BSPLINE-BASED SOLUTIONS OF DIFFERENTIAL EQUATIONS IN BIOENGINEERING: MOTION MONITORING AND IMAGE PROCESSING APPLICATIONS

Abstract

The bachelor¿s thesis objective consists of developing a quintic spline numerical algorithm and study its potential for solving some different bioengineering problems. The numerical algorithm will use collocation methods for finding the approximation of the mathematical analysis problem. This approximation will be estimated with a quintic spline.

In general terms, mathematical analysis problems are found in different areas, where it could be highlighted physical sciences and engineering, moreover it also can be found in social sciences, business, medicine, biology, etc. This method can be used for any of these areas if they fulfil the specific requirements given by the design constraints and by the numerical dynamics.

Although the algorithm will be prepared for solving a great amount of numerical problems, along this work, it will be more focused on the applications related to bioengieneering and medicine. The algorithm is designed to solve problems either formulated as an initial value problem or formulated as boundary value problems. It can be adapted to solve differential equations and non-linear differential equations. It is possible to scale the algorithm for solving multivariate functions and multidimensional function.

The work will have two parts. The first part is oriented to evaluate the algorithm in the different conditions where it is projected to work, this part also will illustrate the way that collocation methods work. It is useful as it will aid to design the hyper-parameters of the rest of the parts. At this part it will be carried some simple synthetic experiments with known solution but with poor applicability in areas of interest.

In this work, a final applied synthetic experiments will be carried out. This experiment will approximate the Gompertz model to estimate the evolution of the population of tumour cells along the time from an initial population of time and with some experimented parameters. As this equation is analytically solved, this experiment is also useful to validate the algorithm as the previous one. During this part of the work, synthetic experiments had been carried out. In the work, we define a synthetic experiment as an approximation of a function that can be fully expressed as a differential equation. Then, it is defined a practical equation as an experiment where an acquired value vary along the time. In this two experiments, acquisition and experimental parameters are needed but do change along the independent variable.

Finally, at the second part of this work, two practical experiments are carried out. One application, in the field of motion monitoring applied to the rehabilitation field and, the second application in the field of image processing applied to laparoscopic liver resection. In rehabilitation, the estimation of the position and it could be done using the angular velocity and the acceleration given by an IMU. It can also be given by an optical position system that in the case of this experiment allow to estimate the performance of an algorithm. In different surgical procedure, pre-operative models are essential to incorporate information from other imaging process to the procedure. The objective of the last application is to represent a tumour preoperative model as a derivable spline.