Journal of Psychosomatic Res

Predictive value of alexithymia in patients with eating disorders:

A 3-year prospective study

Mario Speranzaa,b,4, Gwenole Loasc, Jenny Wallierd, Maurice Corcosb,d

aDepartment of Child Psychiatry, Centre Hospitalier de Versailles, Versailles, FrancebINSERM U669, Paris, France

cDepartment of Psychiatry, Hopital Pinel, Amiens, FrancedDepartment of Adolescent and Young Adult Psychiatry, Institut Mutualiste Montsouris, Paris, France

Received 3 January 2007; received in revised form 27 February 2007; accepted 6 March 2007

Abstract

Objective: Several cross-sectional studies have reported high

levels of alexithymia in populations with eating disorders.

However, only few studies, fraught with multiple methodological

biases, have assessed the prognostic value of alexithymic features

in these disorders. The aim of the present study was to investigate

the long-term prognostic value of alexithymic features in a sample

of patients with eating disorders. Methods: Within the framework

of a European research project on eating disorders (INSERM

Network No. 494013), we conducted a 3-year longitudinal study

exploring a sample of 102 DSM-IV eating disorder patients using

the Toronto Alexithymia Scale (TAS-20) and the Beck Depression

Inventory. Results: At the 3-year assessment, 74% (n=76) of the

sample still presented a syndromal or subsyndromal eating

0022-3999/07/$ – see front matter D 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.jpsychores.2007.03.008

4 Corresponding author. Centre Hospitalier de Versailles, Child

Psychiatry, 78157 Le Chesnay, France. Fax: +33 1 39 63 94 25.

E-mail address: mario.speranza@noos.fr (M. Speranza).

disorder (unfavorable outcome: score of z3 on the Psychiatric

Status Rating Scale for anorexia nervosa or bulimia nervosa). In

logistic and hierarchical regression analyses, the Difficulty

Identifying Feelings factor of the TAS-20 emerged as a significant

predictor of treatment outcome, independent of depressive

symptoms and eating disorder severity. Conclusions: The results

of this study indicate that difficulty in identifying feelings can act

as a negative prognostic factor of the long-term outcome of

patients with eating disorders. Professionals should carefully

monitor emotional identification and expression in patients with

eating disorders and develop specific strategies to encourage

labeling and sharing of emotions.

D 2007 Elsevier Inc. All rights reserved.

Keywords: Alexithymia; Depression; Eating disorders; Longitudinal study; Outcome predictors

Introduction

The identification of variables that predict treatment

outcome in patients with eating disorders is critical if we are

to increase the degree of sophistication with which we treat

these disorders. Understanding predictors of outcome could

theoretically facilitate matching treatments to individuals

based on their clinical profile at presentation. Dirks et al. [1]

have coined the term bpsychic maintenanceQ to describe the

chronic outcome of an illness due to psychological reasons.

Among the several psychological features that have been

proposed to predict treatment outcome in patients with

eating disorders, alexithymia has attracted special interest.

Alexithymia is a personality construct characterized by a

difficulty in identifying and describing feelings, a diminu-

tion of fantasy, and a concrete and externally oriented

thinking style [2]. Several arguments, namely, factor

analyses and longitudinal studies, have supported the view

that alexithymia is a stable personality trait rather than a

state-dependent phenomenon linked to depression or to

clinical status [3,4]. Several studies have reported high

levels of alexithymia in patients with eating disorders,

especially in individuals with anorexia nervosa [5–8]. There

are several reasons to believe that this construct could play a

earch 63 (2007) 365–371

M. Speranza et al. / Journal of Psychosomatic Research 63 (2007) 365–371366

major role in the illness course of eating disorders: due to

their cognitive limitations in emotion regulation, alexithy-

mic individuals with eating disorders may resort to

maladaptive self-stimulatory behaviors such as starving,

bingeing, or drug misuse to self-regulate disruptive emo-

tions [9]. The lack of insight and the externally oriented

thinking style of alexithymic subjects may also interfere

with their capacity to benefit from psychotherapeutic

interventions. However, in spite of the clinical relevance

of this issue, clear data on the prognostic value of

alexithymic features in eating disorders are still lacking.

Two studies conducted on individuals with bulimia have

failed to demonstrate a specific impact of alexithymia on the

outcome of these patients. However, these studies presented

some methodological limitations: samples were relatively

small; the longitudinal time period was too short (10

weeks); both studies used an earlier version of the Toronto

scale to assess alexithymic features; and finally, the outcome

measures did not account for the degree of clinical change

between baseline and follow-up [6,10].

The aim of the present study was to investigate the long-

term prognostic value of alexithymic features in a large

sample of patients with eating disorders, taking into account

the limitations of previous studies.

Method

Participants and procedures

The subjects of this study were derived from a multi-

center research project investigating the psychopathological

features of eating disorders (Inserm Network No. 494013).

The overall design of the Network was a cross-sectional

investigation, with only a subset of research centers

involved in a prospective follow-up study. The recruitment

centers were academic psychiatric hospitals specialized in

adolescents and young adults (age range of reception: 15–30

years). For this study, only female participants who had

requested care for a disorder of eating behavior were

screened for inclusion. At the first assessment and 3 years

later, patients included in the sample completed a research

protocol, which consisted of a clinical interview (for

sociodemographic and diagnostic data) and a self-report

questionnaire eliciting psychopathological features (namely,

alexithymia and depression). Eating disorder diagnoses,

whether of anorexia nervosa or bulimia nervosa, were made

by a psychiatrist or a clinical psychologist specialized in the

field of eating disorders using DSM-IV diagnostic criteria

[11]. Diagnostic assessment was made using the Mini

International Neuropsychiatric Interview, which is a struc-

tured, validated diagnostic instrument that explores each

criterion necessary for the establishment of current and

lifetime DSM-IV Axis I main diagnoses (anxiety and

depressive disorders, substance-related disorders) [12,13].

In relation to the main purpose of the study, which was to

investigate the predictive power of alexithymia in eating

disorders, we excluded patients presenting a comorbid

diagnosis of current major depressive episode (MDE)

(n=11) and patients presenting a current alcohol or drug

dependency (n=6). MDEs were excluded to raise the chance

of detecting a significant relationship between alexithymia

and outcome, which would have been reduced by an

excessive overlap between alexithymic and depressive

scores. Alcohol and drug dependencies were excluded

because of very few cases, thus creating a more homoge-

neous sample of eating disorders.

Patients were invited to participate in the follow-up

study 3 years later. At 18 months, a reminder letter was

sent to all participants. A second letter was sent just before

contacting them by phone to plan the second assessment.

The protocol was approved by the local ethics committee

(Paris Cochin Hospital). After all the necessary information

was provided, all subjects gave written consent for

participation in the study.

Measures

Alexithymia was rated using the French translation of the

revised Toronto Alexithymia Scale (TAS-20) [14–16],

which is a self-report scale with 20 items rated on a five-

point Likert scale. The items of the TAS are clustered into

three factors: Difficulty Identifying Feelings (DIF), Diffi-

culty Describing Feelings (DDF), and Externally Oriented

Thinking (EOT) [17].

Depression severity was measured with the French

translation of the abridged version of the Beck Depression

Inventory (BDI-13) [18]. The BDI-13 has been developed

by Beck and Beck [19] as a specific tool for epidemiological

studies by selecting within all the items showing a high

correlation (N.90) with the total score of the BDI-21.

The severity of the illness was assessed by the

bseverity of illnessQ item of the Clinical Global Impres-

sion (CGI). The CGI requires the clinician to rate on a 7-

point scale (1=normal to 7=extremely ill) the severity of

the patient’s illness at the time of assessment, relative to

the clinician’s past experience and training with patients

with the same diagnosis.

Outcome criteria

The clinical outcome at 3 years was approached by two

complementary perspectives: categorically, according to the

presence or absence of eating symptoms, and dimensionally,

according to the degree of clinical improvement between

baseline and follow-up. For the categorical approach, eating

disorder symptoms were assessed by the Psychiatric Status

Rating Scale (PSRS) for anorexia nervosa or bulimia

nervosa [20,21]. The PSRS, which is part of the diagnostic

assessment LIFE Eat II, is based on DSM-IV diagnostic

criteria for eating disorders. It defines six levels of severity

according to the presence and the degree of clinical

M. Speranza et al. / Journal of Psychosomatic Research 63 (2007) 365–371 367

symptoms. According to Fichter and Quadflieg [22], we

have defined two categories: a favorable outcome (score of

1 or 2) with a complete disappearance of all eating

symptoms and an intermediate/unfavorable outcome (score

of 3 or more), characterized by the persistency of a

subsyndromal pattern or a complete diagnosis of eating

disorder. For the dimensional approach, following Porcelli

et al. [4], the degree of clinical improvement, according to

the initial clinical level, was calculated as follows: (CGI at

baseline�CGI at follow-up/CGI at baseline)�100 (between

�100% and +100% of change).

Treatments received during follow-up were recorded

(types of treatment, age at the beginning of treatment,

duration). The types of treatment were recoded in dichoto-

mous answers (yes/no) according to Honkalampi et al. [23].

Statistical analysis

To compare the mean scores at baseline and follow-up,

we used a t test for paired samples for the entire group and

according to outcome. The relative stability of alexithymic

features was assessed by test–retest correlations and by the

comparison of the magnitude of changes of the scores of

alexithymia, depression, and clinical severity (Cohen’s D for

paired samples: g2 paired=t2/t2+N1�1). To evaluate the

predictive power of alexithymic features, we performed a

series of stepwise (logistic or hierarchical) regression

analyses with the TAS factors of the baseline assessment.

To assess the extent of the predictive power of alexithymia,

over and above those of depression and/or clinical severity

Table 1

Scores at baseline and follow-up

Variables

T1 T3

Mean S.D. Mean

Entire sample (N=102)

TAS-TOT 56.7 11.5 50.0

TAS-DIF 22.8 5.8 18.3

TAS-DDF 16.7 4.5 14.6

TAS-EOT 17.1 4.7 17.2

BDI 13.4 8.4 8.7

CGI 4.8 1.0 3.6

Unfavorable outcome (n=76)

TAS-TOT 58.1 11.5 52.0

TAS-DIF 23.7 5.5 19.5

TAS-DDF 17.1 4.6 15.1

TAS-EOT 17.3 4.8 17.3

BDI 14.8 8.3 10.0

CGI 4.9 1.0 4.2

Favorable outcome (n=26)

TAS-TOT 52.0 10.4 44.3

TAS-DIF 20.1 5.9 14.5

AS-DDF 15.5 4.1 13.3

TAS-EOT 16.4 4.3 16.5

BDI 9.23 7.4 4.7

CGI 4.6 0.9 1.8

TAS-TOT, TAS-20 total score. T1, initial assessment. T3, follow-up assessment a

4 Pb.01.

44 Pb.05.

(incremental predictivity), we secondarily added to the

regression analyses, with a forced-entry method, the CGI

and the BDI-13 [3]. Differences in treatments delivered

during the study period were taken into account by

maintaining these variables in the models. Results are

presented as meanFstandard deviation. Statistical analyses

were performed with the 10.1 version of the Statistical

Package for Social Sciences.

Results

From the initial sample of the Inserm Network on eating

disorders, 144 patients were selected to be included in the

follow-up study and retraced 3 years later. Among these

subjects, 35 refused to participate or were unavailable. One

hundred nine subjects were directly interviewed, of whom

seven had to be excluded because they did not completely

answered the self-questionnaire. Statistical analyses were

performed on a final sample of 102 subjects (72.8%). Sixty-

three had an initial diagnosis of anorexia nervosa, while 39

had an initial diagnosis of bulimia. No significant differ-

ences were found between the patients who did not

participate in the follow-up and those who completed the

study with regard to sociodemographic variables, eating

disorder subtypes, severity of the eating disorder (CGI), and

levels of alexithymia (TAS-20) and depression (BDI-13) at

baseline. The sample was composed mostly of young

women (meanFS.D. age: 21.5F5 years), with a medium

to high level of education.

Statistics

S.D. t Cohen D R

12.7 5.654 0.24 .534

6.3 6.934 0.32 .404

5.1 4.244 0.15 .494

4.7 �0.07 0.00 .614

7.9 5.424 0.22 .424

1.4 7.654 0.37 .2044

12.4 4.684 0.23 .554

6.2 5.514 0.29 .374

5.2 3.594 0.15 .524

4.7 �0.03 0.00 .614

7.6 4.964 0.25 .444

1.0 4.294 0.20 .294

11.7 3.134 0.28 .4044

5.1 4.284 0.42 .26

4.7 2.1544 0.16 .3844

4.9 �0.09 0.00 .594

7.8 2.3144 0.18 .13

0.8 12.64 0.86 .18

t 3 years.

Table 2

Hierarchical regression predicting the clinical improvement of eating

disorders

Steps/Variables R2 F df

Final model

B S.E. P b

1 .09 5.424 1, 99

TAS-DIF 0.32 0.14 .02 �.222nd .14 7.144 2, 98

TAS-DIF 0.25 0.16 .05 �.17BDI-13 0.13 0.16 .15 �.15Dependent variable: improvement of the clinical severity (%).

Predictors: Treatments (enter method); TAS-DIF, TAS-DDF, TAS-EOT

(stepwise method); BDI (enter method).

R2=Nagelkerke R2. Change was produced by adding the BDI: R2chg=.04,

F(1)=2.38, P=ns.

4 Pb .01.

M. Speranza et al. / Journal of Psychosomatic Research 63 (2007) 365–371368

Outcome results

On the basis of the outcome criteria, the sample showed

an overall negative outcome: 47% (n=48) of the sample still

had a moderate to severe diagnosis of eating disorder (a

score of 5 or 6 on the PSRS). One third of patients (n=26;

28%) had an intermediate outcome (persistency of clear

symptoms without filling the diagnostic criteria: a score of 3

or 4 on the PSRS). Only 25% (n=26) had a favorable

outcome with a complete disappearance of any eating

disorder symptoms (a score of 1 or 2 on the PSRS).

Concerning the degree of clinical change, there was an

overall improvement of the sample, as well as major

interindividual differences (mean: 22%; from �67% to

+83% improvement). The outcome was independent from

the initial diagnosis of anorexia or bulimia. The majority of

the patients had at least one form of therapeutic intervention

between baseline and follow-up. Psychotherapy was the

most common (57%), followed by antidepressants (40%).

Since patients with anorexia and bulimia had similar clinical

outcomes and did not show any differences in alexithymic

scores at baseline, statistical analyses were performed on the

entire sample.

Table 1 displays the means and standard deviations of the

entire sample, in accordance with the favorable or unfav-

orable outcome for the TAS-20, BDI-13, and CGI at

baseline and follow-up. In the entire sample, there was a

significant improvement between baseline and follow-up

scores for alexithymia [TAS-TOT: t(100)=�5.65, Pb.01],paralleled by a similar improvement in clinical severity

[CGI: t(100)=�5.42, Pb.01] and depression [BDI:

t(100)=�7.65, Pb.01]. Improvement concerned all subjects

independent of the clinical outcome of the eating disorder.

The magnitude of change (calculated via Cohen’s D) for

alexithymic scores was similar to BDI and smaller than

CGI. Moreover, although data failed to show an absolute

stability between baseline and follow-up for alexithymic

scores, a relative stability was observed in the correlation

coefficients between the two assessments, thus suggesting

that alexithymia may serve as a predictor of treatment

outcome. Correlations were moderate to strong for all

alexithymic factors. The issue of the relative weight of trait

and state effects on alexithymia scores (and the influence of

depression and clinical severity) was further explored with a

structural equation modeling procedure. The results (data

not shown) highlight that a model combining a state and a

trait effect has the best adjustment to the data (Speranza

et al., in preparation).

Predicting treatment outcome

Regression analyses were performed on the entire sample

of patients with eating disorders in the absence of differ-

ences in outcome between patients with anorexia and

patients with bulimia. The subjects/variables ratio of 17

was large enough for statistical power. Outliers were

eliminated and non-multicollinearity was verified. For all

the analyses, we used standardized scores [24].

Favorable versus unfavorable outcome

Seventy-six patients (74%) were classified as having an

unfavorable outcome. Logistic regression analysis showed

that the best model contained the DIF factor of the TAS

(TAS-DIF). TAS-DIF was a significant predictor of an

unfavorable outcome of eating disorders [Exp(B)=1.76,

P=.02]. This model correctly predicted belongingness to the

clinical categories in 76.5% of the cases [�2log L=103.3,

Model F(1)=12.5, Pb.002] and explained 17% of the

variance (Nagelkerke R2=.17). A second model integrating

the BDI and the CGI was still significant [�2logL=103.3,Model F(3)=16.1, Pb.003]. It slightly improved the

explained variance compared to the first model (Nagelkerke

R2 changes from 17 to 21; prediction from 76.5% to 77.5%).

Although BDI and CGI reduced the predictive power of the

TAS-DIF [Exp(B)=1.32, P=.04], they did not erase it

completely. TAS-DIF continued to be significant predictive

factors of outcome, whereas CGI [Exp(B)=1.05, P=ns] and

BDI [Exp(B)=1.76, P=ns] were not.

Degree of change in eating disorders

The logistic regression according to the clinical outcome

categorized the subjects independently of the initial clinical

status. This procedure can artificially raise the relationships

between the predicting variables and the subjects, presenting

a more severe clinical presentation at baseline. To assure

that the model predicted the real change of clinical severity

between the assessments, we performed a hierarchical

regression analysis using the degree of improvement on

the CGI as the dependent variable. This variable integrated

the level of initial severity. Since the CGI was one of the

predicting variables, the baseline CGI was discarded from

the predictors in the final model [4].

Results of hierarchical regression were close to those of

the logistic regression. The TAS-DIF was significantly

M. Speranza et al. / Journal of Psychosomatic Research 63 (2007) 365–371 369

associated to the outcome of patients with eating disorders.

It was negatively correlated to the clinical improvement

(TAS-DIF: b=�.22). The model explained a limited part of

the variance of change of the clinical severity (9%). Adding

BDI to the model slightly reduced the significance of the

TAS-DIF, which remained a significant predictor of the

clinical improvement at follow-up (see Table 2).

Discussion

The results of our study indicate that one of the facets

of the alexithymia construct, the difficulty in identifying

feelings, is a negative prognostic factor for the long-term

outcome of patients with eating disorders. Patients with the

greatest difficulties at identifying emotions at baseline are

more often symptomatic at follow-up and show a less

favorable clinical improvement. The relative stability

shown by alexithymia over time legitimates its use as a

potential prognostic factor in eating disorders. Further-

more, this result highlights how a multidimensional

approach can be more informative than a unidimensional

one because it allows to explore the impact of each facet

of the construct on other psychopathological variables

[25]. It must be acknowledged that the predictive power of

this factor is limited (it explains a small amount of the

variance of the clinical outcome); however, it remains

significant even after having taken into account the impact

of received treatments and the influence of the initial

clinical severity and depressive symptoms. This is espe-

cially notable because the long duration of the study

(3 years) inevitably introduces an important number of

noncontrollable factors.

Among the many correlation studies exploring alexithy-

mic features in eating disorders, not one has longitudinally

investigated the predictive value of alexithymia on the long-

term clinical outcome of these patients. Therefore, our

results cannot be easily compared with those of the

two earlier negative studies. These studies were limited by

their small recruitment of patients with bulimia (31 and

41 subjects) over a short period (10 weeks maximum), by

their analysis of only the total score of the TAS, and by their

use of outcome measures that did not account for the initial

clinical severity [6,10].

Our results are close to those carried out with patients

presenting functional somatic symptoms [4,26], which share

many features with eating disorders [4,27]. Bach and Bach

[26] observed that alexithymia slightly predicted the

persistency of functional somatizations 2 years later,

independently of depressive symptoms. The limited pre-

dictive value of alexithymia in our study contrasts with the

results of the study of Porcelli et al. [4] who found that

alexithymia was a strong negative predictor of the evolution

of gastrointestinal functional disorders. It must be noted that

Porcelli et al. used the total score of the TAS-20. Three

points can explain the different results: the duration of the

studies (6 months vs. 3 years), the low levels of depression

of patients with functional disorders, and finally, a largely

favorable response of these disorders to any kind of

therapeutic intervention compared to patients with eating

disorders (Porcelli, personal communication, May, 2004).

There are several ways in which alexithymia can affect

the clinical outcome of eating disorders: via the negative

influence it exerts on the clinical expression of the disorders

and on the response to therapeutic interventions.

First, the difficulty in identifying feelings may reduce the

capacity of patients with eating disorders to adapt to

stressful situations [28]. Such situations generate an emo-

tional overflow that alexithymic subjects apprehend less by

emotional and cognitive features than by their associated

somatic indexes [29]. This uncertainty between feelings and

bodily sensations reminds us of the interoceptive confusion

proposed by Hilde Bruch [30,31], which continues to be

pertinent from a clinical point of view. Luminet et al. [32]

have experimentally observed a dissociation of the compo-

nents of the emotional response of alexithymic subjects (a

physiological hyperreactivity to emotional stimuli associ-

ated to a deficit at the level of the cognitive experience),

which illustrate the functioning of patients with eating

disorders. Faced with the physiological arousal induced by

emotional demands, these patients may show poor adaptive

strategies. They may resort to restricted patterns of repetitive

and automated behaviors, such as the hyperactivity of

anorexic individuals or the binges/purge cycles of bulimic

subjects, which temporarily relieve their feeling of dis-

comfort and restore their inner equilibrium [33,34] but

generate, in the long term, a positive reinforcement of the

eating disorder.

Second, alexithymia may be related to a chronic course

of eating disorders by its relationship with other patholog-

ical behaviors, especially with addictive disorders. We have

shown in previous studies that alexithymia is associated

with addictive behaviors in patients with bulimia [35].

Patients with eating disorders may resort to addictive

behaviors to relieve the anxious and depressive feelings

elicited by their negative perceptions of themselves [36].

Finally, alexithymia can worsen the outcome of eating

disorders by limiting the compliance to care and the efficacy

of therapeutic strategies. This hypothesis has been the object

of many theoretical reflections [37–39] but of very few

empirical studies. Recently, McCallum et al. [39] have

found that, for both short-term group therapy and short-term

individual therapy, alexithymic features were associated

with the worst treatment outcome.

There are several limitations to this study that reduce the

generalizability of results. First, the sample was composed

of young women with a medium to high level of education

recruited from university hospitals specialized in adoles-

cents and young adults. It is possible that the sample

contained patients with specific clinical and sociodemo-

graphic profiles. However, the degree of clinical improve-

ment was individually assessed, and being female and

M. Speranza et al. / Journal of Psychosomatic Research 63 (2007) 365–371370

having a high level of education are known to be typical

characteristics of patients with eating disorder. One must

also be cautious in generalizing because we excluded from

the sample patients presenting an MDE and/or an alcohol or

drug dependency. However, the number of excluded cases

was limited, and the sample seems to reflect a homogenous

population of patients with eating disorders consulting in the

centers participating in the study. It is also hard to compare

our results with those of other studies because none of the

previous studies on eating disorders and alexithymia

accounted for comorbid addictive disorders. Future research

on this topic should better describe the clinical profile and

the comorbidity of the research samples.

Second, the study had a naturalistic design, with

therapeutic interventions freely chosen on the basis of usual

practices. Differences in treatments received by patients

may have influenced the evolution of alexithymia over time

(although we controlled treatments in all statistical analy-

ses). If this can be considered a limitation of the study, it

also underscores the fact that clinicians choose therapeutic

strategies on the basis of several factors, among which

alexithymia might be an important one that needs to be

better formalized. We project to further analyze the data of

our naturalistic study concerning the treatments given to

patients according to their alexithymia levels.

Third, the rate of follow-up was not high, with a certain

amount of refusals explained by the young age at inclusion

and the long duration of the study. However, we made the

choice of directly collecting the data and retaining only

patients with complete files because the aim of this research

was the study of psychopathological features associated

with eating disorders; it did not aim to conduct an

exhaustive epidemiological survey. The bias of losing

observations must not be overvalued because we did not

find any differences in clinical and demographic variables

between followed-up and lost patients.

Aside from these limitations, the results of our study

indicate that difficulty in identifying feelings can act as a

negative prognostic factor of the long-term outcome of

patients with eating disorders.

Professionals should carefully monitor emotional identi-

fication and expression in patients with eating disorder and

develop specific strategies to encourage emotion labeling and

sharing, such as group therapy that is focused on emotional

communication [40] or writing disclosure techniques [41],

which have already proven their efficacy in medical and

surgical patients. Controlled studies on the impact of these

techniques in eating disorders are of major interest.

Acknowledgments

This work was conducted within the clinical research

project called bDependence Network 1994–2000,Q which

has received the support of the Institut National de la Sante

et de la Recherche Medicale (Reseau Inserm No. 494013)

and of the Fondation de France. The promoter of the project

is the Institut Mutualiste Montsouris. We would like to

thank Olivier Luminet of the University of Louvain for his

helpful advice on this research.

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