Solving inverse problems for medical applications

It is essential to have an accurate feedback system to improve the navigation of surgical tools. This thesis investigates how to solve inverse problems using the example of two medical prototypes. The first aims to detect the Sentinel Lymph Node (SLN) during the biopsy. This will allow the surgeon to remove the SLN with a small incision, reducing trauma to the patient. The second investigates how to extract depth and tissue characteristic information during bone ablation using the emitted acoustic wave.
We solved inverse problems to find our desired solution. For this purpose, we investigated three approaches: In Chapter 3, we had a good simulation of the forward problem; namely, we used a fingerprinting algorithm. Therefore, we compared the measurement with the simulations of the forward problem, and the simulation that was most similar to the measurement was a good approximation. To do so, we used a dictionary of solutions, which has a high computational speed. However, depending on how fine the grid is, it takes a long time to simulate all the solutions of the forward problem. Therefore, a lot of memory is needed to access the dictionary.
In Chapter 4, we examined the Adaptive Eigenspace method for solving the Helmholtz equation (Fourier transformed wave equation). Here we used a Conjugate quasi-Newton (CqN) algorithm. We solved the Helmholtz equation and reconstructed the source shape and the medium velocity by using the acoustic wave at the boundary of the area of interest. We accomplished this in a 2D model. We note, that the computation for the 3D model was very long and expensive. In addition, we simplified some conditions and could not confirm the results of our simulations in an ex-vivo experiment.
In Chapter 5, we developed a different approach. We conducted multiple experiments and acquired many acoustic measurements during the ablation process. Then we trained a Neural Network (NN) to predict the ablation depth in an end-to-end model. The computational cost of predicting the depth is relatively low once the training is complete. An end-to-end network requires almost no pre-processing. However, there were some drawbacks, e.g., it is cumbersome to obtain the ground truth.
This thesis has investigated several approaches to solving inverse problems in medical applications. From Chapter 3 we conclude that if the forward problem is well known, we can drastically improve the speed of the algorithm by using the fingerprinting algorithm. This is ideal for reconstructing a position or using it as a first guess for more complex reconstructions. The conclusion of Chapter 4 is that we can drastically reduce the number of unknown parameters using Adaptive Eigenspace method. In addition, we were able to reconstruct the medium velocity and the acoustic wave generator. However, the model is expensive for 3D simulations. Also, the number of transducers required for the setup was not applicable to our intended setup. In Chapter 5 we found a correlation between the depth of the laser cut and the acoustic wave using only a single air-coupled transducer. This encourages further investigation to characterize the tissue during the ablation process.