All methods were implemented according to relevant guidelines and regulations.
We distributed a recruitment form to recruit elementary school boys who can volunteer from a soccer club near the facility to which the author belongs. Prior to measurements, the purpose of this study was explained in writing and orally to the participants and their guardians, and written consent was obtained. This study was conducted after receiving approval (approval number: 2020-242) from the "Ethics Review Committee for Research Involving Human Subjects" of Waseda University. The exclusion criteria were as follows: those who did not agree with the purpose of the study, those who had pain in the lower limbs, those who could not participate in all three measurements, those with missing data, those who were unable to wear the arch support for an average of at least 1 hour per day during the intervention period, and those with no decline in the MLA were excluded （Fig．１）.
2. Outcome measurement
2.1. Classification of foot morphology
The foot posture index-6 (FPI) was used to classify foot morphology . The FPI is an evaluation tool developed considering its simplicity and convenience. One or more feet in the MLA shape item were defined as flat feet with low MLA and were included in the intervention. A physical therapist with sufficient experience performed the measurements.
2.2．Foot morphology evaluation
An automatic three-dimensional foot measuring machine (Dream GP) was used to evaluate foot morphology (Fig．２). Measurements were taken in static standing (on both feet) and sitting positions after marking the scaphoid bone length. The measurements included foot length (from the back of the calcaneus to the tip of the longest toe), foot width (from the first metatarsal to the fifth metatarsal head), foot circumferences (circumference at the first metatarsal head - fifth metatarsal head), and navicular height (from the floor surface to the lowest end of the navicular rough surface). The difference in navicular height between the upright and seated positions, the navicular drop, was expressed as an index of arch morphology and also as arch height ratio （Standing navicular height divided by foot length times 100）. The measurements were performed by a person with sufficient measurement experience.
2.3. Muscle cross-sectional area evaluation of intrinsic and extrinsic foot muscles
The intrinsic and extrinsic foot muscles were measured using an ultrasound imaging system (SonoSite Edge II, Fujifilm Corporation) in the B-mode. A probe (Linear Probe HFL38xp, Fujifilm Corporation) with a frequency of 6-13 MHz was used for the measurements. The intrinsic muscles of the foot flexor hallucis brevis (FHB) muscle, abductor hallucis (ABH) muscle, and flexor digitorum brevis (FDB) muscle were placed in a dorsal recumbent position with the knee in slight flexion and the ankle in slight plantar flexion. The extrinsic ankle muscles—the flexor digitorum longus (FDL) muscle, flexor hallucis longus (FHL) muscle, peroneal (PER) muscle, and tibialis posterior (TP)—were placed in an end-sitting position with the ankle joint in the mid-position, the knee joint in 90° flexion, and the hip joint in 90° flexion. The participants were instructed to relax their feet without applying pressure to the lower leg, and then the measurements were performed. The FHB and FDB muscles were measured by applying a probe in the short-axis direction to the proximal portion of the first metatarsal head , ABH and FDB to the medial aspect  and plantar surface  of the foot between the navicular tuberosity and medial tubercle of the calcaneus, respectively (Fig. ３-a). The FDL, FHL, PER, and TP were imaged by applying a short-axis probe to the proximal 50%  of the medial end of the tibial plateau and the inferior end of the medial end of the medial tibial plateau, the 60%  of the proximal end of the fibular head and the inferior end of the external capsule, the 50%  of the proximal end of the fibular head and the inferior end of the external capsule, and the 30%  of the proximal end of the lateral knee joint cleft and the inferior end of the external capsule, respectively (Fig. ３-b). A fully experienced physical therapist performed the measurements and analyzed the data after confirming that the measurements were highly reliable.
2.4．Assessment of developmental age
Since the level of foot growth differs depending on the biological maturity , the peak height velocity age (PHVA), an index of biological maturity, was calculated using the BTT method and examined. PHVA, the age at which height increases the most, was estimated from the history of each participant's height data using a dedicated software AUXAL3.1 (Scientific Software International). Developmental age is the difference between age and PHVA at the time of measurement and is an indicator of maturity .
To assess the developmental effects, the intervention study was conducted as an 18-week crossover study (Fig. １). The study protocol included a pre-intervention session prior to the start of the experiment, in which an overview of the experiment, the wearing method of the arch supporter, and precautions were explained. Obtained consent before conducting the experiment. The participants were randomly divided into two groups: the first-half supporter group received a supporter intervention period of 9 weeks in the first half and an observation period of 9 weeks in the second half. The second-half supporter group received an observation period of 9 weeks in the first half and a supporter intervention period of 9 weeks in the second half. The intervention and observational phases were switched at 9 weeks after the intervention. Since a previous report showed that the intervention was effective after 8 weeks in adults , the intervention period was 9 weeks. The allocation was made from a list table randomly arranged by one examiner, and another examiner randomly divided into two groups in the order of the list.Measurements were taken at three time points: pre-intervention, midterm, and post-intervention. For the results, the difference between the second measurement and the first measurement was defined as the period I change, and the difference between the third measurement and the second measurement was defined as the period II change.
For the arch support, an arch supporter (Solvo-Tate Arch Supporter, manufactured by Sanjin Sangyo Co Ltd.) with an arch pad made of a viscoelastic polymer material attached to a stretchy knit was used (Fig．4). The small (S) or large (L) size of the supporter was selected according to the participant’s foot length. The height of the arch pad was 8 mm for size S and 10 mm for size L, and the cloth thickness was 1 mm). The method of wearing the supporter was explained using the manufacturer’s instructions, according to which the pads attached to the supporter were aligned with the medial arch of the foot and worn barefoot with socks worn over the top. The participants were instructed to wear shoes for as long as possible, except during strenuous exercise, sleeping, and bathing. They were also instructed to record the time for which the arch supporters were worn. An in-person site visit was conducted between the 4th and 5th weeks of intervention where the examiner evaluated the wearing method of the arch supporter and the wearing time using questionnaires. Additional instructions were provided as needed to increase compliance with the arch supporter use.
2.6. Statistical analysis
We compared the means of the two groups’ physical characteristics by using either an unpaired t-test or the Mann–Whitney U test. To examine the effect of the intervention, the sum of the means of the change in the first half of the supporter group in period I and the change in the second half of the supporter group in period II and the change in the first half of the supporter group in period II and the change in the second half of the supporter group in period I were compared using a corresponding t-test for items for which normality was found and a t-test for items for which normality was not found. The Wilcoxon signed-rank sum test was used for items that were not identified. For items that were indicated to be significant by the corresponding t-test or Wilcoxon signed-rank sum test, an additional analysis was performed using analysis of covariance (ANCOVA) with the developmental age as a covariate to take into account the effects of growth. The significance level was set at less than 5%, and the effect size was calculated using Cohen's d (small: 0.20, medium: 0.50, and large: 0.80). A statistical software (SPSS Statistics 27, IBM, USA) was used for the statistical analysis.