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## Optimization and Data Mining in Epilepsy Research

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**Optimization and Data Mining in Epilepsy Research**W. Art Chaovalitwongse Assistant Professor Industrial and Systems Engineering Rutgers University**Acknowledgements**• Comprehensive Epilepsy Center, St. Peter’s University Hospital • Rajesh C. Sachdeo, MD • Deepak Tikku, MD • Brain Institute, University of Florida • Panos M. Pardalos, PhD • J. Chris Sackellares, MD • Paul R. Carney, MD • Bioengineering, Arizona State University • Leonidas D. Iasemidis, PhD**Agenda**• Background: Epilepsy • Electroencephalogram (EEG) Time Series • Chaos Theory: Dimensionality Reduction • Seizure Prediction • Feature Selection • Process Monitoring • Concluding Remarks**Facts About Epilepsy**• At least 2 million Americans and other 40-50 million people worldwide (about 1% of population) suffer from Epilepsy. • Epilepsy is the second most common brain disorder (after stroke) • The hallmark of epilepsy is recurrent seizures. • Epileptic seizures occur when a massive group of neurons in the cerebral cortex suddenly begin to discharge in a highly organized rhythmic pattern.**Epileptic Seizures**• Seizures usually occur spontaneously, in the absence of external triggers. • Seizures cause temporary disturbances of brain functions such as motor control, responsiveness and recall which typically last from seconds to a few minutes. • Seizures may be followed by a post-ictal period of confusion or impaired sensorial that can persist for several hours.**Rationale**• Based on 1995 estimates, epilepsy imposes an annual economic burden of $12.5 billion in the U.S. in associated health care costs and losses in employment, wages, and productivity. • Cost per patient ranged from $4,272 for persons with remission after initial diagnosis and treatment to $138,602 for persons with intractable and frequent seizures.**How To Fight Epilepsy**• Anti-Epileptic Drugs (AEDs) • Mainstay of epilepsy treatment • Approximately 25 to 30% remain unresponsive • Epilepsy surgery • Require long-term invasive EEG monitoring • 50% of pre-surgical candidates do not undergo respective surgery • Multiple epileptogenic zones • Epileptogenic zone located in functional brain tissue • Only 60% of surgery cases result in seizure free • Electrical Stimulation (Vagus nerve stimulator) • Parameters (amplitude and duration of stimulation) arbitrarily adjusted • As effective as one additional AED dose • Side Effects • Seizure Prediction?**Open Problems**• Is the seizure occurrence random? • If not, can seizures be predicted? • If yes, are there seizure pre-cursors preceding seizures? • If yes, what measurement can be used to indicate these pre-cursors? • Does normal brain activity during differ from abnormal brain activity?**Electroencephalogram (EEG)**• …is a tool for evaluating the physiological state of the brain. • …offers excellent spatial and temporal resolution to characterize rapidly changing electrical activity of brain activation • …captures voltage potentials produced by brain cells while communicating. • In an EEG, electrodes are implanted in deep brain or placed on the scalp over multiple areas of the brain to detect and record patterns of electrical activity and check for abnormalities.**From Microscopic to Macroscopic Level (Electroencephalogram**- EEG)**ROF**LOF LTD RST LST LOF LST RTD LTD Depth and Subdural electrode placement for EEG recordings**Goals of Research**• Test the hypothesis that seizures are not a random process. • Employ data mining techniques to differentiate normal and abnormal EEGs • Employ quantitative analysis to identify seizure pre-cursors • Demonstrate that seizures could be predicted • Develop a closed-loop seizure control device (Brain Pacemaker)**10-second EEGs: Seizure Evolution**Normal Pre-Seizure Post-Seizure Seizure**Dimensionality Reduction**• The brain is a non-stationary system. • EEG time series is non-stationary. • With 200 Hz sampling, 1 hour of EEGs is comprised of 200*60*60*30 = 21,600,000 data points = 43.2MB (assume 16-bit ASCI format) • 1 day = 1 hour*24 • 1 week = 1 hour*168 • 20 patients = 1 hour*3360 → Terabytes → Gigabytes → Megabytes Kilobytes**Dimensionality Reduction Using Chaos Theory**• Chaos in Brain? • Chaos in Stock Market? • Chaos in Foreign Exchanges (Swedish Currency)? • Measure the brain dynamics from EEG time series. • Apply dynamical measures (based on chaos theory) to non-overlapping EEG epochs of 10.24 seconds = 2048 points. • Maximum Short-Term Lyapunov Exponent • measures the average uncertainty along the local eigenvectors and phase differences of an attractor in the phase space • Measures the chaoticity of the brain waves**Embed the data set (EEG). Xi =**(x(ti),x(ti+τ),…,x(ti+(p-1)τ))T whereτ is the selected time lag between the components of each vector in the phase space, p is the selected dimension of the embedding phase space, and ti [1,T-(p-1) τ]. • Pick a point x(t0) somewhere in the middle of the trajectory. Find that point's nearest neighbor. Call that point z0 (t0). • Compute |z0 (t0) - x(t0)| = L0. • Follow the ``difference trajectory" -- the dashed line -- forwards in time, computing |z0 (ti) - x(ti)| = L0(i) and incrementing i, until L0(i) > ε. Call that value L0' and that time t1. • Find z1 (t1), the “nearest neighbor” of x(t1), and go to step 3. Repeat the procedure to the end of the fiduciary trajectory t = tn, keeping track of the Li and Li' . where M is the number of times we went through the loop above, and N is the number of time-steps in the fiduciary. NΔt = tn - t0**2-D Example: Circle of initial conditions evolves into an**ellipse.**Pre-Ictal**Ictal Post-Ictal STLmax Profiles**Then, we calculate the average value, ,and the**sample standard deviation, , of . How similar are they?Statistics to quantify the convergence of STLmax By paired-T statistic: Per electrode, for EEG signal epochs i and j, suppose their STLmax values in the epochs (of length 60 points, 10 minutes) are The T-index between EEG signal epochs i and j is defined as**IID (Independent and Identically Distributed) Test**Assumption 1: Within a window of 30 STLmax points, the differences of STLmax values (Dij) between two electrode sites i and j are independent. To verify this assumption, Employ “portmanteau” test of white noise developed by Ljung and Box. Assumption 2: Within a wt window of 60 points, the differences of STLmax values between two electrode sites i and j are normally distributed. To verify this assumption, Employ To check this assumption, we employed the Shapiro-Wilk W test, which is is a well-established and powerful test of departure from normality.**Models**Homoclinic Chaos (Silnikov’s Theorem): Rössler systems, Lorentz systems, population dynamical systems (1) w, a, b and g are intrinsic parameters. e and e’ are directional coupling strengths. N = number of oscillators (2) (3)**Not every electrode site shows the convergence.**Feature Selection: Select the electrodes that are most likely to show the convergence preceding the next seizure. Why Feature Selection?**Optimization Problem**• Optimization: • We apply optimization techniques to find a group of electrode sites such that … • They are the most converged (in STLmax) electrode sites during 10-min window before the seizure • They show the dynamical resetting (diverged in STLmax) during 10-min window after the seizure. • Such electrode sites are defined as “critical electrode sites”. • Hypothesis: • The critical electrode sites should be most likely to show the convergence in STLmax again before the next seizure.**Multi-Quadratic Integer Programming**• To select critical electrode sites, we formulated this problem as a multi-quadratic integer (0-1) programming (MQIP) problem with … • objective function to minimize the average T-index among electrode sites • a linear constraint to identify the number of critical electrode sites • a quadratic constraint to ensure that the selected electrode sites show the dynamical resetting**Notation and Modeling**• x is an n-dimensional column vector (decision variables), where each xi represents the electrode site i. • xi= 1 if electrode i is selected to be one of the critical electrode sites. • xi= 0 otherwise. • Qis an (nn) matrix, whose each element qijrepresents the T-index between electrode i andj during 10-minute window before a seizure. • bis an integer constant. (the number of critical electrode sites) • Dis an (nn) matrix, whose each element dijrepresents the T-index between electrode i andj during 10-minute window after a seizure. • α = 2.662*b*(b-1), an integer constant. 2.662 is the critical value of T-index, as previously defined, to reject H0: “`two brain sites acquire identical STLmax values within 10-minute window”**Conventional Linearization Approach for Multi-Quadratic 0-1**Problem**KKT Conditions Approach**• Consider the quadratic 0-1 programming problem • eT = (1,1,…,1) • Relax x ≥ 0, we then have the following KKT conditions: Q is an (nn) matrix. b is an integer constant x is an n-dimensional column vector**KKT Conditions Approach**• Add slack variables a and define s = u.e + a • Minimizing slack variables, we can formulate this problem as: • Note that this problem formulation is an efficient approach, as n increases, because it has the SAME number of 0-1 variables (n), and 2n additional continuous variables. Fix x{0,1}**Connections Between QIP problems and MILP problems**• For any matrix Q where qij≥0 • We want to prove that P and P are equivalent: Equivalent**Theoretical Results:MILP formulation for MQIP problem**• Consider the MQIP problem • We proved that the MQIP program is EQUIVALENT to a MILP problem with the SAME number of integer variables. Equivalent**Reference:**• P.M. Pardalos, W. Chaovalitwongse, L.D. Iasemidis, J.C. Sackellares, D.-S. Shiau, P.R. Carney, O.A. Prokopyev, and V.A. Yatsenko. Seizure Warning Algorithm Based on Spatiotemporal Dynamics of Intracranial EEG. Mathematical Programming, 101(2): 365-385, 2004.**Empirical Results:Performance on Larger Problems**• Reference: • W. Chaovalitwongse, P.M. Pardalos, and O.A. Prokopyev. Reduction of Multi-Quadratic 0-1 Programming Problems to Linear Mixed 0-1 Programming Problems. Operations Research Letters, 32(6): 517-522, 2004.**Hypothesis Testing - Simulation**• Hypothesis: • The critical electrode sites should be most likely to show the convergence in STLmax (drop in T-index below the critical value) again before the next seizure. • The critical electrode sites are electrode sites that • are the most converged (in STLmax ) electrode sites during 10-min window before the seizure • show the dynamical resetting (diverged in STLmax ) during 10-min window after the seizure • Simulation: • Based on 3 patients with 20 seizures, we compare the probability of showing the convergence in STLmax (drop in T-index below the critical value) before the next seizure between the electrode sites, which are • Critical electrode sites • Randomly selected (5,000 times)**Automated Seizure Warning System**ASWA Monitor the average T-index of the critical electrodes Continuously calculate STLmax from multi- channel EEG. Select critical electrode sites after every subsequent seizure EEG Signals Give a warning when: T-index value is greater than 5, then drops to a value of 2.662 or less**Performance Evaluation for ASWS**• To test this algorithm, a warning was considered to be true if a seizure occurred within 3 hours after the warning. • Sensitivity = • False Prediction Rate = average number of false warnings per hour