t. This paper surveys some of the work our group has done in electrical impedance tomography.

Electrical Impedance Tomography

There has been tremendous advances in our ability to produce images of human brain function. Applications of functional brain imaging extend from improving our understanding of the basic mechanisms of cognitive processes to better characterization of pathologies that impair normal function. Magnetoencephalography (MEG) and electroencephalography (EEG) (MEG/EEG) localize neural electrical activity using noninvasive measurements of external electromagnetic signals. Among the available functional imaging techniques, MEG and EEG uniquely have temporal resolutions below 100 ms. This temporal precision allows us to explore the timing of basic neural processes at the level of cell assemblies. MEG/EEG source localization draws on a wide range of signal processing techniques including digital filtering, three-dimensional image analysis, array signal processing, image modeling and reconstruction, and, blind source separation and phase synchrony estimation. We describe the underlying models currently used in MEG/EEG source estimation and describe the various signal processing steps required to compute these sources. In particular we describe methods for computing the forward fields for known source distributions and parametric and imaging-based approaches to the inverse problem.

Electromagnetic brain mapping

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

/pdf/inverse-problems-a-bayesian-perspective-1dknxttxjb.pdf

Inverse problems: A Bayesian perspective

Now in its second edition, this accessible text presents a unified Bayesian treatment of state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models. The book focuses on discrete-time state space models and carefully introduces fundamental aspects related to optimal filtering and smoothing. In particular, it covers a range of efficient non-linear Gaussian filtering and smoothing algorithms, as well as Monte Carlo-based algorithms. This updated edition features new chapters on constructing state space models of practical systems, the discretization of continuous-time state space models, Gaussian filtering by enabling approximations, posterior linearization filtering, and the corresponding smoothers. Coverage of key topics is expanded, including extended Kalman filtering and smoothing, and parameter estimation. The book's practical, algorithmic approach assumes only modest mathematical prerequisites, suitable for graduate and advanced undergraduate students. Many examples are included, with Matlab and Python code available online, enabling readers to implement algorithms in their own projects.

Bayesian Filtering and Smoothing

A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology.

Diffuse optical tomography (DOT) is a noninvasive imaging technology that is sensitive to local concentration changes in oxyand deoxyhemoglobin. When applied to functional neuroimaging, DOT measures hemodynamics in the scalp and brain that reflect competing metabolic demands and cardiovascular dynamics. Separating the effects of systemic cardiovascular regulation from the local dynamics is vitally important in DOT analysis. In this paper, we use auxiliary physiological measurements such as blood pressure and heart rate within a Kalman filter framework to model physiological components in DOT. We validate the method on data from a human subject with simulated local hemodynamic responses added to the baseline physiology. The proposed method significantly improved estimates of the local hemodynamics in this test case. Cardiovascular dynamics also affect the blood oxygen dependent (BOLD) signal in functional magnetic resonance imaging (fMRI). This Kalman filter framework for DOT may be adapted for BOLD fMRI analysis and multimodal studies.

https://www.nmr.mgh.harvard.edu/DOT/PDF/2005/Diamond_LNCS_3750_649_2005.pdf

Physiological system identification with the kalman filter in diffuse optical tomography

Diffuse optical tomography is a highly unstable problem with respect to modeling and measurement errors. During clinical measurements, the body shape is not always known, and an approximate model domain has to be employed. The use of an incorrect model domain can, however, lead to significant artifacts in the reconstructed images. Recently, the Bayesian approximation error theory has been proposed to handle model-based errors. In this work, the feasibility of the Bayesian approximation error approach to compensate for modeling errors due to unknown body shape is investigated. The approach is tested with simulations. The results show that the Bayesian approximation error method can be used to reduce artifacts in reconstructed images due to unknown domain shape.

Compensation of modeling errors due to unknown domain boundary in diffuse optical tomography.

In this paper, a model-based optimization method is derived to control the thermal dose in biological tissues for ultrasound surgery. The optimization method uses the bioheat equation as a system model and quadratic cost criteria for the desired thermal dose. Time-harmonic quasi-stationary ultrasound fields are used as the heat source. In this method the optimal phase and the amplitude trajectories are found directly by minimizing the associated cost function. The approach also allows for maximum input amplitude constraints. The method is based on the Hamiltonian form of the system and results in a large dimensional nonlinear optimization problem which is solved with a gradient-type iterative scheme. The performance of the optimization method is tested with 2D simulations and it is shown that the approach is able to yield a feasible nominal solution. This nominal evolution would then eventually be sought to be maintained with the help of a feedback controller during the actual sonication.

Thermal dose optimization method for ultrasound surgery

https://iopscience.iop.org/article/10.1088/0266-5611/30/11/110301/pdf

Inverse problems in the Bayesian framework

In electrical impedance tomography (EIT) an approximation for the internal resistivity distribution is computed based on the knowledge of the injected currents and measured voltages on the surface of the body. In some applications the resistivity changes may be so fast that the information on the time evolution of the resistivity distribution is either lost or severely blurred. In this paper we study with phantom experiments the capabilities of the earlier proposed approaches, Kalman filter and Kalman smoother. We show that in “two-dimensional” phantom these approaches are capable of reconstructing a sequence of absolute images of a moving object. By an absolute image we mean that no additional reference voltage measurement is needed for the image reconstruction. An image is obtained after each current injection. Also, when compared to traditional reconstructions, the dynamic approaches reveal much more information about the dynamic behaviour of the object.

Jari P. Kaipio

Papers

Physiological system identification with the kalman filter in diffuse optical tomography

Compensation of modeling errors due to unknown domain boundary in diffuse optical tomography.

Thermal dose optimization method for ultrasound surgery

Inverse problems in the Bayesian framework

Dynamic electrical impedance tomography - phantom studies