## 5 Discussion  
Even though SPH and Shape Matching had already been used to simulate viscoelastic fluids, to our knowledge, this was the first time a viscoelastic fluid was used in conjunction with a biophysical model, specifically, the monodomain model, to simulate biological tissue. Additionally, the use of SPH to provide an alternative mean to solve the monodomain model was implemented and used in the presented simulations. To our knowledge, this was also the first time the monodomain model was solved with SPH. According to the results obtained, the following contributions of this research can be highlighted:  
* **Biological tissue simulation. ** Most of the previous work related to the simulation of biological tissue used methods such as FEM. However, the definition and pre-processing of the mesh, in addition to the high computational cost, even with the use of GPGPU, made the method less than ideal for real time simulations. Here, an alternative mean to simulate biological tissue was presented. In this case, a skeletal muscle was simulated by using a skeletal muscle geometry and a specific activation model. By changing the geometry and the constitutive model, different biological tissues could be simulated. Since SPH was used to simulate the tissue, an integration with other particle-based models is also possible; for example, the inclusion of blood in the tissue. * **Solution of the monodomain model with SPH. ** The monodomain model was usually solved with FEM or the FDM. The use of GPUs was also explored to speed the simulations up. Here, the model was solved with SPH, which is also paralellizable with GPUs, and the results were similar to the ones of a simulation with the FDM. These results show that biophysical models can be solved with the SPH method, and that the solved properties can be easily included in a more complex model. * **Tissue deformation. ** The focus of this work was on the deformation of the muscle tissue by using a biophysical model. In order to apply the transmembrane potential to the tissue, and deform it, the potential was considered as a force of pressure that acted on the fluid. Since the fluid, and in this case, the tissue, flows from regions of high pressure to regions of low pressure, the added pressure made it so that the tissue contracted in a given direction when a stimulus current was applied, or it relaxed to its original shape when the current was removed. If the model were to be used in simulations, additional external forces could be applied to the tissue, and it would appropriately respond because it preserves volume, and the internal pressure force would change and deform the tissue. To our knowledge, this was the first time that such a model was used. The proposed method was able to achieve a contraction of around 23%, which is similar to the achieved contraction of a muscle. Additionally, the RMSE for the simulation was low when compared to a FEM simulation that has proven to be reliable to simulate skeletal muscles. Figure 15: Average speedups for smoothed particle hydrodynamics (SPH) kernels when compared to FEM using CPU. * **Achieved real time simulations. ** In order for the model to be viable in interactive simulations, it had to be able to be simulated in real time, at least 30 FPS. The CPU version, with around 9888 particles, was able to run in real time; at an average of 27.5 FPS for the mesh-based simulation. If 18,475 particles were simulated in CPU, an average of 3.7 FPS were obtained for the mesh-based simulation. In order to get a more detailed tissue, and to be able to simulate more than one, the use of GPGPU was proposed. The GPU version, with 18,475 particles, ran at 70.125 FPS for the mesh-based simulation. Even though the achieved speedups for FPS were not as high, speedups of more than 250 were achieved for specific parts of the method. Additionally, different techniques, such as reducing the neighbor search space, were used to reduce the computational complexity of the model.