The Huangguoshu International Interdisciplinary Conference on Biomedical Mathematics 2008

Dr. Ashok Amin (Carilion Health System): Translating genomic analysis of complex human diseases:
arthritis into personalised medicine. (abstract)

A convergence of technological breakthroughs has taken genomic biomarkers discovery to a new level. Between advances in sequencing, evolving array designs and bioinformatics, a more sophisticated understanding of the genome architecture and its marriage to emerging technologies of micro and nano CT scans augments the perception of personalized medicine. Gene expression arrays demonstrate mRNA expression representing the phenotypic response to a pathophysiological condition at in a particular cell by scanning the total human genome. These “cluster biomarkers” come in several flavors, which include disease classification and detection, treatment response prediction, treatment efficacy, and prognosis. Breast cancer already capitalizes on these technologies, having FDA approval for both diagnosis and prognosis.

Human osteoarthritis (OA) or rheumatoid arthritis (RA) disease process affects not only the cartilage, but also the synovial membrane and subchondral bone. Within these tissues metabolic changes transpire, which include the production of growth factors, pro-inflammatory cytokines and inflammatory mediators. We hypothesized that peripheral blood mononuclear cells (PBMC) obtained from patients with OA or RA traffic through diseased tissues of the diarthrodial joint, and therefore exposed to characteristic cytokines and other stimuli that specifically alter their gene expression. Thus giving it a distinct phenotypic signature at a defined time of the disease, as compared to normals (devoid of clinical symptoms of arthritis).

Our experiments demonstrate that: 1) OA-PBMC are “activated”, and express known inflammatory genes ( e.g. COX-2, IL-1) also observed in RA-PBMC; 2) selected genes are differentially expressed in OA-PBMC and OA cartilage and synovium; and 3) OAPBMC are “primed” to produce inflammatory mediators, such as PGE2, in response to stimulation. 4) Normals, OA and RA-PBMCs have distinct gene expression array signatures. Thus, OA or RA-PBMCs act as sensors of inflammatory processes within the joint. Further studies using pharmacological intervention with anti-TNF drugs in patients with RA, indicated that the outcome of the prognosis (inflammation resolution) could be predicted three weeks earlier with gene expression arrays of PBMC as compared to the classical clinical symptoms defined by ACR. Future emerging technologies suggest that the phenotypic signatures of the mRNA can be translated to structural changes in “sensor cells” and be monitored by micro and nano CT scans.

Professor Simon Arridge (University College London): Model reduction and data selection in optical
tomography. (abstract)

Image reconstruction based on model-fitting often requires numerical methods involving solution of a partial differential equation for the forward problem and inversion of a large dimensional data set for the inverse problem. Model reduction that reduces the computational time of the forward problem introduces modeling errors leading to significant reconstruction artifacts, whereas undersampling of data can lead to an increase of the dimension of the problem null-space. Optical diffusion tomography (ODT) is an example in which the typical required accuracy for the forward problem solver leads to computational times that may be unacceptable, and the employment of very large data sets based on camera detectors leads to conventional inversion schemes that are impractical. In this paper we review the approximation error theory and investigate the interplay between the mesh density, measurement accuracy, and data sampling in the case of optical diffusion tomography. We show that if the approximation errors are estimated and employed, it is possible to derive computationally acceptable schemes.

Professor Guillaume Bal (Columbia University): Inverse transport problems. (abstract)

Inverse transport consists of reconstructing optical parameters from boundary measurements. Several types of measurements may be considered depending on technological feasibility: one may either perform angularly resolved or angularly averaged measurements. One may also perform time dependent measurements (or similarly measurements with time harmonic sources) or steady-state measurements. I will present recent uniqueness and stability results showing which parts of the optical parameters may be reconstructed in a stable fashion under these measurement scenarios. I will then show how these stability results degrade in the presence of physical noise and present open problems of interest in medical imaging applications of inverse transport such as optical tomography and optical molecular imaging.

Professor James Balter (University of Michigan): The dynamic model in radiation oncology: sparse
sampling, surrogates, and complexity in maintaining sufficient precision. (abstract)

The relentless pursuit of precision in Radiation Oncology treatment has moved firmly into the areas of modifying treatment to accommodate short and long term changes due to patient shape, articulation, and motion. The leading tools are position adjustment, followed by field shape/fluence modifications and extending to alteration of an initially optimized plan. While technologically feasible, the intention to respond to changes is not fully supported by sufficient measurement-based knowledge of the modified patient state, degrees of freedom to actually enact a meaningful change, or clinical evidence that such changes can have palpable impact from implementation. Given a chain of uncertainties, a deeper investigation of methods to better understand the dynamic patient and their interaction with the radiotherapy process is warranted.

To better resolve our uncertainty in rapid modification of plans to a changing patient, we typically rely on surrogates (such as coordinates of implanted markers or external indices of patient breathing state or shape) to clarify part of our picture of the patient at any given state. The limited but nonetheless valuable information that surrogates give can be augmented using prior information. The treatment planning process is an important opportunity to extract sufficient prior information, including higher resolution models of the patient, as well as initial estimates of their variability.

Given sufficient prior knowledge, the challenge remains to best estimate the pose, configuration, and motion of the patient as treatment progresses. The data sampled for such estimation is by nature sparser than prior knowledge from the treatment planning process, especially as temporal demands increase.

Here, we discuss issues related to sparse sampling. The typical scales of change and their temporal nature are reviewed, as well as the sampling techniques and resulting data streams available. Examples of the impact of varying residual uncertainties are briefly discussed. A few instances of using prior knowledge are shown, including a reference model for estimating motion from projections, a calibrated deformation map to aid knowledge gained from implanted markers, and augmentation to better estimate the current and near future breathing state.

Professor Heinz Bauschke (University of British Columbia): The lions-mercier projection method and
bregman resolvents. (abstract)

Professor Adi Ben-Israel (The State University of New Jersey): Clustering, classification and contour
approximation of data. (abstract)

“Clustering is the classification of objects into different groups, or more precisely, the partitioning of a data set into subsets (clusters), so that the data in each subset (ideally) share some common trait - often proximity according to some defined distance measure. Data clustering is a common technique for statistical data analysis, which is used in many fields, including machine learning, data mining, pattern recognition, image analysis and bioinformatics", see [1].

The work reported here uses soft assignments of points to clusters, with membership probabilities assumed to depend on distances, for example, p(x|k)d(x|K)=D(x), for all data points x, where p(x|k) is the (subjective) probability that x belongs to the k^th cluster, d(x|K) its distance from (the center of) that cluster, and D(x) is independent of k. The function D(x), called the joint distance function (JDF), measures the classifiability of the data in question (it is, up to a constant, the harmonic mean of the distances d(x|K).) All results, in particular, the centers' update, are stated in terms of the distances, and the probabilities p(x|k) are not needed computationally.

Results:

1)

A new distance clustering method [4], [5]

2)

A viable alternative to the EM method for demixing distributions.

3)

Using the JDF to determine the "right" number of clusters for a given data set.

4)

Contour approximation, and compact representation, of data, [3], [6].

5)

A generalization of the Weiszfeld method for solving the Fermat-Weber location problem with several centers.

6)

Semi supervised clustering, using prior information (labels), is introduced and applied to asses the value of prior information. I show in particular that prior information plays a negligible role in the much researched Wisconsin breast cancer data set, [2].

References:

[1]

http://en.wikipedia.org/wiki/Data_clustering

[2]

Breast Cancer Wisconsin (Diagnostic) Data Set

[3]

M. Arav, Contour approximation of data and the harmonic mean, Math. Ineq. & Appl (to appear).

[4]

A. B-I and C. Iyigun, Probabilistic distance clustering, J. Classification 25(2007), 7-21.

[5]

C. Iyigun and A. B-I, Probabilistic distance clustering adjusted for cluster size, Probability in Engineering and Informational Sciences (to appear).

[6]

C. Iyigun and A. B-I, Contour approximation of data: A duality theory.

[7]

M. Teboulle, A unified continuous optimization framework for center-based clustering methods", J. of Machine Learning 8(2007), 65-102.

Professor James A. Brink (Yale University): Multi-slice CT: optimizing the use of intravenous contrast
material. (abstract)

Various complex pharmacokinetic interrelationships affect the use of intravenous contrast material for computed tomography (CT). These include factors related to each patient’s unique body habitus and degree of health, such as the patient’s age, gender, height, weight, and cardiovascular status. Of equal importance are factors related to the contrast material injection. The volume, concentration, and rate of injection all affect the degree of enhancement that is achieved with an intravenous injection of contrast material. In addition, the injection technique, whether the contrast is infused with a constant injection rate (uniphasic injection) or whether the rate is altered during the injection (multiphasic injection) also affects the magnitude and duration of contrast enhancement.

Iodinated contrast material for CT diffuses rapidly from the intravascular to the extravascular compartment following intravenous infusion. Computer-based simulation of the human circulatory system allows for pharmacokinetic modeling of contrast enhancement with iodinated contrast material using a two-compartment model. Modeling may be performed for parenchymal organs, such as the liver, and vascular organs, such as the aorta. Computer simulations predict the impact of altering injection-related parameters, such as the injection flow rate and concentration of contrast material, and the impact of variations in patient-related parameters, such as patient weight and cardiac output.

Although high injection flow rate and high concentration of contrast material are desirable for arterial imaging, physical properties preclude one from increasing these parameters substantially. Thus, for a given flow rate and concentration, the injection duration is the most important determinant of contrast enhancement magnitude for multi-slice CT angiography. However, the speed of the multi-slice CT scanner affects its ability to record image data during peak enhancement, and timing mechanisms must be used to insure that image acquisition coincides with the desired phase of contrast enhancement. This is typically achieved with use of a test bolus of contrast material, or with bolus tracking. Both techniques predict when contrast material will arrive in the target organ or blood vessel, and scan timing is determined from this parameter, taking into consideration the speed of the scanner and the desired imaging application.

Professor Yair Censor (University of Haifa): Projection methods and their application in image
reconstruction and in intensity-modulated radiation therapy. (abstract)

Projection methods are iterative algorithms that use projections onto sets while relying on the general principle that when a family of (usually closed and convex) sets is present then projections onto the given individual sets are easier to perform than projections onto other sets (intersections, image sets under some transformation, etc.) that are derived from the given individual sets.

Projection methods have different algorithmic structures, of which some are particularly suitable for parallel computing, and they demonstrate nice convergence properties and/or good initial behavior patterns.

The main advantage of projection methods, which makes them successful in real-world applications, is computational. They commonly have the ability to handle huge-size problems of dimensions beyond which other, more sophisticated currently available, methods cease to be efficient. This is so because the building bricks of a projection algorithm are the projections onto the given individual sets (assumed and actually easy to perform) and the algorithmic structure is either sequential or simultaneous (or in-between, i.e., block-iterative, or of the string-averaging variety). Sequential algorithmic structures cater for the row-action approach while simultaneous algorithmic structures favor parallel computing platforms.

We review algorithmic structures and various projection operators in this vast field and present some recent applications in image reconstruction from projections and in intensity-modulated radiation therapy (IMRT).

Dr. Bruno De Man (CT Systems and Applications Laboratory, GE Global Research): Iterative
reconstruction for x-ray computed tomography. (abstract)

Hounsfield’s first CT scanner used the so-called algebraic reconstruction technique (ART). The introduction of filtered backprojection (FBP) by Shepp in 1974 meant the start of a new class of much faster CT reconstruction algorithms, which has seen enormous advances and fine-tuning for many years, and until today is the reconstruction method used by all commercial CT manufacturers. Ironically, there recently is a lot of renewed interest in applying IR for CT. Functional imaging (PET and SPECT) has already adopted statistical iterative reconstruction (SIR) for many years, driven by the IQ needs of these essentially quantum noise limited modalities. In CT, the high computational cost of IR and the high requirements for throughput and time-to-first-image are the reasons that IR has not yet been adopted commercially. To the best of our knowledge, the first research on SIR for real CT data started in the late 90’s, and was inspired by its capability to model the physics of the CT acquisition and its robustness to missing data.

The core of an IR algorithm is its forward model and more specifically the projector-backprojector. Popular approaches include the intersection length method, bilinear interpolation, the distance-driven method, the solid voxel model, and blob-based approaches. The key is to find the best tradeoff between accuracy and computational efficiency. The forward model can also include physical effects such as the X-ray spectrum, the finite beam width, and X-ray scatter.

A second aspect of SIR is the noise model. While energy-integrating detectors are well approximated by a compound Poisson quantum noise model and a Gaussian electronic noise model, most researchers approximate this by a simple Poisson or Gaussian model. The incremental benefit of a more accurate noise model has not been demonstrated to be worthwhile.

The prior or regularization term defines a desire for voxels to deviate minimally from their neighbors or from predefined values. Edge-preserving priors such as the Huber or Geman priors are usually preferred over the simple quadratic prior. All the above elements are combined into a cost function through which they define what the final image will look like. Not only does SIR typically result in a much improved noise-resolution tradeoff, but the noise has a different texture, the spatial resolution is strongly object- and statistics-dependent, and dedicated IQ metrics are desired for SIR.

An optimization method is applied to the above cost function, resulting in an actual SIR algorithm. Iterative coordinate descent (ICD) optimizes one voxel at a time. Conjugate gradients (CG) optimizes all voxels simultaneously. Order subsets (OS) is a widespread acceleration technique that computes updates based on alternating subsets of measurements. The challenge is to find a technique that converges in few iterations and has a low computation cost per iteration. One factor for computation cost is the hardware platform. Researchers have recently explored PC clusters, FPGA’s, GPU’s, cell processors, and multi-core multi-processor solutions for IR.

The main driver for SIR is undoubtedly its reduced noise (or dose), but other potential benefits include its robustness in the case of missing data (metal artifact reduction), sparse data (sparse view sampling) and its ability to deal with object motion (cardiac CT). While it is hard to suddenly replace FBP, a reconstruction technique with decades of history, by a completely new reconstruction technique like IR, a lot of progress has been made to show the potential clinical impact of SIR, and it is only a matter of time before SIR will be routinely available for clinical application.

Professor Tommy Elfving (Linköping University): Iterative methods in image reconstruction:
block-methods, stopping rules, and choice of iteration parameters. (abstract)

Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. A well known example is image reconstruction, which can be modeled using the Radon transform. After expanding the solution into a finite series of basis functions a large, sparse and ill-conditioned linear system arises. When applying an iterative solver to such an ill-posed set of linear equations the error typically initially decreases but after some iterations (depending on the amount of noise in the data, and the degree of ill-posedness) it starts to increase. This phenomenon is called semi-convergence.

One way to control the propagated error is to devise stopping rules aimed at finding the iteration number where the error is minimal. Another way is to specify the iteration parameters such that the semi-convergence behavior becomes less pronounced.

Block-iteration was introduced already in the early days of scientific computing. The purpose was to accelerate the rate of convergence. More recently it has been used to enhance computational efficiency using parallel processors. In the reconstruction literature it is referred to both as block-iteration and ordered subset methods.

We consider a class of Landweber type algorithms which has been applied in image reconstruction. The class include both simultaneous, sequential and block-iteration. We review some results on block-iteration, semi-convergence, choice of iteration parameters and stopping rules.

Professor Jeffrey Fessler (University of Michigan): Model-based image reconstruction in MRI. (abstract)

The inverse fast Fourier transform (FFT) has served the MR community very well as the conventional image reconstruction method for k-space data with full Cartesian sampling. And for well-sampled non-Cartesian data, the “gridding” method with appropriate density compensation factors is fast and effective. But when only under-sampled data is available, or when non-Fourier physical effects like field inhomogeneity are important, then gridding/FFT methods for image reconstruction are suboptimal, and iterative algorithms based on appropriate models can improve image quality, at the price of increased computation. I will summarize a variety of situations where model-based MR image reconstruction is proving to be useful and has seen increasing interest recently. I will also synopsize some of the issues that arise when using iterative algorithms in MRI, including methods for accelerating computation and methods for regularization and regularization parameter selection.

Professor Horst W. Hamacher (Universität Kaiserslautern): Simultaneous intensity and realization
optimization in radiation therapy. (abstract)

Mathematical optimization has contributed in the last 10 years considerably to the development of high quality radiation plans concentrating mainly on three classes of problems,

•

the geometry problem,

•

the intensity problem, and

•

the realization problem.

In general, these three problems are solved sequentially. In the geometry problem, a best selection of angles is identified from which radiation is delivered. The output of the geometry problem is used (and never changed) to determine an intensity profile for each of the angles. In the last problem, the chosen intensity profiles are realized using, for instance, multileaf collimators.

The talk consists in two parts: We will first present a brief survey of methods which are used to tackle the three problems. This material is based on a survey paper of the authors together with Mathias Ehrgott and Lizhen Shao, University of Auckland, to appear in the journal 4OR. Particular emphasis is given in this part to possibilities of extending current methods to the simultaneous treatment of all or at least two of these problems. The second - and larger - part will be devoted to the latter issue by describing in more detail a combinatorial optimization approach dealing with the simultaneous intensity and realization problem.

Professor Ming Jiang (Peking University) : X-ray phase-contrast imaging with 2D grating interferometry.
(abstract)

X-ray imaging is of paramount importance for clinical and pre-clinical applications but it is fundamentally restricted by the attenuation-based contrast mechanism, which has remained essentially the same since Roentgen’s discovery a century ago. Recently, based on the Talbot effect, groundbreaking work was reported using 1D gratings for x-ray phase-contrast imaging with a hospital-grade x-ray tube instead of a synchrotron or micro-focused source. In this paper, we report an extension of our earlier 2D-grating-based work to the case of Gaussian beams. This 2D-grating-based approach has the potential to reduce the imaging time, increase the spatial coherence, and enhance the accuracy and robustness compared to current 1D-grating-based phase-contrast imaging techniques.

Professor Alexander Katsevich (University of Central Florida): Motion compensated image
reconstruction in tomography. (abstract)

Dr. Seung Wook Lee (Korea Atomic Energy Research Institute): An investigation of the grating based
phase imaging technique using cold neutrons. (abstract)

It has recently been reported that the new phase imaging technique utilizing silicon gratings fabricated by nanofabrication process can be applied to neutrons as well as x-rays. To fully explore this new technique for potential future cold neutron imaging stations at National Institute of Standards and Technology and Korea Atomic Energy Research Institute, we have developed a prototype phase imaging station at a monochromatic cold neutron beamline in the NCNR neutron guide hall. We have designed the silicon gratings for a neutron wavelength of 0.44 nm and fabricated them using the state-of-the-art nanofabrication facility at the NIST Center for Nanoscale Science and Technology. It has been demonstrated that the higher sensitivity of the phase imaging technique enables one to visualize the small density changes where the contrast of the conventional transmission image barely allows us to see them. In addition, the new phase imaging technique provides three different information with one identical measurement, which are the one dimensional gradient image of the neutron phase shift, the conventional transmission image, and the dark field image caused by the local small angle scattering. Here we will discuss the experimental apparatus, as well as the recent experimental results obtained with this prototype apparatus. We will also present the future improvement plan and discuss the possibility of biological tissue imaging.

Professor Tony Lomax (The Paul Scherrer Institute): The importance of starting conditions in the
optimization of intensity modulated proton therapy. (abstract)

Intensity Modulated Proton Therapy (IMPT) can be considered to be the proton equivalent of IMRT with photons. It consists of the simultaneous optimization of all Bragg peaks from all field directions, with or without additional dose constraints to neighboring critical structures. However, given that the Bragg peak characteristic of the proton depth-dose curve is well localized in space in three dimensions (unlike the photon depth dose curve, which falls off slowly with a more or less exponential form), with IMPT we literally have a third dimension which can be modulated. Put another way, for each individual field, the optimization problem consists of determining the optimum weight of a set of energy (and therefore depth) modulated Bragg peaks, with each energy layer consisting of a two dimensional distribution of proton pencil beams. If we assume an in-plane pencil beam separation of 5x5mm and an in-depth separation of Bragg peaks of 5mm, then typically about 8000 such pencil beams will be required to cover a liter volume from each field direction. This can be compared to the 400 or so photon pencil beams required to calculate an equivalent sized photon field. Thus, on a field-by-field basis, the optimization problem for IMPT is at least an order of magnitude larger than for photons. It has been previously shown that this additional complexity leads to large degrees of degeneracy in the problem, in that very different patterns of Bragg peak weights and positions can lead to almost identical dose distributions. In this work, we will investigate the importance of the starting conditions for IMPT optimization. Firstly, we will show how the selection of inappropriate starting conditions can result in sub-optimal plans, and move on to demonstrate how manipulation of the starting conditions can be used to ‘steer’ the optimization to results that can be somewhat defined by the user. Although mathematical based optimization techniques for exploiting IMPT degeneracy are also being developed by other groups, we believe that starting condition based optimization as proposed here can provide a useful tool for making the optimization of IMPT plans transparent and understandable to the planner.

Professor Atsushi Momose (The University of Tokyo): Theoretical aspects of x-ray phase imaging with
talbot interferometry. (abstract)

X-ray phase imaging by X-ray Talbot interferometry [1] and subsequently X-ray Talbot-Lau interferometry [2] is becoming reality in combination with a conventional X-ray generator, enabling various practical applications. Authors are conducting a project with the support by Japan Science and Technology Agency (JST) for the development of medical apparatus in collaboration with micro-fabrication specialists (Hyogo Univ.), who are responsible for high-aspect ratio gratings used for Talbot interferometry, and a medical campany (Konica Minolta Medical & Graphic).

Using synchrotron radiation, the development of X-ray phase-sensitive microscope is attractive because the X-ray Talbot interferometer functions with a spherical wave. High-speed phase imaging and four-dimensional phase tomography is also expected using white synchrotron radiation because the X-ray Talbot interferometer functions with X-rays of a broad energy band.

The theoretical understanding of the system of X-ray Talbot interferometry, including the imaging process, is important for utilizing the technique efficiently and determining the specification of the system. We will present the theoretical description for the image formation, phase retrieval, and tomographic image reconstruction by the Talbot interferometry.

References:

[1]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi , K. Takai , and Y. Suzuki, Jpn J. Appl. Phys. 42, L866-L868 (2003).

[2]

F. Pfeiffer, T. Witkamp, O. Bunk, and C. David, Nat. Phys. 2, 258-261 (2006).

Professor Uwe Oelfke (German Cancer Research Center): Optimization of physical and biological dose
distributions in hadron therapy. (abstract)

Intensity modulated particle therapy (IMPT) therapy is one of the most challenging and promising radiotherapy techniques of the future. The complexity of IMPT arises from the fact that dose contributions of up to a few hundred thousand pencil beams have to be combined in order to achieve the optimal accumulated dose within the patient. Furthermore, heavy particle beams are expected to show superior radiobiological properties in comparison to photons. To fully exploit the physical and biological advantages of hadron beams, however, one has to account for the inherent risks and uncertainties of their clinical application already at the planning stage.

For IMPT with proton beams, we will discuss a risk adapted planning strategy based on a so 'called' worst case optimization that accounts for range and patient setup uncertainties. Furthermore, a detailed analysis on different aspects of optimizing spot-scanning dose delivery techniques is also introduced, e.g., concepts for deriving an optimal scanning path or energy selection grid during dose delivery will be presented.

For IMPT with heavy-ion beams the main advantage in addition to protons is expected to arise from their radiobiological properties, i.e., the effects of a locally varying radiobiological effectiveness (RBE) have to be calculated and included into the inverse treatment planning process. First, the concept for the inclusion of RBE-effects in the inverse planning process is presented for the case of proton beams. Next, a detailed overview about strategies for the 'biological optimization' of heavy-ion therapy will conclude the talk.

Professor Franz Pfeiffer (Paul Scherrer Institut & Ecole Polytechnique Fédérale de Lausanne): Phase-
contrast and dark-field x-ray imaging with synchrotron and x-ray tube sources. (abstract)

X-ray computed tomography is an invaluable three-dimensional imaging method used in medical, biological, and industrial applications. The technique yields excellent results where highly absorbing structures are embedded in a matrix of relatively weakly absorbing material (like bones and tissue in the human body). If, however, the differentiation of weakly absorbing materials (like pathologies in the soft tissue structure) is the subject of interest, then the contrast achieved by the currently existing, absorption-based methods is limited, unless contrast agents are used.

Here we present how a recently developed grating-based method can overcome these limitations by using the phase contrast instead of the absorption contrast for three-dimensional tomographic imaging of centimeter sized macroscopic objects. As opposed to existing techniques, our approach can be used with low-brilliance, lab-based x-ray sources and thus is of interest to a wide range of applications in medicine, biology, and non-destructive testing. Several examples of phase-contrast and dark-field imaging for biomedical and material science applications will be given in this presentation.

References:

[1]

T. Weitkamp et al., Opt. Express 13, 6296 (2005).

[2]

F. Pfeiffer et al., Nature Physics, 2, 258 (2006).

[3]

F. Pfeiffer et al., Phys. Rev. Lett. 98, 108105 (2007).

[4]

F. Pfeiffer, et al., Phys. Med. Biol. 52, 6923 (2007).

[5]

F. Pfeiffer et al., Nature Materials 7, 134 (2008).

[6]

B. Schwarzschild, Physics Today, Vol. 3, 12 (2008).

Professor Brian W. Pogue (Thayer School of Engineering at Dartmouth): 3-D multi-modality MRI-guided
optical imaging: computation methods and challenges. (abstract)

The field of near-infrared (NIR) tomography is increasingly moving towards multi-modality systems by combining anatomical structure from MRI, CT or X-ray with optical information to improve resolution. Multi-modality systems provide micron level resolution for functional quantitative spectroscopic information of tissues which has application in breast cancer diagnosis and treatment monitoring. This paper describes computational methods and challenges for obtaining 3-D MRI-guided NIR spectroscopic (NIRS) estimates. Conventional FEM methods can be used to solve the diffusion equation for light propagation in tissue, using MR information as spatial priors. These methods however require volume discretization of arbitrary shapes which is a complex problem that does not lend easily to automation for processing large clinical data sets. An alternate technique, the boundary element method (BEM) has been investigated for MRI-guided NIRS using only surface discretization which is faster and more reliable to generate. BEM based multi-spectral 3-D image-guided NIRS estimates were up to 72% faster than FEM based methods for two-region domains. In phantom experiments, 3-D spectral BEM-based spectroscopy recovered the oxygen dissociation curve with mean error of 6.6% and tracked variation in total hemoglobin linearly. Parallel processing using 4 processors has improved performance of the BEM-based algorithm giving 60% improvement in time. Multi-spectral recovery of tumor-simulating multi-layered phantom showed 94% accuracy in reconstructed parameters. This method is currently being applied to clinical data from subjects with abnormalities and for monitoring neoadjuvant chemotherapy and provides a feasible and computationally efficient method for providing 3-D image-guided NIRS estimates in-vivo in a fast and reliable manner.

Dr. Kui Ren (University of Texas at Austin): Numerical reconstruction methods for transport-based
medical imaging. (abstract)

We review here numerical methods for inverse problems related to the following radiative transport equation:

θ•▽u(x,θ) + ∑_t (x) u(x,θ) = ∑_s (x)∫_S^d-1 k(θ•θ') u(x,θ')dθ' + f(x)

u(x,θ) = g(x,θ), (x,θ)∈Γ_,

where Γ_≡ {(x,θ)∈ ∂Ω × S^d-1 s.t. θ•ν(x)<0 }, Ω ⊂ R^d is the physical domain with boundary ∂Ω and S^d-1 is the unit sphere in R^d. The objective of the inverse transport problems is to reconstruct physical parameters ∑_t (x) and ∑_s (x) and the source ( f(x) ) from the knowledge of the solution at the boundary. The main applications of those inverse problems are optical tomography and optical molecular imaging.

Due to the fact that the transport equations are posed in phase space, numerical solutions of those inverse transport problems are usually extremely expensive. Special simplifications have to be made to reduce computational costs.

We will review numerical reconstruction algorithms for the inverse problems, including methods based on linearization, PDE-constrained optimization and other Newton-type iterative reconstruction schemes for optical tomography problem and some Bayesian type of reconstruction method for the inverse source problem. We will also review methods that allow one to use extremely large data sets while keeping the computational costs reasonable. Numerical examples with synthetic data will be shown to demonstrate the performance of those methods.

Professor John Schotland (University of Pennsylvania): Inverse scattering series in optical tomography.
(abstract)

The inverse scattering problem (ISP) for diffuse waves consists of recovering the spatially-varying absorption of the interior of a bounded domain from measurements taken on its boundary. The problem has been widely studied in the context of optical tomography—an emerging biomedical imaging modality which uses near-infrared light as a probe of tissue structure and function. More generally, diffusion of multiply-scattered waves is a nearly universal feature of wave propagation in random media.

The ISP in optical tomography is usually formulated as a nonlinear optimization problem. At present, the iterative methods which are used to solve this problem are not well understood mathematically, since error estimates and convergence results are not known. In this talk we will show that, to some extent, it is possible to fill this gap. In particular, we will study the solution to the ISP which arises from inversion of the Born series. In previous work we have utilized such series expansions as tools to develop fast, direct image reconstruction algorithms. Here we characterize their convergence, stability and reconstruction error. Numerical results which validate the theory will be presented.

This is joint work with Shari Moskow.

Professor Reinhard W. Schulte (Loma Linda University Medical Center): The application of projection
methods in image reconstruction of proton computed tomography. (abstract)

Proton computed tomography (CT) based on energy loss measurements of protons passing through an object was first suggested by Nobel Laureate Allan M. Cormack. Despite promising experimental results obtained during the 1970s, the technique was never fully developed because of the success of x-ray CT and the lack of suitable proton CT equipment. However, the interest in proton CT in the medical and scientific community has recently increased due to the fact that proton therapy is rapidly evolving as an advanced form of radiation therapy and the prospect of proton CT providing more accurate information for proton treatment planning and image guidance than x-ray CT.

A major challenge of proton CT is to find adequate methods of image reconstruction. Since protons do not pass the object on straight lines, uncritical use of reconstruction methods based on an inversion of the Radon transformation yields images that are unacceptably blurred.

From a mathematical standpoint, the image reconstruction problem in proton CT is the solution of a very large sparse system of linear equations, where every row of the system matrix corresponds a single proton history and every column contains the information on the intersection of a given object voxel by the protons. The projection vector contains the integrated relative electron density along each proton path, which is derived from proton energy loss measurements. The problem is inconsistent due to the additive Gaussian noise of the energy measurement.

Projection methods are very suitable to handle the reconstruction of proton CT image sets as shown in initial work done with the classical Algebraic Reconstruction Technique (ART). However, much remains to be investigated. In this presentation, we review the range various projection methods that can be applied in image reconstruction and show how methods may be optimized in terms of computational speed, robustness, and image quality.

Dr. Belinda Seto (National Institute of Biomedical Imaging and Bioengineering): Modelling and
computational science in predictive medicine. (abstract)

Applied mathematics provides quantitative dimensions to biomedical research in enhancing our understanding of fundamental biological processes with precise measurements and models. Such interdisciplinary approaches to the development of predictive, computational models of disease prevention, diagnosis and treatment are the focus of research supported by the National Institute of Biomedical Imaging and Bioengineering (NIBIB). These models span multiple biological scales including atomic, molecular, cellular, tissues, organs, organisms, and populations. Models, developed with mathematics and engineering principles, may simulate the intricate networks of cellular and molecular interactions that underlie normal human physiology, health and diseases and to make predictions about clinical outcomes. These topics of mathematics and computational science extend well beyond the core application areas in biomedical image reconstruction, image process, data and image analysis, etc.

This presentation will provide an overview of NIBIB support in biomedical modeling and bio-computational research, using examples to illustrate how such approaches may advance predictive medicine.

Dr. Subhadra Srinivasan (Thayer School of Engineering at Dartmouth): 3-D multi-modality MRI-guided
optical imaging: computation methods and challenges. (abstract)

The field of near-infrared (NIR) tomography is increasingly moving towards multi-modality systems by combining anatomical structure from MRI, CT or X-ray with optical information to improve resolution. Multi-modality systems provide micron level resolution for functional quantitative spectroscopic information of tissues which has application in breast cancer diagnosis and treatment monitoring. This paper describes computational methods and challenges for obtaining 3-D MRI-guided NIR spectroscopic (NIRS) estimates. Conventional FEM methods can be used to solve the diffusion equation for light propagation in tissue, using MR information as spatial priors. These methods however require volume discretization of arbitrary shapes which is a complex problem that does not lend easily to automation for processing large clinical data sets. An alternate technique, the boundary element method (BEM) has been investigated for MRI-guided NIRS using only surface discretization which is faster and more reliable to generate. BEM based multi-spectral 3-D image-guided NIRS estimates were up to 72% faster than FEM based methods for two-region domains. In phantom experiments, 3-D spectral BEM-based spectroscopy recovered the oxygen dissociation curve with mean error of 6.6% and tracked variation in total hemoglobin linearly. Parallel processing using 4 processors has improved performance of the BEM-based algorithm giving 60% improvement in time. Multi-spectral recovery of tumor-simulating multi-layered phantom showed 94% accuracy in reconstructed parameters. This method is currently being applied to clinical data from subjects with abnormalities and for monitoring neoadjuvant chemotherapy and provides a feasible and computationally efficient method for providing 3-D image-guided NIRS estimates in-vivo in a fast and reliable manner.

Professor George Starkschall (The University of Texas): Towards a new paradigm for radiation
treatment planning. (abstract)

In the past 40 years radiation treatment planning has evolved from two-dimensional (2D) planning to three-dimensional (3D) conformal planning to 3D inverse planning. With the introduction of multislice helical computed tomography (CT) scanners combined with the use of respiratory monitoring systems, we are now able to explicitly account for intrafractional respiratory-induced motion both in the targeting of the moving tumor as well as in the calculation of radiation dose distributions. Four-dimensional (4D) dose distributions can be calculated by initially generating a 3D treatment plan on a reference phase of a 4D CT data set. The beam configuration is copied onto the remaining phases that constitute the 4D data set and doses are calculated on each phase of the data set. Methods of deformable image registration are used to deform the dose distribution from each phase to the reference phase, and doses are summed on the reference phase to generate the 4D dose distribution. A study was undertaken comparing 3D dose distributions with 4D dose distributions for 15 patients, all of whom were diagnosed with Stage III non-small cell lung cancer (NSCLC). The study indicated that in 4/15 patients, the clinical target volume (CTV) dose coverage was different in 3D as compared to 4D by as much as 5 Gy, whereas in 8/15 patients, the planning target volume (PTV) dose coverage was different. In all cases, normal structure doses were comparable. The effects of interfractional variation on the accumulated dose distribution can be explicitly accounted for in a similar manner, provided on-treatment CT data sets are appropriately aligned with the planning data set using external fiducials, bony anatomy, or the actual tumor volume. Finally, setup uncertainty can be explicitly accounted for in several ways, including brute-force displacement of isocenter for each fraction as well as convolving the dose distribution with a probability distribution function that accounts for setup uncertainties. By explicitly accounting for interfractional motion, interfractional variation, and setup uncertainty in the dose calculation, we have developed a paradigm for radiation treatment planning in which the PTV is identified as an entity for beam targeting purposes and treatment plans are evaluated solely based on the dose received by the CTV, which is an accurate reflection of the radiation actually delivered to tumor cells.

Professor Wolfgang Tome (University of Wisconsin): On motion induced underdose and temporal dose
variations as a function of fraction size, fraction number, dose rate, and number of IMRT fields.
(abstract)

The object of this work is to elucidate the relationship of interplay effects with dose rate, fractionation size, fraction number, and the number of IMRT fields as well as strategies to mitigate these interplay effects. The MLC settings from 5, 9, and 11 field IMRT plans for three non small cell lung cancer patients have been used to simulate the dose to a moving target. Static planar dose distributions were determined for each treatment field. Motion of the tumor was determined from segmentation of 4D-CT scans with maximum amplitudes between 1.5 and 1.75 cm in the cephalad-caudad direction for three patient cases. We found that the mean underdose within the tumor planar projections did not significantly change with various dose rates or fractionation schedules. However, the range in underdose values and the temporal dose variation decreased with decreasing dose rate and increasing dose per fraction. Therefore, the Interplay Effect does not appear to have a bigger effect on hypofractionation compared to traditional fractionation. Computer simulations show that temporal dose variations can be mitigated by dose rate and dose per fraction and individual fields can be identified where interplay effects are the largest. The 9- and 11-field IMRT plans had greater predicted relative tumor underdose and temporal dose variation than the 5-field IMRT plan.

Dr. Alexei V. Trofimov (Harvard Medical School): Patient-adaptive approach to treatment planning,
delivery and risk evaluation in thoracic radiotherapy. (abstract)

Despite the positive trends in lifestyles and prevention, lung disease remains the single largest cause of cancer mortality around the world. While intensity-modulated radiation therapy improved the dose conformality to the tumor and reduced the acute side effects of treatment, the relative benefits with respect to disease control are still inconclusive. Target dose escalation has been identified as means of achieving gains in disease control and patient survival. However, the relative radiosensitivity of the healthy lung tissue, and mobility of lung targets pose a challenge to delivery of necessary target dosage. Additionally, limitations in treatment set-up reproducibility have been documented extensively, and substantial changes in the solid tumor volumes during the treatment course have also been reported. Addressing these issues requires adoption of increasingly individualized case-specific solutions. We focus on two possible directions toward development of such individualized treatment courses.

Motion-adaptive robust treatment planning takes into consideration the characteristics of individual respiratory patterns specific to a patient, as well as the breathing variability, both intra- and inter-fractionally, in addition to immobilization errors. The objective is to yield a plan that, despite the delivery uncertainties, would ensure the desired target dosage, provided that the assumed uncertainty model remains valid throughout the treatment. Robust treatment plans can be optimized for a specific image-guided delivery strategy, such as respiratory gating or tumor tracking.

Further, improved treatment plan design, and reduction of complication risk may be achieved by considering the regional functionality variations in the lung. Functional differentiation of the underlying tissue, which is ignored in common dose-volume histograms, can be utilized to minimize damage to the functionally important regions of the lung, and potentially allow for dose escalation while avoiding toxicity. The information about distribution of ventilation and perfusion activity within the lung can be obtained directly with functional imaging, or estimated using respiratory-correlated computed tomography data and deformable image registration.

Individualized patient-adaptive approaches offer the promise of further improving the targeting and precision of the dose delivery in thoracic patients, and, consequently, increasing the effectiveness of radiation therapy.

Professor Ge Wang (Virginia Polytechnic Institute & State University): Interior tomography: practical
application. (abstract)

The conventional wisdom states that the interior problem (reconstruction of an interior region of interest (ROI) only from projection data along lines through that region) is not uniquely solvable. While it remains correct, recent theoretical and numerical results demonstrated that this interior problem can be solved in a theoretically exact and numerically stable fashion if a sub-region within the interior ROI is known. In contrast to the well-established λ - tomography, the studies on this type of exact interior reconstruction are referred to as “interior tomography”. In this paper, we will discuss the latest development of interior tomography algorithms and applications. The essence of interior tomography is to find the unique solution from highly truncated projection data via analytic continuation. Such an extension can be done either in the filtered backprojection or backprojection filtration formats. The key issue for the exact interior reconstruction is how to invert the truncated Hilbert transform. We have developed a projection onto convex set (POCS) algorithm and a singular value decomposition (SVD) and produced excellent numerical and experimental results in various major preclinical and clinical applications.

Dr. Xiaoyun Xiong (National Natural Science Foundation of China): Imaging research funding by the
NSFC. (abstract)

Professor Lei Xing (Stanford University School of medicine): From 4D imaging to 4D radiation therapy.
(abstract)

The evolution of radiation therapy has been strongly correlated with the development of imaging techniques. Recent technical advances in planning and delivering IMRT provide an unprecedented means for producing exquisitely shaped radiation doses that closely conform to the tumor dimensions while sparing sensitive structures. The utility of modern radiation technologies, such as 3D CRT and IMRT, cannot be fully exploited without eliminating or significantly reducing these uncertainties. The need to improve targeting in radiation treatment has recently spurred a flood of research activities in image-guided radiation therapy (IGRT). Many IGRT solutions have been proposed to attack various aspects of the problem. Briefly, IGRT developments are focused in four major areas: (1) biological imaging tools for better definition of tumor volume; (2) time-resolved (4D) imaging techniques for modeling the intra-fraction organ motion; (3) on-board imaging system or imaging devices registered to the treatment machines for inter-fraction patient localization; and (4) new radiation treatment planning and delivery schemes incorporating the information derived from the new imaging techniques. In this talk I will highlight the recent developments of various available imaging techniques. After hearing the talk, it is hoped that the audience will have an overall picture of IGRT, find it easier to navigate themselves through the vast literatures of IGRT, and get a brief idea on how to implement the new IGRT techniques in their clinics.

Professor Yangbo Ye (University of Iowa): Interior tomography: mathematical analysis. (abstract)

Interior tomography is a novel reconstruction scheme that only uses projection data associated with lines through a region of interest (ROI) or volume of interest (VOI) to be reconstructed, as opposite to the traditional CT reconstruction that does not allow two-side data truncation. By the classical theory (Natterer 1986), the interior problem has numerous solutions. As a result, -tomography has been extensively studied over the past decades. Assuming prior knowledge of an object density function on a small subregion D inside the ROI or VOI, interior reconstruction becomes possible. It was proved that interior tomography aided by the prior knowledge on D has a unique, stable solution. The proof uses a theorem by Gel’fang-Graev (1991), Rullgard (2004), and Zou-Pan (2004) that the backprojected differential projection data can be expressed as a Hilbert transform, the inversion formula for truncated Hilbert transform by Tricomi (1951), and analytic continuation of functions of a complex variable. The stability was proved using the Nevanlina principle or the theory of integrals of Cauchy’s type.

In this study, to relax the data requirements further for interior tomography we assume that an x-ray beam scans an object only over a limited source angle. By a formula of Park-Noo-Clackdoyle (2005), the projection data over the limited angle can be expressed as a difference of two Hilbert transforms in two directions. By taking one direction not pointing toward the object, one can use the other Hilbert transform to reconstruct a portion of the object. Thus, the interior reconstruction approach still relies on the inversion of truncated Hilbert transform and complex analytic continuation. Our main result is the stability analysis using linear algebra based on discretization.

Professor Jinming Yu (Shandong Cancer Hospital): Current functional imaging in radiation oncology.
(abstract)

During the last decade, technological advances in imaging have transformed the way image-guided radiation therapy is performed. Anatomical imaging (computed tomography, magnetic resonance imaging) greatly improved the accuracy of delineating target structures while three-dimensional conformal radiation therapy (3DCRT) and intensity-modulated radiotherapy (IMRT) become common practice. However, radiation oncology is beginning to shift toward a more biological and molecular approach as advances in functional imaging of physiological processes in tumors. This review provides an overview of the role of current functional imaging strategies in radiation oncology, with a focus on positron emission tomography (PET), as it relates to staging and molecular profiling (glucose metabolism, cellular proliferation, angiogenesis, hypoxia, receptor status) of tumors, defining radiation target volumes, and assessing therapeutic response.

In addition, our team work such as imaging-pathological validation, optimal timing of scans, clinical outcome studies are introduced, including the tracers of FDG, 11C-PD153035, FMISO or FETNIM, and FLT to image glucose metabolism, epidermal growth factor receptor (EGFR) status, hypoxia and cellular proliferation, respectively.

Dr. Yantian Zhang (National Institute of Biomedical Imaging and Bioengineering): Recent trend in
biomedical optical imaging research – a perspective view from funding support of optical imaging at
the NIBIB/NIH. (abstract)

NIBIB is a US federal funding agency that supports biomedical research. It is a component of the US National Institutes of Health. NIBIB funds and supports biomedical imaging research through many of its extramural science programs. This talk will offer a general review of optical imaging research supported by the NIBIB, and will attempt to reflect upon recent trends in biomedical optical imaging research based on funding support of optical imaging research at the NIBIB.

In biomedical imaging research and applications, applied math constitutes an integral part of the imaging method development (e.g., in data acquisition design, in image reconstruction, and in post acquisition data/image processing), plays an important part in delivering imaging solutions in specific biomedical research applications, and also represents an inescapable part of major challenges facing the development and optimization of novel imaging technologies. We’ll briefly review applications and challenges of applied math in the area of optical imaging, again, from the perspective of NIBIB funded biomedical optical imaging research.

Dr. Zhaotian Zhang (National Natural Science Foundation of China): Imaging research funding by the
NSFC. (abstract)

Professor Hongkai Zhao (University of California, Irvine): A fast forward solver for radiative transfer
equation with applications in optical imaging. (abstract)

Radiative transport equation (RTE) is one of the most fundamental equations for modeling particle, such as neutron and photon, transport in complex media. Although the RTE is linear, the main challenge for numerical computing comes from high dimensionality of space plus phase variables and the scattering among different phase directions.

In this talk I will present an efficient forward solver for steady-state RTE on both structured and unstructured grid. First we introduce a novel angular discretization in phase space that can handle scattering with strong forward peaking more efficiently. Second, for a steady-state RTE, a large coupled linear system has to be solved efficiently. We develop a Gauss-Seidel iterative scheme that incorporates dominant scattering and angular dependent ordering effectively. The iterative method is then used naturally as an efficient relaxation scheme for multigrid solver in both spatial and angular space. I will demonstrate both efficiency and accuracy of our forward solver by extensive tests and applications in various optical imaging regimes.

Dr. Guangying Zhu (Peking University School of Oncology): Outline of cancer and radiotherapy.
(abstract)

Cancer is the leading cause of death for population in rural and urban areas around the world. Nowadays the medical scientists believe that the cancer comes from the gene mutation and cell anaplasia, cancerization. Lots of mathematical models concerning cancer cell biological behavior and radiotherapy have been established. They are now used in the predicting the cancer growth, therapy response and treatment related complications. But the population deaths caused by tumor is still larger, even little decrease has been made in recent 50 years in the world. Many drugs have been used in clinical treatment to control metastases. Although many new techniques of radiotherapy have been developed, such as Tomotherapy, Rapid Arc Radiotherapy, but defining the margin of the tumor and giving higher does to tumor with tolerable escalation of does to neighboring normal organ at risk are still two problems in modern radiotherapy to be resolved.

Dr. Yu Zou (Toshiba Medical Research Institute USA, Inc.): A physics-based approach to scatter
correction in cone beam CT. (abstract)

A number of computed tomography (CT) systems with large cone angles have been developed in recent years, including various scanners based on flat panel detectors typically based on C-arm gantries and diagnostic multi-slice CT scanners with large curved detectors. The flux of scattered photons rises as the volume covered by X ray beams increases. These photons contribute to the measured signal significantly and the images reconstructed show various artifacts. Since the scatter always enhances the measured intensities, it leads to underestimate the attenuations of imaged subjects and results in darker images or darker regions.

Based on the single scatter, forward, and monochromatic approximations, we have developed an analytical scatter correction approach. The scatter intensities were estimated by convolving a forward function that is primary intensity multiplied by its logarithm with a double Gaussian kernel. The kernel is formed by a narrow Gaussian function overlapped with a wide one. The Rayleigh (coherent) and Compton (incoherent) scattering are taken into account by the narrow Gaussian kernel and the wide Gaussian kernel, respectively. The multiple scatter was included by an additional multiplier on the wide Gaussian kernel. From the Klein-Nishina formula of the Compton differential cross section, we derived the wide Gaussian kernel in the forward scatter approximation. The narrow Gaussian kernel was related to the form factor of the Rayleigh scatter. Additional factors of the Gaussian distribution were introduced to count the solid angles and the antiscatter grid. Monte Carlo simulations were performed for water cylinders of various sizes at different photon energies. The physical parameters in the double Gaussian kernel were adjusted to the simulated scatter intensities. To test the model and the parameters, scatter intensities were measured for the AAPM phantom, the Tommy head phantom, a body phantom with the beam-blocker method. The scatter intensities from the model agree reasonably with those from experiments. To evaluate the model, a 3D image of the body phantom was reconstructed. The dark band artifacts in the image were reduced significantly with the scatter correction combined with a beam hardening correction.

In conclusion, the double Gaussian convolution approach can generate scatter intensities with reasonable accuracy. The approach is fast and stable. Since the approach processes the data view by view, it can also be applied to tomosynthesis imaging and digital x-ray radiography. The ad hoc multiple scatter factor may be further investigated with a diffusion model.

The Huangguoshu International Interdisciplinary Conference on
Biomedical Mathematics
Promising Directions in Imaging, Therapy Planning and Inverse Problems

November 3-9, 2008, The Huangguoshu National Park of China, Guizhou

The Huangguoshu International Interdisciplinary Conference on Biomedical Mathematics Promising Directions in Imaging, Therapy Planning and Inverse Problems

November 3-9, 2008, The Huangguoshu National Park of China, Guizhou

The Huangguoshu International Interdisciplinary Conference on
Biomedical Mathematics
Promising Directions in Imaging, Therapy Planning and Inverse Problems