A hybrid multi-particle approach to range assessment-based treatment verification in particle therapy

Particle therapy (PT) used for cancer treatment can spare healthy tissue and reduce treatment toxicity. However, full exploitation of the dosimetric advantages of PT is not yet possible due to range uncertainties, warranting development of range-monitoring techniques. This study proposes a novel range-monitoring technique introducing the yet unexplored concept of simultaneous detection and imaging of fast neutrons and prompt-gamma rays produced in beam-tissue interactions. A quasi-monolithic organic detector array is proposed, and its feasibility for detecting range shifts in the context of proton therapy is explored through Monte Carlo simulations of realistic patient models and detector resolution effects. The results indicate that range shifts of [Formula: see text] can be detected at relatively low proton intensities ([Formula: see text] protons/spot) when spatial information obtained through imaging of both particle species are used simultaneously. This study lays the foundation for multi-particle detection and imaging systems in the context of range verification in PT.

Core Imaging Library - Part II: multichannel reconstruction for dynamic and spectral tomography

The newly developed core imaging library (CIL) is a flexible plug and play library for tomographic imaging with a specific focus on iterative reconstruction. CIL provides building blocks for tailored regularized reconstruction algorithms and explicitly supports multichannel tomographic data. In the first part of this two-part publication, we introduced the fundamentals of CIL. This paper focuses on applications of CIL for multichannel data, e.g. dynamic and spectral. We formalize different optimization problems for colour processing, dynamic and hyperspectral tomography and demonstrate CIL's capabilities for designing state-of-the-art reconstruction methods through case studies and code snapshots. This article is part of the theme issue 'Synergistic tomographic image reconstruction: part 2'.

Core Imaging Library - Part I: a versatile Python framework for tomographic imaging

We present the Core Imaging Library (CIL), an open-source Python framework for tomographic imaging with particular emphasis on reconstruction of challenging datasets. Conventional filtered back-projection reconstruction tends to be insufficient for highly noisy, incomplete, non-standard or multi-channel data arising for example in dynamic, spectral and in situ tomography. CIL provides an extensive modular optimization framework for prototyping reconstruction methods including sparsity and total variation regularization, as well as tools for loading, preprocessing and visualizing tomographic data. The capabilities of CIL are demonstrated on a synchrotron example dataset and three challenging cases spanning golden-ratio neutron tomography, cone-beam X-ray laminography and positron emission tomography. This article is part of the theme issue 'Synergistic tomographic image reconstruction: part 2'.

Comparing D-bar and common regularization-based methods for electrical impedance tomography

Objective:

To compare D-bar difference reconstruction with regularized linear reconstruction in electrical impedance tomography.

Approach:

A standard regularized linear approach using a Laplacian penalty and the GREIT method for comparison to the D-bar difference images. Simulated data was generated using a circular phantom with small objects, as well as a ‘Pac-Man’ shaped conductivity target. An L-curve method was used for parameter selection in both D-bar and the regularized methods.

Main results:

We found that the D-bar method had a more position independent point spread function, was less sensitive to errors in electrode position and behaved differently with respect to additive noise than the regularized methods.

Significance:

The results allow a novel pathway between traditional and D-bar algorithm comparison.

Three Dimensional Polarimetric Neutron Tomography of Magnetic Fields

Through the use of Time-of-Flight Three Dimensional Polarimetric Neutron Tomography (ToF 3DPNT) we have for the first time successfully demonstrated a technique capable of measuring and reconstructing three dimensional magnetic field strengths and directions unobtrusively and non-destructively with the potential to probe the interior of bulk samples which is not amenable otherwise. Using a pioneering polarimetric set-up for ToF neutron instrumentation in combination with a newly developed tailored reconstruction algorithm, the magnetic field generated by a current carrying solenoid has been measured and reconstructed, thereby providing the proof-of-principle of a technique able to reveal hitherto unobtainable information on the magnetic fields in the bulk of materials and devices, due to a high degree of penetration into many materials, including metals, and the sensitivity of neutron polarisation to magnetic fields. The technique puts the potential of the ToF time structure of pulsed neutron sources to full use in order to optimise the recorded information quality and reduce measurement time.

Characterization of objects by electrosensing fish based on the first order polarization tensor

Weakly electric fish generate electric current and use hundreds of voltage sensors on the surface of their body to navigate and locate food. Experiments (von der Emde and Fetz 2007 J. Exp. Biol. 210 3082-95) show that they can discriminate between differently shaped conducting or insulating objects by using electrosensing. One approach to electrically identify and characterize the object with a lower computational cost rather than full shape reconstruction is to use the first order polarization tensor (PT) of the object. In this paper, by considering experimental work on Peters' elephantnose fish Gnathonemus petersii, we investigate the possible role of the first order PT in the ability of the fish to discriminate between objects of different shapes. We also suggest some experiments that might be performed to further investigate the role of the first order PT in electrosensing fish. Finally, we speculate on the possibility of electrical cloaking or camouflage in prey of electrosensing fish and what might be learnt from the fish in human remote sensing.

Temporal sparsity exploiting nonlocal regularization for 4D computed tomography reconstruction

X-ray imaging applications in medical and material sciences are frequently limited by the number of tomographic projections collected. The inversion of the limited projection data is an ill-posed problem and needs regularization. Traditional spatial regularization is not well adapted to the dynamic nature of time-lapse tomography since it discards the redundancy of the temporal information. In this paper, we propose a novel iterative reconstruction algorithm with a nonlocal regularization term to account for time-evolving datasets. The aim of the proposed nonlocal penalty is to collect the maximum relevant information in the spatial and temporal domains. With the proposed sparsity seeking approach in the temporal space, the computational complexity of the classical nonlocal regularizer is substantially reduced (at least by one order of magnitude). The presented reconstruction method can be directly applied to various big data 4D (x, y, z+time) tomographic experiments in many fields. We apply the proposed technique to modelled data and to real dynamic X-ray microtomography (XMT) data of high resolution. Compared to the classical spatio-temporal nonlocal regularization approach, the proposed method delivers reconstructed images of improved resolution and higher contrast while remaining significantly less computationally demanding.

Employing temporal self-similarity across the entire time domain in computed tomography reconstruction

There are many cases where one needs to limit the X-ray dose, or the number of projections, or both, for high frame rate (fast) imaging. Normally, it improves temporal resolution but reduces the spatial resolution of the reconstructed data. Fortunately, the redundancy of information in the temporal domain can be employed to improve spatial resolution. In this paper, we propose a novel regularizer for iterative reconstruction of time-lapse computed tomography. The non-local penalty term is driven by the available prior information and employs all available temporal data to improve the spatial resolution of each individual time frame. A high-resolution prior image from the same or a different imaging modality is used to enhance edges which remain stationary throughout the acquisition time while dynamic features tend to be regularized spatially. Effective computational performance together with robust improvement in spatial and temporal resolution makes the proposed method a competitive tool to state-of-the-art techniques.

Multimodal Image Reconstruction Using Supplementary Structural Information in Total Variation Regularization

In this paper, we propose an iterative reconstruction algorithm which uses available information from one dataset collected using one modality to increase the resolution and signal-to-noise ratio of one collected by another modality. The method operates on the structural information only which increases its suitability across various applications. Consequently, the main aim of this method is to exploit available supplementary data within the regularization framework. The source of primary and supplementary datasets can be acquired using complementary imaging modes where different types of information are obtained (e.g. in medical imaging: anatomical and functional). It is shown by extracting structural information from the supplementary image (direction of level sets) one can enhance the resolution of the other image. Notably, the method enhances edges that are common to both images while not suppressing features that show high contrast in the primary image alone. In our iterative algorithm we use available structural information within a modified total variation penalty term. We provide numerical experiments to show the advantages and feasibility of the proposed technique in comparison to other methods.

Mutual information as a measure of image quality for 3D dynamic lung imaging with EIT

We report on a pilot study of dynamic lung electrical impedance tomography (EIT) at the University of Manchester. Low-noise EIT data at 100 frames per second were obtained from healthy male subjects during controlled breathing, followed by magnetic resonance imaging (MRI) subsequently used for spatial validation of the EIT reconstruction. The torso surface in the MR image and electrode positions obtained using MRI fiducial markers informed the construction of a 3D finite element model extruded along the caudal-distal axis of the subject. Small changes in the boundary that occur during respiration were accounted for by incorporating the sensitivity with respect to boundary shape into a robust temporal difference reconstruction algorithm. EIT and MRI images were co-registered using the open source medical imaging software, 3D Slicer. A quantitative comparison of quality of different EIT reconstructions was achieved through calculation of the mutual information with a lung-segmented MR image. EIT reconstructions using a linear shape correction algorithm reduced boundary image artefacts, yielding better contrast of the lungs, and had 10% greater mutual information compared with a standard linear EIT reconstruction.

Shape deformation in two-dimensional electrical impedance tomography

Electrical impedance tomography (EIT) uses measurements from surface electrodes to reconstruct an image of the conductivity of the contained medium. However, changes in measurements result from both changes in internal conductivity and changes in the shape of the medium relative to the electrode positions. Failure to account for shape changes results in a conductivity image with significant artifacts. Previous work to address shape changes in EIT has shown that in some cases boundary shape and electrode location can be uniquely determined for isotropic conductivities; however, for geometrically conformal changes, this is not possible. This prior work has shown that the shape change problem can be partially addressed. In this paper, we explore the limits of compensation for boundary movement in EIT using three approaches. First, a theoretical model was developed to separate a deformation vector field into conformal and nonconformal components, from which the reconstruction limits may be determined. Next, finite element models were used to simulate EIT measurements from a domain whose boundary has been deformed. Finally, an experimental phantom was constructed from which boundary deformation measurements were acquired. Results, both in simulation and with experimental data, suggest that some electrode movement and boundary distortions can be reconstructed based on conductivity changes alone while reducing image artifacts in the process.

Impact of model shape mismatch on reconstruction quality in electrical impedance tomography

Electrical impedance tomography (EIT) is a low-cost, noninvasive and radiation free medical imaging modality for monitoring ventilation distribution in the lung. Although such information could be invaluable in preventing ventilator-induced lung injury in mechanically ventilated patients, clinical application of EIT is hindered by difficulties in interpreting the resulting images. One source of this difficulty is the frequent use of simple shapes which do not correspond to the anatomy to reconstruct EIT images. The mismatch between the true body shape and the one used for reconstruction is known to introduce errors, which to date have not been properly characterized. In the present study we, therefore, seek to 1) characterize and quantify the errors resulting from a reconstruction shape mismatch for a number of popular EIT reconstruction algorithms and 2) develop recommendations on the tolerated amount of mismatch for each algorithm. Using real and simulated data, we analyze the performance of four EIT reconstruction algorithms under different degrees of shape mismatch. Results suggest that while slight shape mismatch is well tolerated by all algorithms, using a circular shape severely degrades their performance.

In vivo impedance imaging with total variation regularization

We show that electrical impedance tomography (EIT) image reconstruction algorithms with regularization based on the total variation (TV) functional are suitable for in vivo imaging of physiological data. This reconstruction approach helps to preserve discontinuities in reconstructed profiles, such as step changes in electrical properties at interorgan boundaries, which are typically smoothed by traditional reconstruction algorithms. The use of the TV functional for regularization leads to the minimization of a nondifferentiable objective function in the inverse formulation. This cannot be efficiently solved with traditional optimization techniques such as the Newton method. We explore two implementations methods for regularization with the TV functional: the lagged diffusivity method and the primal dual-interior point method (PD-IPM). First we clarify the implementation details of these algorithms for EIT reconstruction. Next, we analyze the performance of these algorithms on noisy simulated data. Finally, we show reconstructed EIT images of in vivo data for ventilation and gastric emptying studies. In comparison to traditional quadratic regularization, TV regularization shows improved ability to reconstruct sharp contrasts.

Validation of a finite-element solution for electrical impedance tomography in an anisotropic medium

Electrical impedance tomography is an imaging method, with which volumetric images of conductivity are produced by injecting electrical current and measuring boundary voltages. It has the potential to become a portable non-invasive medical imaging technique. Until now, implementations have neglected anisotropy even though human tissues such as bone, muscle and brain white matter are markedly anisotropic. We present a numerical solution using the finite-element method that has been modified for modelling anisotropic conductive media. It was validated in an anisotropic domain against an analytical solution in an isotropic medium after the isotropic domain was diffeomorphically transformed into an anisotropic one. Convergence of the finite element to the analytical solution was verified by showing that the finite-element error norm decreased linearly related to the finite-element size, as the mesh density increased, for the simplified case of Laplace's equation in a cubic domain with a Dirichlet boundary condition.

Temporal image reconstruction in electrical impedance tomography

Electrical impedance tomography (EIT) calculates images of the body from body impedance measurements. While the spatial resolution of these images is relatively low, the temporal resolution of EIT data can be high. Most EIT reconstruction algorithms solve each data frame independently, although Kalman filter algorithms track the image changes across frames. This paper proposes a new approach which directly accounts for correlations between images in successive data frames. Image reconstruction is posed in terms of an augmented image x and measurement vector y, which concatenate the values from the d previous and future frames. Image reconstruction is then based on an augmented regularization matrix R, which accounts for a model of both the spatial and temporal correlations between image elements. Results are compared for reconstruction algorithms based on independent frames, Kalman filters and the proposed approach. For low values of the regularization hyperparameter, the proposed approach performs similarly to independent frames, but for higher hyperparameter values, it uses adjacent frame data to reduce reconstructed image noise.

A narrow-band level set method applied to EIT in brain for cryosurgery monitoring

In this paper, we investigate the feasibility of applying a novel level set reconstruction technique to electrical imaging of the human brain. We focus particularly on the potential application of electrical impedance tomography (EIT) to cryosurgery monitoring. In this application, cancerous tissue is treated by a local freezing technique using a small needle-like cryosurgery probe. The interface between frozen and nonfrozen tissue can be expected to have a relatively high contrast in conductivity and we treat the inverse problem of locating and monitoring this interface during the treatment. A level set method is used as a powerful and flexible tool for tracking the propagating interfaces during the monitoring process. For calculating sensitivities and the Jacobian when deforming the interfaces we employ an adjoint formula rather than a direct differentiation technique. In particular, we are using a narrow-band technique for this procedure. This combination of an adjoint technique and a narrow-band technique for calculating Jacobians results in a computationally efficient and extremely fast method for solving the inverse problem. Moreover, due to the reduced number of unknowns in each step of the narrow-band approach compared to a pixel- or voxel-based technique, our reconstruction scheme tends to be much more stable. We demonstrate that our new method also outperforms its pixel-/voxel-based counterparts in terms of image quality in this application.

Absolute conductivity reconstruction in magnetic induction tomography using a nonlinear method

Magnetic induction tomography (MIT) attempts to image the electrical and magnetic characteristics of a target using impedance measurement data from pairs of excitation and detection coils. This inverse eddy current problem is nonlinear and also severely ill posed so regularization is required for a stable solution. A regularized Gauss-Newton algorithm has been implemented as a nonlinear, iterative inverse solver. In this algorithm, one needs to solve the forward problem and recalculate the Jacobian matrix for each iteration. The forward problem has been solved using an edge based finite element method for magnetic vector potential A and electrical scalar potential V, a so called A, A - V formulation. A theoretical study of the general inverse eddy current problem and a derivation, paying special attention to the boundary conditions, of an adjoint field formula for the Jacobian is given. This efficient formula calculates the change in measured induced voltage due to a small perturbation of the conductivity in a region. This has the advantage that it involves only the inner product of the electric fields when two different coils are excited, and these are convenient computationally. This paper also shows that the sensitivity maps change significantly when the conductivity distribution changes, demonstrating the necessity for a nonlinear reconstruction algorithm. The performance of the inverse solver has been examined and results presented from simulated data with added noise.

Uses and abuses of EIDORS: an extensible software base for EIT

EIDORS is an open source software suite for image reconstruction in electrical impedance tomography and diffuse optical tomography, designed to facilitate collaboration, testing and new research in these fields. This paper describes recent work to redesign the software structure in order to simplify its use and provide a uniform interface, permitting easier modification and customization. We describe the key features of this software, followed by examples of its use. One general issue with inverse problem software is the difficulty of correctly implementing algorithms and the consequent ease with which subtle numerical bugs can be inadvertently introduced. EIDORS helps with this issue, by allowing sharing and reuse of well-documented and debugged software. On the other hand, since EIDORS is designed to facilitate use by non-specialists, its use may inadvertently result in such numerical errors. In order to address this issue, we develop a list of ways in which such errors with inverse problems (which we refer to as 'cheats') may occur. Our hope is that such an overview may assist authors of software to avoid such implementation issues.

Reconstruction of spatially inhomogeneous dielectric tensors through optical tomography

A method to reconstruct weakly anisotropic inhomogeneous dielectric tensors inside a transparent medium is proposed. The mathematical theory of integral geometry is cast into a workable framework that allows the full determination of dielectric tensor fields by scalar Radon inversions of the polarization transformation data obtained from six planar tomographic scanning cycles. Furthermore, a careful derivation of the usual equations of integrated photoelasticity in terms of heuristic length scales of the material inhomogeneity and anisotropy is provided, resulting in a self-contained account about the reconstruction of arbitrary three-dimensional, weakly anisotropic dielectric tensor fields.

EIT reconstruction algorithms: pitfalls, challenges and recent developments

We review developments, issues and challenges in electrical impedance tomography (EIT) for the 4th Conference on Biomedical Applications of Electrical Impedance Tomography, held at Manchester in 2003. We focus on the necessity for three-dimensional data collection and reconstruction, efficient solution of the forward problem, and both present and future reconstruction algorithms. We also suggest common pitfalls or 'inverse crimes' to avoid.

Uniqueness and reconstruction in magnetic resonance-electrical impedance tomography (MR-EIT)

Magnetic resonance-electrical impedance tomography (MR-EIT) was first proposed in 1992. Since then various reconstruction algorithms have been suggested and applied. These algorithms use peripheral voltage measurements and internal current density measurements in different combinations. In this study the problem of MR-EIT is treated as a hyperbolic system of first-order partial differential equations, and three numerical methods are proposed for its solution. This approach is not utilized in any of the algorithms proposed earlier. The numerical solution methods are integration along equipotential surfaces (method of characteristics), integration on a Cartesian grid, and inversion of a system matrix derived by a finite difference formulation. It is shown that if some uniqueness conditions are satisfied, then using at least two injected current patterns, resistivity can be reconstructed apart from a multiplicative constant. This constant can then be identified using a single voltage measurement. The methods proposed are direct, non-iterative, and valid and feasible for 3D reconstructions. They can also be used to easily obtain slice and field-of-view images from a 3D object. 2D simulations are made to illustrate the performance of the algorithms.

Generation of anisotropic-smoothness regularization filters for EIT

In the inverse conductivity problem, as in any ill-posed inverse problem, regularization techniques are necessary in order to stabilize inversion. A common way to implement regularization in electrical impedance tomography is to use Tikhonov regularization. The inverse problem is formulated as a minimization of two terms: the mismatch of the measurements against the model, and the regularization functional. Most commonly, differential operators are used as regularization functionals, leading to smooth solutions. Whenever the imaged region presents discontinuities in the conductivity distribution, such as interorgan boundaries, the smoothness prior is not consistent with the actual situation. In these cases, the reconstruction is enhanced by relaxing the smoothness constraints in the direction normal to the discontinuity. In this paper, we derive a method for generating Gaussian anisotropic regularization filters. The filters are generated on the basis of the prior structural information, allowing a better reconstruction of conductivity profiles matching these priors. When incorporating prior information into a reconstruction algorithm, the risk is of biasing the inverse solutions toward the assumed distributions. Simulations show that, with a careful selection of the regularization parameters, the reconstruction algorithm is still able to detect conductivities patterns that violate the prior information. A generalized singular-value decomposition analysis of the effects of the anisotropic filters on regularization is presented in the last sections of the paper.

Krylov subspace iterative techniques: on the detection of brain activity with electrical impedance tomography

In this paper, we review some numerical techniques based on the linear Krylov subspace iteration that can be used for the efficient calculation of the forward and the inverse electrical impedance tomography problems. Exploring their computational advantages in solving large-scale systems of equations, we specifically address their implementation in reconstructing localized impedance changes occurring within the human brain. If the conductivity of the head tissues is assumed to be real, the pre-conditioned conjugate gradients (PCGs) algorithm can be used to calculate efficiently the approximate forward solution to a given error tolerance. The performance and the regularizing properties of the PCG iteration for solving ill-conditioned systems of equations (PCGNs) is then explored, and a suitable preconditioning matrix is suggested in order to enhance its convergence rate. For image reconstruction, the nonlinear inverse problem is considered. Based on the Gauss-Newton method for solving nonlinear problems we have developed two algorithms that implement the PCGN iteration to calculate the linear step solution. Using an anatomically detailed model of the human head and a specific scalp electrode arrangement, images of a simulated impedance change inside brain's white matter have been reconstructed.