cm². The analytical calculated wall strength for wall thickness 0.2 cm and aorta radius 1 cm was 24.5 N/cm². FEA showed PWS of 57 N/cm² and wall stress of 21.2 N/cm² at the anterolateral wall of AAA in the area of harvested tissue [71].

Figure 1.1 (a) Cartoon schema. (b) Real laboratory model. Laboratory model consists of the mechanical pump (3) with Crebs–Ringer solution (4), heater (1) and heat exchanger (2), tissue container (6), pressure sensor (5), and transducer with camera (9). Camera and pressure transducer system were connected by USB connection with the laptop serving as a control unit and data collector. After heating the Crebs–Ringer solution in the system to 37 °C (7 – temperature sensor), the pump was turned on and the pressure was gradually increased exposing aortic tissue (6) to the maximum pressure until the moment of tissue rupture that was recorded by a webcam placed above the tissue. Pressure value in the moment of rupture was known due to the dedicated software defining tissue (failure) strength.

1.5 Material Properties of the Aorta Wall

      It is very important to use arterial wall with proper material model because the wall stiffness increases when lumen diameter increases and calcification and medial sclerosis occur with aging and disease. The aneurysmal wall has been found to be mechanically anisotropic [74]. Isotropic properties assumption is a reasonable assumption in most cases. A more accurate constitutive model is needed to describe the properties of the wall.

The flat arterial tissue layer is assumed to be a fiber‐reinforced material with relatively stiff collagenous fibers embedded in a homogeneous isotropic (soft) ground matrix [75]. The assumption of strain energy functions holds good only for single continuous medium tissues which is not the case with arterial tissue [76]. The mechanical properties of soft biological tissues depend greatly on their microstructure integration attained in the constitutive model [49]. Taghizadeh et al. [77] proposed a new biaxial constitutive model based on microstructural properties as opposed to the simple uniaxial tests carried out by Sokolis et al. [78] and Karimi et al. [79]. Wall thickness is also very important. Since aneurysmal rupture occurs at a specific site of the aortic lumen, the properties of the wall affect the computed solution. The aortic wall thickness in computational methods is assumed to be in the range of 1.5–2 mm. While this may be largely accurate for most simulations, it has been acknowledged as a major limitation СКАЧАТЬ