Epikpsy
Res., 8 (1991) 153-165
153
Elsevier EPIRES 00382
Temporal distribution of seizures in epilepsy
Erik Taub@lla, Arvid Lundervoldb and Leif Gjerstada “Department of Neurology, Ribhospitalet -
The National Hospital, University of Oslo, and bNorwegian Computing Center, Blindern, Oslo (Norway)
(Received 26 June 1990; accepted 1 October 1990) Key words: Epilepsy; Biorhythms; Stochastic processes; Seizure occurrence; Catamenial epilepsy
A major problem in epileptology is why a seizure occurs at a particular moment in time. An initial step in solving this problem is a detailed analysis of the temporal distribution of seizures. Using methods and theories of stochastic processes, seizure patterns in a group of epileptic outpatients were examined for stationarity, randomness, dependency and periodicity in a prospective study. Sixteen of the 21 seizure diaries included in the study showed stationarity; 2 were non-stationary and 3 inconclusive. Eleven of the 16 stationary diaries were non-Poisson (P < O.OOS),indicating that in the majority of patients seizures did not occur randomly. The most frequently encountered phenomenon was seizure clustering. Clustering was considered when the diaries fulfilled all three criteria: (1) a positive R-test (P < 0.001); (2) deviation from the fitted Poisson distribution towards clustering; and (3) the feature of an autoregressive process in the autocorrelogram plot. Dependency between seizure events was demonstrated in 8 of the 16 stationary diaries, computing first order transition probabilities. A detailed analysis of seizure occurrence is a major step towards a better understanding of the mechanisms underlying seizure precipitation. This is exemplified by our finding of a relation between seizure frequency and the menstrual cycle
INTRODUCTION Through neurobiological research, much has been learnt about the basic mechanisms of epileptogenesis. However, the reason why a specific seizure occurs in a patient at a given moment in time has seldom been elucidated. An initial step in exploring this problem is a detailed analysis of seizure occurrence in patients with epilepsy. A rhythmic appearance of seizures might indicate periodic variations in basal physiological parameters of importance for the precipitation of seizures. Randomly occurring clusters of epileptic atCorrespondence
Rikshospitalet,
to: Erik Taubbll, Dept. of Neurology, University of Oslo, 0027 Oslo 1, Norway.
0920-1211/91/$03.50 0 1991 Elsevier Science Publishers B.V.
tacks might be indicative of exogenous seizureprovoking factors. Seizure clustering might also imply that secondary alterations due to a seizure facilitate the precipitation of a second attack in the manner of a positive feedback mechanism. An excess of long inter-seizure intervals would, on the other hand, suggest a negative feedback mechanism. One of the most widely accepted factors affecting seizure frequency is the menstrual cycle. This relationship often merely reflects a clinical impression rather than statistical findings (for a review, see 3, 19), and even the phase of the menstrual cycle with the highest seizure frequency is disputed3. Rhythmic occurrence of seizures or seizure clus-
154
tering may be used to study basic seizure-precipitating factors. In a single patient, information about seizure occurrence makes it possible to provide the patient with a better understanding of seizure-provoking factors and thereby how to minimize them. If seizure-provoking factors can be identified, modifying these may decrease seizure frequency in nearly 50% of patients with poorly controlled epilepsy2. Knowing the ‘normal’ temporal distribution of seizures also provides the background for a better analysis of the effect of antiepileptic drug treatment. Before account can be taken of the temporal distribution of seizures in the understanding of precipitating factors and in therapy, one has to establish appropriate statistical models for these patterns. The aim of the present investigation has been to examine seizure occurrence more thoroughly by model fitting and parameter estimation in a group of epileptic outpatients. PATIENTS AND METHODS P&en ts Seizure diaries from 17 patients (9 female and 8 male) were studied. Their mean age was 32.5 years with a range of 4-52 (Table I). All were socially well-adapted outpatients at different hospitals who responded to a request in the periodical of the Norwegian Chapter of the International Bureau for Epilepsy, to participate in a study of factors affecting seizure occurrence. Five additional patients responded to the initial request, but did not complete their diaries. All patients used stable antiepileptic medication (Tabie I) with a serum concentration within the reference range. The onset of epilepsy ranged from 2 to 47 years prior to the start of this investigation, with a median of 15 years. There was no evidence of symptomatic epilepsy due to cerebral tumor, congenital or vascular ma~ormations in any of the patients, and there was no history of mental illness or retardation. The patients were asked to register on a daily basis, the number and type of seizures. They also had to indicate any deviation from normal daily routine, with special emphasis on physical and psychological stress, alcohol, lack of sleep or food, missed medication, intercurrent disease and the
TABLE I Clinical data
forall patients
CBZ, carbamazepine; CPS, complex partial seizures; CZP, clonazepam; GTC, generalized tonic-clonic seizures; NZP, nitrazepam; PB, phenobarbital; PHT, phenytoin; PRIM, primidone; SPS, simple partial seizure; VPA, Na-valproate. 2’ GTC, secondary generalized tonic-clonic seizure. In Patient 5, CZP was introduced during the study (at the beginning of subdiary 5b: see Fig. 1C). Patient NO. .._1
2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 ___~_
Sex --
M F M M F M M F M F M F F F M F F _~~
Age __
48 40 28 31 36 29 4 40 22 34 14 21 26 34 44 52 49 ~_.
Seizure type .-.
.~_~~ ~-
GTC CPS -t 2” GTC CPS CPS Absence CPS SPS + 2” GTC GTC CPS + 2” GTC SPS + 2” GTC CPS + 2” GTC CPS + 2” GTC SPS CPS CPS CPS CPS ..___ -.. _~~_ -
Medication -.
PTH CBZ CBZ f VPA PHT + CBZ PRIM + PB + (CZP) CBZ + PHT CBZ + PHT PHT VPA PHT CBZ + VPA + NZP CZP CBZ PHT + PB + CBZ PHT f PB + CBZ PRIM + CBZ PHT ---.___ -
use of additional drugs. Care was taken not to introduce the concept of rhythmicity. Seizure diaries were collected prospectively, except for two cases (patients 2 and 15), in whom parts of the retrospective seizure diaries were also used, Two males (patients 6 and 15) were included, although a rhythmic seizure pattern could be anticipated from their medical histories. Classified according to their main seizure type, 14 patients had partial seizures (11 complex, 3 simple), and 3 patients had primarily generalized seizures (Table I). Three patients had their diary split, resulting in a total of 21 single seizure diaries that were analyzed individually. Patient 5 had her diary split into 3 sub-diaries because of a change in medication, and a period of 53 days with missed registration. Patient 6 had his diary split into 2 sub-diaries due to an 8 month stop in registration. Finally, patient 13 had her diary divided into 2 because of intervening neurosurgery (temporal lobectomy).
155 Time series analysis and stochastic modeling
Our modeling was aimed at making the following clinical important questions more precise: (1) Is there an upward or downward trend in seizure occurrence with time? (2) Are seizures in a particular patient occurring at random? (3) Do seizure events show a clustering tendency in some patients? (4) Do seizures occur in a.periodic manner over several weeks or months of observation? (5) What kind of changes in seizure pattern do we observe after change in antiepileptic therapy? (6) Is it possible to predict seizure events in a patient, if one has knowledge of his or her previous seizure diary? In order to provide a statistical setting for describing the features of seizure occurrences during time, we defined a seizure diary d to be a sequence of seizure events indexed in discrete time. Thus, each seizure diary was recognized as a discrete stochiastic process, d = {Zr}tET. For each diary d we took the index set to be T = {tit = 0, 1, . . . , Td - 1)) where t denotes number of days after start of diary registration, and Td the duration of the diary registration. The random variable Z, denotes the number of seizure events registered on day t. However, in some of the models considered, we made a ‘hard limiting transformation’ of Z, resulting in a binary series, X, such that X, = 1 if at least one seizure occurred on day t, and otherwise X, = 0. An important class of stochastic processes is those which are stationary, in that most of the probability theory of time series assumes stationarity. In simple terms, stationarity means that the probability laws governing the seizure events do not change with time (i.e. the mean and autocovariance function is time independent). We have concluded non-stationarity of a diary if either the autocorrelogram, the non-parametric Cox and Stuart test’ (a sign test variant) or the plot of cumulated seizures indicates a trend. As an aid in this study, one of us (A.L.) wrote a toolbox of C code, portable between UNIX work stations (SUN 3, DECstation) and a PC running MS-DOS. Steps in the analysis of a given diary. Our analytical approach, which is related to a single diary at
the time, can be summarized in the following steps: (1) Registration of the diary data on a text file. (2) Inspection of the recorded number of seizures over time by: making a plot of the cumulated number of seizures versus days” generating a 2-dimensional gray level raster image. (3) Compute descriptive statistics. (4) Check for trend and stationarity: inspection of plot of cumulated number of seizures over time” Cox and Stuart test for trend’ inspection of the autocorrelogram’. (5) Check for randomness: testing for a Poisson process of number of seizure occurrences” testing for a Poisson process of interseizure intervalsl’. (6) Check for clustering: applying the R-test for seizure clusteringl’j (binarized diary) check for feature of clustering in Poisson model” inspecting autocorrelogram for short-term correlation’. (7) Check for periodicity: inspecting plot of cumulated number of seizures inspecting autocorrelogram for oscillations’ computing the periodogram22. (8) Check for dependency: computing chi-square test statistic from estimated transition probabilities15. (9) Choose a plausible/tentative model which can aid in seizure prediction: first-order Markov process13; parameters estimated from the first half of the binarized diary. After registration of seizure events, the first step in the time series investigation was careful scrutiny of the recorded data plotted with time and a plot of the corresponding autocorrelogram, as these could suggest the method of analysis and the statistics to be used to summarize the information in the data. We have thus displayed each of the 21 sample diaries graphically by plotting the cumulated sum of the random variable on the ordinate and the time scale on the abscissa. Furthermore, the
156
autocorrelation function for each diary is plotted for lag 1 up to lag 60 days. Fig. 1 shows the cumulated number of seizures versus day for three of the diaries, and a gray level raster image representation of a diary exhibiting clustering. We used the Poisson model to test the randomness of seizure events in stationary diaries. Poisson processes have simple probability assumptions about the occurrence of time events, and have been reported by other investigators to be well adapted to the majority of seizure diaries studied”. We therefore investigated whether a diary d was random or not by doing a goodness-of-fit test with the model specified by the null hypothesis, Pr(Z, = i) = e-“*A’/;!for i = O,l, . . . , maxi. For a stationary diary {Z,}, we estimated the Poisson probability rate of occurrence of seizure events, i to be the maximum-likelihood estimate a15. In order to explore further any deviation from a Poisson process, we plotted the logarithm of the proportion of interseizure intervals longer than i days versus i (1 d i d maxinterseizure interva,),as nonlinearity of this plot is inconsistent with a Poisson processi3. Another model for randomness is a white noise process. We evaluated the diaries with respect to white noise by inspecting their autocorrelograms, as the autocorrelation coefficients of a white noise process is almost zero for all non-zero lags’. To characterize the concept of seizure centering, the following 3 criteria were considered. (1) Computing the total number, R, of ‘l-runs’ and ‘O-runs’ in the clipped diary {X,}, where we reject the hypothesis of randomness (i.e., ‘d is a sequence of Td binomial trials’) when the number of runs is too small, indicating that too many long runs occur16. (2) There are significantly more days with a high number of seizures, or more days with no seizures than expected from the Poisson model”.(3) The autocorrelogram exhibits an ‘autoregressive feature’ in that a fairly large value of the autocorrelation coefficient at lag 1 is detected, followed by a few more coefficients which, while greater than zero, tend to become successively smaller7. In using these clustering criteria, only the second criterion assumes stationarity. If a diary exhibited~erjod~ci~, this feature could be detected either directly in the cumulated plot of
events or in the autocorrelogram. If any periodicity was suspected by these means, further spectral analysis was done by computing the periodogram2* foralllagsm= 1,. . . , 60. The periodogram will disclose the power spectrum of different rhythms within the time span of observation. Periodicity was concluded if any of the sinusoidal components contributed to a major part of the total power. In an attempt to model some of the diaries for the purpose of seizure prediction, we prepared the following arguments leading up to a simple model which was then tested on a sub-sequence of some of the diaries. If a diary exhibits stationarity and we assume that the presence or absence of seizures at time t is largely determined by the presence or absence of a high susceptibility state (i.e., seizures) at times t l,t-2, . . ) t - k, this suggests that a stationary binary kth order Markov chain with state space S = {OJ} may be taken as a rough model to represent the data. To simplify matters we have only considered a 1st order Markov process to model this kind of dependency. Introducing common terminology, the probability of Xt+i being in state j given that X, is in state i is called the one-step trunsition probability, denoted by pji. Thus, pji = WX,+ 1 = i 1 X, = i} where we assume time-independence of the transition probabilities because of stationarity. Making the Markov chain assumption, a diary d is completely described by the transition probability matrix
wherepro + plo = pal + pI1 = 1, together with the probability distribution of the initial state X0, i.e. Pr{X, = 0} = .n, and Pr{X, = l} = n,. For all nonrandom stationary diaries we have computed the maximum likelihood estimates of the probability p = Pr {X, = 1> and the transition probabilities pji according to Kedem13; these values are given in Table III. In the attempt to make prediction according to the Markov model, we have made use of the following results from Kedem”. Given a stationary binary time series or just a segment of size n where the dependence is Markovian of either 2nd or 1st
157
the premenstrual, the menstrual and the non-menstrual parts of the cycle. The premenstrual period was defined as the last 4 days before onset of menstruation, menstruation was said to occur during the next 5 days, and the non-menstrual period spanned the rest of the cycle. A Wilcoxon signed rank test was used to evaluate possible differences in seizure occurrence between these 3 periods of a full menstrual cycle. Secondly, the cross-correlation coefficients at time lags exceeding the cycle length was computed for each of the 9 female patients.
order, or where there is no dependence, then it is possible to prove a simple algorithm that computes the probabilities, ~(k,n) of having k days with seizures during a period of FZdays, k = O,l, . . . , n. We have implemented this algorithm, and under the given assumptions we have estimated the input parameters from the first half of a given diary and computed the chance that the patient will have 0,l or a specific number of days with seizures within a period of 7 days. These predicted probabilities are then compared with the respective frequencies actually observed during successive periods of length 7 days throughout the rest of the diary (Table IV).
RESULTS Seizure events related to menstruation. A possible relationship between seizures and the menstrual cycle was investigated in the 9 women, First of all, number of seizures/day was calculated for TABLE
The descriptive data for the 21 seizure diaries is summarized in Table II. A total of 3229 seizures were registered over a time span of 19.7 patient
II
Descriptive diary data Diary
Tot. reg. time, Td
Days with seizure, S
Total No. of seizures, n
Seizure frequency,ii
Cox -I-Stuart test, F-value
Stationarity’ NA +
1
205
6
7
0.034
0.656
2
574
145
320
0.558
0.860
3
182
28
42
0.231
0.076
+
4
300
11
13
0.043
0.113
+
Sal
128
122
1183
9.242
0.052
+
5a2 5b
79 61
56 36
172 94
2.177 1.541
0.309 0.007
+ L
6a 6b
144 226
20 31
68 62
0.472 0.274
0.412 0.001
+ f
7
482
11
11
0.023
0.033
NA
8
174
7
7
0.040
0.500
+
9
293
41
82
0.280
0.304
+
10
323
16
17
0.053
0.806
+
11
180
34
49
0.272
0.925
+
12
182
6
17
0.093
0.688
NA
13a 13b
174 146
51 23
58 43
0.333 0.244
0.030 0.995
+
14
724
86
133
0.184
0.938
+ +
15
2360
243
768
0.325
0.924
+
16
166
50
71
0.428
0.971
+
17
185
12
12
0.065
0.073
+
7288
1035
Total Mean Median Range
347 183 61-2360
49 34 6-243
3229 (1780b) 154 (99b) 58 (43b) 7-1183
+,16 0.805 (0.220b) 0.274 (0.20gb) 0.023-9.242
72 ;A,3
* + Stationary time series; -, non-stationary time series; NA, not applicable (i.e., too few days with seizures, defined as $ 6 l/30). b Data from patient 5 excluded.
d
158
ary according to the Cox and Stuart test for trend. However, these two patients (patients 5 and 6) also had sub-diaries which were stationary. For patient 5, this was due to the introduction of clonazepam at the beginning of sub-diary 5b (Fig. IC). The reason for the non-stationarity in diary 6b is probably that the time between the first 2 clusters was about twice the nearly constant inter-cluster interval during the rest of the sub-diary. The remaining 3 diaries (patients 1, 7, 12) were inconclusive and not suited to further analysis because of a small number of days with seizures com-
years. If we exclude patient 5, who had daily absence seizures, this gives a mean seizure frequency of 0.206 in this sample of 17 patients. Except for this female patient, there were no major differences between the sexes. Stutionarity
For 16 of the 21 seizure diaries there was a linear increase in the cumulated number of seizures versus time, indicating a stationary process for seizure occurrence (R* >0.85, F-statistics, P < 0.001). Two diaries (5b,6b) were found to be non-station-
A
B
1
100
200
300
400
500
600
Days
Days D
g
1400
'i .s 1200$ 6
1000-
2
600.
E a 2
600.
i 0
50
100
150
200
250
Days Fig. 1. Cumulated number of seizures plotted against time for three different patients together with a gray level raster representation of a diary. (A) Stationary seizure pattern in patient 2. (B) Stationary pattern in diary 6a, exhibiting marked seizure clustering. (C) Non-stationary pattern due to change in medication (introducing Clonazepam) on day 127. The diary (5a) was later split into two stationary sub-diaries (5al and 5a2). (D) A gray level raster image representation of diary 6a with features of clustering (isolated, high intensity areas along the rows). Each pixel corresponds to a particular day of observation, where the diary is represented by a row by row arrangement making up the rectangular image. Each row equals an epoch of 16 days. The intensity value of each pixel represents the number of seizures on that particular day.
159 pared to the length of the diary (i.e., less than one seizure day per month). Fig. 1 shows the cumulated number of seizures plotted against time for 3 different patients, illustrating stationarity (Fig. lA), seizure clustering (Fig. 1B) and non-stationarity due to a change in medication (Fig. 1C). The typical feature of clustering is depicted in the gray-level raster representation (Fig. 1D).
tribution was due to an excess of days with no seizures, resulting in a few days with more seizures than expected (Fig. 2A), this indicates seizure clustering (denoted by a + in the Non-Poisson column in Table III). In five patients (8, 10, 13a, 16, 17) we could not exclude a Poisson pattern. However, 3 of these diaries were characterized by a very small seizure intensityb < 0.07). A frequency plot for one of the ‘Poisson patients’ is given in Fig. 2B. One diary (No. 4) could not be excluded from being just a random sequence of Td = 300 binomial trials with Pr(seizure) = 0.04 and Pr(no seizure) = 0.96 according to the R-test (P = 0.367). Two diaries (5a2 and 13b) could not be excluded from being white noise processes by our analysis.
Randomness Considering a Poisson process for random distribution of seizures, 11 of the 16 stationary diaries (69%) were non-Poisson (P c 0.005) as shown in Table III. When the deviation from a Poisson disTABLE III Randomness,
clustering, periodic&y and transition probabilities of seizure occurrence
The 5 non-stationary
diaries were not analyzed (denoted by *). Plots of the autocorrelograms are evaluated for the lag interval (0 d m of the Poisson model and the transition probabilities are given as their maximum likelihood estimates. A periodogram was only performed when periodicity was indicated from the correlogram. NA, not applicable; NS, not studied; DEP, dependency between events ‘seizures’/‘no seizures’. pm, probability of remaining seizure-free on a given day on the assumption that no seizure(s) occurred on the preceding day. pII, probability of having a seizure on a given day on the assumption that a seizure(s) did occur on the preceding day. s 60). The parameters
Diary
Clustering
Randomness Poisson process A
P
Transition probabilities
Periodic&y
White
R-test
Non-
Correlo-
Correlo-
Periodo-
noise in
p-value
Pokson
gram
gram
gram
i
Res., 8 (1991) 153-165
153
Elsevier EPIRES 00382
Temporal distribution of seizures in epilepsy
Erik Taub@lla, Arvid Lundervoldb and Leif Gjerstada “Department of Neurology, Ribhospitalet -
The National Hospital, University of Oslo, and bNorwegian Computing Center, Blindern, Oslo (Norway)
(Received 26 June 1990; accepted 1 October 1990) Key words: Epilepsy; Biorhythms; Stochastic processes; Seizure occurrence; Catamenial epilepsy
A major problem in epileptology is why a seizure occurs at a particular moment in time. An initial step in solving this problem is a detailed analysis of the temporal distribution of seizures. Using methods and theories of stochastic processes, seizure patterns in a group of epileptic outpatients were examined for stationarity, randomness, dependency and periodicity in a prospective study. Sixteen of the 21 seizure diaries included in the study showed stationarity; 2 were non-stationary and 3 inconclusive. Eleven of the 16 stationary diaries were non-Poisson (P < O.OOS),indicating that in the majority of patients seizures did not occur randomly. The most frequently encountered phenomenon was seizure clustering. Clustering was considered when the diaries fulfilled all three criteria: (1) a positive R-test (P < 0.001); (2) deviation from the fitted Poisson distribution towards clustering; and (3) the feature of an autoregressive process in the autocorrelogram plot. Dependency between seizure events was demonstrated in 8 of the 16 stationary diaries, computing first order transition probabilities. A detailed analysis of seizure occurrence is a major step towards a better understanding of the mechanisms underlying seizure precipitation. This is exemplified by our finding of a relation between seizure frequency and the menstrual cycle
INTRODUCTION Through neurobiological research, much has been learnt about the basic mechanisms of epileptogenesis. However, the reason why a specific seizure occurs in a patient at a given moment in time has seldom been elucidated. An initial step in exploring this problem is a detailed analysis of seizure occurrence in patients with epilepsy. A rhythmic appearance of seizures might indicate periodic variations in basal physiological parameters of importance for the precipitation of seizures. Randomly occurring clusters of epileptic atCorrespondence
Rikshospitalet,
to: Erik Taubbll, Dept. of Neurology, University of Oslo, 0027 Oslo 1, Norway.
0920-1211/91/$03.50 0 1991 Elsevier Science Publishers B.V.
tacks might be indicative of exogenous seizureprovoking factors. Seizure clustering might also imply that secondary alterations due to a seizure facilitate the precipitation of a second attack in the manner of a positive feedback mechanism. An excess of long inter-seizure intervals would, on the other hand, suggest a negative feedback mechanism. One of the most widely accepted factors affecting seizure frequency is the menstrual cycle. This relationship often merely reflects a clinical impression rather than statistical findings (for a review, see 3, 19), and even the phase of the menstrual cycle with the highest seizure frequency is disputed3. Rhythmic occurrence of seizures or seizure clus-
154
tering may be used to study basic seizure-precipitating factors. In a single patient, information about seizure occurrence makes it possible to provide the patient with a better understanding of seizure-provoking factors and thereby how to minimize them. If seizure-provoking factors can be identified, modifying these may decrease seizure frequency in nearly 50% of patients with poorly controlled epilepsy2. Knowing the ‘normal’ temporal distribution of seizures also provides the background for a better analysis of the effect of antiepileptic drug treatment. Before account can be taken of the temporal distribution of seizures in the understanding of precipitating factors and in therapy, one has to establish appropriate statistical models for these patterns. The aim of the present investigation has been to examine seizure occurrence more thoroughly by model fitting and parameter estimation in a group of epileptic outpatients. PATIENTS AND METHODS P&en ts Seizure diaries from 17 patients (9 female and 8 male) were studied. Their mean age was 32.5 years with a range of 4-52 (Table I). All were socially well-adapted outpatients at different hospitals who responded to a request in the periodical of the Norwegian Chapter of the International Bureau for Epilepsy, to participate in a study of factors affecting seizure occurrence. Five additional patients responded to the initial request, but did not complete their diaries. All patients used stable antiepileptic medication (Tabie I) with a serum concentration within the reference range. The onset of epilepsy ranged from 2 to 47 years prior to the start of this investigation, with a median of 15 years. There was no evidence of symptomatic epilepsy due to cerebral tumor, congenital or vascular ma~ormations in any of the patients, and there was no history of mental illness or retardation. The patients were asked to register on a daily basis, the number and type of seizures. They also had to indicate any deviation from normal daily routine, with special emphasis on physical and psychological stress, alcohol, lack of sleep or food, missed medication, intercurrent disease and the
TABLE I Clinical data
forall patients
CBZ, carbamazepine; CPS, complex partial seizures; CZP, clonazepam; GTC, generalized tonic-clonic seizures; NZP, nitrazepam; PB, phenobarbital; PHT, phenytoin; PRIM, primidone; SPS, simple partial seizure; VPA, Na-valproate. 2’ GTC, secondary generalized tonic-clonic seizure. In Patient 5, CZP was introduced during the study (at the beginning of subdiary 5b: see Fig. 1C). Patient NO. .._1
2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 ___~_
Sex --
M F M M F M M F M F M F F F M F F _~~
Age __
48 40 28 31 36 29 4 40 22 34 14 21 26 34 44 52 49 ~_.
Seizure type .-.
.~_~~ ~-
GTC CPS -t 2” GTC CPS CPS Absence CPS SPS + 2” GTC GTC CPS + 2” GTC SPS + 2” GTC CPS + 2” GTC CPS + 2” GTC SPS CPS CPS CPS CPS ..___ -.. _~~_ -
Medication -.
PTH CBZ CBZ f VPA PHT + CBZ PRIM + PB + (CZP) CBZ + PHT CBZ + PHT PHT VPA PHT CBZ + VPA + NZP CZP CBZ PHT + PB + CBZ PHT f PB + CBZ PRIM + CBZ PHT ---.___ -
use of additional drugs. Care was taken not to introduce the concept of rhythmicity. Seizure diaries were collected prospectively, except for two cases (patients 2 and 15), in whom parts of the retrospective seizure diaries were also used, Two males (patients 6 and 15) were included, although a rhythmic seizure pattern could be anticipated from their medical histories. Classified according to their main seizure type, 14 patients had partial seizures (11 complex, 3 simple), and 3 patients had primarily generalized seizures (Table I). Three patients had their diary split, resulting in a total of 21 single seizure diaries that were analyzed individually. Patient 5 had her diary split into 3 sub-diaries because of a change in medication, and a period of 53 days with missed registration. Patient 6 had his diary split into 2 sub-diaries due to an 8 month stop in registration. Finally, patient 13 had her diary divided into 2 because of intervening neurosurgery (temporal lobectomy).
155 Time series analysis and stochastic modeling
Our modeling was aimed at making the following clinical important questions more precise: (1) Is there an upward or downward trend in seizure occurrence with time? (2) Are seizures in a particular patient occurring at random? (3) Do seizure events show a clustering tendency in some patients? (4) Do seizures occur in a.periodic manner over several weeks or months of observation? (5) What kind of changes in seizure pattern do we observe after change in antiepileptic therapy? (6) Is it possible to predict seizure events in a patient, if one has knowledge of his or her previous seizure diary? In order to provide a statistical setting for describing the features of seizure occurrences during time, we defined a seizure diary d to be a sequence of seizure events indexed in discrete time. Thus, each seizure diary was recognized as a discrete stochiastic process, d = {Zr}tET. For each diary d we took the index set to be T = {tit = 0, 1, . . . , Td - 1)) where t denotes number of days after start of diary registration, and Td the duration of the diary registration. The random variable Z, denotes the number of seizure events registered on day t. However, in some of the models considered, we made a ‘hard limiting transformation’ of Z, resulting in a binary series, X, such that X, = 1 if at least one seizure occurred on day t, and otherwise X, = 0. An important class of stochastic processes is those which are stationary, in that most of the probability theory of time series assumes stationarity. In simple terms, stationarity means that the probability laws governing the seizure events do not change with time (i.e. the mean and autocovariance function is time independent). We have concluded non-stationarity of a diary if either the autocorrelogram, the non-parametric Cox and Stuart test’ (a sign test variant) or the plot of cumulated seizures indicates a trend. As an aid in this study, one of us (A.L.) wrote a toolbox of C code, portable between UNIX work stations (SUN 3, DECstation) and a PC running MS-DOS. Steps in the analysis of a given diary. Our analytical approach, which is related to a single diary at
the time, can be summarized in the following steps: (1) Registration of the diary data on a text file. (2) Inspection of the recorded number of seizures over time by: making a plot of the cumulated number of seizures versus days” generating a 2-dimensional gray level raster image. (3) Compute descriptive statistics. (4) Check for trend and stationarity: inspection of plot of cumulated number of seizures over time” Cox and Stuart test for trend’ inspection of the autocorrelogram’. (5) Check for randomness: testing for a Poisson process of number of seizure occurrences” testing for a Poisson process of interseizure intervalsl’. (6) Check for clustering: applying the R-test for seizure clusteringl’j (binarized diary) check for feature of clustering in Poisson model” inspecting autocorrelogram for short-term correlation’. (7) Check for periodicity: inspecting plot of cumulated number of seizures inspecting autocorrelogram for oscillations’ computing the periodogram22. (8) Check for dependency: computing chi-square test statistic from estimated transition probabilities15. (9) Choose a plausible/tentative model which can aid in seizure prediction: first-order Markov process13; parameters estimated from the first half of the binarized diary. After registration of seizure events, the first step in the time series investigation was careful scrutiny of the recorded data plotted with time and a plot of the corresponding autocorrelogram, as these could suggest the method of analysis and the statistics to be used to summarize the information in the data. We have thus displayed each of the 21 sample diaries graphically by plotting the cumulated sum of the random variable on the ordinate and the time scale on the abscissa. Furthermore, the
156
autocorrelation function for each diary is plotted for lag 1 up to lag 60 days. Fig. 1 shows the cumulated number of seizures versus day for three of the diaries, and a gray level raster image representation of a diary exhibiting clustering. We used the Poisson model to test the randomness of seizure events in stationary diaries. Poisson processes have simple probability assumptions about the occurrence of time events, and have been reported by other investigators to be well adapted to the majority of seizure diaries studied”. We therefore investigated whether a diary d was random or not by doing a goodness-of-fit test with the model specified by the null hypothesis, Pr(Z, = i) = e-“*A’/;!for i = O,l, . . . , maxi. For a stationary diary {Z,}, we estimated the Poisson probability rate of occurrence of seizure events, i to be the maximum-likelihood estimate a15. In order to explore further any deviation from a Poisson process, we plotted the logarithm of the proportion of interseizure intervals longer than i days versus i (1 d i d maxinterseizure interva,),as nonlinearity of this plot is inconsistent with a Poisson processi3. Another model for randomness is a white noise process. We evaluated the diaries with respect to white noise by inspecting their autocorrelograms, as the autocorrelation coefficients of a white noise process is almost zero for all non-zero lags’. To characterize the concept of seizure centering, the following 3 criteria were considered. (1) Computing the total number, R, of ‘l-runs’ and ‘O-runs’ in the clipped diary {X,}, where we reject the hypothesis of randomness (i.e., ‘d is a sequence of Td binomial trials’) when the number of runs is too small, indicating that too many long runs occur16. (2) There are significantly more days with a high number of seizures, or more days with no seizures than expected from the Poisson model”.(3) The autocorrelogram exhibits an ‘autoregressive feature’ in that a fairly large value of the autocorrelation coefficient at lag 1 is detected, followed by a few more coefficients which, while greater than zero, tend to become successively smaller7. In using these clustering criteria, only the second criterion assumes stationarity. If a diary exhibited~erjod~ci~, this feature could be detected either directly in the cumulated plot of
events or in the autocorrelogram. If any periodicity was suspected by these means, further spectral analysis was done by computing the periodogram2* foralllagsm= 1,. . . , 60. The periodogram will disclose the power spectrum of different rhythms within the time span of observation. Periodicity was concluded if any of the sinusoidal components contributed to a major part of the total power. In an attempt to model some of the diaries for the purpose of seizure prediction, we prepared the following arguments leading up to a simple model which was then tested on a sub-sequence of some of the diaries. If a diary exhibits stationarity and we assume that the presence or absence of seizures at time t is largely determined by the presence or absence of a high susceptibility state (i.e., seizures) at times t l,t-2, . . ) t - k, this suggests that a stationary binary kth order Markov chain with state space S = {OJ} may be taken as a rough model to represent the data. To simplify matters we have only considered a 1st order Markov process to model this kind of dependency. Introducing common terminology, the probability of Xt+i being in state j given that X, is in state i is called the one-step trunsition probability, denoted by pji. Thus, pji = WX,+ 1 = i 1 X, = i} where we assume time-independence of the transition probabilities because of stationarity. Making the Markov chain assumption, a diary d is completely described by the transition probability matrix
wherepro + plo = pal + pI1 = 1, together with the probability distribution of the initial state X0, i.e. Pr{X, = 0} = .n, and Pr{X, = l} = n,. For all nonrandom stationary diaries we have computed the maximum likelihood estimates of the probability p = Pr {X, = 1> and the transition probabilities pji according to Kedem13; these values are given in Table III. In the attempt to make prediction according to the Markov model, we have made use of the following results from Kedem”. Given a stationary binary time series or just a segment of size n where the dependence is Markovian of either 2nd or 1st
157
the premenstrual, the menstrual and the non-menstrual parts of the cycle. The premenstrual period was defined as the last 4 days before onset of menstruation, menstruation was said to occur during the next 5 days, and the non-menstrual period spanned the rest of the cycle. A Wilcoxon signed rank test was used to evaluate possible differences in seizure occurrence between these 3 periods of a full menstrual cycle. Secondly, the cross-correlation coefficients at time lags exceeding the cycle length was computed for each of the 9 female patients.
order, or where there is no dependence, then it is possible to prove a simple algorithm that computes the probabilities, ~(k,n) of having k days with seizures during a period of FZdays, k = O,l, . . . , n. We have implemented this algorithm, and under the given assumptions we have estimated the input parameters from the first half of a given diary and computed the chance that the patient will have 0,l or a specific number of days with seizures within a period of 7 days. These predicted probabilities are then compared with the respective frequencies actually observed during successive periods of length 7 days throughout the rest of the diary (Table IV).
RESULTS Seizure events related to menstruation. A possible relationship between seizures and the menstrual cycle was investigated in the 9 women, First of all, number of seizures/day was calculated for TABLE
The descriptive data for the 21 seizure diaries is summarized in Table II. A total of 3229 seizures were registered over a time span of 19.7 patient
II
Descriptive diary data Diary
Tot. reg. time, Td
Days with seizure, S
Total No. of seizures, n
Seizure frequency,ii
Cox -I-Stuart test, F-value
Stationarity’ NA +
1
205
6
7
0.034
0.656
2
574
145
320
0.558
0.860
3
182
28
42
0.231
0.076
+
4
300
11
13
0.043
0.113
+
Sal
128
122
1183
9.242
0.052
+
5a2 5b
79 61
56 36
172 94
2.177 1.541
0.309 0.007
+ L
6a 6b
144 226
20 31
68 62
0.472 0.274
0.412 0.001
+ f
7
482
11
11
0.023
0.033
NA
8
174
7
7
0.040
0.500
+
9
293
41
82
0.280
0.304
+
10
323
16
17
0.053
0.806
+
11
180
34
49
0.272
0.925
+
12
182
6
17
0.093
0.688
NA
13a 13b
174 146
51 23
58 43
0.333 0.244
0.030 0.995
+
14
724
86
133
0.184
0.938
+ +
15
2360
243
768
0.325
0.924
+
16
166
50
71
0.428
0.971
+
17
185
12
12
0.065
0.073
+
7288
1035
Total Mean Median Range
347 183 61-2360
49 34 6-243
3229 (1780b) 154 (99b) 58 (43b) 7-1183
+,16 0.805 (0.220b) 0.274 (0.20gb) 0.023-9.242
72 ;A,3
* + Stationary time series; -, non-stationary time series; NA, not applicable (i.e., too few days with seizures, defined as $ 6 l/30). b Data from patient 5 excluded.
d
158
ary according to the Cox and Stuart test for trend. However, these two patients (patients 5 and 6) also had sub-diaries which were stationary. For patient 5, this was due to the introduction of clonazepam at the beginning of sub-diary 5b (Fig. IC). The reason for the non-stationarity in diary 6b is probably that the time between the first 2 clusters was about twice the nearly constant inter-cluster interval during the rest of the sub-diary. The remaining 3 diaries (patients 1, 7, 12) were inconclusive and not suited to further analysis because of a small number of days with seizures com-
years. If we exclude patient 5, who had daily absence seizures, this gives a mean seizure frequency of 0.206 in this sample of 17 patients. Except for this female patient, there were no major differences between the sexes. Stutionarity
For 16 of the 21 seizure diaries there was a linear increase in the cumulated number of seizures versus time, indicating a stationary process for seizure occurrence (R* >0.85, F-statistics, P < 0.001). Two diaries (5b,6b) were found to be non-station-
A
B
1
100
200
300
400
500
600
Days
Days D
g
1400
'i .s 1200$ 6
1000-
2
600.
E a 2
600.
i 0
50
100
150
200
250
Days Fig. 1. Cumulated number of seizures plotted against time for three different patients together with a gray level raster representation of a diary. (A) Stationary seizure pattern in patient 2. (B) Stationary pattern in diary 6a, exhibiting marked seizure clustering. (C) Non-stationary pattern due to change in medication (introducing Clonazepam) on day 127. The diary (5a) was later split into two stationary sub-diaries (5al and 5a2). (D) A gray level raster image representation of diary 6a with features of clustering (isolated, high intensity areas along the rows). Each pixel corresponds to a particular day of observation, where the diary is represented by a row by row arrangement making up the rectangular image. Each row equals an epoch of 16 days. The intensity value of each pixel represents the number of seizures on that particular day.
159 pared to the length of the diary (i.e., less than one seizure day per month). Fig. 1 shows the cumulated number of seizures plotted against time for 3 different patients, illustrating stationarity (Fig. lA), seizure clustering (Fig. 1B) and non-stationarity due to a change in medication (Fig. 1C). The typical feature of clustering is depicted in the gray-level raster representation (Fig. 1D).
tribution was due to an excess of days with no seizures, resulting in a few days with more seizures than expected (Fig. 2A), this indicates seizure clustering (denoted by a + in the Non-Poisson column in Table III). In five patients (8, 10, 13a, 16, 17) we could not exclude a Poisson pattern. However, 3 of these diaries were characterized by a very small seizure intensityb < 0.07). A frequency plot for one of the ‘Poisson patients’ is given in Fig. 2B. One diary (No. 4) could not be excluded from being just a random sequence of Td = 300 binomial trials with Pr(seizure) = 0.04 and Pr(no seizure) = 0.96 according to the R-test (P = 0.367). Two diaries (5a2 and 13b) could not be excluded from being white noise processes by our analysis.
Randomness Considering a Poisson process for random distribution of seizures, 11 of the 16 stationary diaries (69%) were non-Poisson (P c 0.005) as shown in Table III. When the deviation from a Poisson disTABLE III Randomness,
clustering, periodic&y and transition probabilities of seizure occurrence
The 5 non-stationary
diaries were not analyzed (denoted by *). Plots of the autocorrelograms are evaluated for the lag interval (0 d m of the Poisson model and the transition probabilities are given as their maximum likelihood estimates. A periodogram was only performed when periodicity was indicated from the correlogram. NA, not applicable; NS, not studied; DEP, dependency between events ‘seizures’/‘no seizures’. pm, probability of remaining seizure-free on a given day on the assumption that no seizure(s) occurred on the preceding day. pII, probability of having a seizure on a given day on the assumption that a seizure(s) did occur on the preceding day. s 60). The parameters
Diary
Clustering
Randomness Poisson process A
P
Transition probabilities
Periodic&y
White
R-test
Non-
Correlo-
Correlo-
Periodo-
noise in
p-value
Pokson
gram
gram
gram
i
