I am an NSF postdoctoral fellow at the Mathematical Biosciences Institute in Columbus Ohio. I am generally interested in using stochastic processes, partial differential equations, and adapting/extending methodology used in theoretical physics with the aim of explainings physical processes occuring in biological systems.

I have previously worked on inverse problems using methods of statistical physics with applications to imaging and partial differential equations. On the biological end, I have paid particular attention to ion and oxygen homeostasis in the brain, with the goal of gaining an understanding of a phenomenon known as cortical spreading depression.

From 2008-2012, I was a graduate student in the UCLA Department of Biomathematics (Department of Mathematical and Computational Biology), an applied mathematics program with emphasis on biological applications. My graduate advisors were Tom Chou, KC Brennan, and Van Savage. Additionally, I previously had completed an MS in Statistics under the tutelage of Rick Schoenberg, and Jan de Leeuw.

## Recent Research Output

### Field-theoretic formulation of inverse problems

Joshua C. Chang, Van Savage, and Tom Chou. A path-integral approach to Bayesian inference for inverse problems using the semiclassical approximation, Undergoing peer review, [arXiv]

Abstract: We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (in nite dimensional) coefficient functions from ordinary or partial di erential equations (ODE, PDE), a problem which is typically ill-posed. Regularization of these problems using $\ell^2$ function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem namely whether the subjective choice of regularization is compatible with prior knowledge. The path-integral approach, while offering an alternative to standard computational methods of inversion which typically involve the use of Markov-Chain-Monte-Carlo (MCMC) methods, also gives way naturally to Monte-Carlo methods that can be used to refine perturbative approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theory involving the Poisson equation.

### Mathematical modeling of perfusion in spreading depression

Joshua C. Chang, K.C. Brennan, Dondong He, H. Huang, R.M. Miura, Phillip L. Wilson, J.J. Wylie. A mathematical model of the metabolic and perfusion effects on cortical spreading depression. PLoS One, August 2013. [link] [arXiv preprint] [Siam 2012 Poster]

Spreading depression is a wave of depolarization and subsequent depression that travels in gray matter in the brain. It involves drastic changes in ionic balances and brain metabolism. Previously, the connection between these two aspects of this phenomenon were not clear. Using a lumped vascular model that is controlled by extracellular ion concentrations, and a model for oxygen use and delivery, we examined the connection.

## Past Research Projects

### Using shape priors in image segmentation

Joshua C. Chang, and Tom Chou. Iterative graph cuts for image segmentation with a nonlinear statistical shape prior, Journal of Mathematical Imaging and Vision, 2013, DOI:10.1007/s10851-013-0440-9. [arXiv preprint] [Springer] [MBI IP meeting poster] [source]

Usually, when seeking to identify objects in images, one knows the general shapes of the objects that one is looking for. When this knowledge is available as a set of possible templates, a natural technique for representation of said knowledge is through kernel density estimation. With the prior knowledge, one can frame segmentation as Bayesian inference, resulting in an energy minimization problem that is nonlinear. This nonlinearity can be ameliorated through the use of majorization minimization to find an algorithm such that image segmentations can be found quickly using graph cuts.

### Moving interface tracking

Joshua C. Chang, K.C. Brennan, and Tom Chou. Tracking monotonically advancing boundaries on image sequences using graph cuts and recursive kernel shape priors, IEEE Transactions in Medical Imaging 31(5): 1008-1020.

This project extended particle filtering, a stochastic variant of the Kalman filter, to the tracking of moving interfaces in image sequences. The fast marching method is used as a model for the evolution of the interface, and Gaussian random fields are used to estimate and predict the future speed and position of the interface for use in regularization.

### Spectroscopic investigation of brain oxygenation during spreading depression

Spreading depression is a large metabolic event, where the dicharge of many ions into the extracellular space requires the large use of ATP and hence oxygen for recovery. Furthermore, blood flow in the brain is disrupted as the vascular system reacts to local increases in extracellular potassium. This study investigated these changes using second-derivative optical spectroscopy. Second-derivative spectroscopy shows changes in blood oxygenation level because of spectroscopic signatures of oxy- and deoxy- hemoglobin are different. Furthermore, both spectra have high curvature, so second derivative spectroscopy is able to control for background absorbers while maintaining the blood signal. In this study, we also found a prolonged second phase of imbalance lasting approximately an hour and starting approximately five minutes after the initial spreading depression event. This phase of the phenomenon is poorly understood and may be highly relevant to brain disorder.

### Point-process modeling of emergency medical events

Modeling ambulance dispatch events as a spatial-temporal point-process, one is able to identify clustering in ambulance response time. To some degree, this clustering is expected since ambulances are a finite resource. If a large amount of EMS calls are concentrated in a single area, it adversely affects the ability of the system to respond to future calls, at least while the current ambulances are still busy servicing previous calls. Point-process modeling allows one to identify areas that are performing well with adversely high loads, and areas that are more vulnerable to extraordinary emergencies.

## Peer-reviewed publications

1. Joshua C. Chang, K.C. Brennan, Dondong He, H. Huang, R.M. Miura, Phillip L. Wilson, J.J. Wylie. A mathematical model of the metabolic and perfusion effects on cortical spreading depression. PLoS One, August 2013. [link] [arXiv preprint] [Siam 2012 Poster]
2. Joshua C. Chang, and Tom Chou. Iterative graph cuts for image segmentation with a nonlinear statistical shape prior, Journal of Mathematical Imaging and Vision, 2013, DOI:10.1007/s10851-013-0440-9. [arXiv preprint] [Springer] [MBI IP meeting poster] [source]
3. Joshua C. Chang, K.C. Brennan, and Tom Chou. Tracking monotonically advancing boundaries on image sequences using graph cuts and recursive kernel shape priors, IEEE Transactions in Medical Imaging 31(5): 1008-1020. [link] [preprint] [supplemental].
4. Joshua C. Chang, Lydia L. Shook, Jonathan Biag, Elaine N. Nguyen, Arthur W. Toga, Andrew C. Charles, and Kevin C. Brennan. Biphasic direct current shift, haemoglobin desaturation and neurovas- cular uncoupling in cortical spreading depression, Brain (2010) 133(4): 996-1012. [link]
5. Joshua C. Chang MS and Frederic P. Schoenberg ScB, PhD (2009) "A Statistical Analysis of Santa Barbara Ambulance Response in 2006: Performance Under Load", Western Journal of Emergency Medicine: Vol. 10: No. 1, Article 12. [link]

## Undergoing peer-review

1. Joshua C. Chang, Van Savage, Tom Chou. A path-integral approach to Bayesian inference for inverse problems using the semiclassical approximation. Submitted, December 2013. [arXiv preprint]

## Collaborators

In addition to my faculty advisors listed above, I have the following collaborators:

Robert Miura, NJIT Mathematical Sciences
Huaxiong Huang, York University Department of Mathematics and Statistics
Jonathan Wylie, City U of Hong Kong, Department of Mathematics
Phil Wilson, University of Canterbury, Department of Mathematics

## Colleagues

I have had the priviledge of engaging in intelligent conversation with my many fine collegues over the years. Some of these people include David Alexander, Forrest Crawford, Jennifer Tom, John Ranola, Moses Wilks, Mitchell Johnson, Gabriela Cybis, Connie Phong, Lydia Shook, Terence Tong.