- Diffusion Imaging
Diffusion-weighted imaging (DWI) is currently the only imaging method for non-invasively measuring the local diffusion characteristics of water molecules in vivo. Clinically, diffusion MRI is useful for the diagnoses of conditions (e.g., stroke) or neurological disorders (e.g., Multiple Sclerosis), and helps better understand the connectivity of white matter axons in the central nervous system.
DWI sequences can successfully measure diffusion in the order of micrometers, even though the resolution of the resulting images are usually in the millimeter range. Because of this high sensitivity to motion even in the micrometer range, bulk motion can create phase accruals that result in significant ghosting artifacts. Refocusing reconstruction  corrects these artifacts by rephasing the unaliased component of each acquisiton via multiplication with the navigator phase conjugate, thereby making complete use of the 2D navigator data.
Although the most widely used imaging technique for DWI is echo-planar imaging (EPI), the SSFP technique is also shown to provide high quality DWI images. This method is especially suitable for DWI of short-T2 tissues as in the knee , which are difficult to image with traditional DWI methods like EPI.
- Miller KL, et al., "Nonlinear Phase Correction for Navigated Diffusion Imaging", Magn Reson Med. 2003 Aug;50(2):343-53.
- Miller KL, et al., "Steady-State Diffusion-Weighted Imaging of In Vivo Knee Cartilage", Magn Reson Med. 2004 Feb;51(2):394-98.
- Flow Imaging
Quantitative velocity measurements are widely used in the diagnosis of cardiovascular disease. In mitral and aortic valves, the presence of a stronger peak velocity pressure gradient can indicate the degree of stenosis. Pulmonary hypertension can be detected by the presence of tricuspid regurgitation jets. In the ascending aorta, velocity pressure gradients can be used to diagnose coarctation. Doppler ultrasound (US) has the ability to measure normal and jet velocities with high temporal resolution. However, the technique is sensitive to transducer positioning and the angle of insonation.
The development of high-resolution imaging sequences to depict coronary vessel structure and measure cardiac function demonstrates that MR is a compelling modality for obtaining clinically useful information that is beyond the capabilities of US. The integration of a tool that is the MR equivalent of Doppler US into a real-time MR system will give more accurate prescription capability through real-time localization, acquisition, and display. The addition will also result in a comprehensive cardiac evaluation tool and could revolutionize patient examination and monitoring.
- Hu BS, et al., "Localized real-time velocity spectra determination", Magn Reson Med 1993; 30: 393-398.
- Irarrazabal P, et al., "Spatially resolved and localized real-time velocity distribution", Magn Reson Med 1993; 30: 207-212.
- Pat GT, et al., "One-shot spatially resolved velocity imaging", Magn Reson Med 1998; 40: 603-613.
- DiCarlo JC, et al., "Varialbe-density one-shot Fourier velocity encoding", Magn Reson Med 2005; 54: 645-655.
- Image Reconstruction
Scan time is always a critical factor in a clinical exam: reducing it reduces motion artifacts, alleviates the burden on the patient, and increases MRI scanner throughput. Many approaches might be taken to reduce scan time without impacting SNR.
In almost all product sequences, the k-space trajectory is Cartesian, making the reconstruction a straightforward 2D or 3D DFT. However, this is not always the optimal trajectory. Projection Reconstruction (PR) and spiral trajectories go through k-space in a more time-efficient way. More advanced reconstruction methods are required with such trajectories.
- PR data can be converted into sinograms, analog to the ones obtained in Computed Tomography (CT). We adapted CT reconstruction methods to accurately reconstruct truncated PR-MRI data, obtained for example when coronal and sagittal views of limbs are acquired.
- Spiral data is reconstructed by gridding, which requires the computation of a kernel that can be cumbersome. We developed dedicated and faster ways of reconstructing arbitrary trajectories [1,2].
Another option to reduce scan time is to use parallel imaging. Multiple receiver coils have been introduced since 1999. Two algorithms (SENSE and GRAPPA) are generally used with such acquisitions. Modifying them, we are working on reconstruction methods which can handle any k-space trajectory.
The recent introduction of the theory of compressed sensing (CS) offers a great opportunity to reduce the scan time even further. According to CS, compressible images can be recovered from a significantly smaller number of measurements, well below the Nyquist rate. The recovery is possible as long as the measurements are incoherent, the image is compressible, and the reconstruction uses a special non-linear procedure. MRI images in general, and dynamic images in particular, are often highly compressible. In addition, MRI data are collected in the spatial frequency domain, not as pixels in the image domain. MR provides considerable flexibility in choosing sampling patterns which enables incoherent sampling by pseudo randomly undersampling of k-space. As a result, MRI is a natural CS hardware system. MRI scans can be significantly accelerated by obtaining fewer samples and exploiting the compressibility of the underlying images for reconstruction [3,4]. We have been applying CS for various applications, including dynamic cardiac imaging [4,5], contrast enhanced and flow independent angiography , and hyperpolarized 13C spectroscopy imaging .
- Jackson J, et al., "Selection of a Convolution Function for Fourier Inversion using Gridding", IEEE Trans Med Imaging. 1991;10:473-478.
- Beatty PJ, et al., "Rapid Gridding Reconstruction with a minimal oversampling Ratio", IEEE Trans Med Imaging. 2005;24(6):799-808.
- Lustig M, et al., "Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging", Magnetic Resonance in Medicine, 2007; 58(6):1182-1195
- Lustig M, et al., "Compressed Sensing MRI", IEEE Signal Processing Magazine, 2008;25(2):72-82
- Lustig M., et al., "k-t Sparse: High Frame-Rate Dynamic MRI Exploiting Spatio-Temporal Sparsity", Proc 13th ISMRM, Seattle, 2006. p 2420.
- Cukur T., et al., "Improving Non-Contrast Enhanced SSFP Angiography with Compressed Sensing", Proc 14th ISMRM, Toronto, 2008, p 726.
- Hu S., et al., "Compressed Sensing for Resolution Enhancement of HyperPolarized (13)C flyback 3D-MRSI", J Magn Reson, 2008; 192(2):258-64
MR Systems Hardware
Keeping pace with modern MR imaging techniques requires advanced and innovative hardware. Our research targets novel approaches to MR systems design, from excitation and acquisition, to amplifiers and coils. We have developed and demonstrated complete imaging systems such as our Prepolarized MRI Scanner, which offers competitive image quality at low cost.
Current work is focused on development of our flexible and scalable Medusa Console for parallel imaging applications, as well as a vector modulation array system for parallel transmit and array coil decoupling. Challenges include analog RF and high-speed digital circuit design, syncronization, data management, and control and user-interface software development.
- Scott G, et al., "A Prepolarized MRI Scanner", Proc 9th ISMRM, Glasgow, 2001. p. 610.
- Matter N, et al., "A Fast Recovery Pulsed Readout Power Supply for Prepolarized MRI", Proc 9th ISMRM, Glasgow, 2001. p. 1152.
- Scott G, et al., "Wireless Transponders for RF Coils: Systems Issues", Proc 13th ISMRM, Miami, 2005. p. 330.
- Grafendorfer T, et al., "Optimized Litz Coil Design for Prepolarized Extremity MRI", Proc 14th ISMRM, Seattle, 2006. p. 2613.
- Stang P, et al., "A Scalable Prototype MR Console Using USB", Proc 14th ISMRM, Seattle, 2006. p. 1352.
- Scott G, et al., "Signal Vector Decoupling for Transmit Arrays", Proc 15th ISMRM, Berlin, 2007. p. 168.
- Stang P, et al., "A High Speed Vector Modulation Array System", Proc 15th ISMRM, Berlin, 2007. p. 169.
- Stang P, et al., "MEDUSA: A Scalable MR Console for Parallel Imaging", Proc 15th ISMRM, Berlin, 2007. p. 925.
- Real-Time Interactive MRI
While respiratory and cardiac motion have complicated traditionally long MR scans, recently developed real-time interactive MRI techniques appear robust. Entire image acquisitions are completed within a fraction of a cardiac cycle (usually < 150 ms), with minimal motion artifacts. We have been developing real-time interactive imaging systems, as well as methods for providing different types of image contrast and information in real time.
Particular projects include:
- Color flow MRI, in which spin velocity and spin density information is acquired and displayed simultaneously, with the goal of visualizing cardiac flow and evaluating regurgitant valves.
- black blood MRI, in which blood signal is suppressed to enhance visualization of the myocardium and vessel walls.
- multislice stress MRI, in which multiple slices are imaged at high frame rates thus enabling whole-heart functional and perfusion assessment in a single short examination.
- coronary localization, in which sub-millimeter resolution cardiac images are acquired in real-time, enabling accurate localization and possible initial screening of the coronary arteries.
Clinically, we are also involved in the clinical validation of real-time MRI for the purposes of left ventricular function assessment, small-bowel imaging, valve evaluation, and intravascular imaging.
- Riederer SJ, et al., "MR Fluoroscopy: Technical Feasibility", Mag. Reson. Med. 1988:8:1-15.
- Kerr AB, et al., "Real-Time Interactive MRI on a Conventional Scanner", Mag. Reson. Med. 1996:35:734-740.
- Nayak KS, et al., "Real-Time Color Flow MRI", Mag. Reson. Med. 2000:43:251-258.
- Nayak KS, et al., "Real-Time Black-Blood MRI using Spatial Presaturation", J. Magn. Reson. Imag. (in press).
- Nayak KS, et al., "Rapid Ventricular Assessment using Real-time Interactive Multi-slice MRI, Magn. Reson. Med. (in press).
- Yang PC, et al., "New Real-Time Interactive Magnetic Resonance Imaging Complements Echocardiography", J. Am. Coll. Cardiol. 1998:32:2049-2056.
- Chan FP, et al., "Real-Time Interactive MR Imaging of the Small Bowel", RSNA Electronic Journal 1997:1.
- RF Pulse Design
Exciting spins is at the core of MRI. Controlling this excitation in a precise way is necessary to get the desired image. Radio-frequency (RF) pulses can be shaped in order to excite specific ROIs or given spectral bands . They are limited in amplitude and area, both by the hardware, and for safety reasons. Pulse design tries to optimize the excitation within those constraints. The Shinnar-LeRoux algorithm reduces pulse design to digital filter design, opening a wide window of what can be done [1,3].
- Pauly JM, et al., "Parameter Relations for the Shinnar-Le Roux Selective Excitation Pulse Design Algorithm", IEEE Trans Med Imaging. 1991;10:53-65.
- Pauly JM, et al., "Independent Dual-Band Spectral-Spatial Pulses", Proc 11th ISMRM, Toronto, 2003. p 966.
- Larson PEZ, et al., "Designing Long-T2 Suppression Pulses for Ultrashort Echo Time Imaging", Magn Reson Med. 2006 Jul;56(1):94-103.
- Coronary Artery Imaging
Noninvasive coronary artery imaging is one of the most sought after goals in medical imaging. Despite significant advances in preventive measures, coronary artery disease remains the major cause of mortality and morbidity in the industrialized world. Over the past few years, significant advances have occurred in coronary magnetic resonance angiography, resulting in considerable optimism that this goal is achievable.
Our current approach to coronary artery imaging consists of three strategies:
Breath-held 2D multislice spiral imaging
The spiral k-space trajectory  is compelling for coronary imaging due to its properties, which include highly time- and SNR-efficient sampling of k-space, excellent temporal resolution, and good immunity to flow/motion effects. We have designed spiral sequences for imaging at 3 T to take advantage of the higher available signal , and incorporated variable-density sampling  to achieve higher spatial resolution in a short scan time.
Real-time interactive imaging
Real-time interactive (RTI) imaging merges very rapid image acquisition with real-time image reconstruction and interactive control of the scanner . With RTI imaging, real-time localization of an appropriate scan plane can be achieved rapidly, and the operator can switch to a high-resolution imaging sequence for enhanced detail [5, 6].
Free-breathing 3D imaging
Free-breathing scans generally provide higher SNR compared to breath-held scans, and they simplify the scan protocol for both the operator and the patient. We use a 3D cones k-space trajectory  for high resolution whole-heart coverage. The 3D cones trajectory offers many desirable properties for coronary imaging: robust motion properties in all three dimensions, high readout duty cycle, and favorable trade-off for scan time reduction. To contend with respiratory motion effects in a free-breathing scan, we use an efficient method called the diminishing variance algorithm (DVA) for data acquisition and reconstruction [8, 9].
- Meyer CH, et al., "Fast Spiral Coronary Artery Imaging", Magn Reson Med. 1992 28:208-13.
- Santos JM, et al., "Single Breath-Hold Whole-Heart MRA Using Variable-Density Spirals at 3T", Magn Reson Med. 2006 55:371-9.
- Tsai CM, et al., "Reduced Aliasing Artifacts Using Variable-Density k-space Sampling Trajectories", Magn Reson Med. 2000 43:452-8.
- Santos J, et al., "Real-time Cardiac Imaging Using Balanced SSFP with Ultra-short Variable-Density Spiral Readouts", In: Proc. 8th SCMR, 2005. pp 295-6.
- Nguyen P, et al., "Real-time MR Coronary Angiography at 3T", In: Proc. 12th ISMRM, 2004. p 1877.
- Santos J, et al., "Real-time High-resolution Coronary MRA", In: Proc. ISMRM Workshop on Real-Time MRI, 2006.
- Gurney PT, et al., "Design and Analysis of a Practical 3D Cones Trajectory", Magn Reson Med. 2006 55:575-82.
- Sachs TS, et al., "The Diminishing Variance Algorithm for Real-time Reduction of Motion Artifacts in MRI", Magn Reson Med. 1995 34:412-22.
- Schaffer R, et al., "Improved Accuracy of Shift Calculations Through Appropriate Filtering of High Resolution Navigators", In: Proc. 14th ISMRM, 2006. p 2974.
- Breath-held 2D multislice spiral imaging
- Functional Brain Imaging
Recently, the balanced SSFP technique has been adapted for functional MRI in order to provide improved SNR and reduced distortion. Two different contrast mechanisms have been investigated: transition-band SSFP fMRI, which detects the bulk frequency shift from BOLD using the frequency selectivity of the SSFP profile, and pass-band SSFP fMRI, which primarily measures T2 changes from the diffusion effect. We have investigated the signal characteristics of SSFP fMRI and developed several important techniques to make high resolution and low distortion fMRI possible.
- Miller KL, et al., "Functional brain imaging using a blood oxygenation sensitive steady state", Magn Reson Med. 2003 Aug;50(2):675-683.
- Miller KL, et al., "High-resolution fMRI at 1.5T using balanced SSFP", Magn Reson Med. 2006;55:161-170.
- Lee J, et al., "Respiration-induced B0 field fluctuation compensation in balanced SSFP: Real-time approach for transition-band SSFP fMRI", Magn Reson Med. 2006;55:1197-1201.
- Lee J, et al., "Complex data analysis in high-resolution SSFP fMRI", Magn Reson Med. 2007;57:905-917.
- Lee JH, et al., "Blood oxygenation (BOX) level dependnet functional brain imaging using steady-state free precession", Proc 14th ISMRM, Seattle, 2006. p. 3291.
- Lee JH, et al., "BOX fMRI using multiple-acquisition steady-state free precession imaging for full-brain coverage", Proc 14th ISMRM, Seattle, 2006. p. 3297.
- Lee JH, et al., "Full-brain coverage and high-resolution imaging capabilities of passband SSFP fMRI at 3T", Proc 14th ISMRM, Berlin, 2007. p. 694.
- Kim TS, et al., "Analysis of the BOLD Signal Characteristics in Balanced SSFP fMRI: A Monte-Carlo Simulation", Proc 15th ISMRM, Berlin, 2007. p. 696.
- Interventional MRI
Open bore magnets enable the use of MRI during surgery. While MRI can be used to visualize anatomy during surgical interventions, there are many other compelling applications. Our research focuses on development of hardware and imaging methods for specialized interventional procedures. This includes monitoring of RF heating  and imaging [2, 3] of MRI guidewires, transmit and receive hardware for interventions [4, 5], and current density imaging for RF ablation [6, 7, 8].
- Venook R, et al., "Reducing and Monitoring Resonant Heating in MR Guidewires", Proc 14th ISMRM, Seattle, 2006. p. 2037.
- Scott G, et al., "Resistively Coupled Interventional Device Visualization", Proc 14th ISMRM, Seattle, 2006. p. 266.
- Overall W, et al., "Phase refocusing for improved visualization of interventional guidewires", Proc 15th ISMRM, Berlin, 2007. p. 1117.
- Venook R, et al., "Comparison of Surface Coil and Automatically-tuned, Flexible Interventional Coil Imaging in a Porcine Knee", Proc 12th ISMRM, Kyoto, 2004. p. 2639.
- Overall W, et al., "Optically Coupled Op Amp Transmitter for Use with Interventional Microcoils", Proc 13th ISMRM, Miami, 2005. p. 2655.
- Scott G, et al., "RF Ablation Electrode Current Imaging by MRI", Proc 12th ISMRM, Kyoto, 2004. p. 989.
- Scott G, et al., "RF Ablation Current Visualization at 0.5T", Proc 13th ISMRM, Miami, 2005. p. 151.
- Shultz KM, et al., "Feasibility of Full RF Current-Vector Mapping for MR Guided RF Ablations", Proc 15th ISMRM, Berlin, 2007. p. 1131.
- Musculoskeletal Imaging
MRI has become an extremely valuable tool in diagnosis and evaluation of the musculoskeletal system because of its excellent soft tissue contrast. We have been exploring new pulse sequences, parallel imaging, real-time imaging, and clinical applications such as articular cartilage imaging.
Evaluation of articular cartilage is valuable in diagnosis of osteoarthritis, which is the leading cause of disability in the developed nations. We have developed and tested steady-state and driven equilibrium sequences for cartilage imaging and compared them to various other methods [1,2]. Clinically, we are finding a 3D gradient-echo imaging in conjunction with the IDEAL method for water-fat separation  useful for evaluating cartilage thickness and volume. We have also used T2-selective imaging to isolate the cartilage signal .
It is clinically valuable to have 3D, isotropic resolution images in the knee and ankle in a reasonable scan time, which can be achieved using parallel imaging . This allows for excellent reformats and 3D renderings of the images.
We have been using real-time MRI to accurately measure in vivo joint kinematics to understand normal and pathological joint mechanics . This allows for easy and direct measurements of the motion.
Balanced steady-state free precession (SSFP) imaging offers high SNR in short scan times, and we have continued to develop and explore enhanced fat suppression and banding artifact reductions in SSFP [7-9].
More structured and solid tissues, such as tendons, the menisci, and cortical bone, have no signal with conventional MRI techniques. We have been using ultra-short echo time (UTE) pulse sequences, both in 2D and 3D, to obtain signal from these types of tissues. By using specialized RF pulse schemes  and also the 3D cones acquisition method , we can obtain high contrast images in reasonable scan times.
- Hargreaves BA, et al., "MR imaging of articular cartilage using driven equilibrium", Magn Reson Med. 1999; 42:695-703.
- Hargreaves BA, et al., "Comparison of new sequences for high-resolution cartilage imaging", Magn Reson Med. 2003; 49:700-709.
- Reeder SB, et al., "Iterative decomposition of water and fat with echo asymmetry and least-squares estimation (IDEAL): application with fast spin-echo imaging", Magn Reson Med. 2005 Sep;54(3):636-644.
- Vidarsson L, et al., "Linear combination filtering for T2-selective imaging of the knee", Magn Reson Med. 2006 May;55(5):1191-1196.
- Gold GE, et al., "Isotropic MRI of the knee with 3D fast spin-echo extended echo-train acquisition (XETA): initial experience", AJR Am J Roentgenol 2007 May;188(5):1287-1293.
- Draper CE, et al., "Accuracy of Using Real-Time MRI for Joint Motion Measurements: A Phantom Study", Proc 15th ISMRM, Berlin, 2007, p. 2680.
- Hargreaves BA, et al., "Fat-suppressed steady-state free precession imaging using phase detection", Magn Reson Med. 2003; 50:210-213.
- Cukur T, et al., "Enhanced Spectral Shaping in Steady-State Free Precession Imaging", Mag Reson Med. 2007 Dec;58(6):1216-1223.
- Cukur T, et al., "Multiple-Profile Homogeneous Image Combination: Application to Phase-Cycled SSFP and Multicoil Imaging", Mag Reson Med. 2008 Sep;60(3):732-738.
- Larson PEZ, et al., "Designing long-T2 suppression pulses for ultrashort echo time imaging", Magn Reson Med. 2006 Jul;56(1):94-103.
- Gurney PT, et al., "Design and analysis of a practical 3D cones trajectory", Magn Reson Med. 2006 Mar;55(3):575-582.
- Peripheral Arterial Disease Imaging
The hospitalization treatment of patients with acute and chronic peripheral arterial disease totals more than 100,000 per year in the United States alone. Annually, the amputation rate for peripheral artery disease is more than 10,000. Despite the high prevalence of this disease, conventional evaluations of patients with this condition often lack the sensitivity and specificity needed to make major therapeutic decisions or to monitor potentially new treatment options such as stem cell or growth factor therapy. Thus, there is a need for new noninvasive techniques for monitoring and assessment of this disease. We are currently developing MR imaging tools to assess lower extremity anatomy and blood flow [1-3], global and regional perfusion [4,5], and skin anatomy and physiology [6,7]. These methods will be relevant to the study of patients with deep-tissue leg ischemia and superficial skin pathologies. We believe that this set of information, which is not available through any other means, will, with low risk, enable the identification and risk stratification of patients with peripheral arterial disease.
- Brittain JH, et al., "Three-Dimensional Flow-Independent Peripheral Angiography", Magn Reson Med. 1997; 38:343-354.
- Cukur T, et al., "Non-Contrast-Enhanced Flow-Independent Peripheral MR Angiography with Balanced SSFP ", Magn Reson Med. 2009; to appear.
- Park JB, et al., "Rapid Measurement of Time-Averaged Blood Flow Using Ungated Spiral Phase-Contrast", Magn Reson Med. 2003;49:322-328.
- Kaji S, et al., "Quantification of regional human leg muscle perfusion using first-pass magnetic resonance imaging", Proc 9th ISMRM, 2001. p. 252.
- Lee JH, et al., "Fast 3D imaging using variable-density spiral trajectories with applications to limb perfusion", Magn Reson Med. 2003;50:1276-1285.
- Lee JH, et al., "3D High Resolution Skin Imaging", Proc 12th ISMRM, 2004. p. 2094.
- Dicarlo J, et al., "SNR Comparison of RF Coil Size for Ischemic Skin Imaging", Proc 12th ISMRM, 2004. p. 2641.
- Prepolarized MRI for Imaging Near Metal
Prepolarized MRI for Imaging Near Metal Implants
Prepolarized MRI is a novel method of MRI that uses two inexpensive electromagnets to make an MRI study of the human body. By using inexpensive electromagnets, this new method allows for high quality imaging at nearly an order-of-magnitude cost reduction relative to conventional MRI. The total scanner hardware cost is about $50,000, including manufacturer's materials, labor and profit. Image quality is now about as good as a 0.5 T conventional MRI scanner. The scanner also allows for some unique applications. First, imaging near any metallic implant is much more effective with Prepolarized MRI than conventional MRI or CT. Second, we can provide quantitative maps of protein concentration.
Here we show some preliminary results with the scanner developed at Stanford in Electrical Engineering. Project leaders are Prof. Steve Conolly, now at UC Berkeley (sconolly at berkeley dot edu), and Stanford EE's Dr. Greig Scott (greig at mrsrl dot stanford dot edu) and Prof. Albert Macovski.
- Macovski A, et al., "Novel approaches to low-cost MRI", MRM, 30(2):221?30, 1993.
- Morgan P, et al., "A readout magnet for prepolarized MRI", MRM, 41:1221, 1996.
- Xu H, et al., "Homogeneous magnet design using linear programming", IEEE Transactions on Magnetics, 36(2):476?483, 2000.
- Ungersma S, et al., "Shim coil design using a linear programming algorithm", MRM, 52(3):619?627, August 2004.
- Ungersma S, et al., "Magnetic resonance imaging with T1 dispersion contrast", MRM, 55(6):1362?1371, 2006.
- Matter N, et al., "Rapid polarizing field cycling in magnetic resonance imaging", IEEE Transactions in Medical Imaging, 25(1):84?93, 2006.
- Venook R, et al., "Prepolarized magnetic resonance imaging around metal orthopedic implants", MRM, 56(1):177?186, 2006.
- Matter N, et al., "Three-dimensional prepolarized magnetic resonance imaging using rapid acquisition with relaxation enhancement", MRM, 56(5):1085?1095, 2006.
- Matter N, et al., "Noise Performance of a Precision Pulsed Electromagnet Power Supply for Magnetic Resonance Imaging", Accepted to IEEE Transactions in Medical Imaging, 2007.
Please contact sconolly at berkeley dot edu or greig at mrsrl dot stanford dot edu for reprints.