Sébastien Weibel
email: weibelse@gmail.com

Anne Giersch

Stanislas Dehaene

Caroline Huron

Running head: UNCONSCIOUS TASK SET PRIMING

Keywords: Consciousness, Masking, Cognitive control, Task switching, Semantic judgment, Phonological judgment

Whether unconscious stimuli can modulate the preparation of a cognitive task is still controversial. Using a backward masking paradigm, we investigated whether the modulation could be observed even if the prime was made unconscious in 100% of the trials. In two behavioral experiments, subjects were instructed to initiate a phonological or semantic task on an upcoming word, following an explicit instruction and an unconscious prime. When the SOA between prime and instruction was sufficiently long (84 ms), primes congruent with the task set instruction led to speedier responses than incongruent primes. In the other condition (36 ms), no task set priming was observed. Repetition priming had the opposite tendency, suggesting the observed task set facilitation cannot be ascribed solely to perceptual repetition priming. Our results therefore confirm that unconscious information can modulate cognitive control for currently active task sets, providing sufficient time is available before the conscious decision.

Unconscious Task Set Priming With Phonological and Semantic Tasks

Introduction

It is widely acknowledged that information not consciously perceived can influence our perception and behavior, and subliminal priming has been demonstrated at visual, semantic and motor levels (for a recent review, see [START_REF] Dehaene | Experimental and theoretical approaches to conscious processing[END_REF]. Non-conscious information can modulate performance in many cognitive tasks, eg. object recognition [START_REF] Stoerig | Blindsight in man and monkey[END_REF], extraction of the meaning of words [START_REF] Gaillard | Nonconscious semantic processing of emotional words modulates conscious access[END_REF]Naccache & Dehaene, 2001;[START_REF] Van Den Bussche | Masked primes can be genuinely semantically processed: a picture prime study[END_REF], categorization (Van den [START_REF] Van Den Bussche | Masked priming effects in semantic categorization are independent of category size[END_REF], emotional processing [START_REF] Whalen | Masked Presentations of Emotional Facial Expressions Modulate Amygdala Activity without Explicit Knowledge[END_REF], action planning and execution [START_REF] Binsted | Visuomotor system uses target features unavailable to conscious awareness[END_REF]. Recently, it has been reported that monetary rewards affected subjects' motivation in a force task [START_REF] Pessiglione | How the Brain Translates Money into Force[END_REF], a finger-tapping task [START_REF] Bijleveld | Unconscious reward cues increase invested effort, but do not change speed-accuracy tradeoffs[END_REF], 2012) and a switch task [START_REF] Capa | Long-lasting effects of performance-contingent unconscious and conscious reward incentives during cued taskswitching[END_REF] even though participants were unaware of the reward.

However, the extent to which non-conscious stimuli influence high-order control functions remains controversial in cognitive psychology. Cognitive control processes have traditionally been considered to be based on voluntary control and to depend on conscious decision-making and awareness. As such, they have been contrasted with unconscious, automatic information activation. According to the global neuronal workspace framework [START_REF] Dehaene | Conscious, preconscious, and subliminal processing: a testable taxonomy[END_REF], top-down strategic processes can influence unconscious processing [START_REF] Merikle | Measuring the relative magnitude of unconscious influences[END_REF][START_REF] Naccache | Unconscious masked priming depends on temporal attention[END_REF][START_REF] Van Den Bussche | Conscious and unconscious proportion effects in masked priming[END_REF], but the possibility of an effect of non-conscious stimuli on cognitive control processes is not explicitly included in the model (Dehaene & Naccache, 2001). 1 Recent findings suggest subliminal stimuli can affect high-order cognitive processes such as inhibitory control or task-setting. In a go/no go task, for instance, non-conscious "stop" signals slow down motor responses. This inhibitory control, which occurs unbeknown to the subject, is associated with a frontal activity typically related to response inhibition in both electroencephalographic [START_REF] Hughes | Unconscious priming of a no-go response[END_REF][START_REF] Van Gaal | Frontal cortex mediates unconsciously triggered inhibitory control[END_REF] and functional magnetic resonance imaging (fMRI) [START_REF] Van Gaal | Unconscious activation of the prefrontal no-go network[END_REF] studies. These findings are evidence that inhibitory control can be triggered unconsciously.

Another set of studies has addressed the activation of task sets by masked stimuli that do not reach consciousness [START_REF] Lau | Unconscious activation of the cognitive control system in the human prefrontal cortex[END_REF][START_REF] Mattler | Priming of mental operations by masked stimuli[END_REF][START_REF] Reuss | Unconscious activation of task sets[END_REF]. The concept of task set assumes we can adopt a particular configuration of our cognitive system to perform a given task [START_REF] Rogers | Costs of a predictable switch between simple cognitive tasks[END_REF]. Mattler et al. instructed participants to indicate either a sound's timbre (piano versus marimba) or its pitch (high versus low). A shape (task cue) indicated which task they were to perform, with diamonds denoting the pitch task and squares the timbre task. Before the fully visible task cue was presented, there was a prime in the shape of either a diamond or square. The task cue acted as a metacontrast for the prime so that the latter was either visible or invisible depending on the delay between prime and task cue onsets. The results signaled a congruency effect for subliminal primes: participants responded faster when the shapes of the prime and cue were the same than when they were different. These findings suggested that subliminal primes triggered the establishment of a cognitive task set which shortened the preparation time for the task when the visible cue occurred.

Using a similar paradigm in a fMRI study, [START_REF] Lau | Unconscious activation of the cognitive control system in the human prefrontal cortex[END_REF] reported that, relative to congruent trials, in incongruent trials increased activity was observed in brain regions associated with the task cued by the subliminal prime, whereas reduced activity was reported in brain regions associated with the task cued by the visible instruction. These results demonstrated that the task-related network can be modulated by subliminal information.

There are methodological concerns surrounding these important studies which could undermine their conclusions. Firstly, in all of them the absence of awareness of masked primes cannot be taken for granted in all subjects. In the Mattler study (2003), for example, participants' performance when they had to identify the shapes of the subliminal primes was at best only marginally different from chance, around 55 % (d'=0.28), and in the worst case, close to 60 % (i.e. above-chance). In the Lau and Passingham study (2007), discriminability in the low-visibility condition was lower (d'=0.05), but the difference in relation to the conscious condition was quite small (d'=0.26), with the latter result equivalent to the Mattler unconscious condition. It is possible these results are due to the choice of masking by metacontrast. Secondly, the presence of conscious primes in the same block as masked primes might have had an effect on the visibility of the prime, by creating an expectation of a stimulus, which has been shown to speed information processing [START_REF] Vangkilde | Great expectations: Temporal expectation modulates perceptual processing speed[END_REF], and might have facilitated the priming effect [START_REF] Naccache | Unconscious masked priming depends on temporal attention[END_REF]. All in all, it still seems necessary to check whether task-set priming effects can be replicated in strict conditions where subjects are unaware of the primes. To that end, we used a backward masking paradigm in which a single letter displayed only briefly was followed 24 ms later by a mask consisting of letters surrounding the previous letter location [START_REF] Del Cul | Brain dynamics underlying the nonlinear threshold for access to consciousness[END_REF][START_REF] Del Cul | Preserved subliminal processing and impaired conscious access in schizophrenia[END_REF]. It has previously been shown that both objective measures (proportion of primes correctly identified) and the subjective visibility of the primes reported by participants indicated they were not aware of the presentation of the primes under these experimental conditions. Furthermore, unlike all previous studies, which combined trials with conscious primes with trials with masked primes, we did not include any trials with conscious primes.

Another methodological issue concerns the double dissociation between priming effect and visibility reported in the study by [START_REF] Lau | Unconscious activation of the cognitive control system in the human prefrontal cortex[END_REF]: the priming effect was maximal when the prime visibility was minimal but disappeared when participants consciously perceived the prime (see also [START_REF] Schmidt | Criteria for unconscious cognition: three types of dissociation[END_REF]. The absence of an effect of a visible prime on the task set selection could be due to an excessively too short delay between the prime and the task cue in the conscious condition (16 ms) compared with the subliminal condition (83 ms). It has already been shown that the efficiency of visible primes generally increases as a function of the prime-target delay [START_REF] Kouider | Levels of processing during non-conscious perception: a critical review of visual masking[END_REF]. Two factors were therefore confounded: the visibility of the prime and the delay between prime and instruction. The confound is due to the use of metacontrast masking characterized by a U-shaped visibility curve as the delay between prime and instruction increases. In the present study we examine the impact of this delay manipulation while keeping constant the SOA between prime and mask.

The purpose of the present study was to provide more evidence proving that a subliminal prime could initiate a task set. We designed a different masking procedure to overcome the methodological issues raised by the study by [START_REF] Lau | Unconscious activation of the cognitive control system in the human prefrontal cortex[END_REF]. First of all, we were intent on making sure the participants really were unaware of the prime. Even if it is difficult to demonstrate statistically that visibility is exactly zero, it is known that metacontrast masking rarely produces complete masking at any level of SOA [START_REF] Francis | Cortical dynamics of lateral inhibition: Metacontrast masking[END_REF].

Here, we used backward masking which allows no visibility of the prime, as previously shown by objective performance and subjective ratings [START_REF] Del Cul | Brain dynamics underlying the nonlinear threshold for access to consciousness[END_REF][START_REF] Bach | The Freiburg Visual Acuity Test-Variability unchanged by post-hoc reanalysis[END_REF]. Secondly, primes were subliminal in all trials to avoid any clue that might have modified the participants' attentional focus. Thirdly, to study the impact of the delay between the subliminal prime and the mask, we compared two intervals between prime and instruction in two different experiments. Fourthly, to gain a better understanding of the links between the repetition priming and task set priming, we used both tasks in the same experimental procedure. Each participant performed a task set priming task, followed by a repetition priming task (motor response priming).

Experiment 1

2.1 Methods 2.1.1 Participants. Twenty students from the University of Strasbourg (16 women, 4 men), aged between 19 and 44 (M=24.7, SD=5), took part in the experiment. Their level of education ranged from 12 to 16 years (M=14.8, SD=1.47). All of them had normal or corrected-to-normal visual acuity, verified with the help of the Freiburg Visual Acuity Test [START_REF] Bach | The Freiburg Visual Acuity Test-Variability unchanged by post-hoc reanalysis[END_REF]. Each participant took part in one 90 minute session. Their informed written consent was obtained prior to the study, in accordance with the recommendations set out in the Helsinki Declaration.

Equipment.

Participants were seated 60 cm from the stimulus presentation screen in a dimly lit room for the duration of the experimental session. The behavioral tasks were presented using E-prime version 1.1 (Psychology Software Tools, Inc. Sharpsburg, PA). The screen refresh rate was set at 85 Hz (screen refresh every 11.8 ms). Responses were collected with a serial response box (Psychology Software Tools, Inc. Sharpsburg, PA).

Stimuli.

In each trial, the prime was displayed first, and then followed by a mask, the instruction letter, and the target word. The prime was a letter (A or S, Arial font, size:

1.25°) presented in one of four positions (2.3° above or below and 2.3° to the right or left of the central fixation cross).

The mask consisted of 4 letters surrounding the previous position of the prime (two E above and below, and two M on left and right). The mask served to render the prime invisible.

The instruction consisted of a letter (A or S), the same size as the prime, but in bold font and colored blue. It was presented in the square delimited by the mask furthest from the fixation cross. NB: As a result, the instruction letter never appeared in exactly the same location as the prime (Figure 1).

2.1.4

Procedure. The participants completed a three-phase procedure. They started with a task set priming task, which involved applying one of two strategies according to an instruction letter that could change with every trial. Secondly, they performed a repetition priming task, where they had to identify the letter representing the instruction, which in this task is the target letter. Thirdly, they were told about the presence of a congruent or incongruent invisible prime before performing a prime identification task.

Task set priming. An instruction letter presented on the screen indicated the type of judgment the subjects would have to emit on the following word. In the case of an "S" instruction (like "Syllabic") the participants had to make a phonological judgment: they had to decide whether or not the word was bisyllabic. In the case of an "A" instruction (like "Animate"), they had to make a semantic judgment: they had to decide whether the word corresponded to something living or non-living. The volunteers were told they had to prepare the task set quickly once they had seen the instruction. They were not told about the presence of an invisible prime.

Figure 1 illustrates the experimental stimuli and procedure. The fixation cross was displayed throughout the session. In each trial, the prime was presented for 12 ms. After a 24 ms delay during which the screen remained blank, the mask and instruction letter appeared together for 152 ms, in the same quarter of the screen as the preceding prime. Thus, the stimulus onset asynchrony (SOA) between prime and mask, as well as between prime and instruction, was 36 ms. After a delay of 156 ms during which the screen remained blank (344 ms after the onset of the prime) the target word was presented at the center of the screen for 750 ms. The next trial began 3 seconds after the word offset.

The volunteers responded "yes" by pressing a key with their left index finger and "no" by pressing a key with their right index finger. They were instructed to respond as accurately and as quickly as possible, within 3 seconds of the word onset.

Each participant completed four blocks of 96 trials each (384 trials). In half the trials, the task was phonological, and in the other half, semantic. Within each of these conditions, the prime was congruent with the instruction in half the trials, and incongruent in the other half. In our pilot studies, switching instructions between consecutive trials seemed to be an important factor to control and was taken into account in the study design. Trials with and without task switching between consecutive trials (switch vs. no-switch) were displayed in equal proportions. The fillers, i.e. the first trial in a sequence of two trials, were not included in the statistical analyses. The participants were not told about the fillers, and the temporal course of fillers and target trials was exactly the same. Conditions of task, congruency, task switching, and the position of the stimuli (a total of 32 possible combinations) were represented in the same proportions. The same word was never repeated between two consecutive trials.

Twenty-four different French nouns were used. Their occurrence ranged from 0.68 to 56 per million (Lexique Database: [START_REF] New | Une base de données lexicales du français contemporain sur internet: LEXIQUE TM //A lexical database for contemporary french: LEXIQUE TM[END_REF]. The words were either bi-or trisyllabic, and denoting something either animate or inanimate, in equal proportions. Words that were ambiguous in terms of syllables or meaning were excluded. We chose a small number of words to accelerate decision-making and reduce the variability of RTs.

To avoid unwanted response strategies, the responses were systematically incongruent in both tasks: all trisyllabic words corresponded to something animate, and bisyllabic words to something inanimate. Unbeknown to the participants, the selection of words with intrinsic response incongruence ("yes" for one task and "no" for the other) avoids automatic responses based solely on their memory of stimulus-response pairs. We checked to make sure subjects did not consciously employ a strategy of using the same judgment in all cases and reversing the responses according to the instructions given. A questionnaire filled in at the end of the procedure was designed specifically to hunt out such a strategy. The questions were open to begin with, and then steadily more targeted to focus on the strategies used during the procedure. Three subjects had noticed that bisyllabic words were inanimate and trisyllabic words animate, leading them to use the afore-mentioned strategy. These three participants were excluded from the analyses.

The experiment was divided into four 12-minute runs, between which the participants were allowed a 5-minute rest to ensure their sustained attention. A training session beforehand, consisting of 100 trials, allowed us to check whether or not the instructions were properly understood. The instructions were presented on the computer screen and oral explanations were also given.

Repetition priming. The stimuli (prime, mask and instruction, and word) and their temporal course were identical to the task set priming procedure, including the use of filler trials. Only the task was different. Participants were asked to focus their attention on the instruction letter, which is the target in this phase, and to press, as quickly as possible, the right button in the case of an "A" letter and the left button in the case of an "S" (counterbalanced between subjects). Participants completed one block of 128 trials.

Measurement of prime awareness. Following the two priming tasks, participants were asked whether they had noticed anything other than the E and the M in the mask. They were then told that a prime had been presented and were asked once again whether they had noticed anything before the mask and the instruction letter. Afterwards, they were shown the exact stimulus sequence in slow motion and were asked if they recognized having seen any of the primes during the priming task. Finally, the prime identification task was conducted, consisting of one block of 128 trials. The stimuli and time course were the same as in the two previous tasks. Participants were asked to identify the prime, and to respond by pressing one of the two buttons (counterbalanced between subjects), even if they were unable to see it (forced choice). Lastly, they were asked whether or not they felt they were able to see the masked primes during the prime identification task.

< Insert Figure 1 about here > 2.1.5 Statistical analyses. We conducted ANOVAs with repeated measures, prime congruency (prime and target identical vs. different), task (semantic vs. syllabic), and switch (task instruction identical to the previous trial vs. different) as within-group factors for the task set priming session, and prime congruency and switch as within-group factors for the repetition priming session. Analyses were performed on median correct response times. For the prime detection task, accuracy was analyzed with respect to prime recognition and compared to the chance level of 50 % using a t-test. The signal detection theory was used to calculate the d' index (index of signal detectability).

Results

Three subjects pointed out that bisyllabic words were inanimate or vice versa, which had lead them to respond always with a semantic task set and to reverse their response in the case of an "S" instruction. These three participants were excluded from the analyses. NB: For the sake of clarity, the results are presented in reverse order to the order in which the three tasks were administered.

Awareness of primes.

None of the 17 participants reported having seen the prime letter before the target during the priming experiments. When informed about the presence of the primes, 3 subjects thought they might have seen a slight blink, but were unsure, and no one thought it was a letter. No participants reported having seen the prime in the prime identification task. They spontaneously reported that they were not confident about their estimations and were only guessing. Their performance in the forced-choice task ranged from 41% to 59% and averaged 49.7%, which was not significantly different from chance, t(16)=-0.24, p=0.81. The mean value of d' was -0.013, which was not significantly different from 0 (t(16)=-0.24, p=0.82). Mean accuracy for the 3 subjects who detected a flash was 50% and d' was 0.01.

Repetition priming.

Figure 2 shows the median RTs averaged across subjects for each repetition priming condition. We found a significant main effect of prime congruency on reaction times (F(1,16)=64.5, p<0.001,  2 =0.80). Subjects were quicker to identify the target letter (instruction) when the prime letter was the same as the instruction (442.2 ms versus 482.8 ms when it was different). The mean size of the effect was 40.6 ms (8.7% of the median RT). All subjects displayed a repetition priming effect. A significant interaction between congruency and switch (F(1,16)=4.7, p<0.05,  2 =0.23) indicated that the priming effect was greater when the instruction letters were the same between two subsequent trials (53.8 ms), than when there was a switch (31.2 ms).

Task set priming. Figure 2 displays the median RTs for correct responses

averaged across subjects for each condition of congruency, task and switch. A significant main effect of task (F(1,16)=22.1, p<0.001,  2 =0.58) was observed, with faster responses when participants performed a semantic judgment. A significant effect of the switch (F(1,16)=13.1, p<0.005,  2 =0.45) was also observed, with faster responses when two consecutive trials involved the same task. There was neither a main effect of prime congruency (F(1,16)=0.27, p=0.6,  2 =0.017), nor a significant interaction with any other factor.

A similar analysis of response accuracy resulted in a significant main effect of switch (F(1,16)=13.6, p<0.005). The overall accuracy level was high (96.5%), indicating that participants performed the task properly.

< Insert Figure 2 about here >

Experiment 2

We found no task set priming in Experiment 1, when the stimulus onset asynchrony between the prime and the instruction was set at 36 ms. In the experiment conducted by [START_REF] Lau | Unconscious activation of the cognitive control system in the human prefrontal cortex[END_REF], the task set priming was found when the SOA was 84 ms, but not when it was 36 ms, even though the prime was then conscious. These results suggest the interval between the prime and the instruction may be decisive for task set priming. In a second experiment, we used an interval similar to Lau and Passingham in their experiment (2007) to ascertain whether task set priming occurs in these conditions.

Methods

Participants.

Twenty students from the University of Strasbourg (10 women, 10 men), aged from 23 to 30 (M=23.6, SD=2.0), took part in the experiment. Their level of education ranged from 12 to 16 years (M=14.8, SD=1.47). As in the previous experiment, all of the participants had normal or corrected-to-normal vision. They all took part in one 90minute session. Their informed written consent was obtained before the study in accordance with the recommendations set out in the Helsinki Declaration.

Experimental procedure.

The only difference in relation to Experiment 1 was the time course of the mask and instruction letter as shown in Figure 1. To increase the SOA between prime and instruction, without modifying the masking procedure, the instruction appeared not in the same time as the mask, but 48 ms after the mask onset. The instruction duration (154 ms) was the same as in Experiment 1. The mask was displayed until the instruction offset. Thus, the SOA between prime and mask remained at 36 ms, guaranteeing an excellent invisibility of the primes, but this time the SOA between prime and instruction letter was increased to 84 ms. We also ensured there was a constant interval between the onsets of the prime and the target word by slightly shortening the blank separating the instruction and the word (Figure 1). As a result, the word appeared 344 ms after the prime, like in Experiment 1.

Results

Awareness of primes.

Four subjects pointed out that bisyllabic words were inanimate or vice versa, leading them to respond always with a semantic task set and to reverse their response in the case of an "S" instruction. One participant's results were incoherent, with abnormally long and variable RTs (median RT=1265 ms, SD=690; compared to 727 ms and mean SD of 283 for the other 15 subjects). These five participants were excluded from the analyses.

As in Experiment 1, none of the 15 participants reported having seen the prime letter before the target during the priming experiments. When told about the presence of the primes, 2 subjects said they thought they might have seen a slight blink but were unsure, and none thought it was a letter. Performance in the forced-choice task ranged from 42% to 60% and averaged 49.8%, which was not significantly different from chance, t(14)=-0.15, p=0.88. The mean value of d' was 0. 075 (not significantly different from 0: t(14)=0.784, p=0.45). Mean accuracy for the 2 subjects who detected a flash was 49% and mean d' was -0.051.

Repetition priming.

The results of the repetition priming are presented in Figure 4. The ANOVA revealed a significant main effect of prime congruency (F(1,14)=6.9, p<0.05,  2 =0.33) and switch (F(1,14)=8.2, p<0.05,  2 =0.37). No significant interaction was observed between prime congruency and switch (F(1,14)=0.028, p=0.87,  2 =0.002).

Participants responded more quickly if the prime was the same as the instruction (437.6 ms) than if it was different (459.3). The mean extent of the priming effect was 21.7 ms.

Participants also responded faster if the instruction was the same rather than different for two consecutive trials (437.9 ms versus 456.2 ms).

Task set priming.

Figure 3 shows the median RTs for correct responses averaged across subjects for each condition in the task set priming procedure. A significant main effect of task (F(1,14)=14.9, p<0.005,  2 =0.52) was observed, with faster responses when participants made a semantic judgment. A significant effect of the switch (F(1,14)=15.2, p<0.005,  2 =0.52) was also observed, with faster responses when the same instruction was used for two consecutive trials. Crucially, we now observed a significant main effect of prime congruency (F(1,14)=9.44, p<0.01,  2 =0.40), evidence that a subliminal prime identical to the conscious instruction produced a faster response than when the prime differed from the instruction. The only significant interaction was a switch-by-task interaction (F(1,14)=8.76, p<0.05,  2 =0.38): in the phonological task, the switch-related slowdown was more pronounced.

Accuracy was high (98.2 %), and no significant main effects or interaction were observed for response accuracy.

 < Insert Figure 3 about here >

To check whether behavioral priming effects were greater in the case of participants who performed better in the prime visibility task, correlations were calculated between task set priming and prime identification performance. The correlations were based on the use of raw scores in the forced-choice task used to explore the awareness of primes on the one hand, and on the extent of task set priming, i.e. the difference between RTs for congruent and incongruent trials, in the different conditions of switch and task on the other hand. There were no significant correlations either for the whole group (r=-0.27, p=0.33), or when the analyses included only the 8 participants whose prime identification performance was greater than 0.5 were included in the analyses (r=0.19, p=0.66).

We conducted another correlation analysis, to check whether the repetition priming was correlated to the task set priming. The extent of task set priming was not correlated with that of repetition priming, i.e. the difference between RTs for congruent and incongruent trials during the repetition priming task (r=0.145, p=0.61).

Comparison with Experiment 1.

In order to compare the two experiments, we conducted ANOVAs with repeated measures with the same within-group factors, and with the experiment as a between-group factor. There was no significant difference between the mean accuracy or d' value in prime identification (F(1.30)=0.008, p=0.93,  2 =0). However, in the repetition priming task, we observed a significant interaction between congruency and experiment (F(1,30)=8.87, p<0.005,  2 =0.23). The repetition priming effect we observed was greater when the SOA prime-instruction was 36 ms rather than 84 ms. The same analysis conducted on the task set priming task revealed a triple interaction between the experiment, congruency and switch factors (F(1,30)=4.28, p<0.05,  2 =0.12). There was no other significant interaction involving the experiment factor. The significant interaction between the experiment, congruency and switch factors was decomposed by means of sub-analyses. The effect of congruency was significantly different across experiments in the no-switch condition, as shown by an ANOVA (F(1,30)=5.71, p<0.05,  2 =0.16). The task set priming effect was greater in the no-switch condition for Experiment 2 (24 ms) than for Experiment 1 (-8 ms), i.e. when the SOA between prime and instruction was 84 ms rather than 36 ms.

Discussion

The primary aim of the present study was to investigate whether control processes, and in particular those involved in establishing task sets, are modulated by unconscious stimuli.

We explored this question in experimental conditions where a strong masking procedure ensured the prime was not consciously perceived, and where there were no trials involving conscious primes that attracted attention to their existence. The second aim was to clarify whether the task-set priming effect requires some delay before it can occur. The results showed that in strict experimental conditions (i) participants were faster to make a semantic/syllabic judgment when the instruction was preceded by a congruent prime in Experiment 2 but not in Experiment 1, and (ii) the prime congruency accelerated identification of the target letter in both experiments, but this repetition priming effect was greater in Experiment 1 than Experiment 2.

Both subjective and objective measures showed that priming effects occurred in conditions where participants were not aware that a prime had been presented before the explicit cue telling them which task to perform. No participant reported having identified or even perceived a prime, and the d' resulting from the prime identification task were at chance level. Moreover, there was no correlation between the priming effect and the prime awareness. Taken together, these results suggest the task set priming effect observed in Experiment 2 does not arise from a minimal level of conscious prime perception by some of the participants in the study. However, it could be due to a repetition priming effect, which would mean that participants identify the instruction that has already been subliminally presented more quickly and thus they respond faster to the task instruction. If this were the case, faster reaction times would reflect perceptual priming rather than task set priming.

Before discussing this alternative, we shall first turn our attention to the differences in repetition priming effects observed between Experiment 1 and 2 and compare them to results previously reported in the literature.

Consistent with previous studies [START_REF] Arguin | Priming and Response Selection Processes in Letter Classification and Identification Tasks[END_REF][START_REF] Neumann | Motor responses to nonreportable, masked stimuli: Where is the limit of direct parameter specification[END_REF], the subliminal primes were found to affect identification of the conscious instruction letter in the repetition priming procedure in both experiments, with shorter RTs when the prime was congruent, as opposed to incongruent, with the instruction. However, the extent of this repetition priming effect was greater in Experiment 1 than Experiment 2, when the lag between the prime and target was shorter (36 ms and 84 ms in Experiment 1 and 2, respectively). The suggestion is that the repetition priming effect dwindles over time. In a word naming task, [START_REF] Ferrand | The masked repetition priming effect dissipates when increasing the interstimulus interval: Evidence from word naming[END_REF] also showed that the masked repetition priming effect dissipated when the prime-target interval increased. However, this decrease occurred with prime-target intervals of 500 ms and 1000 ms whereas the priming effect remained strong with 50 ms and 150 ms intervals. With shorter intervals, in the range of those used in the present study, [START_REF] Vorberg | Different time courses for visual perception and action priming[END_REF] reported a heightening of the priming effect when the prime-target interval increased. It is to be noted, however, that they used a metacontrast masking in which the mask is also the target. The increase in the prime-target interval was therefore associated with an increase in the delay between the prime and the mask, which could have induced deeper processing of the prime. Indeed, according to Vorberg et al., even if the participants were never able to accurately discriminate the shape from the prime, as the prime-mask interval increased they became better at detecting the presence of a prime. Other studies that used prime-target intervals as short as those used in the present experiments reported results consistent with our findings. For example, Lignau and [START_REF] Lingnau | The time course of response inhibition in masked priming[END_REF], used a masked priming task in which the interval between the prime and the mask remained constant (70 ms) and showed that the priming effect decreased when the prime-target SOA increased from 70 ms to 154 ms. [START_REF] Schlaghecken | The influence of subliminally presented primes on response preparation[END_REF] observed the same tendency with pattern masking, when the prime and the target were peripheral. The decrease in the repetition priming effect between our two experiments could also be related to differences in the mask duration, which was longer in Experiment 2 (200 ms) than in Experiment 1 (152 ms) [START_REF] Hashimoto | Backward-masking: the effect of the duration of the second stimulus on recognition of the first stimulus[END_REF]. Whatever the explanation for the changes in the repetition priming effect depending on the prime-target SOA, an important finding from our study was the dissociation between the repetition priming effect and the task set priming effect. When the prime-instruction interval increased from 36 ms to 84 ms, the repetition priming effect decreased whereas the task set priming effect increased (Figure 4). This dissociation argues against the hypothesis that these two priming effects are subtended by the same mechanism. It is further supported by the lack of correlation between the repetition priming effect and the task set priming effect, the suggestion being that faster identification of the letter induced by the prime was not a determining factor in the task set priming. To sum up, our findings from Experiments 1 and 2 are evidence that task set activation can be affected by a prime that participants do not consciously perceive, independently of a possible perceptual repetition priming effect. This conclusion is consistent with a study conducted by [START_REF] Reuss | Unconscious activation of task sets[END_REF] which showed that a task set can be activated by subliminal information even when perceptual priming of the task cue cannot take place. In their study, Reuss et al. dropped the prime and manipulated the visibility of the task cue so that participants consciously perceived it in some trials but not in others. Stimuli were one-digit numbers. Task cues (w or b) indicated to participants that they were to perform either a parity task (odd or even) or a magnitude task (less or greater than 5).

When cues were rendered invisible, subjects were instructed to choose freely which task they carried out. The results showed that even when they were not aware of the task cue, participants chose the cued task more often than the non-cued task and performed it more quickly.

< Insert Figure 4 about here >

The task set priming effect occurred in Experiment 2 but not in Experiment 1.

Participants were equally unaware of the presence of a prime in both experiments, which means the explanation cannot reside in differences in conscious perception of the prime from one experiment to the next. In contrast, the time between the onset of the prime and the onset of the instruction increased from 36 ms to 84 ms between Experiments 1 and 2. [START_REF] Mattler | Priming of mental operations by masked stimuli[END_REF] showed that priming of cognitive control operations did not occur when the prime-cue SOA was set at 34 ms or 51 ms but increased linearly when the SOA increased from 68 ms to 119 ms. Taken together, these findings suggest the time interval between the prime and the task cue has to be long enough to allow a task set priming. This interpretation is also consistent with the results obtained by Lau and Passingham, which showed no task set priming effect when the prime was consciously perceived but the delay between the prime and the instruction was as short as 16 ms.

We have to consider a limitation to this interpretation which stems from differences across our two experiments in the interval between the consciously perceived instruction letter and the word to be processed. When the SOA between prime and instruction letter increased from 36 ms to 84 ms, the interval between the instruction letter and the word changed in the opposite direction (308 ms in Experiment 1 versus 260 ms in Experiment 2). Therefore, the possibility that the task priming effect also occurred in Experiment 1 but then dissipated because the word occurred later relative to when the task instruction was presented cannot be ruled out. In [START_REF] Lau | Unconscious activation of the cognitive control system in the human prefrontal cortex[END_REF], however, the interval between the instruction and the word remained constant even when the interval between the prime and the instruction changed. Thus, this second explanation cannot account for the absence of a task priming effect in their short prime-instruction condition. Consequently, it seems unlikely that a decrease in the instruction-word interval is the sole explanation for all of our results. Our own data also suggest otherwise. When the interval between the instruction letter and the word is shorter, there is less time to initiate the task set. This could explain the greater influence of the unconscious prime in Experiment 2. However, if the instruction-word delay is crucial, the task set priming effect should be greater when the task set is the most difficult to initiate after the instruction. In fact, difficulty initiating the task set should reduce the instruction-word delay still further. In our case, this situation is typical of the phonological task: to count the number of syllables when reading is not a natural task. The task set should be even more difficult in the switch condition, i.e. when the phonological task has to be initiated after a semantic task trial, but, in reality, the results are not consistent with this prediction. On the contrary, they show that the task set priming tends to be greater when the instruction-induced task setting is the easiest, i.e. in the repeated (no switch) trials and in the semantic task. This suggests the instruction-word delay is not the key factor.

The task set priming effect we observed was small (15 ms) and, in particular, smaller than the effect reported by [START_REF] Lau | Unconscious activation of the cognitive control system in the human prefrontal cortex[END_REF], which they estimated at 100 ms. Even if the priming effect reported by [START_REF] Mattler | Priming of mental operations by masked stimuli[END_REF], 45 ms, is more similar to our own, the difference in relation to the effect found by Lau and Passingham is important. The first explanation for it could be that in Lau and Passingham study participants were not completely unaware of the presentation of the prime since their d' measurements were higher than ours.

The attention the participants gave the primes could be another critical factor. In our paradigm, they were not told about the presence of primes until the end of the task set priming experiment and, in contrast to the study conducted by Lau and Passingham, there were no trials with conscious primes. It has previously been shown that attention can influence the effect of unconscious primes [START_REF] Greenwald | Long-term semantic memory versus contextual memory in unconscious number processing[END_REF][START_REF] Naccache | Unconscious masked priming depends on temporal attention[END_REF]. For instance, [START_REF] Naccache | Unconscious masked priming depends on temporal attention[END_REF] showed that unattended primes might fail to elicit priming effects. Therefore, it could be argued that the presence of trials in which primes were perceived consciously, as in the study by Lau and Passingham, caused participants to attend to the primes, and thus yielded larger priming effects than in the present study.

Importantly, in our experiments, participants were trained and prepared to perform frequent task switching. The prime may thus have prompted them to switch between two task sets which were simultaneously active. In other words, the prime may not have triggered the whole preparation of a task set, but merely the choice whether or not to initiate a task switch [START_REF] Meiran | Modeling cognitive control in task-switching[END_REF]. Cognitive control has been modeled as consisting of several nested components, with the frontal cortex organized functionally as a cascade of control processes [START_REF] Koechlin | The Architecture of Cognitive Control in the Human Prefrontal Cortex[END_REF]. The higher level is related to episodic control, which enables the subject to establish new task sets, according to previous events or ongoing internal goals. Because both task sets were already consciously prepared, and indeed perhaps routinized, this level would not be the one activated subliminally in our paradigm. Instead, our paradigm would implement a contextual control stage [START_REF] Koechlin | The Architecture of Cognitive Control in the Human Prefrontal Cortex[END_REF]. The unconscious task set priming we observed might be related to the selection of one of the two task sets already activated. This suggests the need to split cognitive control into distinct components differentially related to conscious processing. While task set and goal changing decisions may be partially initiated subliminally, the present results leave open the question of whether other cognitive control operations are influenced by unconscious stimuli.

Conclusion

Our results confirmed that unconscious task set priming is possible for high level cognitive processes. By controlling unconscious conditions better, with a shorter prime, we were able to reproduce and extend the results obtained by [START_REF] Lau | Unconscious activation of the cognitive control system in the human prefrontal cortex[END_REF]. The extent of our task set priming was smaller but significant. In future, it would be useful to combine our paradigm with high temporal resolution cerebral recordings in order to identify the precise neural mechanisms of task set priming. Even if our findings confirm the existence of an unconscious modulation of cognitive control, it remains to be seen whether all prefrontal cognitive control systems can be activated unconsciously, without concomitant conscious control.

Footnotes

1 Although the formulation in Dehaene & Naccache (2001) is ambiguous, the workspace model does not preclude rule out the possibility that automatic bottom-up effects of an unconscious stimulus T1 may bias the choice of a cognitive strategy applied, in turn, to a second target T2. What is ruled out is that an unconscious stimulus T1 changes the strategy applied to itself, as this would imply a closed bottom-up and top-down loop, which, in the global neuronal workspace model, is deemed to imply reverberating ignition and therefore conscious perception. Thus, the present data do not strictly imply rejection of the global workspace model. experiments differed in the interval between the prime and the instruction letter. The participants performed three tasks in succession (in three blocks). First, they performed the "task set priming" task, in which they had to make either a phonological judgment (bisyllabic or not) or semantic judgment (animate/inanimate) on the following word, according to the instruction letter. Then, they performed the "repetition priming" task, in which they had to identify the instruction letter, and, then, finally, they attempted to identify the prime letter, so that we could check whether or not they were aware of the prime ("prime detection"). A main effect of task type (p<0.001) and a main effect of the switch (p<0.005) were observed.

However, there was no significant difference in response times between when the subliminal prime was congruent and when it was incongruent with the explicit instruction. Same format as Figure 2.

A main effect of task set type (p<0.005) and a main effect of the switch (p<0.005) were observed, as in Experiment 1. Moreover, there was a significant reduction in response times when the subliminal prime was congruent with the explicit instruction, relative to the incongruent condition (p<0.01). (SOA=36ms) and 2 (SOA=84ms). As the prime-instruction stimulus onset asynchrony (SOA) lengthens from 36 ms to 84 ms, the repetition priming effect decreases, whereas the task set priming increases.

Figure 1 .

 1 Figure 1. Stimuli and procedure of Experiments 1 and 2. The stimuli consisted of a prime

Figure 2 .

 2 Figure 2. Median response time for correct responses (in ms) averaged across subjects in the

Figure 3 .

 3 Figure 3. Median response time for correct responses (in ms) averaged across subjects in the

Figure 4 .

 4 Figure 4. Comparison of the priming effect (incongruent RTcongruent RT) for the


Camille Maumet

Pierre Maurel

Jean-Christophe Ferré

Béatrice Carsin

Christian Barillot

Patient-specific detection of perfusion abnormalities combining within-subject and between-subject variances in Arterial Spin Labeling

Keywords: Arterial Spin Labeling, Hypo-perfusion, Hyper-perfusion, General Linear Model, Within-subject variance, Heteroscedasticity

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Arterial Spin Labeling (ASL), a Magnetic Resonance Imaging (MRI) technique introduced in the early 1990's, permits the measurement of the level of perfusion through a quantitative index: the cerebral blood flow (CBF). Contrary to standard perfusion imaging, including Positron Emission Tomography (PET) and Single Photon Emission Computed Tomography (SPECT) in nuclear medicine, or Dynamic Susceptibility weighted Contrast (DSC) in MRI, ASL is completely non-invasive and does not require the injection of an exogenous contrast agent. In the context of this paper, we aim at outlining areas of abnormal perfusion in the perfusion map of a subject of interest by comparison to a group of healthy controls.

The most widespread approach to compare voxel-wise maps in neuroimaging is the massively univariate General Linear Model (GLM). To detect differential patterns between two groups with repeated measurements a sub-type of GLM is employed: a mixed-effect hierarchical twosample t-test with two levels: subject and group. In this context, two variance components are of interest: the withinsubject variance (or the measurement error, estimated from the repeated ASL acquisitions of a single subject) and the between-subject variance.

In the functional MRI community, where similar statistical models are applied, two approaches are currently in use to solve hierarchical GLMs. On the one hand, the homoscedastic approach, also termed "summary statistics" [START_REF] Penny | Random-Effects Analysis[END_REF], or referred as "ordinary least square estimation" [START_REF] Mumford | Estimation efficiency and statistical power in arterial spin labeling fMRI[END_REF][START_REF] Mumford | Simple group fMRI modeling and inference[END_REF] or "conventional group analysis" [START_REF] Chen | FMRI group analysis combining effect estimates and their variances[END_REF], assumes homogeneous within-subject variances or negligible within-subject variances by comparison to between-subject variance. On the other hand, the heteroscedastic approach, also referred as "full mixedeffect" [START_REF] Friston | Mixed-effects and fMRI studies[END_REF][START_REF] Thirion | Analysis of a large fMRI cohort: Statistical and methodological issues for group analyses[END_REF][START_REF] Poldrack | Handbook of functional MRI data analysis[END_REF], "mixed-effect model" [START_REF] Chen | FMRI group analysis combining effect estimates and their variances[END_REF] or "weighted least square estimation" [START_REF] Mumford | Estimation efficiency and statistical power in arterial spin labeling fMRI[END_REF][START_REF] Mumford | Simple group fMRI modeling and inference[END_REF]) models heterogeneous within-subject variances. There is indeed a large panel of homonyms to refer to these two approaches, in current practice they are also sometimes referred as "mixed-effect" and "random-effect" approaches [START_REF] Lindquist | Estimating and testing variance components in a multi-level GLM[END_REF]. However, in a statistical sense both the homoscedastic and the heteroscedastic approaches are mixed-effect models. That is why, in the context of this paper, we chose to term the two approaches "homoscedastic model" and "heteroscedastic model" to identify them according to their main difference: the homoscedasticity (constant within-subject variance across subjects) assumption.

The homoscedastic model is theoretically less efficient in the presence of heterogeneous within-subject variances. However, in practice, the true variance components are unknown and the superiority of the heteroscedastic model is therefore questioned. Whether modeling heterogeneous variances should be preferred over the homoscedastic model is still under discussion in the fMRI community. Several authors outlined the benefits of heteroscedastic models including [START_REF] Worsley | A general statistical analysis for fMRI data[END_REF][START_REF] Beckmann | General multilevel linear modeling for group analysis in FMRI[END_REF][START_REF] Woolrich | Multilevel linear modelling for FMRI group analysis using Bayesian inference[END_REF][START_REF] Mumford | Estimation efficiency and statistical power in arterial spin labeling fMRI[END_REF][START_REF] Mériaux | Combined permutation test and mixed-effect model for group average analysis in fMRI[END_REF][START_REF] Thirion | Analysis of a large fMRI cohort: Statistical and methodological issues for group analyses[END_REF][START_REF] Chen | FMRI group analysis combining effect estimates and their variances[END_REF]. However, [START_REF] Mumford | Simple group fMRI modeling and inference[END_REF] showed that the homoscedastic approach is still valid with near optimal sensitivity in the context of one-sample t-tests. Nonetheless, the same authors acknowledged that, in two-sample t-tests, appropriate modeling of heterogeneous within-subject variances might be crucial [START_REF] Mumford | Simple group fMRI modeling and inference[END_REF][START_REF] Poldrack | Handbook of functional MRI data analysis[END_REF]. Recently the conclusions regarding onesample t-tests were revisited leading to opposite conclusions [START_REF] Chen | FMRI group analysis combining effect estimates and their variances[END_REF]. In the different software packages currently available to deal with fMRI data both approaches are represented: SPM 1 favors the homoscedastic approach [START_REF] Holmes | Generalisability, Random Effects & Population Inference[END_REF] while FSL 2 [START_REF] Woolrich | Multilevel linear modelling for FMRI group analysis using Bayesian inference[END_REF], AFNI 3 [START_REF] Cox | AFNI: software for analysis and visualization of functional magnetic resonance neuroimages[END_REF] and fmristat 4 [START_REF] Worsley | A general statistical analysis for fMRI data[END_REF] use the heteroscedastic model.

In this paper, we focus on quantitative detections of pathological brain perfusion abnormalities at the individual level. To this aim, we employ and compare two GLMbased models: the homoscedastic and heteroscedastic approaches. We test whether the assumptions underlying the homoscedastic approach are verified in pulsed ASL datasets. We furthermore study how the homoscedastic and heteroscedastic approaches behave in terms of specificity and sensitivity in the detection of patient-specific perfusion abnormalities.

A quantitative validation is performed on a dataset of 25 patients diagnosed with brain tumors. The model of normal perfusion is computed out of the data of 36 healthy volunteers.

Section 2 begins with a presentation of the homoscedastic and heteroscedastic models employed in the detection of patient-specific brain perfusion abnormalities with 1 http://www.fil.ion.ucl.ac.uk/spm 2 http://fsl.fmrib.ox.ac.uk/fsl/fslwiki 3 http://afni.nimh.nih.gov/afni 4 http://www.math.mcgill.ca/keith/fmristat ASL. Section 3 presents the datasets under study: the acquisition and pre-processing are described. Then, in section 4, the experiments designed to test the assumptions of the homoscedastic model and measure sensitivity and specificity are described. The results are presented in section 5. Section 6 gives a discussion and concludes.

Theory

In this section, the homoscedastic and heteroscedastic models used in the context of patient-specific detection of perfusion abnormalities in ASL are presented. In the standard approach, the GLM is defined for each voxel, which is why this approach is often termed "massively univariate". Without loss of generality to the multi-voxel procedure, we will consider the case of a single voxel in the remainder of this section. The voxel index is therefore omitted. In the context of this paper, we focus on a subtype of GLM: a one-versus-many mixed-effect two-sample t-test. As several measurements are available for each subject, a hierarchical model is defined with 2 levels: subject and group.

Subject level (First level)

In ASL, at the subject level, the data under study is a 4D volume of CBF maps containing r volumes (1 per repeated acquisition). The observations are therefore repeated measurements of the same underlying value. The subject level is therefore defined by:

Y s =    1 . . . 1    β s + s
where Y s is a vector of length r containing the observations at the given voxel and β s is the subject parameter to be estimated. Assuming Gaussian noise of the errors, each element of s follows a normal distribution: v s ∼ N (0, σ 2 s ). While in fMRI temporal autocorrelation must be accounted for [START_REF] Aguirre | Empirical analyses of BOLD fMRI statistics. II. Spatially smoothed data collected under null-hypothesis and experimental conditions[END_REF], in ASL, thanks to the subtraction process between control and labeled scans, we can reasonably assume white noise [START_REF] Aguirre | Experimental design and the relative sensitivity of BOLD and perfusion fMRI[END_REF][START_REF] Mumford | Estimation efficiency and statistical power in arterial spin labeling fMRI[END_REF]. The subject parameter β s is thus estimated by Ordinary Least Squares (OLS):

βs = 1 r r i=1 y s,i , (1)
where y s,i is the i th element of vector Y s . Similarly, the sampling variance of βs is estimated by:

Var( βs ) = σ2 s r where σ2 s = 1 r -1 r i=1 y s,i -βs 2 . (2)

Group level (Second level)

Given a group of n subjects, without loss of generality we assume that subjects 1 to n -1 are part of the control group and subject n is the patient of interest. Then, the one-versus-many second-level model is defined by:

     β1 . . . βn-1 βn      =      1 0 . . . . . . 1 0 0 1      β controls β patient + γ GC . (3)
The error term γ GC contains two combined sources of variations: the error measurement on the subject parameters (also termed within-subject variance) and the between-subject variance. Each element of γ GC therefore follows a normal distribution:

γ s G C ∼ N 0, σ 2 G + σ 2 s r
. Linear combinations of the group parameters can be calculated using a particular contrast. Here we are interested in the patient versus control group contrast:

b = [1 -1] β controls β patient = β controls -β patient . (4)
Two approaches have been proposed in the neuroimaging literature to solve the system (3) and find an estimate of the patient versus control group contrast, b, and its associated sampling variance, Var( b).

The homoscedastic model is based on the assumption that the within-subject variance is either negligible by comparison to the between-subject variance (i.e. σ 2 s σ 2 G , ∀ 1 ≤ s ≤ n) or, roughly constant across subjects (i.e.

σ 2 s ≈ σ 2 SU B , ∀ 1 ≤ s ≤ n).
Then each element of γ GC follows a normal distribution:

γ s G C ∼ N (0, σ 2 G C ) (5)
where σ 2 G C is the combined within-and between-subject variance. Depending on the assumption, we have σ 2

G C ≈ σ 2 G + σ 2 SU B r or σ 2 G C ≈ σ 2
G and the combined within-and between-subject variance is therefore constant across subjects. The sphericity assumption (no heteroscedasticity and no autocorrelation) hence holds and the system (3) is then solved by OLS. In the heteroscedastic model, heterogeneous variances are accounted for. Due to the nonsphericity of the measurement errors (as a consequence of heteroscedasticity), the system (3) is solved by using a weighted least square.

For a detailed calculation of the homoscedastic and heteroscedastic estimates, the interested reader is referred to previous developments of this methodology in the fMRI literature (for example [START_REF] Mumford | Simple group fMRI modeling and inference[END_REF]; [START_REF] Beckmann | General multilevel linear modeling for group analysis in FMRI[END_REF]). Tables 1 and2 provides a summary of the final estimates for the homoscedastic and heteroscedastic models. Table 1 gives the estimates of the control group ( βcontrols ) and patient ( βpatient ) effects along with their sampling variances ( Var( βcontrols ) and Var( βpatient )). Table 2 provides If the data are homoscedastic then it can easily be proven that the heteroscedastic model reduces to the homoscedastic model. However, in the presence of heteroscedasticity, the heteroscedastic model has two main advantages by comparison to the homoscedastic model:

1. In the control group, observations with high withinsubject variances are downweighted in order to provide a more efficient estimate of the control group parameter, βcontrols . 2. While with the homoscedastic model the sampling variance of the patient group estimate, Var( βpatient ), depends solely on variance estimations performed in the control group, the heteroscedastic sampling variance takes advantage of both the control group (estimation of σ 2 G ) and the patient (estimation of σ 2 n ) data.

The impact on the efficiency of the control group estimate (point 1) might be subtle and leads to no substantial improvement in one-sample t-tests if the homoscedasticity assumption is not overly altered [START_REF] Mumford | Simple group fMRI modeling and inference[END_REF]. However, the impact on the variance estimate of the patient parameter (point 2) can be large if the withinsubject variance of the patient of interest is very different from the control subjects. In ASL studies, the large influence of within-subject variance has been described in [START_REF] Viviani | Components of variance in brain perfusion and the design of studies of individual differences: the baseline study[END_REF]. Furthermore, patients are known to be less cooperative than control subjects which could potentially lead to within-subject variance inflation due to increased movement.

Hypothesis testing

Under the null hypothesis:

H 0 : β controls = β patient , (7)
the estimated patient versus control group contrast, b, divided by its estimated sampling standard deviation ( Var( b))

1 2

follows a t-distribution with n -1 degrees of freedom.

A probability under the null hypothesis can therefore be calculated for each voxel with:

P (X < x) where X = b Var( b) ∼ T n-1 , (8)
and x is the value taken by X at the voxel of interest. Equation ( 8) gives the probability to have a hyperperfusion, similarly, hypo-perfusions can be detected by substituting P (X < x) by P (X > x). The thresholding of this probability map gives the detections.

Table 1: Control group and patient estimates along with their sampling variances for the homoscedastic and heteroscedastic models.

βcontrols βpatient V ar βcontrols V ar βpatient Homoscedastic 1 n -1 n-1 s=1 βs βn σ2 G C n -1 σ2 G C Heteroscedastic 1 n-1 j=1 1 σ2 G +σ 2 j n-1 s=1 1 σ2 G + σ2 s βs βn 1 n-1 s=1 1 σ2 G +σ 2 s σ2 G + σ2 n
Table 2: Control group versus patient estimate along with its sampling variance for the homoscedastic and heteroscedastic models.

b

V ar b

Homoscedastic 1 n -1 n-1 s=1 βs -βn σ2 G C 1 n -1 + 1 Heteroscedastic 1 n-1 j=1 1 σ2 G +σ 2 j n-1 s=1 1 σ2 G + σ2 s βs -βn 1 n-1 s=1 1 σ2 G +σ 2 s + σ2 G + σ2 n 3. Materials
This section starts with a presentation of the datasets under study in 3.1. The pre-processing steps applied to the data before the statistical analysis are then described in 3.2.

Data acquisition

25 patients diagnosed with brain tumors and 36 healthy volunteers were involved in this study. One control subject and four patients were excluded because of strong borderzone signs [START_REF] Zaharchuk | Arterial spin-label imaging in patients with normal bolus perfusionweighted MR imaging findings: pilot identification of the borderzone sign[END_REF]. The final dataset therefore included 21 patients (13 males, 8 females, age: 55.2 ± 14.1 years) and 35 healthy volunteers (16 males, 19 females, age: 27.7 ± 6.4 years).

Data acquisition was performed on a 3T Siemens Verio MR scanner with a 32-channel head-coil. Patients were scanned in the context of clinical practice. The imaging protocol included a 3D T1-weighted anatomical sequence (T1w) (TR: 1900 ms, TE: 2.27 ms, FOV: 256 mm × 256 mm × 176 mm, flip angle: 9 • , resolution: 1 mm × 1 mm × 1 mm), a PICORE Q2TIPS sequence [START_REF] Wong | Quantitative imaging of perfusion using a single subtraction (QUIPSS and QUIPSS II)[END_REF] with crusher gradients (TR: 3000 ms, TE: 18 ms, FOV: 192 mm × 192 mm, flip angle: 90 • , resolution: 3 mm × 3 mm, slice thickness: 7 mm, inter-slice gap: 0.7 mm, TI: 1700 ms, TI wd : 700 ms, 60 repetitions, mSENSE parallel imaging with accelerating factor of 2). In addition to these sequences, the patients also underwent a 3D T1w post gadolinium (T1w-Gd) sequence (TR: 1900 ms, TE: 2.27 ms, flip angle: 9 • , FOV: 250 mm × 250 mm ×176 mm, resolution: 1 mm × 1 mm × 1 mm) and a 2D T2w FLAIR sequence (TR: 9000 ms, TE: 90 ms, FOV: 220 mm × 199.4 mm, flip angle: 150 • , resolution: 0.69 mm × 0.69 mm, slice thickness: 4 mm).

Out of the 22 patients, 17 subjects also underwent a DSC sequence (GRE EPI, TR: 1500 ms, TE: 30 ms, FOV: 230 mm× 230 mm, flip angle: 90 • , in plane resolution: 1.8 mm × 1.8 mm, slice thickness: 4 mm, inter-slice gap: 1.2 mm).

Pre-processing 3.2.1. CBF estimation with ASL

Image pre-processing was performed using SPM8 (Statistical Parametric Mapping 8, Wellcome Department of Imaging Neuroscience, University College, London) in Matlab R2012a (Mathworks, Natick, MA). The anatomical image of each subject was segmented, bias corrected and normalized into the ICBM-452 T1 template space using the unified segmentation [START_REF] Ashburner | Unified segmentation[END_REF]. The normalization was performed using an affine linear transformation followed by a non-linear transformation based on discrete cosine transform functions (default SPM normalization scheme). A subject-specific anatomical brain mask was created, excluding voxels with less than 50% of brain tissue in subsequent statistical analyses. A sixparameter rigid-body registration of the ASL volumes was carried out in order to reduce undesired effects due to subject motion. Rigid coregistration onto the whole-brain anatomical map was then performed based on mutual information. The average of unlabeled volumes was used to estimate the geometrical transformation to apply to each volume. Pair-wise subtraction of the control and labeled scans was then computed. A standard kinetic model [START_REF] Buxton | A general kinetic model for quantitative perfusion imaging with arterial spin labeling[END_REF] was applied in order to get ASL CBF, according to the following equation:

f = 6000 × λ ∆M 2 M 0 α TI wd exp -(TI +idx sl * TI sl )/T1 b (9)
where f is the 4D CBF map in mL.100g -1 .min -1 , M 0 the acquired M 0 map (first volume of the ASL series), λ the blood/tissue water partition coefficient, α the labeling efficiency, ∆M the 4D perfusion-weighted map, TI = 1.7 s the inversion time [START_REF] Ferré | Improving quality of arterial spin labeling MR imaging at 3 tesla with a 32-channel coil and parallel imaging[END_REF], idx sl the slice index (0 for the first slice), TI sl = 0.045 s the readout time for one slice, TI wd = 0.7 s the temporal width of the bolus, TI b the T1 of blood. According to [START_REF] Wang | Regional reproducibility of pulsed arterial spin labeling perfusion imaging at 3T[END_REF], we assumed: λ = 0.9 mL.g -1 , TI b = 1.5 s and α = 0.95. The 6000 factor allows the conversion from mL.g -1 .s -1 to mL.100g -1 .min -1 which is the standard unit for CBF. We bring the attention of the reader to the fact that, contrary to what is usually done in ASL pre-processing, ∆M represents the set of perfusion-weighted maps (one volume per repetition) instead of a single perfusion-weighted map obtained by averaging across the repetitions. This is necessary in order to allow for the measurement of the within-subject variance.

Spatial normalization parameters estimated during the segmentation step were then applied to the T1 and ASL CBF maps in order to normalize the subjects into a common template space. Normalisation in intensity was then applied to each ASL CBF map in order to reduce the intersubject variability [START_REF] Aslan | On the sensitivity of ASL MRI in detecting regional differences in cerebral blood flow[END_REF]. The normalisation parameter was calculated as the average CBF in grey matter similarly to [START_REF] Petr | Template-based approach for detecting motor task activation-related hyperperfusion in pulsed asl data[END_REF].

CBF estimation with DSC

The DSC images were processed using MR manufacturer software by manually choosing an arterial input function to calculate: the CBF, cerebral blood volume, and mean transit time maps. The method is based on a deconvolution algorithm as described in [START_REF] Ostergaard | High resolution measurement of cerebral blood flow using intravascular tracer bolus passages. Part I: Mathematical approach and statistical analysis[END_REF]. Similarly to ASL, DSC CBF maps were coregistered on anatomical maps and spatially normalized.

Ground truth

Based on clinical knowledge, we used a semi-automatic procedure that took advantage of the complementary anatomical (T1w-Gd, T2w FLAIR) and perfusion (DSC) information to get an estimation of the ground truth.

True positives. In order to get an estimation of the ground truth, we implemented a method inspired by the hotspot technique [START_REF] Noguchi | Perfusion imaging of brain tumors using arterial spinlabeling: correlation with histopathologic vascular density[END_REF] on the DSC CBF map. To this aim, the tumor was first segmented using a semiautomated method based on the T2w and T1w-Gd images and visually inspected by an expert neuro-radiologist. Then, we compared the tissue segmented as part of the tumor to its controlateral counterpart in the DSC CBF map. Voxels overtaking the lower and upper deciles were identified as potential hypo-and hyper-perfusions. Each potential perfusion abnormality was then visually inspected by an expert neuro-radiologist and manually corrected if needed. Special care was taken in order to avoid inclusion of hyper-perfusions related to the presence of arteries.

Due to its low SNR, ASL is not well suited to measure low levels of perfusion [START_REF] Wintermark | Comparative overview of brain perfusion imaging techniques[END_REF]. That is why, we focused on hyper-perfusions for sensitivity estimation. Hypo-perfusions were nevertheless retained for specificity calculations.

Out of the 17 patients included in this study who underwent a DSC sequence, 9 presented hyper-perfusions, 16 hypo-perfusions and 8 both.

False positives. According to clinical knowledge, in the absence of metastasis, the perfusion abnormalities should be confined to the affected tissue (tumor and oedema) identifiable on T1w-Gd and T2w. The proportion of nonaffected tissue detected as a perfusion abnormality was used as a measure of the false positive rate.

In the control group, in which no detections were expected, an additional estimate of the specificity was calculated by leave-one-out cross-validation.

Methods

In 4.1, the implementation of the heteroscedastic model employed in this paper is described. Then, in 4.2, the experiments undertaken in order to test the assumptions of the homoscedastic model and to compare the homoscedastic and heteroscedastic models are presented.

Implementation choice for the heteroscedastic model

In this paper, we computed the estimate of the betweensubject variance, σ2

G , using a recent computationally efficient approach available in the MEMA function of the AFNI software package. This algorithm is based on the method of moments and the restricted maximum likelihood. The estimation method is then selected depending on the data observed at the voxel level [START_REF] Chen | FMRI group analysis combining effect estimates and their variances[END_REF]. It is however worth noting that alternative publicly available algorithms, such as FLAME1 in FSL, are also suited to compute this kind of heteroscedastic approach.

Evaluation framework

Validity of the assumptions of the homoscedastic model

The homoscedastic model makes the assumption that within-subject variance is either negligible by comparison to between-subject variance, or roughly constant across subjects. In order to test these assumptions, we performed two experiments.

Negligible within-subject variance. First, following [START_REF] Chen | FMRI group analysis combining effect estimates and their variances[END_REF], we measured the proportion of total variability that occurred within subjects with the following index, defined at each voxel v for each subject s: Values close to 1 mean that the within-subject variance (σ 2 s,v ) is preponderant compared to between-subject variance (σ 2 G,v ) and values close to 0 mean that betweensubject variance holds the majority of the total variance. This measure is provided as an output of the MEMA function in AFNI [START_REF] Chen | FMRI group analysis combining effect estimates and their variances[END_REF].

λ s,v = σ2 s,v σ2 G,v + σ2 s,v (10)
Constant within-subject variance across subjects. In a second experiment, we focused on the within-subject variance to verify whether it could be assumed roughly constant across subjects. To this aim, we calculated an average within-subject variance for each subject. This index was computed as suggested by [START_REF] Mumford | Simple group fMRI modeling and inference[END_REF] by averaging the within-subject variance for voxels within the interquartile range of the nonzero betweensubject variance. Given this index of within-subject variance, we searched for outliers in the control group with Rosner's test [START_REF] Rosner | Percentage points for a generalized ESD manyoutlier procedure[END_REF], assuming that the distribution of the variance estimate was approximately normal. Then, we checked whether the control and patient groups had significantly different medians with a non-parametric Kruskal-Wallis test.

Comparison of the homoscedastic and heteroscedastic

models Detections at a fixed threshold. In order to assess the difference that heteroscedasticity modeling would induce, we compared the sensitivity and specificity of the homoscedastic and heteroscedastic models. We compared both approaches in a usual setting, where the detections were identified at a threshold p < 0.05 with False Discovery Rate (FDR) correction for multiple comparisons [START_REF] Benjamini | Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing[END_REF]. The sensitivity was estimated as the average sensitivity across the patient group. In each group (patient, control), the specificity was estimated by averaging the specificity across subjects. As the data is usually pre-smoothed with a Gaussian kernel at the end of the pre-processing, we studied 6 kernel sizes within the typical smoothing range, defined by their full-width at half maximum (FWHM): 0 mm 3 (i.e. no smoothing), 4 mm 3 , 6 mm 3 , 8 mm 3 , 10 mm 3 , 12 mm 3 . ROC analysis. We compared the sensitivity and specificity of the homoscedastic and heteroscedastic models with Receiver-Operating-Characteristics (ROC) curves. In order to draw the ROC curves, we used 122 p-values (un-corrected), equally spaced in the logarithmic space. For each p-value, the false positive rate (respectively true positive rate) was calculated as the average false positive rate (respectively true positive rate) over the subjects of the patient group. ROC curves provide a way to measure the performance of a classifier without focusing on a single threshold. We calculated the area under the curve as an indicator of the classification accuracy. Since the size of the expected detections is much smaller than the number of voxels that must not be detected, the area of interest in the ROC curve is the one of high specificity. That is why, as previously proposed in the literature, we focused on the area under the curve corresponding to false positive rates ranging from 0% to 10% [START_REF] Skudlarski | ROC analysis of statistical methods used in functional MRI: individual subjects[END_REF]. As in the previous experiment, different smoothing kernels were studied.

Results

In this section we start with a graphical presentation of the parameter estimates of the homoscedastic and heteroscedastic models in the control group (5.1). Then, we present the results of the experiments regarding the validity of the assumptions of the homoscedastic model (5.2). In a last subsection, we display the results of the quantitative comparison between the homoscedastic and heteroscedastic models (5.3).

ASL template: a model of normal perfusion

In fig. 1, the parameter estimates of the homoscedastic and heteroscedastic models are displayed. Each parameter is defined as a voxelwise map. First, an estimate of the control group parameter, βcontrols , is provided (the heteroscedastic estimate is displayed, the homoscedastic estimate being visually nearly identical). As expected, the CBF is higher in the cortex and the basal ganglia than in white matter. Then, three standard deviation estimates are displayed:

• the combined within-and between-subject standard deviation estimate σG C from the homoscedastic model (1 b);

• the between-subject standard deviation estimate σG from the heteroscedastic model (1 c);

• the root square of the average within-subject variance estimates in the control group ( 1

n-1 n-1 s=1 σ2 s ) 1
2 . This map is not part of the estimated standard deviations but is provided as a visual example of expected within-subject standard deviation in the control group (1 d).

The high variance values observed in the vascular structures, such as the transverse sinus, are in concordance with the findings of [START_REF] Viviani | Components of variance in brain perfusion and the design of studies of individual differences: the baseline study[END_REF]. This pattern is clearly visible in the combined within-and betweensubject variance estimate of the homoscedastic model and captured by the within-subject variance estimate in the heteroscedastic model. The main variations observed in the between-subject variance, as estimated in the heteroscedastic model, are related to inter-subject misregistrations in the cortex. An increased variance is also visible in the occipital lobe, probably related to increased arterial transit times in these regions [START_REF] Macintosh | Assessment of arterial arrival times derived from multiple inversion time pulsed arterial spin labeling MRI[END_REF].

Testing the assumptions of the homoscedastic model

The homoscedastic model stands on one of the following assumptions: either the within-subject variance is negligible by comparison to between-subject variance, or the within-subject variance is roughly constant across subjects.

Relative weights of between-subject and within-subject

variances in the control group In order to verify if the within-subject variance can be assumed negligible by comparison to between-subject variance, fig. 2 presents the histogram of the ratio of withinsubject to total variance in the control group (λ s,v from eq. ( 10)). Overall, a total of 1 094 790 voxels are considered. Values close to one indicate a preponderance of the within-subject variance, whereas values close to zero outline a dominating between-subject variance. Clearly, both components of variance have an important impact as the λ s,v index spans the complete range of values between 0 and 1. The large peak indicating voxels with a zero between-subject variance was also observed in [START_REF] Chen | FMRI group analysis combining effect estimates and their variances[END_REF] and might be the consequence of calculation inaccuracies. While the true between-subject variance might not be exactly zero, these voxels nevertheless present a very small between-subject variance, negligible by comparison to the within-subject variance. The quartiles of the distribution are 0.23 and 0.78, so that the outermost 25% of the voxels (on each side) have either a dominant within-subject variance or a dominant between-subject variance. In the remaining 50% voxels, both variance components have a significant impact. In conclusion, it cannot be assumed that within-subject variance is negligible in comparison to between-subject variance.

Cross-subject comparison of within-subject variances

In this experiment, we tested whether within-subject variance could be assumed roughly constant across subjects. As proposed by [START_REF] Mumford | Simple group fMRI modeling and inference[END_REF], we calculated an average within-subject variance across voxels. This led to an index of within-subject variance per subject. Fig. 3 displays the within-subject variance indexes in the control and patient groups.

In the control group, it is clear that one of the subjects presents an unexpected high variance by comparison to the other controls. Retrospectively, we found out that this subject moved substantially more than other controls and was identified as uncooperative by the MR physicist during the acquisition. As expected from fig. 3, we found one outlier in the control group with Rosner's test.

The within-subject variance was significantly higher in the patient group than in the control group (p < 0.05 with Kruskal-Wallis test). This might be explained by the fact that patients tend to have more difficulties to lie still during the acquisition due to their pathological condition.

The index proposed by [START_REF] Mumford | Simple group fMRI modeling and inference[END_REF] is an average across voxels and therefore focused on global variations of the within-subject variance. However, strong variations, sometimes caused by artefacts, that appear lo-cally can also be a concern. As an example, fig. 4a displays the estimated within-subject standard deviation for a control subject presenting locally atypical patterns. While the high variance induced by the presence of large vessels is a pattern shared across subjects (as previously described in fig. 1, last panel), the high variance observed bilaterally in the frontal lobe is specific of this control subject. We hypothesize that these strong variations are the consequence of motion during the acquisition that was not correctly compensated during the pre-processing. Even if this subject was not previously outlined as an outlier (with an average within-subject variance of 0.0024), these atypical patterns of variance might have a detrimental impact on the analysis if not properly taken into account.

In conclusion, variations across subjects of the withinsubject variance appear to be important in ASL. This might be a consequence of the low SNR of this technique, since small artefacts in the original control and labeled scans can lead to substantial variations in the perfusionweighted map after subtraction. In the next sections we investigate whether modeling heteroscedasticity can improve the detections of perfusion abnormalities. 5.3. Comparison of homoscedastic and heteroscedastic models 5.3.1. Fixed threshold Quantitative analysis. Table 3 presents the sensitivity to detect hyper-perfusions and the specificity both in the patient group and by leave-one-out cross-validation in the control group with the homoscedastic and heteroscedastic models for different smoothing kernels (FWHM = [0,4,6,8,10,12] mm 3 ), at p < 0.05 FDR corrected.

Overall, the heteroscedastic model leads to a decrease in false positive rate. This is particularly noticeable in the patient group where the specificity is improved for each smoothing studied. As expected, in the control group, this effect is also observed but to a lesser extent. This is probably due to the fact that the hypothesis of homoscedasticity is better suited for the control subjects.

Qualitative analysis. At the individual level, as illustrated in fig. 4, local atypical patterns of variance can further lead to a substantial increase in false positives with the homoscedastic model. The unexpected high standard deviation in the frontal lobe for the control subject presented in fig. 4 leads to false positive hyper-perfusions with the homoscedastic model whereas the heteroscedastic model does not get any false positive (smoothing FWHM = 8 mm 3 ). Fig. 5 further illustrates the benefits of the heteroscedastic model compared to the homoscedastic model in a patient subject. The motion artefacts and hyper-signals induced by arteries indeed correspond to regions of high within-subject standard deviation. Modeling heterogeneous variances reduces the artefactual detections in these regions while preserving the quality of the true detections.

To investigate whether, at a same specificity rate, the sensitivity differs between the two approaches, we employed a ROC analysis as described in the next section. Table 3: Sensitivity and specificity in the control (first row) and patient (second and third rows) groups with the homoscedastic and heteroscedastic models for different smoothing kernels (FWHM = [0,4,6,8,10,12] mm 3 ), at p < 0.05 FDR corrected.

Homoscedastic model

Heteroscedastic model 0 4 6 8 10 12 0 4 6 8 10 12 Specificity (controls) 1.00 0.99 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00 Specificity (patients) 0.92 0.89 0.87 0.86 0.85 0.84 1.00 0.99 0.99 0.99 0.99 0.98 Sensitivity (hyper) 0.49 0.59 0.63 0.66 0.67 0.67 0.51 0.59 0.61 0.58 0.57 0.55

ROC analysis

Quantitative comparison. Fig. 6 presents the ROC curves in the patient group for the homoscedastic and heteroscedastic models. The average over the studied smoothing kernels is plotted along with the spread provided by the standard deviation over the 6 measurements. The ROC curve of the heteroscedastic model is substantially closer to the ideal classifier than the homoscedastic curve. Table 4 presents the area under the ROC curves for false positives rates ranging from 0 to 10%, for the homoscedastic and heteroscedastic models with different smoothing kernels. The heteroscedastic model outperforms the homoscedastic model with an increased area under the curve. The best values are 0.72 and 0.49 with the heteroscedastic and homoscedastic model respectively. For both approaches, a maximum area under the curve is reached for a smoothing kernel of 8 mm 3 .

Qualitative comparison. In order to illustrate the advantage of the heteroscedastic over the homoscedastic model, we chose 3 representative subjects and compared the methods at fixed false positive rate and true positive rate. To this aim, we selected the uncorrected p-values that would lead to a pre-specified false positive rate (respectively true positive rate) from the ROC analysis. We worked with data smoothed with a Gaussian kernel of 8 mm 3 that led to the best area under the curve with both models. It is worth noting that in the previous section the sensitivity and specificity were estimated at the group level and that the 3 subjects presented in this part were chosen so that they would best illustrate the group findings. Fig. 7 presents the detections obtained with both methods on the 3 selected patients. The first subject, which presents a small hyper-perfusion, is studied at a true positive rate of 50%. In the two remaining patients, the methods are compared at a false positive rate of 0.1%.

Patient 15 suffers from a gliosarcoma in the left hemisphere close to the parahippocampal region. The lesion displays a small hyper-perfusion in its dorsal part, small hypo-perfusions are seen in the surrounding edema. At a true positive rate of 50%, the homoscedastic model displays a larger number of false positives than the heteroscedastic model. Patient 16 was diagnosed with a high grade tumor in the left temporal lobe. The lesion is characterized by a large hyper-perfusion and a surrounding hypo-perfusion. With a false positive rate of 0.1%, the hyper-perfusion is correctly located with both methods. The extent of the hyper-perfusion is however better cov- ered by the heteroscedastic model. Patient 6 suffers from a meningioma partly hyperperfused. Similarly to patient 16, at a false positive rate of 0.1%, both methods detect the hyper-perfusion but the heteroscedastic model is clearly more sensitive.

These 3 cases illustrate the loss of sensitivity of the homoscedastic model by comparison to the heteroscedastic model at the same specificity level.

Discussion and conclusion

We have compared two approaches to quantitatively outline patient-specific pathological patterns of abnormal perfusion in ASL data based on the massively univariate GLM: the homoscedastic and heteroscedastic models.

We demonstrated that the assumptions underlying the homoscedastic model are not verified in ASL studies. More precisely, the within-subject variance cannot be considered as negligible by comparison to between-subject variance, nor constant across subjects. In fMRI data, small deviations from homoscedasticity have shown to not overly alter the results in one-sample mixed-effects GLM analysis [START_REF] Mumford | Simple group fMRI modeling and inference[END_REF]. Here, we showed that modeling heterogeneous within-subject variances is essential in order to reach a satisfactory level of specificity in a mixed-effect two-sample t-test comparing a patient to a group of controls in ASL. These results are in line with a recent study by [START_REF] Chen | FMRI group analysis combining effect estimates and their variances[END_REF] where heteroscedastic mixed-effects GLM were shown to provide more accurate results in fMRI.

In the context of this paper, we defined a single parameter in the subject-level design matrix and focused on perfusion-weighted images obtained after pair-wise subtraction of the control and labeled scans. Other authors have suggested that taking the complete ASL time-course into account (before subtraction) would lead to more efficient estimates of perfusion in fMRI [START_REF] Mumford | Estimation efficiency and statistical power in arterial spin labeling fMRI[END_REF]. Also, additional regressors can be introduced as nuisance covariates for denoising purposes in the subjectlevel design matrix as suggested in [START_REF] Wang | Improving cerebral blood flow quantification for arterial spin labeled perfusion MRI by removing residual motion artifacts and global signal fluctuations[END_REF]. Modifying the subject-level design matrix would change the estimated subject parameters and their sampling variances. However, an heteroscedastic model would still be appropriate in this setting so that the conclusions of this paper remain valid even with a different subject-level model.

We demonstrated that within-subject variance captures important information regarding the subject-specific spatial distribution of noise in ASL data. We also outlined that patient-specific brain perfusion abnormalities can be correctly detected using ASL if the heterogeneous withinsubject variances are properly modeled. We therefore advise the use of heteroscedastic models in ASL studies.

  the estimate of the control group versus patient effect ( b) along with its sampling variance ( Var( b)), both estimates are obtained by combination of the control group and patient estimates: b = βcontrolsβpatient Var( b) = Var( βcontrols ) + Var( βpatient ) (6)

Figure 1 :

 1 Figure 1: Parameter estimates of the homoscedastic and heteroscedastic model in the control group, computed from 35 healthy subjects. a) Mean perfusion estimate βcontrols . b) Combined within-and between-subject standard deviation estimate from the homoscedastic model σG C . c) Between-subject standard deviation estimate from the heteroscedastic model σG . d) Root square of the average within-subject variance in the control group. Perfusion is expressed in normalized units (ratio to mean grey matter perfusion). Axial slices are displayed in neurological convention.

Figure 2 :

 2 Figure 2: Histogram of the ratio of within-subject variance onto total variance in the control group across subjects and voxels. Both variance estimates have a significant impact depending on the voxels.

Figure 3 :

 3 Figure 3: Average estimated within-subject variances in control and patient groups. For each box, the red line corresponds to the median and the top and bottom lines of the blue square are the upper and lower quartiles of the distribution. The whiskers cover the data points that are located up to 1.5 times the inter-quartile distance away from the lower or upper quartile. Points falling out of the whiskers are displayed with a red cross.

Figure 4

 4 Figure 4: a) Estimated within-subject standard deviation in a control subject presenting locally atypical values. b) T1w map with false positive detections (in red) by leave-one-out cross-validation (smoothing FWHM = 8 mm 3 ) with the homoscedastic model (b) and the heteroscedastic model (c). No false positive detections are observed with the heteroscedastic model.

Figure 5 :

 5 Figure 5: Detections of perfusion abnormalities with the homoscedastic and heteroscedastic models in a patient suffering from a high grade glioma. a) T1w-Gd map, the tumor site is pointed by a black arrow. b) Patient ASL CBF estimate βn . c) Within-subject standard deviation of ASL CBF σ2 n . d) T1w-Gd map with ground truth overlaid. T1w-Gd map with hypo-(blue color-map) and hyper-perfusions (hot color-map) overlaid for the homoscedastic (e) and heteroscedastic (f) models. Modeling heterogeneous variances (heteroscedastic model) reduces the false positive detections while preserving the true detections. Axial slices are displayed in neurological convention.

Figure 6 :

 6 Figure 6: ROC curves for perfusion abnormality detections with the homoscedastic and heteroscedastic models. The average ROC curves across the studied smoothings are plotted in plain line. Dotted lines are plotted one standard deviation away from the average.

Figure 7 :

 7 Figure7: Perfusion abnormalities detections in 3 patients with homoscedastic and heteroscedastic models. From left to right: ground truth hyper-perfusions (red) and hypo-perfusions (blue) overlaid on the T1w-Gd map; detections with the homoscedastic model with a smoothing kernel of FWHM = 8 mm 3 , hyper-perfusions (hot colormap) and hypo-perfusions (blue); detections with the heteroscedastic model with a smoothing kernel of FWHM = 8 mm 3 , hyper-perfusions (hot colormap) and hypo-perfusions (blue).

Table 4 :

 4 Area under the ROC curve for false positive rates ranging between 0% and 10% with the homoscedastic and heteroscedastic models. The heteroscedastic model outperforms the homoscedastic model.

		Homoscedastic model		Heteroscedastic model
	0	4	6	8	10 12	0	4	6	8	10 12
	Area under the ROC curve 0.46 0.49 0.49 0.49 0.48 0.48	0.63 0.70 0.72 0.72 0.69 0.65