Study Of The Numerical Computational Methods In Biomedical Electromagnetic Forward And Inverse Problems

There are many computational electromagnetic problems in biomedical engineering field,which have some common mathematical characteristics.Due to the different relationship ofsource and field for solved problem,the computational electromagnetic problems can be dividedinto forward problem (FP) and inverse problem (IP).The FP is calculating the field distributionbased on the certain source,and it is from reason to result and always be well-posed:theexistence,uniqueness and continuous dependence of the solution are met.While the IP isreconstructing the source based on the field,it is from result to reason and always be ill-posed:the three conditions are not fully met.The FP is the basis of IP,since it provides the relationshipof source and field for the IP,while the latter is the purpose of the former.Different numericalcomputational methods are required for the FP and IP:finite element method (FEM),boundaryelement method (BEM) or finite difference method (FDM) are often applied to compute the FP,which is often direct and easy;while the calculation of IP always needs to be combined with apriori information about the specific problem,and regulatization are often need to be taken toobtain the stable solutions,which is difficult and costs a lot of efforts.The dissertation focuseson numerical computational methods in biomedical electromagnetic forward and inverseproblems and deales with three problems:electrocardiography (ECG) problem,MRI coil designproblem and magnetic induction tomography (MIT) problem.In ECG field,the source is the heart while the field is the body surface potentials (BSPs).Accordingly,the ECG FP is simulating/calculating the BSPs from heart source,while the ECGIP is reconstructing the information of heart source from the BSPs.Therefore,if excluding thevirtual heart study,the ECG FP and IP include the research about cardiac equivalent sources,volume conductor modeling,FP calculation and IP calculation.In this dissertation,the BEM isapplied to calculate the ECG FP.The volume conductor model and virtual heart is the one whichdeveloped before in our lab.The regularization methods are used to solve the ECG IP and theepicardial potentials (EPs) are reconstructed.In the MRI coil design problem,the current passes through the coils of wires and producesthe magnetic field on target domain.The optimal design of the coils of wires is the work in MRIcoils design,which also includes the relationship of source and field:source is the current carried coils of wires and the field is the produced magnetic field.To design of high performance MRIgradient and RF coils is always the hot topic in MRI academical and industry field and lots ofmethods have been proposed to design the coils,which needs certain numerical calculation.Inthis dissertation,the coil design methods are studied.The objective of MIT,like ECG problem,is to develop new imaging modality.The imagingtarget of MIT is the dielectric characteristics like conductivity.Based on the electromagneticinductive theory,MIT provides a noninvasive and non contact imaging technique,which hasmany potential applications in biomedical and industrial field.While this dissertation focuses onthe biomedical application of MIT:the detection and monitoring of Brain Bleeding,mainly inthe simulation study of the MIT system and numerical computational methods concerned.Although the three projects have the individual characteristics,they have many things incommon:they are all the approximation of the Maxwell equations in quasi-static condition;theyhave similar numerical computational methods to solve the forward and inverse problem,et'al.The dissertation has done the following research work:1.Study of the ECG FPThe dissertation analyses and discusses the main content,numerical computational methodsand research progress in the study of ECG FP in detail.After the analysis of the commonnumerical methods in ECG FP,and the importance and complexity of the mesh,the h-adaptiveBEM (h-aBEM) is proposed for the ECG FP.The h-aBEM calculation is an iterativeoptimization mesh procedure,and the optimal mesh is built by adding new nodes andreconnection based on the BEM calculation.H-aBEM is used to calculate the ECG FP and IP inthis dissertaion,and the simulation results show that the application of h-aBEM not onlyimproved the accuray of the solutions of ECG FP and IP,but also has some introduction foroptimal choice of electrodes for BSP measurement.In ordert to save the computational cost forh-aBEM,another h-aBEM called hierarchical h-aBEM is also used.In the implementation ofhierarchical h-aBEM,the hierarchical shape function for linear triangular element is deduced forthe first time.Using the virtual heart model developed in lab,the two h-aBEM and the tworefinement schemes are studied deeply.2.Study ofregularization methods in ECG IPThe dissertation discusses the application of common regularization methods like Tikhonov and TSVD etc.,and compares the different regularization parameter choose methods in ECG IP.Considering the real situation of ECG IP:due to the effect of the volume conductor modelingand heart beating,the two sides of the discretized linear system exist the different error\noises,the regularization methods based on the total least square (TLS) theory:regularization TLS(RTLS) and truncated TLS (TTLS),are proposed to calculate the ECG IP.The proposed methodsdeal with the problem existed in ECG IP for over thirty years,which is process the mearsurementnoise and geometric noise meantime.After the analysis of the basic theory and algorithms ofTTLS and RTLS,the simulation experiments shows that the proposed new regularizationmethods can robustly reconstruct EPs from BSPs and much effective in practical ECG IP.3.Study of MRI coil design methodsAfter analysis and discussion of MRI coils design methods,this dissertation proposes a newscheme,which combines the BEM and regularization techniques,to design the MRI coil.Introducing the the stream function expression of current density source,the source and field arediscretized calculated using BEM and the IP in coil design is calculated by Tikhonov method.Using the proposed scheme,the biplanar transverse gradient coil for open MRI system and theRF phase array coils are designed.The designed coil results show that the proposed scheme isfeasible and easily to be implemented,and it can be further used to design 2D or 3D coils basedon arbitrally geometry.4.Study of MITIn the study of MIT in this dissertation,firstly,the edge FEM is used to calculate the eddycurrent electromagnetic problem and the Jacobian matrix for MIT image reconstruction,thenregularization methods are used to reconstruct the conductivity information.Based on Philips 16channels MIT system,this disseration includes the simulation work in optimal design of MITsystem including coils and size of system optimization,estimation of the electromagnetic salty,the feasibility study of Beagle dog bleeding experiment,etc.Besides,according to thecharacteristics of MIT image reconstruction,the total variation (TV) method is tudy in MIT IPand the simulation/experiment results show that TV method is much better in artifacts removaland edge keeping compared to thel₂ norm constrain methods.