Sebastien Benzekry
email: sebastien.benzekry@inria.fr

Pirmin Schlicke

Pascale Tomasini

Eleonore Simon

Sébastien Benzekry

Eléonore Simon

Mechanistic modeling of brain metastases in NSCLC provides computational markers for personalized prediction of outcome

Keywords: Biological Sciences, Applied Mathematics NSCLC, Brain Metastases, Computational Biomarkers

ABSTRACT

Background:

Intracranial progression after curative treatment of early-stage non-small cell lung cancer (NSCLC) occurs from 10 to 50% and is difficult to manage, given the heterogeneity of clinical presentations and the variability of treatments available.

The objective of this study was to develop a mechanistic model of intracranial progression to predict survival following a first brain metastasis (BM) event.

Methods: Data included early-stage NSCLC patients treated with a curative intent who had a BM as the first and single relapse site (N=31).

We propose a mechanistic mathematical model to estimate the amount and sizes of (visible and invisible) BMs. The two key parameters of the model are 𝛼, the proliferation rate of a single tumor cell; and 𝜇, the per day, per cell, probability to metastasize. The predictive value of these individual computational biomarkers was evaluated.

Findings:

The model was able to correctly describe the number and size of metastases at the time of first BM relapse for 20 patients. Parameters 𝛼 and 𝜇 were significantly associated with overall survival (OS) (HR 1.65 (1.07-2.53) p=0.0029 and HR 1.95 (1.31-2.91) p=0.0109, respectively). Adding the computational markers to the clinical ones significantly improved the predictive value of OS (c-index increased from 0.585 (95% CI 0.569-0.602) to 0.713 (95% CI 0.700-0.726), p<0.0001).

Interpretation:

We demonstrated that our model was applicable to brain oligoprogressive patients in NSCLC and that the resulting computational markers had predictive potential. This may help lung cancer physicians to guide and personalize the management of NSCLC patients with intracranial oligoprogression.

SIGNIFICANCE STATEMENT

Non-small cell lung cancer is difficult to manage when brain metastases are present. This study presents a mathematical model that can be calibrated on individual patients' data early in the treatment course to explain the growth dynamics of brain metastases and demonstrates that the mathematically derived parameters can serve as predictive tool in clinical routine care.

INTRODUCTION

The 5 years survival rate in non-small cell lung cancer (NSCLC) is only 17% [START_REF] Bray | GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries[END_REF][START_REF]Cancer of the Lung and Bronchus -Cancer Stat Facts[END_REF] . This poor prognosis is partly due to the frequent occurrence of brain metastases (BM), occurring in up to 50% of NSCLC patients [START_REF] Chargari | Métastases cérébrales d'un cancer bronchique non à petites cellules : du traitement standardisé au traitement personnalisé[END_REF] and directly responsible for death in 50% cases [START_REF] Tabouret | Recent trends in epidemiology of brain metastases: an overview[END_REF][START_REF] Hall | Long-term survival with metastatic cancer to the brain[END_REF] . Even in initially early-stage NSCLC treated with curative intent, intracranial relapse occurs in 10% of the cases in stage I-II NSCLC and 30 to 50% cases in stage III NSCLC [START_REF] Gauger | Clinical stage I non-small cell lung cancer including FDG-PET Imaging: sites and time to recurrence[END_REF] . Not only is the brain one of the first sites of recurrence after surgery or curative radiotherapy, but it can also be the only site of relapse, then called intracranial oligoprogression [START_REF] Yano | The first site of recurrence after complete resection in non-small-cell carcinoma of the lung[END_REF][START_REF] Sadoyama | Isolated Brain Metastases as the First Relapse After the Curative Surgical Resection in Non-Small-Cell Lung Cancer Patients With an EGFR Mutation[END_REF][START_REF] Figlin | Study Group*. Intracranial Recurrence of Carcinoma after Complete Surgical Resection of Stage I, II, and III Non-Small-Cell Lung Cancer[END_REF] .

Oligoprogression is defined as the progression of a limited number of metastases, ranging from 3 to 5, and possibly treated by local therapies [START_REF] Esmo | Metastatic Non-Small-Cell Lung Cancer: ESMO Clinical Practice Guidelines for diagnosis, treatment and follow-up[END_REF][START_REF] Hellman | [END_REF][START_REF] Laurie | Canadian consensus: oligoprogressive, pseudoprogressive, and oligometastatic non-small-cell lung cancer[END_REF] . Existing data on intracranial oligoprogression are mostly related to palliative situations, and to patients receiving tyrosine kinase inhibitors [START_REF] Schoenmaekers | Central nervous system metastases and oligoprogression during treatment with tyrosine kinase inhibitors in oncogene-addicted non-small cell lung cancer: how to treat and when?[END_REF] . There are no real guidelines for the management of single intracranial oligoprogression in patients with early-stage NSCLC at diagnosis, initially treated with curative intent. Furthermore, the difficulty in establishing guidelines relies on the heterogeneity of this situation. Some patients undergo a single intracranial recurrence, while others suffer from either multiple intracranial relapse or subsequent extra-cranial relapse [START_REF] Luca Ceresoli | Brain metastases in locally advanced nonsmall cell lung carcinoma after multimodality treatment: Risk factors analysis[END_REF] .

Several options are available in the treatment of BM, including local and systemic therapies. Local treatments include neurosurgery, stereotaxic radiosurgery (SRS) and whole brain radiotherapy (WBRT) [START_REF] Zimm | Intracerebral metastases in solid-tumor patients: natural history and results of treatment[END_REF][START_REF] Yamamoto | Stereotactic radiosurgery for patients with multiple brain metastases (JLGK0901): a multi-institutional prospective observational study[END_REF][START_REF] Zindler | Whole brain radiotherapy versus stereotactic radiosurgery for 4-10 brain metastases: a phase III randomised multicentre trial[END_REF][START_REF] Gerosa | Analysis of long-term outcomes and prognostic factors in patients with non-small cell lung cancer brain metastases treated by gamma knife radiosurgery[END_REF][START_REF] Motta | Gamma knife radiosurgery for treatment of cerebral metastases from non-small-cell lung cancer[END_REF] . Local treatment indications have evolved over time and the place of SRS is becoming increasingly important [START_REF] Yamamoto | Stereotactic radiosurgery for patients with multiple brain metastases (JLGK0901): a multi-institutional prospective observational study[END_REF][START_REF] Zindler | Whole brain radiotherapy versus stereotactic radiosurgery for 4-10 brain metastases: a phase III randomised multicentre trial[END_REF][START_REF] Gerosa | Analysis of long-term outcomes and prognostic factors in patients with non-small cell lung cancer brain metastases treated by gamma knife radiosurgery[END_REF][START_REF] Motta | Gamma knife radiosurgery for treatment of cerebral metastases from non-small-cell lung cancer[END_REF][START_REF] Fuentes | Surgery versus stereotactic radiotherapy for people with single or solitary brain metastasis[END_REF] . Nevertheless, WBRT has the advantage of covering all macroscopic and microscopic BM which would not be detected with MRI and could have a prophylactic role on the development of BM [START_REF] Aupérin | Prophylactic Cranial Irradiation for Patients with Small-Cell Lung Cancer in Complete Remission[END_REF][START_REF] Liu | The Role of Prophylactic Cranial Irradiation in Patients With Non-small Cell Lung Cancer: An Updated Systematic Review and Meta-Analysis[END_REF] . However, WBRT is also responsible for neurocognitive decline, leading to a lower quality of life [START_REF] Chang | Neurocognition in patients with brain metastases treated with radiosurgery or radiosurgery plus whole-brain irradiation: a randomised controlled trial[END_REF][START_REF] Soffietti | A European Organisation for Research and Treatment of Cancer phase III trial of adjuvant whole-brain radiotherapy versus observation in patients with one to three brain metastases from solid tumors after surgical resection or radiosurgery: quality-of-life results[END_REF] . On the other hand, systemic treatments include chemotherapy, targeted therapies or immunotherapy [START_REF] Esmo | Metastatic Non-Small-Cell Lung Cancer: ESMO Clinical Practice Guidelines for diagnosis, treatment and follow-up[END_REF][START_REF] Besse | Bevacizumab in Patients with Nonsquamous Non-Small Cell Lung Cancer and Asymptomatic, Untreated Brain Metastases (BRAIN): A Nonrandomized, Phase II Study[END_REF][START_REF] Socinski | Atezolizumab for First-Line Treatment of Metastatic Nonsquamous NSCLC[END_REF][START_REF] Reck | Pembrolizumab versus Chemotherapy for PD-L1-Positive Non-Small-Cell Lung Cancer[END_REF][START_REF] Gandhi | Pembrolizumab plus Chemotherapy in Metastatic Non-Small-Cell Lung Cancer[END_REF] . The intracranial efficacy of systemic treatments is heterogeneous and partly depends on the blood-brain barrier (BBB) crossing of the drugs [START_REF] Sweeney | Blood-Brain Barrier: From Physiology to Disease and Back[END_REF] . Moreover, the unique brain microenvironment likely plays a critical role in treatment response, especially with immunotherapy [START_REF] Cortinovis | Italian Cohort of the Nivolumab EAP in Squamous NSCLC: Efficacy and Safety in Patients With CNS Metastases[END_REF][START_REF] Dudnik | Intracranial response to nivolumab in NSCLC patients with untreated or progressing CNS metastases[END_REF][START_REF] Goldberg | Pembrolizumab for patients with melanoma or non-small-cell lung cancer and untreated brain metastases: early analysis of a non-randomised, openlabel, phase 2 trial[END_REF][START_REF] Gauvain | Intracerebral efficacy and tolerance of nivolumab in non-small-cell lung cancer patients with brain metastases[END_REF] . The combination of systemic treatments such as chemotherapy and immunotherapy seems to induce better intracranial response than immunotherapy alone [START_REF] Leighl | 34: A randomized trial of durvalumab and tremelimumab +/-platinum-based chemotherapy in patients with metastatic (Stage IV) squamous or nonsquamous non-small cell lung cancer (NSCLC)[END_REF][START_REF] Powell | Pembrolizumab (pembro) plus platinum-based chemotherapy (chemo) in NSCLC with brain metastases: Pooled analysis of KEYNOTE-021, 189, and 407[END_REF] . Combinations of local and systemic treatments also seem to have a synergistic effect and lead to better intracranial response [START_REF] Wang | Efficacy and Safety of Radiotherapy Plus EGFR-TKIs in NSCLC Patients with Brain Metastases: A Meta-Analysis of Published Data[END_REF] . Radiotherapy (either SRS or WBRT) could help damage the BBB, allowing the crossing of systemic treatments.

The complexity of multimodal management of BM relies on the diversity of possible treatments (including local and systemic treatments), the difficulty of crossing the BBB for systemic treatments, the specificity of the brain microenvironment and the lack of predictability of the appearance of BM, as well as further extracranial progression, as evidenced by Levy's study of European practices [START_REF] Levy | Diversity of brain metastases screening and management in non-small cell lung cancer in Europe: Results of the European Organisation for Research and Treatment of Cancer Lung Cancer Group survey[END_REF] .

To guide the management of patients with BM, prognostic scores have been established, such as the mol-Lung GPA [START_REF] Gaspar | Recursive partitioning analysis (RPA) of prognostic factors in three radiation therapy oncology group (RTOG) brain metastases trials[END_REF][START_REF] Sperduto | Secondary analysis of RTOG 9508, a phase 3 randomized trial of whole-brain radiation therapy versus WBRT plus stereotactic radiosurgery in patients with 1-3 brain metastases; poststratified by the graded prognostic assessment (GPA)[END_REF][START_REF] Sperduto | Estimating Survival in Patients With Lung Cancer and Brain Metastases: An Update of the Graded Prognostic Assessment for Lung Cancer Using Molecular Markers (Lung-molGPA)[END_REF] . However, these scores do not help predict the occurrence of metastases, even if some predictive factors were identified such as age ≤ 62 years, stage T4 or N2-3, adenocarcinoma subtype, presence of leukoaraiosis, and molecular profile [START_REF] Bajard | Multivariate analysis of factors predictive of brain metastases in localised non-small cell lung carcinoma[END_REF][START_REF] Hayashi | Negative impact of leukoaraiosis on the incidence of brain metastases in patients with lung cancer[END_REF] .

Predicting the course and the appearance of BM would improve the monitoring of patients and offer a more personalized treatment. For this purpose, the contribution of mathematical modeling can be crucial.

The first mathematical models of metastatic occurrence based on experimental observations were established by Liotta in the 1970's [START_REF] Liotta | Stochastic model of metastases formation[END_REF] . Further mathematical models were then established in animal models [START_REF] Hartung | Mathematical Modeling of Tumor Growth and Metastatic Spreading: Validation in Tumor-Bearing Mice[END_REF][START_REF] Baratchart | Computational Modelling of Metastasis Development in Renal Cell Carcinoma[END_REF][START_REF] Benzekry | Modeling Spontaneous Metastasis following Surgery: An In Vivo-In Silico Approach[END_REF] . These models were based on two main phenomena: growth and spread of a tumor. The latter is often successfully described using the Gompertz model [START_REF] Laird | TUMOUR GROWTH: COMPARISON OF GROWTH RATES AND EXTRAPOLATION OF GROWTH CURVE TO ONE CELL[END_REF][START_REF] Norton | A Gompertzian model of human breast cancer growth[END_REF][START_REF] Benzekry | Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth[END_REF] . For metastatic spread, we have developed models based on pioneering work by Iwata et al [START_REF] Baratchart | Computational Modelling of Metastasis Development in Renal Cell Carcinoma[END_REF][START_REF] Benzekry | Modeling Spontaneous Metastasis following Surgery: An In Vivo-In Silico Approach[END_REF][START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF][START_REF] Schlicke | How mathematical modeling could contribute to the quantification of metastatic tumor burden under therapy: insights in immunotherapeutic treatment of non-small cell lung cancer[END_REF] . Our team further developed these mathematical constructs to be applicable in the clinic, for example on breast [START_REF] Nicolò | Machine Learning and Mechanistic Modeling for Prediction of Metastatic Relapse in Early-Stage Breast Cancer[END_REF] or kidney [START_REF] Cooley | Experimental and computational modeling for signature and biomarker discovery of renal cell carcinoma progression[END_REF] cancer. This model was then developed within the framework of NSCLC BMs [START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] . This last work proposed a method to determine the age of the disease according to the pathology and size of the primary tumor, to test different dissemination and colonization scenarios and finally to predict the appearance and growth of BMs over time. However, this preliminary work only included two patients.

The objective of this study was to develop the mathematical model and apply it to a situation of intracranial oligoprogression in multiple patients with early-stage NSCLC, initially treated with a curative intent. The secondary objective was to assess the association of this model with overall survival and progression-free survival.

MATERIALS & METHODS

Data.

We retrospectively investigated all early-stage NSCLC patients treated with a curative intent who had a brain relapse at Public Assistance from Marseille Hospitals (APHM) in the department of Multidisciplinary Oncology and Therapeutic Innovations. We enrolled all patients with BM as the first relapse site. Primary tumor progression or metastatic extracranial progression at the time of the brain oligo-progression were excluded. Patients with ongoing systemic treatment at the time of oligo-progression were also excluded. This study was approved by the institution Data Protection Officer (# 2021-25).

Patients' characteristics (including sex, age, smoking history, performance status), NSCLC data (histology, stage at diagnosis, molecular profile), treatment data (treatment received for the primary tumor, history of BM treatment) and outcome data (size and number of BM at first relapse, longitudinal data of BM, progression-free survival (PFS) and overall survival (OS)) were investigated. OS is the time from diagnosis of the primary tumor to death from any cause (or date of last follow up). PFS was defined as the time between the time point of first metastatic relapse and second metastatic relapse.

Since the model presented in this study is based on numbers of cells, the longitudinal diameter measurements of the data set were converted by assuming spherical shape calculating the volume as 𝑣𝑜𝑙(𝑥) = Model definition. A primary tumor with size 𝑆 𝑃 (𝑡) at time 𝑡 ≥ 0 was assumed to follow Gompertzian growth [START_REF]On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies[END_REF][START_REF] Casey | The Experimental Alteration of Malignancy with an Homologous Mammalian Tumor Material: I. Results with Intratesticular Inoculation[END_REF] . Since patients underwent surgery, we adjusted the growth equation to account for full tumor removal at day of surgery 𝑇 𝑠 > 0 such that 𝑑𝑆 𝑃 (𝑡) 𝑑𝑡 = 𝑔 𝑃 (𝑆 𝑃 (𝑡)) = (𝛼 𝑃 -𝛽 𝑃 ln(𝑆 𝑃 ))𝑆 𝑃 , if 0 < 𝑡 ≤ 𝑇 𝑆 but 𝑆 𝑃 (𝑡) = 0 if 𝑡 > 𝑇 𝑆 . The initial size of a tumor at time 𝑡 = 0 was assumed to be of exactly one cell, i.e., 𝑆 𝑃 (0) = 1 [START_REF] Benzekry | Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth[END_REF] . The parameter 𝛼 𝑃 can be interpreted as the initial specific growth rate of a single tumor cell and the parameter 𝛽 𝑃 is the exponential decay factor of this specific growth rate [START_REF] Vaghi | Population modeling of tumor growth curves and the reduced Gompertz model improve prediction of the age of experimental tumors[END_REF] . The analytical solution of the growth dynamics for the primary tumor and the analogous dynamics for the metastases are given in the supplementary material.

Two additional parameters 𝜇 and 𝛾 were introduced for the dissemination process:

parameter 𝜇 stands for the probability per day and per cell for a tumor cell to succeed going through all steps of the postulated metastatic cascade [START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] . Parameter 𝛾 ∈ [0,1] describes how the geometrical structure of the tumor enables the tumor to seed new metastases. The dissemination rate (number of newly created metastases per day) was therefore assumed to be:

𝑑 (𝑆 𝑝 (𝑡)) = 𝜇𝑆 𝑝 (𝑡) 𝛾 .
The model equations for the number of metastases (visible and invisible) are described in the supplementary material and details to this approach are given in [START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] . Since there is evidence that secondary and thus lately seeded metastases are of minor clinical relevance [START_REF] Bethge | Are Metastases from Metastases Clinical Relevant? Computer Modelling of Cancer Spread in a Case of Hepatocellular Carcinoma[END_REF] , for the purpose of our predictive framework we focused on metastases seeded by the primary tumor only.

Parameter assumptions. Following Bilous et al. [START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] we reduced further the number of free parameters to ensure practical identifiability. We assumed a maximum carrying capacity of 𝑒𝑥𝑝(𝛼/𝛽) = 10 12 cells 62 for both the primary tumor and metastases' growth kinetics.

Since longitudinal pre-diagnosis measurements of primary tumors were not available, we assumed a pathology-dependent doubling time estimated from a meta-analysis of the literature [START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] . Specifically, doubling times were assumed to be 201, 104 and 91 days for adenocarcinoma, squamous cell carcinoma and undifferentiated carcinoma, respectively [START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] . The metastases observed clinically, i.e., the visible BMs, are assumed to be larger than the volume corresponding to a diameter measurement of three mm. The corresponding volumetric measure is denoted by the threshold size 𝑠 𝑣 .

Parameter identification. The procedure to determine the size and number of BMs from the model equations and values of the parameters is described in the supplementary material. We solely used data up to the time of first metastatic relapsedenoted 𝑇 𝐵𝑀and consequently identified the parameters using only one time point with BMs measurements. Preliminary explorations of fitting procedures showed that the parameter identification was very instable due to the unknown metastatic density distribution below the threshold size (𝑠 𝑣 ). We therefore considered two possible scenarios for the density distribution below 𝑠 𝑣 : i) no additional BM, i.e., all existing BM above 𝑠 𝑣 have been detected successfully ('best case') or ii) another BM of size 𝑠 𝑣 , i.e., one BM has been overseen ('worst case'). Our final method was a compromise between i) and ii): we assumed one artificial BM with size 𝑠 𝑣 with an associated number of one-half BM. From this artificial measurement, we were able to control the important variance of the metastatic density below 𝑠 𝑣 .

The doubling time 𝜏 𝑃 (𝑠) of a primary tumor of size 𝑆 𝑃 (𝑇 𝑑 ) at time of diagnosis 0 ≤ 𝑇 𝑑 ≤ 𝑇 𝑆 was used to determine the two growth parameters 𝛼 𝑃 and 𝛽 𝑃 (see supplementary material). Preliminary exploration of the parameter space suggested that a value of 𝛾 = 0.1 was able to describe the data. To ensure practical identifiability of the model parameters we therefore fixed 𝛾 = 0.1 and only considered parameters 𝛼 and 𝜇 as variables during model fitting. The model was implemented in Matlab. A least squares calculation comparing model evaluations against the empirical data of the cumulative metastatic density distribution at the time of first metastatic relapse was performed using the GlobalSearch environment [START_REF]MATLAB Documentary GlobalSearch[END_REF][START_REF] Ugray | Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization[END_REF] , fitting for the two parameters 𝛼 and 𝜇 on biologically reasonable intervals (i.e. positive values), respectively.

Statistical analysis. These two computational parameters 𝛼 and 𝜇 were used as computational biomarkers with potential additional prognosis value than the mere clinical data. We analyzed association of these parameters with overall survival and progressionfree survival using a previously introduced method [START_REF] Benzekry | Development and Validation of a Prediction Model of Overall Survival in High-Risk Neuroblastoma Using Mechanistic Modeling of Metastasis[END_REF] . Log-rank statistical tests were performed on dichotomized groups (with threshold taken at the median value of the computational biomarker) [START_REF] Bland | The logrank test[END_REF] . In addition, multivariable proportional hazard Cox regression models were considered, either using only the clinical covariates or the clinical covariates augmented with 𝛼 and 𝜇. These were implemented using the lifelines python package [START_REF] Davidson-Pilon | lifelines: survival analysis in Python[END_REF] .

The predictive ability of these Cox models was determined calculating the mean value of Harrell's c-index [START_REF] Harrell | Multivariable prognostic models: issues in developing models, evaluating assumptions and adequacy, and measuring and reducing errors[END_REF] of one hundred replications of a three-fold cross-validation procedure.

In a closer examination, only covariates below a significance level p < 0.2 were selected for multivariable analysis. To determine the statistical significance of mean c-indices between Cox models, two-sample t-tests were conducted.

RESULTS

Data presentation and description

We identified 31 patients who had intracranial oligoprogression after initial treatment with a curative intent for early-stage NSCLC. Of these, 10 were women and 21 were men. The data included values of longitudinal diameter measurements determined via computed tomography (CT) and magnetic resonance imaging (MRI) images of primary tumors and BMs as well as the time of relapse and time of death. Representative patient plots of these time series data are reported in Figure 1.

Out of the 31 patients, three patients had no primary tumor measurements at primary diagnosis. In the later analysis, these patients had to be excluded, since the primary tumor growth parameter calibration could not be performed (cf. Mathematical model).

Figure 1. Data: Example patients. Time course of three example patients. Black bullets show the primary tumor size and colored bullets the brain metastases sizes. The time course was shifted such that time zero corresponds to the day of primary diagnosis. Also shown are local and systemic treatments (triangle, cross, flash, circle, colored background), the presence of extra-cerebral metastases (black asterisk, not linked to diameter axis) and the time of the respective patient's death (black cross with dashed line).

Parameter values

As explained above, the remaining parameters subject to identification through data fitting were 𝛼 (growth) and 𝜇 (dissemination). Parameter values are shown in Table 2.

Simulations and outcome

The model was used to simulate history of all BMs, including those not yet clinically detectable at 𝑇 𝐵𝑀 . To ensure parameter identifiability, a specific method was introduced (cf. Methods). The total number of predicted BMs at time 𝑇 𝐵𝑀 (clinically visible + invisible) was compared to the number of BMs detected during the full clinical history of the individual patient (Figure 2).

Individual simulation of BM history for a specific patient is presented in Figure 3.

Analogous simulations for all patients are shown in the supplementary material. Figure 3A shows the data of the clinical history. Panels 3B and 3C compare the model-predicted metastatic distributions at the two time points of primary diagnosis and 𝑇 𝐵𝑀 , respectively.

The whole-time course of primary tumor and metastases sizes is presented in Figure 3D.

The simulated visible metastases in comparison to the clinically detected metastases at time 𝑇 𝐵𝑀 as well as the size distributions of BMs at 𝑇 𝐵𝑀 are presented in Figure 3E.

Compared to the data, the model calculations produced reasonable estimations of size and number of visible BMs at 𝑇 𝐵𝑀 throughout the data set. For 20 patients the calculated theoretical number of BMs at time 𝑇 𝐵𝑀 was greater or equal to the number of BMs observed during the full clinical treatment of the respective patient. We considered this not only as a reasonable estimate but also a good indication that the model-predicted total number and size of BMs proves to be useful in estimating the invisible BMs early in the treatment course.

Figure 2. Population results for the calculated number of BMs (visible + invisible) compared to the clinically observed number over the treatment course.

Estimations that prognosed less BMs than clinically observed are indicated with red crosses. Estimations that prognosed the correct or a larger number of BMs than observed are indicated as blue diamonds. Several points correspond to multiple cases:

(1,1) four times, (1,2) five times, (2,1) two times, (4,1) two times and (3,5) two times.

Figure 3. Simulation of example patient. A. Time series data of an example patient. Data time points used for estimation of the deterministic model parameters are the primary diagnosis (blue dashed line) and the time of first metastatic relapse 𝑇 𝐵𝑀 (red dashed line). The size of the primary tumor at primary diagnosis determines the growth parameters for the primary tumor growth dynamics (cf. text). From the number and size of metastases at first metastatic relapse the two free model parameters were fitted. B. Model calculation allows to compute the metastatic distribution over time for the primary diagnosis. C. Further calculation shows the metastatic distribution at time 𝑇 𝐵𝑀 . Visible metastases are the ones to the right of the imaging detection limit (gray dotted line). D. Growth dynamics can be followed for the primary tumor (blue solid line) and the metastases (red solid line) respectively. E. Comparison of the computed BMs sizes at time 𝑇 𝐵𝑀 (black) with the observed data (gray).

We then compared two patients with the same number of metastases at time 𝑇 𝐵𝑀 and relatively familiar time frames. Interestingly, the model yielded very different profiles for the invisible BMs and therefore might suggest different treatment approaches for the two patients (Figure 4). Even though the clinical data at 𝑇 𝐵𝑀 was very similar for the two patients, the model predicted different outcomes. For instance, the number of visible and invisible BMs at 𝑇 𝐵𝑀 were 5 and 8, respectively. These numbers again appeared reasonable when compared to the full clinical treatment course of each patient.

Figure 4. Comparison of two patients. The upper row shows the full clinical history of the two patients. The estimation of the metastatic distribution at time of first metastatic relapse and the spherical representations of the corresponding metastases' sizes (black) are shown in the lower row for the respective patient, compared to the data of clinically detectable metastases (gray).

Statistical evaluation and prediction models

To assess the predictive power of the computational markers 𝛼 and 𝜇, dichotomized Kaplan-Meier were generated for OS and PFS (Figure 5). For 𝛼, significant differences were observed for OS (𝑝 = 0.0026) and PFS (𝑝 = 0.0108). For 𝜇, a significant difference was found at the 80% quantile for OS (𝑝 = 0.0356) and PFS (𝑝 = 0.0254), thus identifying a fraction of high-risk patients. We further assessed the association with outcome of 𝛼 and 𝜇, as well as the clinical variables in univariate Cox models (Table 3). Here, we found statistically significant hazard ratios for both the computationally determined markers 𝛼 and 𝜇 for OS (𝑝 = 0.0229 and 𝑝 = 0.0011, respectively).

Figure 5. Kaplan-Meier curves for the computational parameters 𝜶 and 𝝁. The survival curves for the two computational parameter groups above and below the population's median value (first row) and the 80% quantile value (second row) with respect to overall survival (first and third column) and progression-free survival (second and fourth column). The corresponding calculated p-value from the log-rank test for dichotomized groups is shown in the right lower corner.

To determine the added value of the computational markers in comparison to the clinical data, we tested their significance in multivariable Cox models built on the clinical data alone or the clinical data and the computational markers (Figure 6 and Table 4).

Table 3: Results of the univariate Cox regression for OS and PFS. The table shows the characteristics of the results of the univariate Cox regression including clinical and computational covariates for overall survival and progression-free survival. HR = hazard ratio, CI = confidence interval, BM = brain metastases, PT

Table 4. Results of the multivariate Cox regression for OS and PFS. The table shows the characteristics of the results of the multivariate Cox regression for the predictive model including clinical and computational covariates for overall survival and

progression-free survival. HR = hazard ratio, CI = confidence interval, BM = brain metastases, PT = primary tumor. For OS, the c-index for those two cases were 0.585 (95% CI 0.569 to 0.602) and 0.713 (95% CI 0.700 to 0.726), respectively (statistically significant increase, p<0.0001). This drastic increase of predictive value also held in reduced models containing only variables with p < 0.2 in the multivariate analysis (0.647 (95% CI 0.631 to 0.664) versus 0.789 (95% CI 0.779 to 0.800), statistically significant increase, p<0.0001). For PFS, the c-indices were 0.560 (95% CI 0.545 to 0.575) and 0.595 (95% CI 0.582 to 0.608), respectively (statistically significant increase, p<0.001). Considering again only variables with p < 0.2 in the multivariate regression yielded c-indices of 0.708 (95% CI 0.697 to 0.718) versus 0.703 (95% CI 0.693 to 0.713).

Hazard

Interestingly in both cases for OS and PFS, the prediction model with clinical variables identifies the number of visible brain metastases as variable with large hazard ratio, but the prediction model with the computational markers selects 𝜇 as having better predictive power. That is, the number of brain metastases at first metastatic relapse loses significance when the prediction model is augmented with 𝜇. For OS, the other selected covariates besides 𝜇 were 𝛼, the age at diagnosis and the summed volume of BMs at first

DISCUSSION

Because of their prevalence and high morbidity, the management of BMs is a major issue in NSCLC. Given the heterogeneity of clinical cases and variability of treatments available, therapeutic decisions following intracranial oligoprogression are complex

Predicting the onset and evolution of BMs in NSCLC patients using mathematical modeling would offer decision support with a personalized follow-up and treatment. The objective of this study was to test a mathematical model previously developed and to apply this model to a situation of intracranial oligoprogression in patients with early-stage NSCLC initially treated with a curative intent.

We first demonstrated that the mathematical model was suitable to describe clinical history of size and number of BMs in most patients. Simulations aligned with the data for 20 patients, while in eight cases the model underestimated the total number of BMs.

However, among these eight cases the simulations correctly identified the presence of multiple BMs. Among the 8 patients, there are 3 squamous cell carcinomas and 5 adenocarcinomas, 3 of which had a mutation of interest: KRAS G12C, STK11, BRAF.

These mutations may lead to a different growth of the primary tumor or a different metastatic dissemination. BRAF and STK11 have been described as genes involved in the metastatic process [START_REF] Azin | STK11 Loss: A Novel Mechanism for Melanoma Metastasis with Therapeutic Implications[END_REF][START_REF] Huang | Mutations of key driver genes in colorectal cancer progression and metastasis[END_REF] . Moreover, one of the squamous cell carcinoma had a PTEN mutation, which has been shown to be associated with BM in breast cancer [START_REF] Wikman | Relevance of PTEN loss in brain metastasis formation in breast cancer patients[END_REF] . Epidermoid histology and molecular biology possibly misled the model.

In addition, predictive models of OS including the computational markers 𝛼 and 𝜇 strongly outperformed models based on clinical variables only. For the metastatic growth rate 𝛼, significant differences were observed for OS and PFS. For the metastatic seeding 𝜇, a significant difference was found at the 80% quantile for OS and PFS, thus identifying a fraction of high-risk patients.

Identification of a high-risk group for recurrence could assist the lung cancer physicians in therapeutic management.

In univariate analysis, WBRT at first relapse had a significant effect on PFS (HR 0.649 (95% CI 0.442 -0.955), p=0.0282) and systemic treatment at first relapse had a minor non-significant effect (HR 0.824 (95% CI 0.546 -1.25), p=0.359). This could be due to undetected brain metastases at 𝑇 𝐵𝑀 , or dormant metastatic cells. Systemic treatment or WBRT may therefore be of value for patients at risk. Indeed, WBRT has the advantage of covering all macroscopic and microscopic BM to control cerebral progression.

However, WBRT does not increase survival [START_REF] Aupérin | Prophylactic Cranial Irradiation for Patients with Small-Cell Lung Cancer in Complete Remission[END_REF][START_REF] Liu | The Role of Prophylactic Cranial Irradiation in Patients With Non-small Cell Lung Cancer: An Updated Systematic Review and Meta-Analysis[END_REF] and leads to neurocognitive decline [START_REF] Chang | Neurocognition in patients with brain metastases treated with radiosurgery or radiosurgery plus whole-brain irradiation: a randomised controlled trial[END_REF][START_REF] Soffietti | A European Organisation for Research and Treatment of Cancer phase III trial of adjuvant whole-brain radiotherapy versus observation in patients with one to three brain metastases from solid tumors after surgical resection or radiosurgery: quality-of-life results[END_REF] .

The effect of systemic treatments seems less obvious given their variable intracranial efficacy. Our model could lead to a computational tool that could assist lung cancer clinical oncologists in identifying patients with high-risk of multiple recurrences that could benefit from systemic treatments.

Other tools exist to help the management of NSCLC patients. Some are based on molecular biology prognostic scores such as the Lung-Mol Score [START_REF] Sperduto | Estimating Survival in Patients With Lung Cancer and Brain Metastases: An Update of the Graded Prognostic Assessment for Lung Cancer Using Molecular Markers (Lung-molGPA)[END_REF] , others rely on radiomics features [START_REF] Lohmann | PET/MRI Radiomics in Patients With Brain Metastases[END_REF][START_REF] Zhao | MRI radiomic signature predicts intracranial progression-free survival in patients with brain metastases of ALK-positive non-small cell lung cancer[END_REF][START_REF] Huang | Radiomics as prognostic factor in brain metastases treated with Gamma Knife radiosurgery[END_REF][START_REF] Zhang | Computer Tomography Radiomics-Based Nomogram in the Survival Prediction for Brain Metastases From Non-Small Cell Lung Cancer Underwent Whole Brain Radiotherapy[END_REF] . For example, Huang et al. demonstrated that radiomics features extracted from pre-treatment T1 MRI was an independent prognostic factor of local control for BM patients who underwent gamma knife radiosurgery [START_REF] Huang | Radiomics as prognostic factor in brain metastases treated with Gamma Knife radiosurgery[END_REF] . The advantage of our mathematical model compared to these tools is, that it is of biologically grounded and mechanistic nature. In turn, this allows to perform simulations of the disease and clinical course combined with predictive power, unlike the prognostic tools mentioned above.

The main limitation of our study is the number of patients included. It will be necessary to confirm our findings on a larger number of patients and to include molecular profiling data.

In addition, we could conduct a prospective study to keep double-blinded design for predictions.

On the other hand, the population chosen in this study (patients with early-stage NSCLC initially treated with a curative intent and presenting with intracranial oligoprogression) limits the application of our results. However, this made it possible to limit bias of systemic treatments and in this situation, lung cancer physicians are facing an unmet need with no existing guidelines.

The modeling approach remains a simplified description of the complexity of the metastatic cascade that is yet incompletely understood [START_REF] Perlikos | Key molecular mechanisms in lung cancer invasion and metastasis: A comprehensive review[END_REF] . However, considering the limited data available, the model complexity had to be reduced to ensure practical identifiability of the parameters. This resulted in only two critical free parameters controlling growth (𝛼) and dissemination (𝜇). An assumption that leaves for improvement is that primary tumor growth parameters were determined from meta-analyses values using only the tumor pathology [START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] . We need further investigation to understand the influence of molecular biology, especially EGFR mutations [START_REF] Nakamura | Epidermal Growth Factor Receptor Mutations: Effect on Volume Doubling Time of Non-Small-Cell Lung Cancer Patients[END_REF][START_REF] Zhang | Volume doubling time of lung adenocarcinomas considering epidermal growth factor receptor mutation status of exon 19 and 21: three-dimensional volumetric evaluation[END_REF] or ALK translocation because these patients have more multiple brain relapses but have better prognosis [START_REF] Robin | Excellent Outcomes with Radiosurgery for Multiple Brain Metastases in ALK and EGFR Driven Non-Small Cell Lung Cancer[END_REF] .

Moreover we need to understand the impact of treatments on the primary tumor, particularly with the advent of neoadjuvant and adjuvant treatments, which could reduce micrometastatic disease and thus risk of relapse [START_REF] Wu | Osimertinib in Resected EGFR -Mutated Non-Small-Cell Lung Cancer[END_REF][START_REF] Forde | Neoadjuvant Nivolumab plus Chemotherapy in Resectable Lung Cancer[END_REF][START_REF] Felip | Adjuvant atezolizumab after adjuvant chemotherapy in resected stage IB-IIIA non-small-cell lung cancer (IMpower010): a randomised, multicentre, openlabel, phase 3 trial[END_REF] .Moreover, our model could be extended to account for more complex processes of the metastatic cascade, such as dormancy [START_REF] Endo | Dormancy in cancer[END_REF] and the possibility that the primary tumor is not the only one that can spread new lesions [START_REF] Bethge | Are Metastases from Metastases Clinical Relevant? Computer Modelling of Cancer Spread in a Case of Hepatocellular Carcinoma[END_REF] . Regarding the relapse site, we must understand the influence of the bloodbrain barrier and the unique brain microenvironment [START_REF] Quail | The Microenvironmental Landscape of Brain Tumors[END_REF] .

However, simplifying the metastatic cascade increases the feasibility of the use of this model in clinical practice. Once the model is validated on a larger scale, applications will be developed for the clinicians to determine the risk of cerebral recurrence for each patient by selecting two or three clinical parameters available in routine.

To our knowledge, this is the first mechanistic modeling approach to predict the occurrence and growth of BMs in NSCLC with data routinely acquired in the clinics. This work could allow to identify patients at high risk of recurrence and to propose a personalized management such as follow-up imaging schedule, addition of systemic or non-systemic therapy, and use of WBRT depending on the patient's risk of multiple recurrences.

S2: Model equation for the metastatic density 𝜌(𝑠, 𝑡) of size 𝑠 and time 𝑡

The full equation corresponding to the approach of [START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] The used parameters are explained in the text.

S3: Parameter estimation: growth parameters

The doubling time 𝜏 𝑃 (𝑠) of a primary tumor of size 𝑆 𝑃 (𝑇 𝑑 ) at time of diagnosis 0 ≤ 𝑇 𝑑 ≤ 𝑇 𝑆 , i.e. the time a tumor of size 𝑠 needs to grow to a size of 2𝑠, is implicitly given by [START_REF] Schlicke | How mathematical modeling could contribute to the quantification of metastatic tumor burden under therapy: insights in immunotherapeutic treatment of non-small cell lung cancer[END_REF] 𝜏 𝑃 (𝑠) = -1 𝛽 𝑃 ln ( 𝛽 𝑃 ln(2𝑆 𝑃 ) -𝛼 𝑃 𝛽 𝑃 ln(𝑆 𝑃 ) -𝛼 𝑃 )

and from 𝑒𝑥𝑝(𝛼 𝑃 /𝛽 𝑃 ) = 10 12 we determined the parameters 𝛼 𝑃 and 𝛽 𝑃 from the primary tumor histology and the primary tumor size at diagnosis. Since on the other hand 𝑒𝑥𝑝(𝛼/𝛽) = 10 12 for the metastases' dynamics, the parameter 𝛽 was determined from the value of parameter 𝛼, which in turn was a fitted parameter (cf. Fitting procedure).

S4: Numerical implementation: calculate metastases

Since we assumed that the primary tumor was the only source for metastases, we applied the procedure from [START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] These numbers are the values that we observed clinically -the visible BMs in the data are assumed to correspond to the sizes of metastases larger than the volume corresponding to a diameter measurement of three mm. Thus, the number of metastases larger than this threshold size 𝑠 𝑣 at time 𝑡 can be calculated as 𝑓(𝑡, 𝑠 𝑣 ).

S5: Model simulations for every individual patient of the data set

The explanation of every subpanel is given in the description of figure 3 of the main text.

3 where

 3 𝑥 is the individual diameter value. These volumetric data were then rescaled into cell numbers using the conversion rule 1mm 3 = 10 6 cells 55 .

Figure 6 .

 6 Figure 6. Hazard ratios for the predictive models. The hazard ratios were determined performing a multivariable Cox regression. The ratios are corresponding to the predictive models for OS (left) and PFS (right) with clinical and computational covariates, i.e., the settings where the determined parameters have been added to the respective patient data.

  modelling the metastatic density 𝜌(𝑠, 𝑡) as function of size 𝑠 and time 𝑡 with boundary and initial conditions neglecting secondary metastasationnot considering the integral in the boundary conditionreads 𝜕𝜌(𝑠, 𝑡) 𝜕𝑡 + 𝜕(𝑔 𝑚 (𝑠)𝜌(𝑠, 𝑡)

  to calculate the absolute number of metastases 𝑁(𝑡) at a certain time 𝑡 from 𝑁(𝑡) = ∫ 𝜇 𝑡 0 𝑆 𝑃 (𝜏) 𝛾 𝑑𝜏 and to introduce the cumulative size distribution 𝑓(𝑡, 𝑠) as the number of metastases larger than a fixed size 𝑠 at a certain time 𝑡 as 𝑓(𝑡, 𝑠) 𝜎)𝑑𝜎.With 𝑡(𝑠) as the time for a tumor to reach size 𝑠 under Gompertz growth[START_REF] Bilous | Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer[END_REF] we receive 𝑓(𝑡, 𝑠) = 𝑁(𝑡 -𝑡(𝑠))

  There were 9 squamous cell carcinomas and 22 non-squamous carcinomas, including 20 adenocarcinomas. At the time of diagnosis, 20 patients were stage I/II and 11 were stage III according to the 8th TNM classification. Number of BMs lied between one and three and the corresponding diameter sizes between 4 mm and 75 mm. Treatment received for the primary tumor, the number, and the treatment of brain metastasis at the first relapse are reported inTable 1. 17 patients had more than one intracranial recurrence. Median OS was 80.98 months (95% CI 37.56 -98.27) and median PFS was 11.73 months (95% CI 4.93 -27.20).

Table 1 . Data description: Patient overview

 1

	Data population	Population N 31 (%)
	Sex
	Female -Male	10 (32.3%) -21 (67.7%)
	Median Age at Diagnostic	61 (39-82)
	Smoking Status
	Smoker	11 (35.5%)
	Former smoker	13 (41.9%)
	No smoker	1 (3.2%)
	Stage at diagnostic
	I/II	20 (65%)
	III	11 (35%)
	Primary tumor pathology subtype

. PT = Primary tumor, BM = Brain metastases, SRS = stereotaxic radiosurgery, WBRT = whole brain radiotherapy

Median time between the first and second relapse (months) 11

  .73 (95%CI 4.93 -27.20)

Table 2 .

 2 Parameter description. Some measures for the fitted parameter values on the population level, std = standard deviation.

	parameter mean	median	std	min	max
	𝛼	4.93 ⋅ 10 -2 4.03 ⋅ 10 -2 3.18 ⋅ 10 -2 1.22 ⋅ 10 -2 1.34 ⋅ 10 -1
	𝜇	3.57 ⋅ 10 -4 2.33 ⋅ 10 -4 2.77 ⋅ 10 -4 8.22 ⋅ 10 -5 1.22 ⋅ 10 -3

  = primary tumor.

		Hazard	p-value	95% CI (OS) Hazard	p-value	95% CI (PFS)
		Ratio	(OS)		Ratio	(PFS)
		(OS)			(PFS)
	Covariate
	Sex	1.1	0.602	0.768 -1.58	0.984	0.936	0.663 -1.46
	Age at diagnosis	1.03	0.900	0.679 -1.55	0.738	0.167	0.479 -1.14
	Number of bm at relapse	1.72	0.00556	1.17 -2.53	1.25	0.255	0.851 -1.84
	Stage at diagnosis	0.866	0.435	0.604 -1.24	1.1	0.674	0.716 -1.67
	Histology of pt	0.792	0.223	0.545 -1.15	0.715	0.117	0.47 -1.09
	Parameter 𝛼	1.65	0.0229	1.07 -2.53	1.25	0.237	0.862 -1.83
	Parameter 𝜇	1.95	0.00109	1.31 -2.91	1.54	0.0729	0.961 -2.46
	Size of PT at diagnosis	1.05	0.781	0.755 -1.45	0.9	0.648	0.574 -1.41
	Size of BM at relapse
	Whole-brain radiotherapy
	after 𝑇 𝐵𝑀
	Systemic treatment after
	𝑇 𝐵𝑀

ACKNOWLEDGEMENTS

PS wants to thank Christina Kuttler for valuable discussions on the mathematical modelling method. Supported by Deutsche Forschungsgemeinschaft (DFG) through TUM International Graduate School of Science and Engineering (IGSSE), GSC 81.

CONTRIBUTIONS

Conceived the research idea: SB, PT. Model setup: PS, SB. Collected the data: ES, PT.

Performed data analysis and presentation: PS. Wrote software to estimate parameters and simulate: PS, SB. The paper was written by ES and PS with editorial input from all authors. All authors read and approved the final manuscript.

AVAILABILITY OF DATA AND MATERIALS

The datasets used and analyzed during this study are available from the corresponding author on reasonable request.

SUPPLEMENTARY MATERIAL

S1: Analytical solution of the primary tumor and metastases' growth dynamics

Variables and parameters are explained in the text. The analytical solution to the primary tumor dynamics can be determined to be

and the metastases' Gompertz growth function is given analogously, without the possible surgery modelled, as 𝑔 𝑚 (𝑠) = (𝛼 -𝛽ln(𝑠))𝑠.

The metastatic Gompertzian growth equation as analytical solution of the equation

with initial condition 𝑠(0) = 1 that describes the metastatic size 𝑠 at time 𝑡 ≥ 0 reads 𝑠(𝑡) = 𝑒𝑥𝑝 ( 𝛼 𝛽 (1 -exp(-𝛽𝑡))).


Mickaël Jury
email: mickael.jury@uca.fr

Aurélie Laurence

Sylvie Cèbe

Caroline Desombre

David Pérez-Jorge

Ana Isabel González

Herrera

María Del

Carmen Rodríguez Jiménez

Keywords: teachers, concerns, attitude, France, inclusive education

The present research investigates teachers' concerns about inclusive education and said concerns' specific links with attitudes teachers have regarding this policy. More precisely, while the literature has already identified that teachers share distinct concerns regarding inclusive education (e.g., lack of resources, increased workload, appropriateness, or the difficulties associated with this policy), the first purpose of this exploratory study was to identify whether and how teachers specifically rank these concerns in terms of priority. Additionally, while teachers' concerns can be negatively associated with attitudes toward inclusive education, the present study also sought to investigate the distinct manner in which teachers' concerns predict these attitudes. A correlational study involving 508 French teachers was conducted indicating that teachers express the most concern about, by order of descending importance, resources, workload, difficulties, and appropriateness. Moreover, despite the fact that the latter factor was deemed least important, it most consistently predicted the voicing of negative attitudes by teachers toward inclusive education. This interesting paradox, as well as the implications and limitations of the present findings, are discussed.

frontiersin.org

Teachers' concerns about inclusive education and the links with teachers' attitudes

Introduction

Enabling every child to benefit from a mainstream education, regardless of their special educational needs (SEN), remains one of the major challenges facing our education systems [START_REF] Ferguson | International trends in inclusive education: the continuing challenge to teach each one and everyone[END_REF]. Indeed, improving social participation and the learning opportunities of students with SEN is all too necessary but appears costly at every level, from the policy level (i.e., investment of money to change the education system) down to that of practitioners (i.e., modifying teaching practices to meet students' needs). It is therefore not surprising that teaching staff should express concerns regarding this policy [START_REF] Sharma | Measuring concerns about integrated education[END_REF]. Our aim with the present exploratory research is twofold. First, we seek to examine the manner in which teachers potentially rank these concerns hierarchically and, second, how said ranking relates to teachers' attitudes toward this policy.

Teachers' concerns about inclusive education

The inclusion of students with special educational needs has brought about a genuine upheaval in teachers' daily routines. Indeed, they have gone from teaching relatively homogeneous groups of students for whom a single pedagogical method was used to deal with a student body for whom a single traditional method is no longer sufficient. In addition, while teachers used to be relatively alone in managing the classroom, they must now collaborate with teaching or students' assistants, special education teachers and even paraprofessionals (e.g., speech or occupational therapists, medical doctors). In other words, teaching has changed, and as with all changes and innovations in education, it has given rise to concerns.

The literature regarding concerns about inclusive education is extensive, as teachers' concerns were rapidly identified as a hindrance to the full implementation of inclusive education. For our purposes, concerns refer to the "questions, uncertainty and possible resistance that teachers may have in response to new situations and/or changing demands" (Yan and Deng, 2019, p. 385). In other words, concerns are negative expectations related to a specific event or behavior. Not surprisingly, such concerns are negatively associated with teachers' attitudes toward inclusive education (i.e., the more concerns teachers have regarding inclusive education, the less they support the implementation of this policy) and intentions to use inclusive practices (i.e., the more concerns teachers have regarding inclusive education, the lesser their intentions to use endorse inclusive teaching practices, see [START_REF] Miesera | Correlation between attitudes, concerns, self-efficacy and teaching intentions in inclusive education evidence from German pre-service teachers using international scales[END_REF].

Regarding types of concern, both qualitative and quantitative studies show that teachers fairly universally and consistently indicate that they the lack resources to include students with SEN. They state that their workload is increasing, that they are experiencing more difficulties with classroom management and/ or that the inclusion of students with SEN reduces teaching/ education quality (see [START_REF] Sharma | Measuring concerns about integrated education[END_REF][START_REF] Bhatnagar | Nearly two decades after the implementation of persons with disabilities act: concerns of Indian teachers to implement inclusive education[END_REF][START_REF] Miesera | Correlation between attitudes, concerns, self-efficacy and teaching intentions in inclusive education evidence from German pre-service teachers using international scales[END_REF][START_REF] Sahli Lozano | The concerns about inclusive education scale: dimensionality, factor structure, and development of a shortform version (CIES-SF)[END_REF][START_REF] Warnes | Mainstream teachers' concerns about inclusive education for children with special educational needs and disability in England under pre-pandemic conditions[END_REF]. Nevertheless, it should be noted that in the reference tool to record these concerns (i.e., the "Concerns about Inclusive Education Scale"-CIES, [START_REF] Sharma | Measuring concerns about integrated education[END_REF], the scope of these concerns is larger since for example, for teachers' concerns about resources, items focus on funding, staff, infrastructure or learning material (see method section for more details regarding the specific tool we will use here). Put differently, the four abovementioned concerns should be considered as quite broad categories.

Since its publication, this tool has been used in a large number of studies throughout the world (e.g., Australia, Korea, and Switzerland), often with the goal of comparing individuals' concerns based on their professional status (i.e., teacher or school principal, [START_REF] Sharma | Measuring concerns about integrated education[END_REF][START_REF] Warnes | Mainstream teachers' concerns about inclusive education for children with special educational needs and disability in England under pre-pandemic conditions[END_REF], country of origin [START_REF] Sharma | In-service teachers' attitudes, concerns, efficacy and intentions to teach in inclusive classrooms: an international comparison of Australian and Italian teachers[END_REF], or training [START_REF] Sokal | How important is practicum to pre-service teacher development for inclusive teaching? Effects on efficacy in classroom management[END_REF][START_REF] Song | Impact of teacher education on preservice regular school teachers' attitudes, intentions, concerns and self-efficacy about inclusive education in South Korea[END_REF]. The main reason behind such interest was, and remains, that better understanding teachers' preoccupations regarding inclusive education could open doors in the development of specific responses (from policymakers to special education teachers) to help teachers overcome negative expectations and allow students with SEN to fully benefit from mainstream education.

Hierarchy of concerns and attitudes toward inclusive education

Quite surprisingly, none of the abovementioned studies specifically and statistically compare teachers' concerns in order to identify a hierarchy among them. It is nevertheless possible that teachers express distinct levels of preoccupation by domain. As a support to this claim, [START_REF] Warnes | Mainstream teachers' concerns about inclusive education for children with special educational needs and disability in England under pre-pandemic conditions[END_REF], in the qualitative part of their study, cited "Resources" as the greatest concern spontaneously expressed by teachers. We have synthesized the available quantitative data from the studies having used the CIES (and identified the same 4-factor structure) within Table 1, which tend to confirm that participants did not rate the four concerns at the same level. More precisely, while their order slightly differs between studies, a grand mean calculation suggests that participants express greater concern regarding the lack of resources -by order of preoccupation-than acceptance of students with SEN, the potential threat to academic standards, and, finally, the projected increase in workload. Such a comparison invites more rigorous study, since identifying how these core concerns are hierarchized among teachers could be additionally relevant to the implementation of inclusive education (by providing more precise knowledge about where to direct the response).

In the same vein, level of concern may be negatively associated with teachers' attitudes toward inclusive education [START_REF] Sokal | Canadian in-service teachers' concerns, efficacy, and attitudes about inclusive teaching[END_REF][START_REF] Miesera | Correlation between attitudes, concerns, self-efficacy and teaching intentions in inclusive education evidence from German pre-service teachers using international scales[END_REF] even if this relationship is not always observed, see for example [START_REF] Yada | Japanese in-service teachers' attitudes toward inclusive education and self-efficacy for inclusive practices[END_REF]. There might also be added value in knowing how specifically distinct concerns are associated with teachers' attitudes in order to develop effective (specific) responses regarding the acceptance of inclusive education. The purpose of this exploratory study is thus to provide preliminary answers to these two questions: (1) Are concerns ranked hierarchically among teachers, and if so, how? And (2) are concerns specifically related to teachers' attitudes toward inclusive education, and if so, how? It should be noted that the present study was conducted with French teachers. The chosen context is particularly relevant to the teachers' concerns, since inclusive education has been one of the main transformations introduced in France's education system over the last decade (with numerous changes within the past 5 years especially). The debate over the legitimacy of this paradigm (and thus the associated preoccupations) remains particularly vivid in the country (see [START_REF] Ployé | L'inclusion scolaire en France, un processus inachevé [inclusive education in France, an unfinished process[END_REF][START_REF] Galle | La scolarisation des élèves en situation de handicap: l'inclusion en question [the schooling of students with disabilities: inclusion in question[END_REF][START_REF] Jellab | Les élèves à besoins éducatifs particuliers à l' épreuve de l' égalité des chances ou ce que la «marge» pourrait faire à la «norme»: vers Une inclusion réciproque au sein des EPLE? [pupils with special educational needs put to the test of equal opportunities or what the "margin" could do to the "norm[END_REF].

This study has been pre-enregistered (AsPredicted #90237, see here: https://aspredicted.org/KTL_7G6) and the associated data and materials are fully accessible here: https://osf.io/fsjcu.

Frontiers in Education 03 frontiersin.org

Materials and methods

Participants

Due to the correlational nature of the study, we used [START_REF] Schönbrodt | At what sample size do correlations stabilize?[END_REF] work on critical sample size to detect a small-to-medium effect size with an 80% confidence level, determining that a sample size of 250 participants should be sufficient to obtain stable correlations. During the period in which the study was promoted on social and professional networks (i.e., 3 weeks), 508 participants completed the questionnaire. This sample included 451 women and 31 men, with 26 missing data. Most participants were over 40 (55.4%) and had more than 10 years of experience as in-service teachers (63%) in elementary schools and kindergartens (86%). Finally, almost all participants had experience with inclusive education (96.3%), but only a slight majority were familiar with disability (56.5%).

Material

Concerns about inclusive education

To measure teachers' concerns about inclusive education, the short-form version of the CIES validated by Sahli Lozano et al. ( 2022) was used. This scale includes 12 items in total, 3 for each concern. For resources, items cover concerns regarding staff and funding. For workload, items seek to capture teachers' concerns regarding the extended time they would need and the few incentives they would get while including students with SEN. It should be noted that in comparison with the original scale developed by [START_REF] Sharma | Measuring concerns about integrated education[END_REF], the factor "Acceptance" has been replaced by "Difficulties." This new factor encompasses 3 items solely focused on the difficulties associated with the implementation of inclusive education (in terms of discipline, equal attention, or emotion). Similarly, the factor "Academic Standard" has been slightly reorganized into an "Appropriateness" factor which only include items pertaining to declining teaching performance in teaching and learning in inclusive classrooms (for full details regarding the changes, see Sahli [START_REF] Sahli Lozano | The concerns about inclusive education scale: dimensionality, factor structure, and development of a shortform version (CIES-SF)[END_REF]. Finally, additional modifications have been made for the purposes of this specific study. First, the research team translated items into French in order to make them accessible to the participants. Second, instead of using a 4-point Likert scale, participants were prompted to indicate the extent to which they agree with each item from 1, Totally disagree to 5, Totally agree. Third and finally, items in the original scale are written in the future tense (e.g., "My workload will increase"). However, since inclusive education is a reality, the research team chose to rewrite them in present tense, specifying the inclusive education context (e.g., "With inclusive education, my workload is increasing," see online material).

If replicating the confirmatory factor analysis conducted by Sahli Lozano et al. ( 2022) translated a good fit of our model (CFI = 0.996; RMSEA = 0.029; SRMR = 0.042), it should be noted that reliability analyses were less satisfactory for the factors "Resources" (ω = 0.59) and "Workload" (ω = 0.64), but good for "Difficulties" (ω = 0.75) and "Appropriateness" (ω = 0.87).

Attitudes toward inclusive education

In order to measure teachers' attitudes toward inclusive education, participants were prompted to indicate the extent to which they agree with six items extracted from the MATIES scale [START_REF] Mahat | The development of a psychometrically-sound instrument to measure teachers' multidimensional attitudes toward inclusive education[END_REF], one of the most psychometrically sound questionnaires regarding this question, [START_REF] Ewing | Teachers' attitudes towards inclusive education: a critical review of published questionnaires[END_REF]. In her original scale, the author proposed a measure to assess the cognitive, affective, and behavioral components of teachers' attitudes toward inclusive education. For the present study, only the six items referring to the cognitive component were used (e.g., "I believe that an inclusive school is one that permits academic progression of all students regardless of their ability"; "I believe that students with disabilities should be educated in special schools"-reverse coded-). Participants were asked to indicate their level of agreement on the same 5-point scale as earlier. The reliability analysis was satisfactory (ω = 0.79). Descriptive statistics and correlations between all variables are presented in Table 2.

Procedure

Data were collected using an online questionnaire sent out to teachers in March 2022 via social and professional networks. Participants were informed that the study was anonymous, that no personal data would be collected, and that they could quit at any time without justification or prejudice. Participants who did not object to participating were then invited to complete the two scales and, finally, socio-demographic data (age, gender, grade level taught, seniority, and inclusive education experience) were collected. At the end of the study, participants were fully debriefed regarding the purpose of the research.

Results

Pre-registered analyses

As indicated in the pre-registration form, three analyses were planned. First, a repeated measure analysis of variance with the distinct concerns as a within factor (i.e., Resources, Difficulties, Appropriateness, and Workload) was conducted in order to identify how teachers prioritized the factors. Then, the attitudes mean score was regressed on a model in which the four concerns were entered as predictors. Finally, a Relative Importance Analysis [START_REF] Tonidandel | Relative importance analysis: a useful supplement to regression analysis[END_REF] on these four predictors assessed the extent to which each predictor contributes to the model.

Hierarchy of concerns

The ANOVA conducted indicated that the levels of preoccupation significantly differ depending on the type of concern, F(3,1,521) = 699.46, p < 0.001, η p 2 = 0.58. More precisely, as indicated in Figure 1 below, French teachers appear to be more concerned about the lack of resources (M = 4.10, SE = 0.03) than, respectively, the workload generated (M = 3.85, SE = 0.04), the perceived difficulties (M = 3.34, SE = 0.04), and the appropriateness of inclusive education (M = 2.39, SE = 0.05; all p Bonferroni < 0.001). This hierarchy is slightly distinct from the one presented earlier and will be discussed further below.

Links between concerns and attitudes toward inclusive education

In order to study how each concern relates to teachers' attitudes toward inclusive education, we conducted an analysis in which the attitude score has been regressed on a model including the 4 concerns. Overall, this model explained nearly 36% of the variance in the attitudinal score (R 2 = 0.359). Regarding the links between each concern, the analysis revealed that, surprisingly, the concerns regarding resources, B = -0.00, SE = 0.04, t(503) = -0.03, p = 0.98, 95% CIs [-0.08; 0.08], or workload, B = -0.06, SE = 0.04, t(503) = -1.29, p = 0.20, 95% CIs [-0.13; 0.03] are not related to teachers' attitudes. However, the more concerned teachers were about the difficulties that inclusive education might generate, the less supportive they were of this policy, B = -0.14, SE = 0.04, t(503) = -3.35, p < 0.001, 95% CIs [-0.22; -0.06]. In the same vein, the more teachers were concerned that inclusive education could cause a decline in teaching performances or academic achievement, the less favorable they were toward this policy, B = -0.29, SE = 0.04, t(503) = -8.27, p < 0.001, 95% CIs [-0.36; -0.22]. Finally, since the predictors are correlated (see Table 2), a regression analysis might not be the best choice to properly partition variance in the predicted criterion (i.e., attitudes). To address this limitation, a relative importance analysis was conducted (for more details, see [START_REF] Tonidandel | Relative importance analysis: a useful supplement to regression analysis[END_REF] through an RWA-Web Shiny App [START_REF] Tonidandel | RWA web: a free, comprehensive, webbased, and user-friendly tool for relative weight analyses[END_REF]. Confidence intervals for individual relative weights and all corresponding significance tests were based on bootstrapping with 10,000 replications (see [START_REF] Tonidandel | Determining the statistical significance of relative weights[END_REF]. The results from this analysis confirm that the weighted linear combination of our four concern variables explained 35.9% of the variance in teachers' attitudes. An examination of the relative weights revealed that all four variables accounted for a statistically significant amount of variance in teachers' attitudes, as none of the 95% CIs for the tests of significance contained zero, with, by order of importance, Appropriateness (RW = 0.186, 51.9%), Difficulties (RW = 0.112, 32.3%), Workload (RW = 0.040, 11.2%), and Resources (RW = 0.020, 5.7%).

The relative weight results differ slightly from those obtained in the regression analysis, since in the analysis presented above, neither Workload nor Resources provided a statistically significant incremental effect in the prediction of teachers' attitudes, holding constant all of the remaining concern variables. The lack of concordance between the results from these two analyses is not uncommon. For [START_REF] Tonidandel | Relative importance analysis: a useful supplement to regression analysis[END_REF], this suggests that both variables are to explain for non-trivial variance in teachers' attitudes, but that the correlations they share with one another (and the other concern variables) means that they account for little unique, incremental variance.

This analysis also revealed that the relative weight of Appropriateness was significantly greater than all of the remaining concern variables (none of the 95% CIs for the tests of significance contained zero). In sum, it appears that the majority of the predicted variance in teachers' attitudes could be attributed to their fear of a potential decline due to inclusive education, a concern that is weaker yet than other concerns in teachers' minds. This interesting paradox will be discussed in the next section.

Discussion

Implementing inclusive education is challenging. Teachers stand at the very core of this new paradigm, and understanding their worries is essential to changing the system and shaping their professional development. Drawing on an abundant literature on teachers' concerns about inclusive education, we conducted an exploratory study seeking to identify a potential hierarchy among these concerns as well as their specific links to teachers' attitudes toward inclusive education.

Our results from this single correlational study involving French teachers indicated a clear pattern as regards concerns. Indeed, it seems that teachers are worried about, by order of French teachers' concerns about inclusive education. Different letters indicate significant differences between groups (p < 0.05). Error bars represent 95% CIs. Jury et al. 10.3389/feduc.2022.1065919 Frontiers in Education 06 frontiersin.org priority, resources, workload, difficulties, and the appropriateness of inclusive education. Unsurprisingly, a lack of resources is clearly the number one preoccupation for teachers when it comes to implementing inclusive education (a result less obvious from the students' , Goldan and Schwab, 2020, or the parents' perspective, [START_REF] Paseka | Parents' attitudes towards inclusive education and their perceptions of inclusive teaching practices and resources[END_REF]. If funding is clearly an essential tool for the implementation of inclusive education [START_REF] Meijer | Financing special needs and inclusive education-from Salamanca to the present[END_REF]) and appears to be, at least in teachers' minds, the solution for the success of this policy, three warnings should, however, be mentioned regarding this issue. First, distinct kinds of resources exist (i.e., human resources, material resources, spatial resources, and training) and more research is needed to identify the priority among them. Second, even in countries (e.g., Italy) where resources have been expended, teachers continue to claim for more investments (e.g., more special education teachers, reduced class sizes, and more training for teachers, [START_REF] Saloviita | Teacher attitudes in Italy after 40 years of inclusion[END_REF]. Third and finally, as mentioned by Sharma and Vlcek (2021, p. 63), "more funding is unlikely to result in better outcomes unless clear mechanisms are in place to identify how funds will be used to support teachers and resource schools. " In other words, if increasing the resources is clearly needed, these warnings suggest that other factors should also explain why implementing inclusive education remains so difficult.

Among the findings, it appears that an increase in workload is an important concern among French teachers. If such result is not surprising given the literature [START_REF] Gunnþórsdóttir | Additional workload or a part of the job? Icelandic teachers' discourse on inclusive education[END_REF], see also [START_REF] Warnes | Mainstream teachers' concerns about inclusive education for children with special educational needs and disability in England under pre-pandemic conditions[END_REF] it should be noted that this concern was less salient in previous studies from other countries (i.e., as in India or Australia, see Table 1 above). This difference might be explained by a cultural factor, since inclusive education in France is somewhat recent and still not always properly understand by teachers [START_REF] Perrin | Mieux comprendre les attitudes à l' égard de l' éducation inclusive: Rôle du type de handicap de l' élève, du statut de l' enseignant et de la familiarité[END_REF]. While multicultural studies often compare teachers' concerns as a function of their countries' policy (see [START_REF] Sharma | In-service teachers' attitudes, concerns, efficacy and intentions to teach in inclusive classrooms: an international comparison of Australian and Italian teachers[END_REF][START_REF] Sahli Lozano | A comparison of Australian and Swiss secondary school teachers' attitudes, concerns, self-efficacy, and intentions to teach in inclusive classrooms: does the context matter?[END_REF], another variable of interest could be the history of inclusive education within the country. One might reasonably assume that teachers from countries with many years of experience including students with disabilities would express fewer concerns regarding projected workloads in light of their professional habits. Such comparisons could also aid in identifying trends in the evolution of concerns over time (even if a crosssectional study cannot replace a longitudinal study).

Finally, and maybe more importantly, the present results raise a paradox that merits attention. Indeed, while Resources and Workload appear to be the most important concerns before Difficulties and Appropriateness, subsequent analyses (i.e., regression and RIA) revealed that the former account for less variance in teachers' attitudes toward inclusive education than the latter. As a matter of fact, if teachers appear to be right to not be overly worried about appropriateness (see [START_REF] Szumski | Academic achievement of students without special educational needs in inclusive classrooms: a metaanalysis[END_REF][START_REF] Kart | Academic and social effects of inclusion on students without disabilities: a review of the literature[END_REF][START_REF] Krämer | Inclusive education of students with general learning difficulties: a meta-analysis[END_REF], the fact that this concern is the strongest (negative) predictor of attitudes toward inclusive education is particularly thought-provoking. This indeed suggests that teachers believe that allowing some -initially excluded -students to participate and offering them the same learning opportunities could pose a threat to their school's standing (by enhancing the risk of lowered academic quality). It seems that this fear of school devaluation could be linked to the very functions of school since, according to the functionalist perspective on education [START_REF] Dornbusch | The social structure of schooling[END_REF], educational institutions notably have a selection function (i.e., sorting students into different curricula to ultimately allot them social positions commensurate with their individual merit, [START_REF] Autin | Social justice in education: how the function of selection in educational institutions predicts support for (non) egalitarian assessment practices[END_REF][START_REF] Autin | The function of selection of assessment leads evaluators to artificially create the social class achievement gap[END_REF][START_REF] Darnon | Achievement goal promotion at university: social desirability and social utility of mastery and performance goals[END_REF]. This selection function, by determining which students are "worthy" of pursuing the most prestigious courses of study [START_REF] Batruch | School selection and the social class divide: how tracking contributes to the reproduction of inequalities[END_REF], should serve the prosperity of society [START_REF] Parsons | The school as a social system: Some of its function in American society[END_REF] and has been identified as incompatible with inclusive education [START_REF] Khamzina | The conflict between inclusive education and the selection function of schools in the minds Jury et al[END_REF]. Therefore, for teachers who inherently contribute to this selection function, welcoming students with SEN could be perceived as a risk of having to devote more time to them, thereby preventing other students from reaching the expected benchmarks (thus, risking a "poorer" selection). While this threat is not always salient due to the low number of students with SEN who are fully included (for an example on students with Autism Spectrum Disorder in France, see [START_REF] Rattaz | School inclusion in children and adolescents with autism Spectrum disorders in France: report from the ELENA French cohort study[END_REF], perhaps also explaining low concern, their numbers appear sufficient to produce negative attitudes. If future studies were to investigate more deeply the origin of this appropriateness concern and its relationship to teachers' attitudes, they could also investigate the consequences of this concern from the perspective of students with SEN. They may indeed be perceived as the source of a threat to teaching quality, and looked upon more negatively among teachers who strongly endorse this preoccupation.

Although these results seem promising and likely to open new perspectives, a few limitations must be mentioned. First, we chose to translate the questionnaire validated by Sahli [START_REF] Sahli Lozano | The concerns about inclusive education scale: dimensionality, factor structure, and development of a shortform version (CIES-SF)[END_REF] without conducting a full translation and validation process. This might explain some mediocre reliability indices, inviting caution regarding the interpretation of the results. It should also be noted that, as evoked in the introduction section, that the four concerns measured through this tool represent general categories that are maybe too broad. Future research can study more precisely the scope of these concerns (i.e., what they cover) and how they could also differ depending on the type of needs or disability faced by students. It would not be surprising that the concern hierarchies identified were different for students with a motor disability versus those with an Autism Spectrum Disorder (for an example regarding attitudes see [START_REF] Jury | Attitudes toward inclusive education: an exploration of the interaction between teachers' status and students' type of disability within the French context[END_REF]. Finally, this study was conducted in a specific context (i.e., the French context), which inherently reduces the generalizability of findings. A cross-cultural study investigating how and whether this hierarchy differs between countries would also contribute greatly to a better understanding of the barriers to inclusive education.

Notwithstanding these limitations, the present results offer contributions to the literature. Indeed, by showing that teachers Frontiers in Education 07 frontiersin.org not only have concerns about inclusive education but assign them a clear hierarchical ranking, our study opens avenues for both new research and interventions to develop more specific responses and, perhaps, more effective action to mitigate these concerns. Additionally, by showing that the relationship between attitudes and inclusive education depends on the type of concerns, the study also helps to understand some of the inconsistent results regarding this specific point, calling on researchers to investigate these relationships more deeply. Finally (and unfortunately), the present research is another illustration of the long road that education systems must still travel before every student can enjoy their right to education.

FIGURE 1

 1 FIGURE 1

TABLE 1

 1 Synthesis of studies that have used the CIES and provide information regarding the means for each concern.

	Study	Country	N	Mean for	Mean for	Mean for	Mean for	Hierarchy
				workload	resources	acceptance	academic	(descriptive)
							standard
	Bhatnagar and Das	India	500	1.99	2.76	2.32	2.33	R > AS>A > W
	(2013)
	Miesera et al. (2019)	Germany	887	2.51	2.94	2.62	2.16	R > A > W > AS
	Sharma and Desai	India	794	1.72	2.79	2.09	2.16	R > AS>A > W
	(2002)
	Sharma and Nuttal	Australia	030	1.68	2.65	2.48	2.2	R > A > AS>W
	(2016)*
	Sharma et al. (2018)	Australia/	309	1.80	2.60	2.09	1.97	R > A > AS>W
		Italy
	Song et al. (2019)*	Korea	116	1.81	2.42	2.33	1.43	R > A > W > AS
	Sokal and Sharma	Canada	137	1.95	2.57	2.00	1.90	R > A > W > AS
	(2014)
	Sokal et al. (2013)*	Canada	060	1.86	2.52	2.44	2.30	R > A > AS>W
	Warnes et al. (2022)	United	093	2.48	2.61	2.12	2.46	R > W > AS>A
		Kingdom
	Grand mean	N/A	2,926	2.06	2.77	2.31	2.14	R > A > AS > W
	calculation

Descriptive mean of the scale was not always provided. The scores presented here are calculated from the information available in the paper. R, Resources; W, Workload; A, Acceptance; AS, Academic standard. *Pre-intervention scores were collected.

TABLE 2

 2 Descriptive statistics and intercorrelations between variables.

	Variables	M	SD	1	2	3	4	5
	1. Resources	4.10	0.76	_
	2. Workload	3.85	0.79	-0.49***	_
	3. Difficulties	3.34	0.98	-0.44***	-0.55***	_
	4. Appropriateness	2.39	1.08	-0.38***	-0.45***	-0.71***	_
	5. Attitudes	3.62	0.74	-0.27***	-0.35***	-0.52***	-0.58***	_

***p < 0.001.

Frontiers in Education

Acknowledgments

The authors wish to thank Aylin Summer for help in data collection.

Funding

This study was funded by the Agence Nationale de la Recherche (ANR-AAPEG2020-DIVISE) and within the project PIA3 "100% IDT: Inclusion, un Défi, un Territoire. "

Data availability statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found at: https://osf.io/fsjcu.

Ethics statement

Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. The patients/participants provided their written informed consent to participate in this study.

Author contributions

AL, CD, MJ, and SC conceived and designed the study and collected the data. MJ analyzed the data and drafted the manuscript. AL, SC, and CD provided critical revisions. All authors contributed to the article and approved the submitted version.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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