In this section we detail the performed work in the modeling of each organ, particularly each constitutive law employed for that purpose.
Liver
The liver is the biggest gland in the human body. It is connected to the diaphragm by the coronary ligament so it seems reasonable to assume it to be constrained at the posterior face by the rest of the organs, while the anterior face is accessible to the surgeon. The inferior vena cava travels along the posterior surface, and the liver is frequently assumed clamped a that location. The literature on the mechanical properties of the liver parenchyma is not very detailed. In [24] a Mooney–Rivlin and an Ogden models are compared to experimental results on deformations applied to a liver. No clear conclusion is obtained, however, given that no in vivo measurements could be performed. In view of that, we have assumed a simplified Kirchhoff-Saint Venant model, with Young’s modulus of 1.60 kPa, and a Poisson coefficient of 0.48, thus nearly incompressible [25]. This constitutes just a simplification that should be validated with the help of experienced surgeons, but remains valid as long as more complex models can be developed within the PGD techniques exposed before without any difficulty [26].
The finite element mesh of the liver, see Fig. 4 consists of 4349 nodes and 21,788 linear tetrahedral elements. The PGD modes obtained in the off-line procedure described before are shown in Fig. 5.
Gallbladder
The gallbladder is a pressurized vesicle attached to the liver and also connected to the duodenum. It contains the bile generated by the hepatocytes [27] provides the most comprehensive constitutive modeling of gallbladder walls. Gallbladder is composed by four different layers: adventitia, muscle, mucosa and epithelium. Of course, the muscle layer is responsible of expelling the bile right after consumption of foods or drinks.
By employing elastography experimental measurements [27], developed a model very closely resembling that of Gasser-Holzapfel-Ogden for arteries, previously employed in some of our previous works [10]. However, it is also stated that a linear elastic model that takes into account the heavy changes in gallbladder wall thickness produces almost the same results. Also a neo-Hookean model [28] has been found to accurately capture most of the deformation patterns of elastin. Therefore, we have adopted again, for simplicity and without loss in generality, a Kirchhoff-Saint Venant model, despite its well-known limitations.
The gallbladder is subjected, in normal conditions, to an internal pressure that has been estimated in [27] on 466.6 Pa. We have assumed a wall thickness on the order of 2.5 mm, and a Young’s modulus on the order of 1.15 kPa, with \(\nu \,=\,\)0.48, and thus nearly incompressible.
The finite element model, see Fig. 6, is composed by 4183 nodes and 17,540 linear tetrahedral elements Fig. 7.
In principle the gallbladder has not been considered as attached to the liver, but to the duodenum only, since during the surgery procedure it needs to be detached from it, by appropriately simulating the scratching process done by the surgeon, more properly related to continuum damage mechanics than to cutting itself. This is currently one of our lines of research.
Pancreas and duodenum
In our model, pancreas and duodenum have been modeled as being attached to each other, see Fig. 8, since indeed they are, on one side, and most likely will be removed together, without detaching one from the other.
Very few papers deal with the constitutive modeling of pancreatic tissue, despite a few simulators exist, see for instance [29] for an example of a web-based navigation system. In [30] elastography is employed to determine the shear stiffness of the tissue, giving some 1.20 kPa at 40 Hz. In our simulations, an almost incompressible character (i.e., \(\nu \,=\,\)0.48) is assumed.
The two free opposite sides of the duodenum are considered clamped. In particular, the proximal one is indeed attached to the stomach, whose distal part is usually removed during this type of surgery. Therefore, in a more advanced version of the simulator, a model of the stomach should also be considered for completeness. In this proof of concept prototype, however, this simplification does not imply a loss of validity of the proposed methodology. In Fig. 9 the first four modes of the PGD approximation to the pancreatic vademecum are depicted.