April 16, 2014




Conference on Mathematics of Medical Imaging
June 20-24, 2011
hosted by the Fields Institute
held at the University of Toronto

Organizing Committee:
Adrian Nachman , University of Toronto
Dhavide Aruliah, University of Ontario Institute of Technology
Hongmei Zhu, York University

Contributed Talks
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Qiong Wu --A Semi-Definite, Nonlinear Model for Optimizing k-Space Sample Separation in Parallel Magnetic Resonance Imaging
Yogesh Chinta Venkateswarao --Sparse Sampling of Velocity MRI
Michael Smith --Moving the best ideas from 1985’s constrained reconstruction techniques into 2011’s compressed sensing reconstruction
Nataliya Portman-- The Modelling of Biological Growth: a Pattern Theoretic Approach
Lakshminarayan V. Chinta --Quantitative Perfusion Estimation from Two Photon Fluorescence Microscopy Microvasculature Maps

Lakshminarayan V. Chinta
Sunnybrook Research Institute and Department of Medical Biophysics, University of Toronto

OBJECTIVE: Many models have been proposed to explain the relationship between neural activity and hemodynamic parameters but in their present forms these models fail to provide a detailed understanding of the neurovascular coupling on the micron scale. Our laboratory has been recently using two photon fluorescence microscopy (2PFM) to image the 3D vascular network close to the epicenter of neural activity elicited by somatosensory stimulation and quantitatively analyzing the 3D vascular morphology to deduce dynamic changes in the cerebral blood volume across the vascular tree. The current work estimates cerebral blood flow and perfusion from the 3D geometry and a 2D time series tracking the bolus passage of an injected fluorescent dextran.

METHODS: Experiments were conducted on male adult Sprague–Dawley rats (140±45g). The animals were anesthetized with isoflurane during surgical preparation and alpha-choralose during imaging. The right femoral artery, femoral vein, and tail vein were cannulated for blood gas analysis, intravenous administration of anesthesia, and fluorescent agents respectively. Stereotaxic surgery was performed to prepare a small cranial window (4mm x 3mm) over the forelimb representation in the primary somatosensory cortex. Two-photon microscopy imaging was performed following three 5 mg/kg bolus of Texas Red dextran (70 kDa). To measure the vascular transit time, the bolus passage was tracked by acquiring a time series of a single ~300 µm2 imaging plane, ~50 µm below the cortical surface, at 0.31 ± 0.07 fps and with a spatial resolution of 1.59 µm/pixel. To estimate vessel-wise volume, we obtained a stack of high-resolution images, 300 slices at 1µm lateral and 3µm axial resolution.

ANALYSIS: The signal intensity curves from bolus passage experiments were integrated over time for all labeled vessels in the bolus plane. A second order plus dead time model [1] was used to estimate dead time (?), damping ratio (?), and natural frequency (?). Laplace domain transfer functions were next calculated and the corresponding impulse response was used to compute onset and peak time. Imaris (Bitplane Scientific Software) was used for semi-automatic segmentation of the 3D vascular network. Next, the 2D bolus passage time series was registered to the segmented 3D network and a standard method applied for estimating the transit time between transected vessels [2]. We identified closed paths between any two vessels of the bolus tracking plane by tracing through the 3D network. For the multiple connecting paths, we devised a methodology to estimate the transit times of the individual segments. We solved for the unknown transit times based on equations of transit times and CBF, where CBF at each segment was calculated from the central volume principle as CBF=CBV/TT. Perfusion was then computed by normalizing CBF by the volume of the irrigated tissue. Based on the 2PFM literature [3], the diffusion distance for oxygen was assumed to be ~55-70 µm in the rat’s somatosensory cortex and the tissue volume irrigated was thus estimated as a convolution of ~65 µm sphere and our vascular subtree centrelines.

RESULTS AND CONCLUSION: The application of our methodology to a sample subject shows a skewed distribution of perfusion in the vascular paths: most (~65%) of the paths exhibit perfusion in the 0.56 ± 0.25 mL/g/min range, ~25% exhibit a mean perfusion of 1.53 ± 0.36 mL/g/min while ~10% show high perfusion of 2.56 ± 0.5 mL/g/min under alpha-chloralose anesthesia in the somatosensory cortex of rats. Recent work using iodo[14C]antipyrine autoradiographic estimated perfusion at ~0.6 mL/g/min [4] and optical coherence tomography measurements suggested 0.51-0.68 mL/g/min [5] in the somatosensory cortex of rats under the same anesthesia protocol. Our results show evidence of heterogeneity in perfusion, we expect this heterogeneity to relate to local vascular density. This work describes a novel methodology to estimate perfusion at the micron level from the 2PFM imaging of cerebral microvasculature, its application to a cohort of subjects should allow detailed investigation of the relationship between cortical microvascular topology and blood flow.

[1] Rangaiah, Chem. Eng. Sci (1996); [2] Kety, Am J Physiol (1945); [3] Masamoto, J Applied Physiology (2007); [4] Nakao, Proc Natl Acad Sci USA. (2001); [5] Srinivasan, Opt Express. (2010)

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Nataliya Portman , Montreal Neurological Institute, McGill University (slides available here)

The Modelling of Biological Growth: a Pattern Theoretic Approach

This presentation will take you for a journey from the beginnings of computational anatomy to mathematical representations of biological laws of growth. I will explore evolution and the power of mathematical ideas that allow biologically meaningful interpretation and understanding of images of natural shapes.

More precisely, I will focus on a pattern theoretic model for a biological growth called GRID (Growth as Random Iterated Diffeomorphisms). It was first introduced by Dr. Ulf Grenander in 2005 in response to the need to study growth and development of human anatomy based on a sequence of images. The GRID model represents a modification of growth models employed in the field of computational anatomy, acknowledging that diffeomorphic transformations induced by growth are dependent on genetic controls within an organism. The genetic control is expressed by a probability law which governs spatial-temporal patterns of cell decisions (cell division/death, enlargement) in such a way that the image of an initial organism becomes continuously transformed into the image of a grown organism.

I will then demonstrate the GRID-based inference algorithm developed in my doctoral thesis that automatically estimates growth characteristics of an organism directly from image data. An example of larval growth of the Drosophila wind disc as seen in confocal micrographs of Wingless gene expression patterns will be considered.

Overall, with implementation of the GRID model one can gain a new insight into the growth-induced shape generation process as controlled by the genes. Namely, one can reveal spatial-temporal patterns of intensity of cell divisions hidden deeper in given observations of growth.
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Michael Smith University of Calgary

Moving the best ideas from 1985’s constrained reconstruction techniques into 2011’s compressed sensing reconstruction

With the recent availability of efficient algorithms for non-linear optimizations, there has been an explosion of interest in applying compressive sensing (CS) techniques in various engineering applications. A key concept behind MR compressed sensing is the gathering of reduced k-space data sets. This concept can be traced back to super-resolution reconstruction (SR) algorithms of 1985 which were an attempt to improve upon the even earlier techniques of partial Fourier transform reconstruction. In this paper, we demonstrate how many successful processes and validation technique from super-resolution can be adapted to compressed sensing reconstruction to considerable advantage. We examine (i) the importance of ensuring that the simulated data used to validate and tune CS algorithms matches the characteristics of experimental data and (ii) techniques to improve the appearance of CS images used for diagnostic purposes. We propose the adaption of two existing SR techniques for use in CS reconstruction (iii) by using a k-space approach to generating a sparser data set by modeling the edges of the data rather the data itself, and (iv) combining the TERA SR algorithm with CS’s sparse sampling regime to remove issues surrounding the truncation of k-space data.

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Yogesh Chinta Venkateswarao
McMaster University
coauthors Dr. Christopher Anand
Sparse Sampling of Velocity MRI

The standard MRI is being used to image objects at rest. In addition to standard MRI images, which measure tissues at rest, Phase Contrast MRI can be used to quantify the motion of blood and tissue in the human body. The current method used in Phase Contrast MRI is time consuming. The development of new trajectories has minimized imaging time, but creates subsampling errors. The proposed method uses regularization of velocities and proton densities to eliminate errors arising from k-space undersampling.

Qiong Wu McMaster University
Coauthor Dr. Christopher Anand
A Semi-Definite, Nonlinear Model for Optimizing k-Space Sample Separation in Parallel Magnetic Resonance Imaging

Parallel MRI, in which Fourier (k-Space) is regularly undersampled, is critical for imaging speed. In our approach, a semi-definite model is built to optimize the pattern of regular data sampling to minimize noise in the reconstructed image. To solve the model, a bi-level strategy is applied.
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