OPTIMAL CONTROL BASED ON NONLINEAR CONJUGATE GRADIENT METHOD IN CARDIAC ELECTROPHYSIOLOGY NG KIN WEI UNIVERSITI TEKNOLOGI MALAYSIA

OPTIMAL CONTROL BASED ON NONLINEAR CONJUGATE GRADIENT METHOD IN CARDIAC ELECTROPHYSIOLOGY NG KIN WEI UNIVERSITI TEKNOLOGI MALAYSIA

OPTIMAL CONTROL BASED ON NONLINEAR CONJUGATE GRADIENT METHOD IN CARDIAC ELECTROPHYSIOLOGY NG KIN WEI A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy (Mathematics) Faculty of Science Universiti Teknologi Malaysia APRIL 2013

iii Specially dedicated to my beloved father, Ng Kwai Weng and my beloved mother, Tai Hee.

iv ACKNOWLEDGEMENT This thesis would not have been completed without the invaluable helps and supports that I received from several individuals. Thus, many individuals deserve acknowledgement in this thesis. First of all, I would like to express my deepest gratitude to my supervisor, Associate Professor Dr. Rohanin bt. Ahmad, who suggested and supervised me during my research. I would like to appreciate Dr. Chamakuri Nagaiah, who is one of the founders of the optimal control problem of monodomain model, for his assistance in helping me to understand the mathematical model as well as the related optimal control problem. Also, thanks to him for checking my MATLAB code and provides suggestions for improving it. Lastly, I would like to thank my family who has provided me love, support and motivation from the first day of my study.

v ABSTRACT Sudden cardiac death is often attributed to cardiac arrhythmia, the situation when normal heart rhythm is disordered. In the context of optimal control of cardiac arrhythmia, it is essential to determine the optimal current required to be injected to the patient for dampening the excitation wavefront propagation resulting from cardiac arrhythmia, in which this process is known as defibrillation. Consequently, this leads to an optimization problem arising from cardiac electrophysiology, namely Optimal Control Problem of Monodomain Model (OCPMM). The OCPMM is a nonlinear programming problem that is constrained by parabolic partial differential equation coupled to a system of nonlinear ordinary differential equations, which turned out to be computationally demanding. The main aim of this research is on discovering more efficient optimization methods for solving OCPMM. First, the original complex problem is decomposed into sub-problems through the operator splitting technique for reducing the complexity of OCPMM. Next, the classical, modified and hybrid nonlinear conjugate gradient methods are employed to solve the split and discretized OCPMM. Numerical results prove that the modified method, namely the variant of the Dai-Yuan (VDY) method as well as the new developed hybrid method, namely the hybrid Ng-Rohanin (hnr) method are very efficient in solving OCPMM. Besides that, this research also studies the effects of control domain on OCPMM using two recognized factors, which are the position and the size. Numerical findings indicate that the control domains should consist of small size domains and located near to the excitation domain, for achieving better defibrillation performance. Lastly, based on the observed effects, an ideal control domain is proposed. Numerical results show that lowest current as well as shortest time are required by the ideal control domain during the defibrillation process. As a conclusion, the ideal control domain is capable of ensuring an efficient and successful defibrillation process.

vi ABSTRAK Kematian kardium mengejut sering dikaitkan dengan aritmia jantung, keadaan apabila rentak jantung yang normal bercelaru. Dalam konteks kawalan optimum aritmia jantung, ia adalah penting untuk menentukan arus elektrik optimum yang diperlukan untuk disuntik kepada pesakit untuk melembabkan penyebaran muka gelombang terangsang yang disebabkan oleh aritmia jantung, di mana proses ini dikenali sebagai defibrilasi. Oleh yang demikian, ini membawa kepada masalah pengoptimuman yang timbul daripada elektrofisiologi jantung, iaitu Masalah Kawalan Optimum Model Monodomain (OCPMM). OCPMM adalah masalah pengaturcaraan tak linear yang dikekang oleh persamaan terbitan separa parabola yang ditambah kepada sistem persamaan terbitan biasa tak linear, ternyata menjadi cabaran dalam pengiraan. Tujuan utama penyelidikan ini adalah dalam pencarian kaedah pengoptimuman yang lebih cekap untuk menyelesaikan OCPMM. Pertama sekali, masalah kompleks asal dipecahkan kepada sub masalah dengan menggunakan teknik pemisahan operator untuk mengurangkan kekompleksan OCPMM. Kemudian, kaedah kecerunan konjugat klasik, terubahsuai dan hibrid digunakan untuk menyelesaikan OCPMM yang telah dipecah dan didiskret. Keputusan berangka membuktikan bahawa kaedah terubahsuai, iaitu kaedah varian Dai-Yuan (VDY) serta kaedah hibrid yang baru dibina, iaitu kaedah hibrid Ng-Rohanin (hnr) adalah sangat cekap dalam menyelesaikan OCPMM. Selain itu, penyelidikan ini juga mengkaji kesan domain kawalan bagi OCPMM dengan menggunakan dua faktor yang dikenalpasti, iaitu kedudukan dan saiz. Keputusan berangka menunjukkan bahawa untuk mencapai prestasi defibrilasi yang baik, domain kawalan harus terdiri daripada saiz domain yang kecil dan terletak berhampiran dengan domain perangsangan. Akhirnya, berdasarkan kesan yang diperhatikan, satu domain kawalan unggul dicadangkan. Keputusan berangka menunjukkan arus elektrik yang terendah serta masa yang tersingkat diperlukan oleh domain kawalan unggul semasa proses defibrilasi. Sebagai kesimpulan, domain kawalan unggul mampu memastikan proses defibrilasi dijalankan dengan cekap dan berjaya.