to inflate the vein; we expect that the deformation field be axially symmetric and hence uniform on the surface. This prompts us to assume a deformation of the form: x = ax 0 + a x 1X + a x 2Y + a x 3Z, y= a y 0 +a y 1X + a y 2Y + a y 3Z, z= a z 0 +a z 1X + a z 2Y + a z 3Z, (2) where (X,Y,Z) are the Cartesian coordinates of the centroid of a marker in the reference configuration and (x,y,z) are the Cartesian coordinates of the centroid of the same marker in the current configuration, ax i, a y i and a z i (for i = {0,1,2,3}) are constants. Knowing the location of five markers in the current and reference configurations, we obtain ax i, a y i and a z i , by solving equation (2) in the least square sense. Then, the matrix components of the deformation gradient, F is computed as: ¸ ¸ ¸ ¹ · ¨ ¨ ¨ © § 3 2 1 3 2 1 3 2 1 z z z y y y x x x a a a a a a a a a F . (3) It is now straightforward to compute the right Cauchy-Green deformation tensor, C = FtF and its invariants: I1 C = tr(C), I 2 C = tr(C-1), I 3 C = det(C). (4) Before proceeding further a few comments have to be made. Since, we are tracking markers in 3D space, and the markers are not on the same plane, we are able to estimate all the nine components of the deformation gradient. However, since all the markers are on the surface of the vein, the determined deformation gradient is for the surface of the vein. Also, though the constants ax i, a y i and a z i could be found using just four markers, it resulted in larger errors because the volume of the tetrahedron with these four markers at their corners is small that the noises get magnified. Consequently, we use five markers to find the unknown constants in the deformation field (2). We observe that comparing the invariants of the Cauchy-Green deformation tensor is necessitated because one cannot ensure that the laboratory coordinates is the same between the specimens that are tested and therefore one cannot compare the components of the deformation gradient. One cannot ensure that the laboratory coordinates is the same between the experiments because the calibrating wedge dimension is small (12 mm x 7.5 mm x 10 mm), it is difficult to orient it accurately in the same direction each time. Of course any invariants could also be used. But there is some merit in using the invariants defined in (4). Any deviation of I3 C from 1 indicates that the material is compressible. Further, if the deformation gradient, F is singular then I3 C would be close to 0 and I 2 C be a large number. The deformation gradient would become singular if the markers selected to find the deformation field happened to lie nearly on a plane. Thus, the validity of the determined deformation field could also be ascertained by using the invariants defined in equation (4). 2.5 Materials Discarded Saphenous veins from patients undergoing coronary artery bypass surgery were collected from Fortis Malar Hospital Chennai and stored at 4oC in normal saline. The experiments were completed within 24 hours from harvest. Details of the specimens tested are tabulated in table 1. Test samples were connected using plexiglass ferrules on both the ends and then connected to 2-way stopcock. This arrangement again connected to the 3-way stopcock. The vein is inflated by pumping saline through the syringe pump at one end and the other end of the sample is connected to the pressure transducer. Markers are placed in the middle portion of the vein using water insoluble eyeliner (see figure 2). Table 1: Details of vein samples tested Sample No Age (Years) Sex Length (mm) Diameter (mm) Wall thickness (mm) 1 67 F 40 4.0 0.5 2 63 M 25 5.0 0.5 1 Figure 2: Arrangement of markers for (a) sample 1 (b) sample 2 Figure 3: Close up view of the vein connected to the plexiglass ferrules with markers in the middle. 3 6 9 10 11 12 7 4 8 5 2 1 (a) 2 3 4 5 6 7 8 9 10 11 12 (b) 82
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