Seizure warning algorithm based on optimization and ...

Math. Program., Ser. B (2004) Digital Object Identifier (DOI) 10.1007/s10107-004-0529-4

Panos M. Pardalos · Wanpracha Chaovalitwongse · Leonidas D. Iasemidis · J. Chris Sackellares · Deng-Shan Shiau · Paul R. Carney · Oleg A. Prokopyev · Vitaliy A. Yatsenko

Seizure warning algorithm based on optimization and nonlinear dynamics Received: March 18, 2002 / Accepted: April 2, 2004 Published online: 7 July 2004 – © Springer-Verlag 2004 Abstract. There is growing evidence that temporal lobe seizures are preceded by a preictal transition, characterized by a gradual dynamical change from asymptomatic interictal state to seizure. We herein report the first prospective analysis of the online automated algorithm for detecting the preictal transition in ongoing EEG signals. Such, the algorithm constitutes a seizure warning system. The algorithm estimates ST Lmax , a measure of the order or disorder of the signal, of EEG signals recorded from individual electrode sites. The optimization techniques were employed to select critical brain electrode sites that exhibit the preictal transition for the warning of epileptic seizures. Specifically, a quadratically constrained quadratic 0-1 programming problem is formulated to identify critical electrode sites. The automated seizure warning algorithm was tested in continuous, long-term EEG recordings obtained from 5 patients with temporal lobe epilepsy. For individual patient, we use the first half of seizures to train the parameter settings, which is evaluated by ROC (Receiver Operating Characteristic) curve analysis. With the best parameter setting, the algorithm applied to all cases predicted an average of 91.7% of seizures with an average false prediction rate of 0.196 per hour. These results indicate that it may be possible to develop automated seizure warning devices for diagnostic and therapeutic purposes.

1. Introduction Epilepsy consists of more than 40 clinical syndromes affecting 50 million people worldwide. At least 30% of patients with epilepsy continue to have seizures despite treatment with antiepileptic drugs. Epileptic seizure occurrences seem to be random and unpredictable. However, recent studies in epileptic patients suggest that seizures are deterministic rather than random. Subsequently, studies of the spatiotemporal dynamics in electroP.M. Pardalos: Departments of Industrial and Systems Engineering and Biomedical Engineering, University of Florida, USA W. Chaovalitwongse: Departments of Industrial and Systems Engineering and Neuroscience, University of Florida, USA L.D. Iasemidis: Departments of Biomedical Engineering and Electrical Engineering, Arizona State University, USA J.C. Sackellares: Departments of Neuroscience, Neurology and Biomedical Engineering, University of Florida, USA D.-S. Shiau: Department of Neuroscience, University of Florida, USA P.R. Carney: Departments of Pediatrics, Neuroscience, and Neurology, University of Florida, USA O.A. Prokopyev: Department of Industrial and Systems Engineering, University of Florida, USA V.A. Yatsenko: Department of Neuroscience, University of Florida, USA Mathematics Subject Classification (1991): 20E28, 20G40, 20C20

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encephalograms (EEG’s), from patients with temporal lobe epilepsy, demonstrated a preictal transition of approximately 21 to 1 hour duration before the ictal onset [20, 22, 34, 36, 53, 54]. This preictal dynamical transition is characterized by a progressive convergence (entrainment) of dynamical measures (e.g. maximum Lyapunov exponents ST Lmax ) at specific anatomical areas in the neocortex and hippocampus. This finding is supported by subsequent works of other investigations [3, 16, 46, 50, 56]. Although the existence of the preictal transition period has recently been confirmed and further defined by other investigators, the characterization of this spatiotemporal transition is still far from complete. Therefore, the development of a model for the mechanism of generation of epileptic seizures remains a difficult task. For example, even in the same patient, different set of cortical sites may exhibit preictal transition from one seizure to the next. In addition, resetting of the entrainment of the normal sites with the epileptogenic focus (critical cortical sites) occurs after each seizure. Therefore, we postulate that complete or partial postictal resetting of preictal entrainment of the epileptic brain, affects the route to the subsequent seizure, contributing to the apparently non-stationary nature of the entrainment process. Even though a complete modeling of the process remains elusive, the dynamical measures we have used have resulted in the development of effective seizure prediction schemes. In a retrospective analysis (i.e., after a seizure occurrence) utilizing ST Lmax as a dynamical measure and a global optimization technique to identify critical electrode sites, we found that the preictal transition preceded more than 91% of the seizures analyzed [20, 34]. The results of these studies confirmed the predictability of seizures. Further, our group has shown the existence of resetting of the brain after seizures’ onset [10, 38, 52], that is, divergence of ST Lmax profiles after seizures. This finding indicated that, if one knows which critical electrode sites will participate in the next preictal transition, it may be possible to detect the transition in time to warn of an impending seizure. To incorporate these findings together, we have to select electrode sites such that they are most entrained prior to the seizure and are disentrained after the seizure onset. In this framework, we employed optimization techniques to solve the electrode selection problem. Such, a problem was formulated as a quadratically constrained quadratic 0-1 programming problem. However, the method previously proposed by our group [20, 34] could not be applied to solve this quadratic programming problem with additional quadratic constraints. We developed a novel linearization technique to reformulate a quadratically constrained quadratic 0-1 programming problem as an equivalent mixed integer programming (MIP) problem. The practical importance of this reformulation is that the number of additional continuous variables is O(n), where n is the number of initial 0–1 variables, and the number of 0–1 variables remains the same. Based on the results from our previous findings, as a criterion to pre-select the critical electrode sites, the proposed optimization techniques were applied to solve the electrode site selection problems for the detection of the preictal transition and the warning of epileptic seizures. We herein report the first prospective analysis of the online automated algorithm for detecting the preictal transition in ongoing EEG signals. Such, the algorithm constitutes a seizure warning system. The algorithm estimates ST Lmax , a measure of the order or disorder of the signal, of EEG signals recorded from individual electrode sites.

Seizure warning algorithm based on optimization and nonlinear dynamics

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The organization of the succeeding sections of this paper is as follows. The background of the automated seizure warning algorithm, the method of estimation of ST Lmax and the spatiotemporal dynamical analysis, is described in section 2. The notation and modeling for selection of critical cortical sites is addressed in section 3. In section 4, the datasets and method of seizure warning algorithm is presented. The performance of the algorithm, sensitivity and false warning rate, applied to 5 patients is presented in section 5. The conclusions and performance, limitation, and possibility to develop devices for diagnostic and therapeutic purposes of this algorithm are discussed in the final section 6.

2. Background 2.1. Estimation of short term largest lyapunov exponents (ST Lmax ) Since the brain is a nonstationary system, algorithms used to estimate measures of the brain dynamics should be capable of automatically identifying and appropriately weighing existing transients in the data. In a chaotic system, orbits originating from similar initial conditions (nearby points in the state space) diverge exponentially (expansion process). The rate of divergence is an important aspect of the system dynamics and is reflected in the value of Lyapunov exponents. The method we developed for estimation of Short Term Largest Lyapunov Exponents (ST Lmax ), an estimate of Lmax for nonstationary data, is explained in detail elsewhere [21, 35, 59]. Herein we will present only a short description of our method. Construction of the embedding phase space from a data segment x(t) of duration T is made with the method of delays. The vectors Xi in the phase space (see Figure 1) are constructed as: Xi = (x(ti ), x(ti + τ ), . . . , x(ti + (p − 1) ∗ τ ))

(1)

where τ 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 ]. If we denote by L the estimate of the short term largest Lyapunov exponent ST Lmax then: L=

Na |δXi,j (t)| 1
log2 Na t |δXi,j (0)|

(2)

i=1

with δXi,j (0) = X(ti ) − X(tj ) δXi,j (t) = X(ti + t) − X(tj + t)

(3) (4)

where – X(ti ) is the point of the fiducial trajectory φt (X(t0 )) with t = ti , X(t0 ) = (x(t0 ), . . . , x(t0 + (p − 1) ∗ τ )). X(tj ) should be a properly chosen vector adjacent to X(ti ). For more detailed discussion see [21, 35].

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Fig. 1. Diagram illustrating the estimation of ST Lmax measures in the state space. The fiducial trajectory, the first three local Lyapunov exponents (L1 , L2 , L3 ), is shown

– δXi,j (0) = X(ti ) − X(tj ) is the displacement vector at ti , that is, a perturbation of the fiducial orbit at ti , and δXi,j (t) = X(ti + t) − X(tj + t) is the evolution of this perturbation after time t. – ti = t0 + (i − 1) ∗ t and tj = t0 + (j − 1) ∗ t, where i, j ∈ [1, Na ] with j = i. – t is the evolution time for δXi,j , that is, the time one allows δXi,j to evolve in the phase space. If the evolution time t is given in sec, then L is in bits per second. – t0 is the initial time point of the fiducial trajectory and coincides with the time point of the first data in the data segment of analysis. In the estimation of L, for a complete scan of the attractor, t0 should move within [0, t]. – Na is the number of local Lmax ’s that will be estimated within a duration T data segment. Therefore, if Dt is the sampling period of the time domain data, T = (Na − 1)Dt = Na t + (p − 1)τ .

We computed the short term largest Lyapunov exponent ST Lmax using the method proposed by Iasemedis et al. [21], which is a modification of the method by Wolf et al. [59]. We call the measure short term to distinguish it from those used to study autonomous dynamical systems studies. Modification of the Wolf’s algorithm is necessary to better estimate of ST Lmax in small data segments that include transients, such as interictal spikes. The modification is primarily in the searching procedure for a replacement vector at each point of a fiducial trajectory. Results from simulation data of known attractors have shown the improvement in the estimates of L achieved by using the proposed modifications [21].

Seizure warning algorithm based on optimization and nonlinear dynamics

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2.2. Spatiotemporal dynamical analysis Having estimated the ST Lmax temporal profiles at individual cortical site, and as the brain proceeds towards the ictal state, the temporal evolution of the stability of each cortical site is quantified. However, the system under consideration (brain) has a spatial extent and, as such, information about the transition of the system towards the ictal state should also be included in the interactions of its spatial components. The spatial dynamics of this transition are captured by consideration of the relations of the ST Lmax between different cortical sites. For example, if a similar transition occurs at different cortical sites, the ST Lmax of the involved sites are expected to converge to similar values prior to the transition. We have called such participating sites "critical sites", and such a convergence "dynamical entrainment". More specifically, in order for the dynamical entrainment to have a statistical content, we have allowed a period over which the difference of the means of the ST Lmax values at two sites is estimated. We have used periods of 10 minutes (i.e. moving windows including approximately 60 ST Lmax values over time at each electrode site) to test the dynamical entrainment at the 0.01 statistical significance level. We employ the T -index (from the well-known paired T-statistics for comparisons of means) as a measure of distance between the mean values of pairs of ST Lmax profiles over time. The T -index at time t between electrode sites i and j is defined as: √ Ti,j (t) = N × |E{ST Lmax,i − ST Lmax,j }|/σi,j (t) (5) where E{·} is the sample average difference for the ST Lmax,i − ST Lmax,j estimated over a moving window wt (λ) defined as:  1 if λ ∈ [t − N − 1, t] wt (λ) = 0 if λ ∈ [t − N − 1, t], where N is the length of the moving window. Then, σi,j (t) is the sample standard deviation of the ST Lmax differences between electrode sites i and j within the moving window wt (λ). The thus defined T -index follows a t-distribution with N-1 degrees of freedom. For the estimation of the Ti,j (t) indices in our data we used N = 60 (i.e., average of 60 differences of ST Lmax exponents between sites i and j per moving window of approximately 10 minute duration). Therefore, a two-sided t-test with N − 1(= 59) degrees of freedom, at a statistical significance level α should be used to test the null hypothesis, Ho : “brain sites i and j acquire identical ST Lmax values at time t". In this experiment, we set α = 0.01, the probability of a type I error, or better, the probability of falsely rejecting Ho if Ho is true, is 1%. For the T -index to pass this test, the Ti,j (t) value should be within the interval [0,2.662].

3. Notation and modeling In this paper we refer to the Sherrington-Kirkpatric Hamiltonian that describes the meanfield theory of the spin glasses where elements are placed on the vertices of a regular lattice, the magnetic interactions hold only for nearest neighbors and every element has

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only two states (Ising spin glasses [2, 4, 5, 15, 44]). One of the most interesting problems about this model is the determination of the minimal-energy states (GROUND STATE problem) [4–6]. Quadratic 0-1 programming has been extensively used to study Ising spin glass models [6]. This has motivated us to use quadratic 0-1 programming to select the critical cortical sites, where each electrode has only two states, and to determine the minimal-average T-index state. We formulated this problem as a quadratic 0-1 knapsack problem with objective function to minimize the average T-index (a measure of statistical distance between the mean values of ST Lmax ) among electrode sites and the knapsack constraint to identify the number of critical cortical sites [20, 34]. The electrode selection problem can be formulated as follows:

T

min f (x) = x Ax, s.t.

n


xi = k, x ∈ {0, 1}n .

(6)

i=1

Let A be n×n matrix, whose each element ai,j represents the T-index between electrode i and j within 10-minute window before the onset of a seizure. Define x = (x1 , ..., xn ), where each xi represents the cortical electrode site i. If the cortical site i is selected to be one of the critical electrode sites, then xi = 1; otherwise, xi = 0. k denotes the number of selected critical electrode sites. Later on, our group has shown dynamical resetting of the brain following seizures [10, 38, 52], that is, divergence of ST Lmax profiles after seizures. Therefore, we want to incorporate this finding with our existing critical electrode selection problem (Eq. (6)). Thus, we have to ensure that the optimal group of critical sites shows this divergence by adding one more quadratic constraint to Eq. (6). The quadratically constrained quadratic 0-1 problem is given by: min x T Ax n
s.t. xi = k

(7) (8)

i=1

x T Bx ≥ Tα k(k − 1) x ∈ {0, 1}n

(9) (10)

Let B be n × n matrix, whose each element bi,j represents the T-index between electrode i and j within 10-minute window after the onset of a seizure. Note that the matrix A = (aij ) is the T-index matrix of brain sites i and j within 10-minute windows before the onset of a seizure. Tα is the critical value of T-index, as previously defined, to reject Ho : “two brain sites acquire identical ST Lmax values within time window wt (λ)". Next we investigate the complexity of the problem Eqs. (7)–(10). We prove that it is NP -hard. To solve it we propose two computational approaches. In the first approach, we applied a conventional linearization technique by introducing a new variable for each product of two variables and adding some additional constraints. In the second approach we developed a novel linearization technique.

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3.1. Complexity of the electrode selection problem Note that the considered problem in Eqs. (7)–(10) is a special case of a multi-quadratic 0–1 programming problem. For the matrices A and B we have that ∀i, j aij ≥ 0, bij ≥ 0 and ∀i aii = 0, bii = 0. Next we present the proof that this restricted case of the multiquadratic 0–1 programming problem remains NP-hard. Consider the following problem min

x∈{0,1}n , eT x=k

x T Qx,

(11)

where ∀i, j qij ≥ 0 and ∀i qii = 0. Theorem 1. Problem (11) is NP-hard. Proof. In [15] it is shown that the maximum clique problem (which is known to be NP-hard) in a graph G = (V , E) with vertex set V = {1, . . . , n} and edge set E is equivalent to min f (x) = −

n
i=1

xi + 2




xi xj = −eT x + 2

(i, j ) ∈ /E i>j




xi xj

(i, j ) ∈ /E i>j

s.t.x ∈ {0, 1}n .

(12)

Obviously, we can solve this problem (12) by solving n + 1 problems
min fk (x) = xi xj (i, j ) ∈ /E i>j

s.t. eT x = k, x ∈ {0, 1}n .

(13)

for each k ∈ [0, n]. Note that problem (13) is a restricted version of problem (11). The solution of the problem (12) will be the one which gives the minimal 2fk (x) − k. Therefore, we can solve the maximum clique problem by solving n + 1 problems (13). Hence, problem (11) is NP-hard.   Since the addition of a quadratic constraint makes the problem more general, the problem stated in Eqs. (7)–(10) is NP-hard. 3.2. Conventional linearization approach For each product xi xj in (7) we introduce a new continuous variable, xij = xi xj (i = j ). Note that xii = xi2 = xi for xi ∈ {0, 1}. The equivalent mixed integer programming problem (MIP) is given by:

min aij xij (14) i

j

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s.t.

n


xi = k,

(15)

i=1

xij ≤ xi , for i, j = 1, ..., n (i = j ) xij ≤ xj , for i, j = 1, ..., n (i = j ) xi + xj − 1 ≤ xij , for i, j = 1, ..., n (i = j )

bij xij ≥ Tα k(k − 1) i

(16) (17) (18) (19)

j

0 ≤ xij ≤ 1 for i, j = 1, ..., n (i = j )

(20)

where xi ∈ {0, 1}, i, j = 1, ..., n. The main disadvantage of this approach is that the number of additional variables we need to introduce is O(n2 ), and the number of new constraints is also O(n2 ). The number of 0–1 variables remains the same. Although we can apply CPLEX to solve the linearized problem, this approach may become computationally inefficient as n increases because the size of the linearized problem increases quadratically.

3.3. A new linearization approach Consider the following two problems: P : min f (x) = x T Qx, x n  s.t. xi = k, i=1

(QP-1) x T Q1 x ≥ α1 , (QP-2) x T Q2 x ≥ α2 , ... ... x T Qm x ≥ αm , (QP-m) where x ∈ {0, 1}n , αi are nonnegative constants (i = 1, . . . , m), Q, Q1 , . . . , Qm are n × n matrices whose every element is nonnegative. n  P : min g(s) = si = eT s, x,y,s,z

s.t.

n  i=1

i=1

xi = k,

Qx − y − s = 0, y ≤ µ(e − x), Q1 x − z1 ≥ 0, ... Qm x − zm ≥ 0, eT z1 ≥ α1 , ... eT zm ≥ αm ,

(P -1) (P -2) (QP -1) ... (QP -m) (AP -1) ... (AP -m)

Seizure warning algorithm based on optimization and nonlinear dynamics

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z1 ≤ µ1 x, (ZP -1) ... ... zm ≤ µm x, (ZP -m) where µi = Qi ∞ (i = 1, . . . , m), µ = Q ∞ , xi ∈ {0, 1} (i = 1, . . . , n), yi ≥ 0 (i = 1, . . . , n), si ≥ 0 (i = 1, . . . , n), zij ≥ 0 (j = 1, . . . , m), e is a vector of all 1’s (i.e., e = (1, . . . , 1)T ). 0 such that Theorem 2. P has an optimal solution x 0 iff there exist y 0 , s 0 , z10 , . . . , zm 0 0 0 0 0 (x , y , s , z1 , . . . , zm ) is an optimal solution to P .

Proof. Sufficiency. Firstly, let us prove the equivalent result from the problem but without constraints (QP-1)–(QP-m), (QP -1)–(QP -m), (AP -1)–(AP -m), (ZP -1)–(ZP -m). Since y ≥ 0 and x ∈ {0, 1}n , we note that Eq. (P -2) is equivalent to the complementarity constraint; that is, y ≤ µ(e − x) ⇐⇒ y T x = 0.

(21)

Since every element in the matrix Q is nonnegative, for every x there always exist y ≥ 0, s ≥ 0 such that (P -1) and (P -2) are satisfied. Let x 0 , y 0 , s 0 be an optimal solution of the problem P . The following constraints are satisfied: Qx 0 − y 0 − s 0 = 0, (y 0 )T x 0 = 0.

(22) (23)

Next, we prove that x 0 is also an optimal solution of the problem P . Multiplying (22) by (x 0 )T , we obtain (x 0 )T Qx 0 − (x 0 )T y 0 − (x 0 )T s 0 = 0. Note that from (23) we have (x 0 )T y 0 = 0. Hence, we have (x 0 )T Qx 0 = (x 0 )T s 0 .

(24)

(x 0 )T s 0 = eT s 0 ,

(25)

If we can prove that then (x 0 , y 0 , s 0 ) is also an optimal solution to P . To prove that (25) holds, it suffices to show that, for any i if xi0 = 0 then si0 = 0. This fact can be proven by contradiction. Assume that for some i, xi0 = 0 and si0 > 0. Define vectors y˜ 0 and s˜ 0 as y˜i0 = yi0 +si0 , s˜i0 = 0 and for j = i y˜j0 = yj0 , s˜j0 = sj0 . It is easy to check that (x 0 , y˜ 0 , s˜ 0 ) also satisfies (22), (23), and eT s˜ 0 < eT s 0 . This contradicts the initial assumption that (x 0 , y 0 , s 0 ) is an optimal solution of the problem P . Therefore, we can conclude that (x 0 )T Qx 0 = (x 0 )T s 0 = eT s 0 and we obtain that x 0 is also an optimal solution of the problem P . Next, we extend the proof for the multi-quadratic case. To extend the proof for the multi-quadratic case we will show that if (x 0 , y 0 , s 0 , z0 ) is an optimal solution of P then the vector x 0 satisfies (QP -1)–(QP -m). For any i = 1, . . . , m we have that Qi x 0 − zi0 ≥ 0, eT zi0 zi0

(26)

≥ αi ,

(27)

≤ µi x , 0

(28)

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Multiplying (26) by x 0 we obtain that (x 0 )T Qi x 0 ≥ (x 0 )T zi0

(29)

0 = 0. Hence, similarly to (25), we Note from (28) that if xj0 = 0 then we have zij have

eT zi0 = (x 0 )T zi0 .

(30)

Therefore, from (27), (29), (30) we obtain (x 0 )T Qi x 0 ≥ (x 0 )T zi0 = eT zi0 ≥ αi

(31)

and (QP -1)–(QP -m) are satisfied. Necessity. The proof is very similar.

 

We have shown that our electrode site selection problem can be rewritten as MIP. From the above reduction, the number of 0-1 variables remains the same (n), the number of new continuous variables is O(mn) and the number of new linear constraints is O(mn). For m = o(n) this approach is more efficient than the conventional one. Note that in our application we have only 1 quadratic constraint (m = 1). Next, we compare the proposed approaches and present some numerical results.

3.4. Numerical results We performed a number of computational experiments with real data obtained from the patients. All calculations were conducted using CPLEX 7.0 on a PC with Pentium IV 1.3 GHz and 1 Gb of RAM. In the Table 1 and Figure 2 we compare the average time to solve the selection problems by using the discussed approaches. The results indicate that the new linearization technique outperforms the conventional linearization, and we employed this technique to solve Eqs. (7)–(10) for the selection of electrodes after every subsequent seizure.

Table 1. Time (in sec.) for solving Electrode Selection Problem using conventional and new linearization approaches. k n 30 32 34 36 38 40

5 CL 2.83 3.05 4.71 4.93 6.39 11.35

6 New 0.65 0.74 0.91 1.15 1.46 1.48

CL 4.28 7.31 7.68 9.65 10.01 13.99

7 New 1.54 1.89 2.05 2.22 2.35 2.48

CL 6.13 8.77 10.07 12.45 13.12 19.04

8 New 3.26 5.76 7.06 6.25 5.09 6.75

CL 7.52 8.97 13.16 17.16 20.40 21.57

New 7.50 5.54 6.48 9.26 9.89 13.00

Seizure warning algorithm based on optimization and nonlinear dynamics

Performance Characteristics of Linearization T echniques (k = 5)

Performance Characteristics of Linearization Techniques (k = 6)

12.00

16.00 14.00

8.00 NEW

6.00

CL

4.00

CPU Tim e (seconds)

10.00 CPU Tim e (seconds)

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8.00

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6.00 4.00 2.00

0.00

0.00

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40

30

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Size of the proble m (n)

Size of the problem (n)

Performance Characteristics of Linearization Techniques (k = 7)

Performance Characteristics of Linearization Techniques (k = 8)

20.00

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25.00

18.00 20.00

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0.00

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40

30

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Size of the problem (n)

Fig. 2. Comparison of the linearization techniques for different values of k and n

4. Materials and methods 4.1. Datasets The datasets consisted of continuous long-term (3 to 12 days) multichannel intracranial EEG recordings that had been acquired from 5 patients with medically intractable temporal lobe epilepsy. The recordings were obtained as part of a pre-surgical clinical evaluation. They had been obtained using a Nicolet BMSI 4000 and 5000 recording systems, using a 0.1 Hz high-pass and a 70 Hz low-pass filter. Each record included a total of 28 to 32 intracranial electrodes (8 subdural and 6 hippocampal depth electrodes for each cerebral hemisphere). A diagram of electrode locations is provided in Figure 3. The characteristics of the recordings are outlined in Table 2. The recorded EEG signals were digitized, using a sampling rate of 200 Hz, and stored on magnetic media for subsequent off-line analysis. In this study, all the EEG recordings have been viewed by two independent board-certified electroencephalographers to determine the number and type of recorded seizures, seizure onset and end times, and seizure onset zones.

4.2. Seizure warning algorithm The Seizure Warning Algorithm is outlined in Figure 4. This algorithm involves the following steps: 1. Continuous ST Lmax calculation: ST Lmax values were iteratively calculated from sequential non-overlapping 10.24 second EEG epochs obtained from each electrode site, utilizing the method described in Section 2.1. The ST Lmax method accomplishes a large data reduction (each 10.24 second EEG epoch becomes a single

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(A) AR

AL

BR

BL

CR

CL

(B)

CL

AL BL

Fig. 3. (A) Inferior transverse and (B) lateral views of the brain, illustrating approximate depth and subdural electrode placement for EEG recordings are depicted. Subdural electrode strips are placed over the left orbitofrontal (AL ), right orbitofrontal (AR ), left subtemporal (BL ), and right subtemporal (BR ) cortex. Depth electrodes are placed in the left temporal depth (CL ) and right temporal depth (CR ) to record hippocampal activity Table 2. Characteristics of analyzed EEG dataset

Patient Gender 1 2 3 4 5

Female Male Female Male Female

Number of Seizures Total Number of Partial Length Range of Age electrodes Complex Secondarily Subclinical of data seizure interarrival Partial Generalized (Hours) time (Hours) 41 32 17 3 0 83.30 0.3 - 14.5 29 28 8 0 7 140.15 0.3 - 70.8 38 32 6 0 0 18.24 1.1 - 4.8 60 28 0 7 0 121.92 2.7 - 78.7 45 28 3 0 6 69.53 0.5 - 47.9

sample in the ST Lmax profiles), and it is applied sequentially to each EEG channel, creating a new multichannel time series that is utilized for subsequent analysis. 2. Optimization (Selection of critical electrode sites): After the onset of the first seizure, critical electrode sites are identified automatically by solving the optimization problems with the novel linearization technique described in Section 3. These critical sites are updated after each subsequent electrographic seizure.

Seizure warning algorithm based on optimization and nonlinear dynamics

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EEG signals No

Observe seizure

Continuous STLmax Calculation

Yes Optimization (Selection of critical electrode sites)

Monitoring T-index curves among selected electrode sites

No Observe preictal entrainment transition

Yes

Warn of an impending seizure

Fig. 4. Flow diagram of the Seizure Warning Algorithm. This diagram illustrates the steps employed in the automated algorithm (see text for an explanation of each step)

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In the optimization problem, we basically aimed to select electrode sites such that they are most entrained prior to the seizure, conditional on the disentrainment after the seizure onset. The optimization problem was formulated as the followings. First, a T-matrix, which corresponds to the 10-minute epoch prior to the seizure onset, was generated and put into the objective function, which needs to be minimized. Second, a T-matrix, which corresponds to the 10-minute epoch after the seizure onset was generated and put into the quadratic constraint, which ensures that the selected electrode sites (solution to the optimization problem) show the disentrainment (divergence in ST Lmax ) after the seizure onset. Third, the linear constraint of the number of critical electrode sites (k) was added in the optimization problem. In this case, k (constant in the optimization problem) is one of two parameters which needs to be tuned up. In the selection of the critical electrode sites step, there are two parameters to be trained in this algorithm: number of sites (k) per group and number of groups (m) to be selected. For every subsequent seizure, we want to find m subsets of electrode that yield the minimum average T-index, the second-minimum average T-index, the third-minimum average T-index, etc. Groups (m) are the subsets of the solutions to the optimization problems in m iterations (conditional on those m groups there must be a combination of different electrodes). In this study, for each patient, we utilize the first half of the seizures to train k(3 ∼ 8) and m(1 ∼ 5). The optimal parameter setting was then identified by generating the ROC curves (receiver operating characteristic) of the seizure warning performance, and was applied in the testing set of seizures in the same patient. The evaluation of the warning performance and the ROC curve are discussed in the next subsection. 3. Monitoring the average T-index curve of selected electrode groups: Once groups of critical electrode sites are chosen, the average T-index value for each of these groups is continuously calculated from ST Lmax profiles, using sequential 10-minute overlapping windows. The average T-index values are continuously compared to a preset threshold value (Tα index), defined as the value below which the average difference of the values of ST Lmax in the corresponding time window is not significantly different from 0 (p > 0.01). When the average T-index of a group of selected sites becomes less than the Tα index, the group is considered to be entrained. 4. Warning of an impending seizure: The objective of the automated seizure warning system is to detect preictal transitions in order to prospectively warn of an impending seizure. A seizure warning is generated when the preictal transition is detected. The onset of the preictal transition is defined as that point in time when at least one of the monitored groups of critical electrode sites is entrained. That is, the average T-index for that group of sites initially above 5 (disentrained) drops to a value of 2.662 or less (entrained). These critical values were chosen based on the following statistical considerations: when the T-index is greater than 5, the average ST Lmax values for electrode pairs are highly significantly different (p-value < 0.00001); when the T-index is less than 2.662, the average ST Lmax values for electrode pairs are not significantly different (p-value > 0.01). Following each successive seizure, new groups of critical electrode sites are reselected and the algorithm is repeated.

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4.3. Evaluation of the seizure warning algorithm In this section, we present criterion to evaluate the performance of the seizure warning algorithm; that is, the length of time horizon which was used in this experiments and determined by physicians to consider if a warning was true or false. To test this algorithm, a warning was considered to be true if a seizure occurred within 3 hours after an entrainment transition was detected. A 3-hour period was chosen for purposes of this analysis, based upon the seizure warning intervals observed in preliminary studies of seizure predictability [36]. If no seizure occurred in that period, the warning was considered to be false. If a seizure occurred without a warning during the preceding 3 hours, the algorithm was judged to have failed to warn of that seizure. Thus, the sensitivity was defined as the total number of seizures accurately predicted divided by the total number of seizures recorded. The false prediction rate was defined as the average number of false warnings per hour. We first applied the algorithm in the training seizure set for each patient to determine the optimal parameter settings (k is the number of critical electrode sites in each group and m is the number of groups of the critical electrode sites) settings. To achieve optimal parameter settings, we used a ROC curve for an individual patient to evaluate the performance of the algorithm for all parameter settings. A ROC curve indicates a trade-off that one can achieve between the false alarm rate (1-Specificity, plotted on X-axis) that needs to be minimized, and the detection rate (Sensitivity, plotted on Y-axis) that needs to be maximized. However, it is insufficient and misleading to present the specificity of the algorithm because this seizure warning algorithm was run on the online prospective long-term analysis, which there was no seizures during most of the recordings. Therefore, we plotted the sensitivity (%) on X-axis and the false alarm rate (per hour) on Y-axis (see Fig. 7). In this case, the false alarm rate is a more meaningful measure for physicians to evaluate the performance of the seizure warning algorithm. An appropriate trade-off or the optimal parameter settings for an individual patient were determined by the physician, which, in this case, were achieved by finding the parameter setting on the ROC curve that is closest to the ideal point (100% sensitivity and 0 false positive rate).

5. Results Figure 5 shows an example of ST Lmax profiles versus time, derived from EEG signals recorded from 5 critical electrode sites. These sites, selected from the first seizure in the series, diverge with respect to the values of ST Lmax after that seizure and converge to a common value prior to the next seizure (preictal transition). After the occurrence of the second seizure, reselection of critical sites is made. Preictal transition and postictal divergence are reflected in the corresponding average T-index curves with the gradual reduction preictally and more rapid rise postictally, as shown in Figure 6. This sequence of dynamical state transitions is repeated after each seizure. Figure 7 shows the ROC curve of each patient. Table 3 summarizes the optimal parameter settings and their seizure warning performance. The criterion for determining the optimal parameter settings is that, for patients 1 and 2 with larger number of training seizures (10 and 7, respectively), the sensitivity must be larger than 80% with the

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Fig. 5. A plot of ST Lmax values calculated from a 250-minute sample of intracranial EEG recording which contains 3 of the complex partial seizures recorded from patient 1. After seizure 8 and 9, 5 critical electrode sites (AR 4, AL 4, BR 2, BR 3 and BL 2 after seizure 8 and BR 1, BR 2, BR 4, CR 2 and CR 8 after seizure 9) were selected by the global optimization algorithm. At this point in time, ST Lmax values for these selected sites are significantly different (disentrained). Prior to seizure 9 and 10, ST Lmax values from these same sites converge to a common value (entrained) and these sites become disentrained after seizure 9 and 10

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Fig. 6. This average T-index profile was calculated from the ST Lmax profiles shown in Fig. 3. When the average T-index drops from a value of 5 or above to a critical value of 2.662, the average T-index for these sites is not significantly different than 0. At that point, the sites are considered to be dynamically entrained and a seizure warning is generated by the system. Seizure warnings are generated approximately 50 minutes before seizure 9 and approximately 70 minutes before seizure 10

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ROC curve for training datasets of 5 patients 1

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Fig. 7. ROC curve for optimal parameter setting of 5 patients

minimum false positive rate per hour. For patients 3, 4 and 5, due to the small number of training seizures (3 for each), the sensitivity must be at least 2/3 with the minimum false positive rate per hour. In these 5 training sets, the percentage of seizures that were correctly predicted ranged from 2/3 (patient 4 and 5) to 100% (patient 3), with an overall sensitivity of 80.77% (21/26). The false warnings occurred at a rate ranging from 0.00 to 0.234 (overall 0.159) false warnings per hour. This corresponds, on average, to a false warning every 6.3 hours. Table 4 summarizes the performance of the algorithm in the 5 testing seizure sets when using the selection parameters from the training seizure sets. The prediction sensitivity ranged from 85.71% (patient 2) to 100% (patient 3, 4 and 5), with an overall sensitivity of 91.67% (22/24). The false warnings rates range from 0.049 to 0.366 (overall 0.196) false warnings per hour. This corresponds, on average, to a false warning every 5.1 hours.

6. Conclusions and discussion The results of this study confirm our hypothesis that it is possible to predict an impending seizure based on optimization and nonlinear dynamics of multichannel intracranial

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Table 3. Performance characteristics of automated seizure warning algorithm with optimal parameter settings of training data Number  of critical  of selected of analyzed electrodes critical Sensitivity seizures in each group groups 1 11 5 3 80.00% (8/10) 2 8 3 2 85.71% (6/7) 3 4 3 4 100.00% (3/3) 4 4 5 2 66.67% (2/3) 5 4 4 1 66.67% (2/3) All patients 31 80.77% (21/26) Patient

False Prediction Average Warning Rate Time (False per Hour) (Minutes) 0.095 (4/46.248) 67.8 ± 20.9 0.234 (20/85.468) 66.2 ± 15.7 0.000 (0/7.140) 71.5 ± 12.3 0.065 (1/15.328) 39.0 ± 20.4 0.151 (9/59.565) 72.7 ± 47.8 0.159 (34/213.749) 63.44 ± 6.22

Table 4. Performance characteristics of automated seizure warning algorithm testing on optimal training parameter settings Patient 1 2 3 4 5 All patients

Number of analyzed seizures 10 8 3 4 4 31

Sensitivity 88.89% (8/9) 85.71% (6/7) 100.00% (2/2) 100.00% (3/3) 100.00% (3/3) 91.67% (22/24)

False Prediction Rate (False per Hour) 0.049 (2/41.134) 0.366 (20/54.685) 0.137 (1/7.278) 0.178 (19/106.59) 0.100 (1/9.967) 0.196 (43/219.654)

Average Warning Time (Minutes) 79.3 ± 13.2 90.2 ± 19.2 79.9 ± 6.2 108.8 ± 7.4 104.9 ± 21.1 92.6 ± 6.2

EEG recordings. Prediction is possible because, for the vast majority of seizures, the spatiotemporal dynamical features of the preictal transition are sufficiently similar to that of the preceding seizure. This similarity makes it possible to identify electrode sites that will participate in the next preictal transition, based on their behavior during the previous preictal transition. Although evidence for the characteristic preictal transition utilized by the seizure prediction algorithm employed in this study was first reported by our group in 1991 [22], further studies were required before a practical seizure prediction algorithm was feasible. Development of a seizure prediction algorithm was complicated by three factors: (1) the cortical sites participating in the preictal transition varied from seizure to seizure, (2) the length of the preictal transition varied from seizure to seizure, and (3) it was not known whether or not this type of spatiotemporal transition was unique to the preictal period. These problems were overcome by the use of our proposed approaches to solve quadratically constrained quadratic 0-1 problem. Because the algorithm selects candidate electrode sites and by analyzing continuous EEG recordings of several days of duration, the computational approach to solve the optimization problem has to be very efficient. At present, with the new technique, the electrode selection problems were solved efficiently and the solutions were optimally attained. However, future technology may allow physicians to implant thousands of electrode sites, n > 1000, in the brain. This procedure will extract more information and allow us to have a deeper understanding about the brain. Therefore, to solve this optimization problem with n > 1000, we may need computationally fast heuristic approaches in the future.

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Acknowledgements. The authors would like to thank the referees for their valuable suggestions that have improved the quality of the paper.

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