Beat Knechtle
email: beat.knechtle@hispeed.ch

Thomas Rosemann

Romuald Lepers

Christoph Alexander Rüst

A comparison of performance of Deca Iron and Triple Deca Iron ultra-triathletes

Keywords: Triathlon, Swimming, Cycling, Running, Ultra-endurance

This study intended to compare the performance of ultra-triathletes competing in a Deca Iron ultra-triathlon (i.e. 10 times 3.8 km swimming, 180 km cycling, and 42.2 km running) with the performance of athletes competing in a Triple Deca Iron ultra-triathlon (i.e. 30 times 3.8 km swimming, 180 km cycling, and 42.2 km running). Split and overall race times of six male finishers in a Deca Iron ultra-triathlon and eight male finishers in a Triple Deca Iron ultra-triathlon were analysed using multiple t-tests, linear and non-linear regression analyses, and analysis of variance. Among the 19 starters (i.e. 17 men and two women) in the Deca Iron ultra-triathlon, six men (i.e. 35.3% of all starters) finished the race. The mean swimming, cycling, running and overall race times of the six finishers across the ten days were 1:19 ± 0:09 h:min, 6:36 ± 0:19 h:min, 6:03 ± 0:47 h:min and 14:44 ± 1:17 h:min, respectively. The times of the split disciplines and overall race time increased linearly across the ten days. Total transition times did not change significantly across the days and were equals to 48 ± 8 min. Among the 22 starters (i.e. 20 men and two women) in the Triple Deca Iron ultra-triathlon, eight men (i.e. 36.4% of all starters) finished. The mean swimming, cycling, running and overall race times of the eight finishers across the 30 days were 1:11 ± 0:07 h:min, 6:19 ± 0:32 h:min, 5:34 ± 1:15 h:min and 13:44 ± 1:50 h:min, respectively. Split and overall race times showed no change across the 30 days. Total transition times showed no change across the days and were equal to 41 ± 11 min. To summarize, the daily performance decreased across the ten days for the Deca Iron ultra-triathletes (i.e. positive pacing) while it remained unchanged across the 30 days for the Triple Deca Iron ultra-triathletes (i.e. even pacing).

Background

Ultra-endurance performance is defined as any endurance performance of six hours or longer in duration [START_REF] Zaryski | A comparison of performance of Deca Iron and Triple Deca Iron ultra-triathletes[END_REF]. An Ironman triathlon covering 3.8 km swimming, 180 km cycling and 42.2 km running with the fastest winner times of ~8 hrs has to be considered as an ultra-endurance performance [START_REF] Lepers | Analysis of Hawaii ironman performances in elite triathletes from 1981 to 2007[END_REF]. Apart from the classical Ironman triathlon held as a single stage race, also longer ultra-triathlons such as multi-stage races with the completion of consecutive Ironman triathlons held for several days are known [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF]Knechtle et al. 2011a).

In a Deca Iron ultra-triathlon -where each day an Ironman triathlon has to be finished for ten consecutive daysperformance decreased with increasing duration of the race [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF]Knechtle et al. 2012a) where the fastest Ironman was achieved on the first day [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF]Knechtle et al. 2008a) and the slowest on the last day [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF]. A multi-stage ultra-triathlon such as a Deca Iron ultra-triathlon is a highly selective race and less than 50% of the starters are able to finish [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF]Knechtle et al. 2012a). The most important predictor variables for a successful finish in a Deca Iron ultra-triathlon were extensive previous experience since the number of finished Triple Iron ultra-triathlons and the personal best time in a Triple Iron ultra-triathlon were related to overall race time [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF].

Since the first edition of a Deca Iron ultra-triathlon in 2006 (Knechtle et al. 2008a), several races of this kind have been held mainly in Monterrey, Mexico [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF]Knechtle et al. 2012a). However, in autumn 2013, ultraendurance triathletes in Lonato des Garda, Italy, intended to go for new limits by organizing a Triple Deca Iron ultratriathlon, where the athletes had to finish each day an Ironman triathlon for 30 consecutive days. In the same race, a second group of athletes competed in a separate Deca Iron ultra-triathlon to finish ten Ironman triathlons within ten consecutive days.

Little is known about the pacing strategy in ultraendurance performance [START_REF] Abbiss | Describing and understanding pacing strategies during athletic competition[END_REF]. Actual evidence suggests that during endurance and ultraendurance performance well-trained athletes tend to adopt a positive pacing strategy, whereby after peak speed is reached, the athlete progressively slows [START_REF] Abbiss | Describing and understanding pacing strategies during athletic competition[END_REF]. The underlying mechanisms influencing the regulation of pace during exercise are currently unclear [START_REF] Abbiss | Describing and understanding pacing strategies during athletic competition[END_REF]. It has been suggested, however, that self-selected exercise intensity is regulated within the brain based on a complex algorithm involving peripheral sensory feedback and the anticipated workload remaining [START_REF] St | The role of information processing between the brain and peripheral physiological systems in pacing and perception of effort[END_REF].

To date, to the best of our knowledge, the Triple Deca Iron ultra-triathlon represents the longest event ever finished by triathletes combining swimming, cycling and running. However, in summer 2013, an athlete completed in Laval, Québec, Canada, for the first time in history in a self-paced race the total distance of 33 Ironman triathlons within 33 consecutive days [START_REF] Knechtle | 33 Ironman triathlons in 33 days-a case study[END_REF]. The athlete finished the total distance of 7,458 km (i.e. 125 km swimming, 5,940 km cycling and 1,393 km running) within 410 h and finished each Ironman triathlon in a mean time of 12:27 h:min. During the 33 days, the athlete became slower in swimming, transition time 1, and transition time 2. However, in cycling, running and overall race time, the athlete was able to maintain his performance during the 33 days [START_REF] Knechtle | 33 Ironman triathlons in 33 days-a case study[END_REF]). The question is now whether athletes competing in an official race covering 30 Ironman triathlons within 30 days would be able to maintain their performance during one month as this athlete showed in 33 days in his self-paced race. The aim of the present study was therefore to compare the changes in performance over days for both Triple Deca Iron ultra-triathletes and Deca Iron ultra-triathletes. Based upon previous findings in field studies for Deca Iron ultra-triathletes and in the case study with the 33 Ironman triathlons it was hypothesized that performance would decrease in Deca Iron ultratriathletes, but not in Triple Deca Iron ultra-triathletes.

Methods

Ethics

This study was approved by the Institutional Review Board of St. Gallen, Switzerland, with a waiver of the requirement for informed consent given that the study involved the analysis of publicly available data.

The races

The Deca Iron ultra-triathlon race consisted in performing one Ironman distance triathlon (i.e. 3.8 km swimming, 180 km cycling and 42 km running) daily for ten consecutive days whereas the Triple Deca Iron ultra-triathlon consisted in performing one Ironman distance triathlon daily for 30 consecutive days. The races were held in and around 'Parco La Quiete' (www.parcolaquiete.it) in Lonato del Garda in the North of Italy and south to Lake Garda. The Triple Deca Iron ultra-triathlon started on September 8th, 2013, and the Deca Iron ultra-triathlon started 20 days later on September 28th, 2013. Swimming was held in a non-heated 25-m out-door pool where wetsuits were allowed. Cycling was performed as a 180-km non-drafting time trial on open roads in the hilly surroundings near 'Parco La Quiete' on laps of 6 km. Running was held around the lake in 'Parco La Quiete' on flat laps of one km on grass (~50%) and stone slabs (~50%). Laps in swimming were counted manually by the staff of the race direction. Laps in cycling and running were counted electronically by using a chip system. On Day 7, cycling laps were counted manually due to problems with the electronic chip system. In the first 21 days, air temperature was at ~25-30°C and water temperature at ~25°C. In the last nine days, weather conditions changed considerably where air temperature continuously dropped to ~14°C and water temperature dropped to ~17°C by the end of the race. In the last three days of the race, rain was continuously falling.

Data collection and data analyses

The data set for this study was obtained from the race director for electronically recorded split times. Overall race times and split times (i.e. 3.8 km swimming, 180 km cycling and 42 km running) of all female and male starters were analysed regarding changes over days. Each set of data was tested for normal distribution (D' Agostino and Pearson omnibus normality test) and for homogeneity of variances (Levene's test) before statistical analyses. Since the change in sex difference in endurance is assumed to be non-linear [START_REF] Reinboud | Linear models can't keep up with sport gender gap[END_REF], we calculated both the linear and the non-linear regression models that fit the data best. We compared the best-fit non-linear models to the linear models using Akaike's Information Criteria (AIC) as well as F-test in order to show whether the linear or the non-linear model would be the most appropriate to explain the trend of the data. Differences in absolute and relative performance between finishers in the Deca Iron ultra-triathlon and in the Triple Deca Iron ultra-triathlon were compared using multiple t-test analyses with individual analysis of each pair of Deca and Triple Deca Iron ultra-triathletes and with Holm-Sidak correction for multiple comparisons. Absolute and relative performance of the Deca Iron ultra-triathletes were compared with the performance of Day 1-10, Day 11-20 and Day 21-30 of the Triple Deca Iron ultra-triathletes using repeated measures one-way analysis of variance (ANOVA) with the Greenhouse-Geisser correction and Tukey's multiple comparison test with individual variances computed for each comparison. Statistical analyses were performed using CurveExpert Professional (Version 2.0.3, Hyams D.G.) and GraphPad Prism (Version 6.01, GraphPad Software, La Jolla, CA, USA). Significance was accepted at p < 0.05 (two-sided for t-tests). Data in the text and figures are given as mean ± standard deviation (SD).

Results

Deca iron ultra-triathlon Among the 19 starters (i.e. 17 men and two women), six men (i.e. 35.3% of all starters) finished the race. Among the non-finishers, six triathletes stopped before the 6th day and seven triathletes stopped between the 6th and the 9th day. The mean swimming, cycling, running and overall race times of the six finishers across the ten days were 1:18 ± 0:08 h:min, 6:35 ± 0:37 h:min, 6:02 ± 1:00 h: min and 14:44 ± 1:44 h:min, respectively. All split times and overall race times increased linearly across the ten days (Figure 1). Total transition times did not change significantly across the days and were equals to 48 ± 8 min. The overall race time of the winner (i.e. the triathlete who had the fastest overall time after the ten days) was 129:33 h:min. His mean swimming, cycling, running and overall race times across the ten days were 1:06 ± 0:01 h:min, 6:14 ± 0:25 h:min, 5:06 ± 0:16 h:min and 12:57 ± 0:35 h:min, respectively.

Triple Deca iron ultra-triathlon Among the 22 starters (i.e. 20 men and two women), eight men (i.e. 36.4% of all starters) finished the 30 Ironman distance triathlon. Among the non-finishers, six triathletes finished between one and ten Ironman distances, five triathletes finished between 11 and 20 Ironman distances and three triathletes finished between 21 and 29 Ironman distances. The mean swimming, cycling, running and overall race times of the eight successful finishers across the 30 days were 1:10 ± 0:07 h:min, 6:18 ± 0:42 h:min, 5:33 ± 1:17 h:min and 13:46 ± 1:57 h: min, respectively. In contrast to the Deca Iron ultratriathlon, the times of the different disciplines and the total time did not change across the 30 days (Figure 2). The changes were non-linearly in swimming (i.e. polynomial regression 10th degree), cycling (i.e. polynomial regression 5th degree), running (i.e. polynomial regression 10th degree), and overall race time (i.e. polynomial regression 5th degree). Total transition times did not change significantly across the days and were equal to 41 ± 11 min. The overall race time of the winner (i.e. the triathlete who had the fastest overall race time after the 30 events) was 356:33 h:min. His mean swimming, cycling, running and overall race times across the 30 days were 1:12 ± 0:03 h:min, 6:12 ± 0:29 h:min, 3:56 ± 0:20 h: min and 11:53 ± 0:46 h:min, respectively.

Comparison between the 10 days in the Deca with the 3 × 10 days in the Triple Deca Figure 3 presents the comparison of swimming, cycling, running, and overall race times for the Deca Iron ultratriathletes with the first ten days of the Triple Deca Iron ultra-triathletes expressed in absolute race times. There were no differences in swimming (Figure 3A) and running (Figure 3C), but Triple Deca Iron ultra-triathletes time for Day 1-10 in the Deca Iron ultra-triathlon with Day 1-10 (Figure 6), Day 11-20 (Figure 7) and Day 21-30 (Figure 8) in the Triple Deca Iron ultra-triathlon. In the first ten days in the Triple Deca Iron ultra-triathlon, no differences were found compared to the ten days in the Deca Iron ultra-triathlon (Figure 6). When the second ten days in the Triple Deca Iron ultra-triathlon were compared to the ten days in the Deca Iron ultratriathlon, the Deca Iron ultra-triathletes were relatively faster on Day 18 in cycling (Figure 7). On Day 21-30, the athletes in the Deca Iron ultra-triathlon were relatively faster in cycling on Day 1, Day 3, Day 5, Day 6 and Day 10 compared to the Triple Deca Iron ultratriathletes (Figure 8).

Discussion

This study intended to analyse the changes in performance over time for Triple Deca Iron ultra-triathletes and Deca Iron ultra-triathletes and it was hypothesized that performance would decrease in Deca Iron ultratriathletes, but not in Triple Deca Iron ultra-triathletes. The most important finding was that the hypothesis was confirmed since performance decreased in Deca Iron ultra-triathletes but remained unchanged across time in Triple Deca Iron ultra-triathletes. The continuous decrease in performance in the Deca Iron ultra-triathletes confirms previous findings observed for Deca Iron ultratriathletes competing in the [START_REF] St | The role of information processing between the brain and peripheral physiological systems in pacing and perception of effort[END_REF], 2007and 2009'World Challenge Deca Iron Triathlon' held in Monterrey, Mexico (Herbst et al. 2011). The unchanged performance in swimming, cycling, running and overall race time in the Triple Deca Iron ultra-triathletes confirms very recent findings in an athlete completing 33 Ironman triathlons within 33 days where the athlete was able to maintain his performance in cycling, running and overall race time during the 33 days [START_REF] Knechtle | 33 Ironman triathlons in 33 days-a case study[END_REF].

The aspect of pacing strategy

An interesting observation was the difference in the comparison between absolute and relative performance. When the absolute daily performances of the Deca Iron ultra-triathletes were compared with the performances on Day 1-10, Day 11-20 and Day 21-30 of the Triple Iron ultra-triathletes, only minor differences were found. However, when the split performances were expressed relatively in percent of overall race times, Deca Iron ultra-triathletes were relatively faster in cycling in five of the ten days compared to the Triple Deca Iron ultra-triathletes in their last ten days. When the mean absolute performances of the ten days in the Deca Iron ultra-triathletes were compared to the performances on Day 1-10, Day 11-20 and Day 21-30 of the Triple Iron ultra-triathletes, the Deca Iron ultra-triathletes were slower in swimming, cycling and running compared to the specific segments of the Triple Deca Iron ultra-triathlon race. However, when the performances in the split disciplines was expressed in percent of overall race time, the Deca Iron ultra-triathletes were relatively faster in swimming, cycling and running compared to the three segments of the Triple Deca Iron ultra-triathlon race. A further interesting finding regarding absolute performances was that the Triple Iron ultra-triathletes were faster in the second segment of their race (i.e. Day 11-20) compared to their first segment (i.e. Day 1-10) in cycling, running, and overall performance. Generally, during ultraendurance events (i.e. endurance performances lasting for longer than six hours), athletes tend to adopt a positive pacing strategy (i.e. the athlete's speed gradually declines throughout the duration of the event) [START_REF] Abbiss | Describing and understanding pacing strategies during athletic competition[END_REF]. The Triple Deca Iron ultra-triathletes, however, were able for a negative pacing in the first 20 days of the race. A potential explanation for the negative pacing (i.e. better performance in cycling, running and overall race times) could be the environmental conditions (i.e. unchanged course and habituation to the course), the habituation to the daily task for both athletes and support crews, and the stable weather conditions (i.e. no rain, constant air temperature of ~25-30°C and constant water temperature of 2 ~5°C in the first 21 days). Mean absolute performances in the second (i.e. Day 11-20) and third segment (i.e. Day 21-30) were not different. Most probably the athletes were then very experienced and negated the deterioration of the environmental conditions (i.e. decrease in air and water temperature).

Overall, the Triple Deca Iron ultra-triathletes showed an 'even pacing' during the 30 days in contrast to the Deca Iron ultra-triathletes with a 'positive pacing' [START_REF] Abbiss | Describing and understanding pacing strategies during athletic competition[END_REF] during their ten days. Most probably the successful Triple Deca Iron ultra-triathletes choose the right constant pace in cycling in the first days to be able to successfully finish the whole race. And they were even able to improve cycling and running speed in the second segment of the race. The athletes in the Triple Deca Iron ultra-triathlon might have become accustomed during the first days to the burden of a daily Ironman. They may have learned to adopt a pacing strategy to finish the race successfully. A potential explanation could be that the successful finishers in the Triple Deca Iron ultra-triathlon went slower in the cycling part in order to save energy for the running split which could be due to higher pre-race experience in Triple Deca Iron ultra-triathletes compared to Deca Iron ultra-triathletes. Alternatively, the Deca Iron ultra-triathletes went too fast in the first days in cycling leading to different problems such as muscular problems, loss in stored energy in the muscles, muscle soreness forcing them to go slower in the following days.

The aspect of previous experience

An explanation for the differences in pacing strategy between the Deca Iron and the Triple Deca Iron ultratriathletes could be the previous experience of the successful finishers. Previous experience is an important predictor variable in an ultra-triathlon [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF][START_REF] Lepers | Analysis of ultra-triathlon performances[END_REF]. It has been shown that race time in a Triple Iron ultra-triathlon was highly predictive for race time in a Deca Iron ultra-triathlon [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF][START_REF] Lepers | Analysis of ultra-triathlon performances[END_REF]. Overall race time in a Deca Iron ultra-triathlon might be predicted by the equation Deca Iron ultra-triathlon race time (min) = 5885 + 3.69 × race time in Triple Iron ultra-triathlon (minutes) [START_REF] Lepers | Analysis of ultra-triathlon performances[END_REF].

Since both the number and the personal best time in a Triple Iron ultra-triathlon were highly predictive for the performance in a Deca Iron ultra-triathlon [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF], we summarized for each finisher the number of completed Triple Iron ultra-triathlons with the personal best time with data available from the website of the International Ultra-Triathlon Association (IUTA). In addition to the data of Triple Iron ultra-triathlon, we inserted also the data from Double Iron ultra-triathlon and longer ultra-triathlons for both finishers in the Deca Iron ultra-triathlon (Table 1) and in the Triple Deca Iron ultra-triathlon (Table 2). In accordance with previous reports [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF]Knechtle et al. 2011b;[START_REF] Lepers | Analysis of ultra-triathlon performances[END_REF]) the winner in the Deca Iron ultra-triathlon had the highest number of finishes in both Triple and Double Iron ultra-triathlon and the fastest personal best times in both Triple and Double Iron ultra-triathlon (Table 1) compared to the other finishers. For the fastest finishers in the Triple Deca Iron ultra-triathlon, however, both the numbers and the personal best times in both Triple and Double Iron ultra-triathlon seemed not of relevance (Table 2). The fastest three finishers had focussed more in their previous races on longer ultratriathlon distances such as Deca Iron and Double Deca Iron ultra-triathlon. Similarly to the athletes in the Triple Deca Iron ultra-triathlon, the athlete completing 33 Ironman triathlons in 33 days in his self-paced race showed a broad experience in ultra-triathlon [START_REF] Knechtle | 33 Ironman triathlons in 33 days-a case study[END_REF]. He had finished 11 Double Iron ultratriathlons with a personal best time of 21:48 h:min, six Triple Iron ultra-triathlons with a personal best time of 36:29 h:min and one Deca Iron ultra-triathlon within 297:42 h:min [START_REF] Knechtle | 33 Ironman triathlons in 33 days-a case study[END_REF].

Limitations

This study analysed the changes in performance in split and overall race times in Deca Iron and Triple Deca Iron ultra-triathletes. Unfortunately, aspects such as nutrition [START_REF] Dempster | Case study: nutrition and hydration status during 4,254 km of running over 78 consecutive days[END_REF]Knechtle et al. 2008b), fluid metabolism (Knechtle et al. 2008b), sleep and sleep deprivation (Knechtle et al. 2012b;[START_REF] Lahart | Challenges in maintaining emotion regulation in a sleep and energy deprived state induced by the 4800Km ultra-endurance bicycle race; The Race Across AMerica (RAAM)[END_REF], recovery [START_REF] Neubauer | Recovery after an Ironman triathlon: sustained inflammatory responses and muscular stress[END_REF], pain tolerance [START_REF] Freund | Ultra-marathon runners are different: investigations into pain tolerance and personality traits of participants of the TransEurope FootRace 2009[END_REF], association between anthropometry and performance [START_REF] Knechtle | Differential correlations between anthropometry, training volume, and performance in male and female Ironman triathletes[END_REF], changes in body composition [START_REF] Herbst | Pacing strategy and change in body composition during a Deca Iron triathlon[END_REF]Knechtle et al. 2008a, b;[START_REF] Mueller | Changes in body composition in triathletes during an Ironman race[END_REF][START_REF] Schütz | Characteristics, changes and influence of body composition during a 4486 km transcontinental ultramarathon: results from the Transeurope Footrace mobile whole body MRI-project[END_REF], and overuse injuries of the lower limbs [START_REF] Freund | The foot in multistage ultra-marathon runners: experience in a cohort study of 22 participants of the Trans Europe Footrace Project with mobile MRI[END_REF] were not included.

Conclusions

This study showed that performance decreased linearly across days for Deca Iron ultra-triathletes (i.e. positive pacing) while performance remained unchanged across days for Triple Deca Iron ultra-triathletes (i.e. even pacing). To be successful in a Triple Deca Iron ultra-triathlon, a high number and a fast personal best time in ultratriathlons shorter than the Triple Deca Iron ultra-triathlon seem mandatory. We assume that ultra-triathletes successfully competing in longer races than a Deca Iron ultratriathlon such as a Triple Deca Iron ultra-triathlon need to gain experience in the long ultra-triathlon distances such as a Deca Iron ultra-triathlon.

Figure 1

 1 Figure 1 Changes in split and overall race times during the Deca Iron ultra-triathlon. (Panel A) swimming, (Panel B) cycling, (Panel C) running, (Panel D) overall race times (n = 6). Some cycling times on day 7 were not available. Values are means ± SD.

  were faster in cycling (Figure3B) on Day 10 than Deca Iron ultra-triathletes. Figure4presents the same comparisons for Day 11-20 in the Triple Deca Iron ultratriathlon with Day 1-10 in the Deca Iron ultra-triathlon. Similarly, the Triple Deca Iron ultra-triathletes were faster in cycling (Figure 4B) on Day 10 than the Deca Iron ultra-triathletes. Figure 5 compares Day 21-30 in the Triple Deca Iron ultra-triathlon with Day 1-10 in the Deca Iron ultra-triathlon. There were no differences in the performance between the finishers. Figures 6, 7 and 8 show the comparisons in swimming, cycling and running expressed in percent of overall race

Figure 2

 2 Figure 2 Changes in split and overall race times during the Triple Deca Iron ultra-triathlon. (Panel A) swimming, (Panel B) cycling, (Panel C) running, (Panel D) overall race times (n = 8). Some cycling times on day 3 and 27 were not available. Values are means ± SD.

Figure 3

 3 Figure 3 Comparison of split and overall race times for Deca Iron ultra-triathletes (i.e. all ten race days) and Triple Deca Iron ultra-triathletes (i.e. Day 1-Day 10) expressed in absolute race times. (Panel A) swimming, (Panel B) cycling, (Panel C) running, (Panel D) overall race times. Values are means ± SD.

Figure 4

 4 Figure 4 Comparison of split and overall race times for Deca Iron ultra-triathletes (i.e. all ten race days) and Triple Deca Iron ultra-triathletes (i.e. Day 11-Day 20) expressed in absolute race times. (Panel A) swimming, (Panel B) cycling, (Panel C) running, (Panel D) overall race times. Values are means ± SD.

Figure 5

 5 Figure 5 Comparison of split and overall race times for Deca Iron ultra-triathletes (i.e. all ten race days) and Triple Deca Iron ultra-triathletes (i.e. Day 21-Day 30) expressed in absolute race times. (Panel A) swimming, (Panel B) cycling, (Panel C) running, (Panel D) overall race times. Values are means ± SD.

Figure 6

 6 Figure 6 Comparison of split and overall race times for Deca Iron ultra-triathletes (i.e. all ten race days) and Triple Deca Iron ultra-triathletes (i.e. Day 1-Day 10) expressed in percent of overall race time. (Panel A) swimming, (Panel B) cycling, (Panel C) running. Values are means ± SD.

Figure 7

 7 Figure 7 Comparison of split and overall race times for Deca Iron ultra-triathletes (i.e. all ten race days) and Triple Deca Iron ultra-triathletes (i.e. Day 11-Day 20) expressed in percent of overall race time. (Panel A) swimming, (Panel B) cycling, (Panel C) running. Values are means ± SD.

Figure 8

 8 Figure 8 Comparison of split and overall race times for Deca Iron ultra-triathletes (i.e. all ten race days) and Triple Deca Iron ultra-triathletes (i.e. Day 21-Day 30) expressed in percent of overall race time. (Panel A) swimming, (Panel B) cycling, (Panel C) running. Values are means ± SD.

Figure 9

 9 presents the comparison in absolute times for swimming, cycling, running and overall race times of the ten days in the Deca Iron ultra-triathlon with Day 1-10, Day 11-20 and Day 21-30 in the Triple Deca Iron ultratriathlon. The athletes in the Deca Iron ultra-triathlon were relatively slower in swimming compared to Day 1-10 (p < 0.001), Day 11-20 (p < 0.001) and Day 21-30 (p < 0.001) in the Triple Deca Iron ultra-triathlon. In cycling, Triple Deca Iron ultra-triathletes were relatively faster than Deca Iron ultra-triathletes considering Day 11-20 (p < 0.05) and Day 21-30 (p < 0.05). The Triple Deca Iron ultratriathletes were relatively faster on Day 11-20 compared to Day 1-10 (p < 0.05). Considering running, Triple Deca Iron ultra-triathletes were relatively faster compared to Deca Iron ultra-triathletes on Day 11-20 (p < 0.05). And Triple Deca Iron ultra-triathletes were relatively faster on Day 11-20 compared to Day 1-10 (p < 0.05). For overall race time, Triple Deca Iron ultra-triathletes were relatively faster on Day 1-10 (p < 0.01) and Day 11-20 (p < 0.05) compared to Deca Iron ultra-triathletes. Again, Triple Deca Iron ultra-triathletes were relatively faster on Day 11-20 compared to Day 1-10 (p < 0.01). Figure 10 presents the comparison for swimming, cycling, and running in percent of overall race time of the ten days in the Deca Iron ultra-triathlon with Day 1-10, Day 11-20, and Day 21-30 in the Triple Deca Iron ultratriathlon. Deca Iron ultra-triathletes were relatively faster in swimming compared to Triple Deca Iron ultra-triathletes on Day 11-20 (p < 0.01) and Day 21-30 (p < 0.001). In cycling, the Deca Iron ultra-triathletes were relatively faster compared to the Triple Deca Iron ultra-triathletes on Day 1-10 (p < 0.001), Day 11-20 (p < 0.001) and Day 21-30 (p < 0.001). Considering running, the Deca Iron ultratriathletes were relatively faster on Day 1-10 (p < 0.01), Day 11-20 (p < 0.001) and Day 21-30 (p < 0.01) compared to the Triple Deca Iron ultra-triathletes.

Figure 9

 9 Figure 9 Comparison of split and overall race times for Deca Iron ultra-triathletes with Day 1-10, Day 11-20 and Day 21-30 in Triple Deca Iron ultra-triathletes in absolute race times. (Panel A) swimming, (Panel B) cycling, (Panel C) running, (Panel D) overall race times. Values are means ± SD.

Figure 10

 10 Figure 10 Comparison of split and overall race times for Deca Iron ultra-triathletes with Day 1-10, Day 11-20 and Day 21-30 in Triple Deca Iron ultra-triathletes in percent of overall race time. (Panel A) swimming, (Panel B) cycling, (Panel C) running. Values are means ± SD.

Table 1

 1 Number of finished ultra-triathlons with personal best time for Double Iron, Triple Iron and longer distances for finishers in the Deca Iron ultra-triathlon

	Rank	Number of finished	Personal best time	Number of finished	Personal best time	Other completed
		Triple Iron	in Triple Iron	Double Iron	in Double Iron	ultra-triathlons
		ultra-triathlons	ultra-triathlon	ultra-triathlons	ultra-triathlon
	1	24	37:18 h:min	22	22:57 h:min	3 Deca Iron with best time 240:55 h:min
	2	-	-	2	25:10 h:min	-
	3	1	47:19 h:min	4	26:34 h:min	1 Deca Iron in 285:40 h:min
	4	1	55:29 h:min	-	-	-
	5	-	-	-	-	-
	6	1	46:45 h:min	-	-	-

Table 2

 2 Number of finished ultra-triathlons with personal best time for Double Iron, Triple Iron and longer distances for finishers in the Triple Deca Iron ultra-triathlon

	Rank	Number of finished	Personal best	Number of finished	Personal best	Other completed
		Triple Iron	time Triple Iron	Double Iron	time Double Iron	ultra-triathlons
		ultra-triathlons	ultra-triathlon	ultra-triathlons	ultra-triathlon
	1	-	-	4	25:47 h:min	1 Deca Iron in 268:12 h:min
	2	1	40:14 h:min	4	23:55 h:min	1 Deca Iron in 222:17 h:min and 1 Double
						Deca Iron in 481:54 h:min
	3	-	-	-	-	1 Double Deca Iron in 497:56 h:min
	4	1	36:27 h:min	3	23:45 h:min	-
	5	-	-	-	-	-
	6	-	-	8	28:32 h:min	1 Deca Iron in 267:05 h:min
	7	1	47:03 h:min	1	29:05 h:min	-
	8	2	47:58 h:min	2	28:56 h:min	-

http://www.springerplus.com/content/3/1/461

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

BK collected all data and drafted the manuscript. TR helped in the interpretation of the data and helped drafting the manuscript. RL and CR performed the statistical analyses. All authors read and approved the final manuscript.


Eric Plouhinec
email: eric.plouhinec@st-cyr.terre-net.defense.gouv.fr

Lestp / Ietr

Bernard Uguen

Meriem Mhedhbi

Stéphane Avrillon

3D UTD Modeling of a Measured Antenna disturbed by a Dielectric Circular Cylinder in WBAN Context

. Indeed, the presented model gives results very close to these measurements. These first validations allow the presented model to be implemented into more complex WBAN (Wireless Body Area Networks) propagation models.

Introduction

This paper is related to WBAN (Wireless Body Area Networks) propagation modeling [START_REF] D'errico | Dynamic Channel Modeling at 2.4 GHz for On-Body Area Networks[END_REF] [START_REF] Cotton | Simulation of Millimetre-Wave Channels for Short-Range Body to Body Communications[END_REF]. It is focused on investigating and modeling the interaction of the antenna with the body, according to its orientation and proximity. One possibility is the deterministic WBAN channel modeling using UTD (Uniform Theory of Diffraction). UTD allows the received electromagnetic field computation associated with ray-tracing. This theory is only used with canonical shapes implying that human body could be modeled by cylinders, which is a widely used approximation [START_REF] Zhao | UWB On-Body Radio Channel Modeling using Ray Theory and Subband FDTD Method[END_REF]- [START_REF] Fort | A Body Area Propagation Model derived from fundamental Principles: Analytical Analysis and Comparison with Measurements[END_REF]. The dielectric cylinder could model, for instance, an arm, the trunk or a leg. In this paper, a 3D deterministic propagation prediction model is presented, specific to a measured antenna disturbed by a 3D circular dielectric cylinder. The used ray-tracing technique and the UTD formulations are detailed. This model has already been validated in 2D with a conducting and a dielectric cylinder using [START_REF] Mcnamara | Introduction to the Uniform Geometrical Theory of Diffraction[END_REF] [START_REF] Pathak | A Uniform GTD Analysis of the Diffraction of Electromagnetic Waves by a Smooth Convex Surface[END_REF] and [START_REF] Hussar | A Uniform GTD Treatment of Surface Diffraction by Impedance and Coated Cylinders[END_REF] respectively. Concerning the 3D approach, it has been validated for a conducting cylinder thanks to [START_REF] Govaerts | Electromagnetic Wave Scattering by a Circular Cylinder of Arbitrary Radius[END_REF]. Consequently, in this paper, we focus on a 3D dielectric cylinder approach. The model is compared with measurements made for the characterization of a body-worn antenna relying on a specific OTA (Over-The-Air) test-bed in anechoic chamber and a near-field antenna measurement chamber [START_REF] Mhedhbi | D2.1 -On-Body Antennas Characterization and Exploitable Radiation Properties[END_REF], for various distances between the antenna and a cylindrical phantom. This document is structured as follows: in Section II, the 3D deterministic model is described: we particularly focus on the specific 3D ray-tracing used for an electromagnetic source in presence of a circular dielectric finite length cylinder and the specific UTD formulations used for the computation of the total received electromagnetic field. Then, in Section III, we describe briefly the measurement campaign. Finally, in Section IV, we focus on the validation of the model, making a 3D comparison between the presented model results and measurements.

3D Deterministic Model Description

Ray-Tracing

First, we have to look for the different rays existing between a Source Point S and a Receiver Point P in presence of a 3D finite length circular cylinder of radius R. The cylinder, S and P could be placed anywhere in the cartesian coordinates system but S and P can not be too close to the cylinder because only UTD "Scattering" case [START_REF] Mcnamara | Introduction to the Uniform Geometrical Theory of Diffraction[END_REF] is studied in this paper. But, to make the computation easier, the cylinder axis is necessarily brought back to z axis and the S ordinate is brought back to the "null" value. We will call A (resp. B) the bottom disk center (resp. the top disk center) of the cylinder, as illustrated, for example, by Fig. 1 (a). So, we could have 2 ray-tracing scenarios: S and P could be in Line Of Sight (LOS) as illustrated by Fig. 1 (a) or in Non Line Of Sight (NLOS) as illustrated by Fig. 1 (b). For a LOS scenario, the ray-tracing finds a maximum of 4 rays, as illustrated by Fig. 1 (a): 1 direct ray (in brown), 1 reflected ray (in blue) with reflection point Q R , and 2 diffracted or "creeping"' rays (in red and green). The "red" (resp. "green") ray, of attachment point Q 1 (resp. Q 3 ) and detachment point Q 2 (resp. Q 4 ) will be called ray 1 (resp. ray 2) in the following. By convention, ray 1 (resp. ray 2) has its xy plane projection "creeping" in the clockwise (resp. counterclockwise) direction. For a NLOS scenario, only a maximum of 2 diffracted rays are found by the ray-tracing, as illustrated by Fig. 1 (b). We have to notice that, even if the treated case is canonical, rays diffracted by the bottom and top disks of the cylinder are not taken into account: indeed, they will not appear in WBAN because these disks are not parts of human bodies. Finding the reflection point Q R in 2D could not be solved by ruler and compass : for this, we have to find the minimum of the scalar product n.

---→ SQ R ---→ SQ R + ---→ Q R P ---→ Q R P
, where n is the circle normal at Q R , using an optimization function. Then, the reflection point height is obtained by an appropriated Thales theorem application. Concerning the attachment and detachment points of the "creeping" part of the diffracted rays, they are easily obtained in 2D using trigonometric formulas [START_REF] Mcnamara | Introduction to the Uniform Geometrical Theory of Diffraction[END_REF]. To obtain these points heights, we have to notice that the "creeping" part of the ray is necessarily a cylinder geodesic. If we "unfold" the cylinder, it becomes a plane and this geodesic becomes a straight line: then, these heights are easily found using the Thales theorem in the appropriated planes. Moreover, we need to verify if reflection, attachment or detachment points are on the cylinder because of its finite length. Finally, to test if S and P are in NLOS, we have to detect if -→ SP is blocked by the A center disk, or the B center disk, or the rectangular cross-section of the cylinder, perpendicular to xy plane and containing Q 1 and Q 3 [START_REF] Govaerts | Electromagnetic Wave Scattering by a Circular Cylinder of Arbitrary Radius[END_REF] [START_REF] Rusch | Observation-Point-Dependent Blocking Shadows on a Reflector Antenna[END_REF]. 2.2 UTD Interactions Modeling

Received Total Field Expression

First, the incident (or direct) field is expressed as:

E i θ,φ (P t) = F e (θ e P t , φ e P t )
e -jks src (P t) s src (P t)

where s src (P t) is the distance between S and point P t, k is the well-known wave number and:

F e (
is the value of the antenna pattern in the (θ e P t , φ e P t ) direction from S to P t.

Secondly, the reflected field is expressed as:

E r θ,φ (P ) = A r M r 2 R M r 1 E i θ,φ (Q R ) e -jks obs (Q R ) (3)
with R = R s 0 0 R h the dyadic matrix containing the reflection coefficients R s and R h for polarization "soft"(s) and "hard"(h) [START_REF] Mcnamara | Introduction to the Uniform Geometrical Theory of Diffraction[END_REF] [11] respectively, s obs (Q R ) the distance between Q R and P , A r the reflection spreading factor [START_REF] Pathak | A Uniform GTD Analysis of the Diffraction of Electromagnetic Waves by a Smooth Convex Surface[END_REF], M r 1 the 2D rotation matrix around [START_REF] Pathak | A Uniform GTD Analysis of the Diffraction of Electromagnetic Waves by a Smooth Convex Surface[END_REF] and M r 2 the 2D rotation matrix from Q R local reflected base [START_REF] Pathak | A Uniform GTD Analysis of the Diffraction of Electromagnetic Waves by a Smooth Convex Surface[END_REF] to (Q R , θ r P , φ r P ) with (θ r P , φ r P ) the direction from Q R to P .

--→ SQ R from (Q R , θ e Q R , φ e Q R ) to Q R local incident base
Thirdly, the diffracted field (due to ray 1 or 2) is:

E d1,2 θ,φ (P ) = A 1,2 d M d1,2 2 T 1,2 M d1,2 1 (4) . dη(Q 1,3 ) dη(Q 2,4 ) E i θ,φ (Q 1,3 ) e -jks obs (Q 2,4 ) with T 1,2 = T 1,2 s 0 0 T 1,2
h the dyadic matrix containing the diffraction coefficients associated with ray 1 or 2, A 1,2 d the diffraction spreading factor [START_REF] Pathak | A Uniform GTD Analysis of the Diffraction of Electromagnetic Waves by a Smooth Convex Surface[END_REF] associated with ray 1 or 2, M d1,2 [START_REF] Pathak | A Uniform GTD Analysis of the Diffraction of Electromagnetic Waves by a Smooth Convex Surface[END_REF] and M d1,2 (θ d1,2 P , φ d1,2 P ) the direction from Q 2,4 to P . Moreover, the dη(Q 1,3 ) dη(Q 2,4 ) factor illustrates the conservation of energy flux in the surface-ray strip from Q 1,3 to Q 2,4 [START_REF] Pathak | A Uniform GTD Analysis of the Diffraction of Electromagnetic Waves by a Smooth Convex Surface[END_REF].

1 the 2D rotation matrix around ---→ SQ 1,3 from (Q 1,3 , θ e Q 1,3 , φ e Q 1,3 ) to Q 1,3 local incident base
Finally, the total received field is computed thanks to the following expression:

E T θ,φ (P ) α (E i θ,φ (P ) + β r E r θ,φ (P )) + 2 i=1 β di E di θ,φ (P ) (5) α = 1 (resp. α = 0) in the LOS (resp. NLOS) case. β r = 1 if Q R is on the cylinder, β r = 0 otherwise. β d1 = 1 (resp. β d2 = 1) if Q 1 and Q 2 (resp. Q 3
and Q 4 ) are on the cylinder, β d1 = 0 (resp. β d2 = 0) otherwise. We have also to notice that even if (s, h) polarization is generally used for the UTD computation [START_REF] Mcnamara | Introduction to the Uniform Geometrical Theory of Diffraction[END_REF] [11], we obtain the received field in the (θ, φ) polarization which is directly related to the well-know spherical coordinates system used for the following measurements.

Interaction Coefficients

First, we have to define the curvature parameter m(P t) expressed as:

m(P t) = kR c (P t) 2 1 3 (6)
where R c (P t) is the cylinder radius of curvature along the geodesic containing point P t, and the distance parameter L(P t 1 , P t 2 ) expressed as:

(L(P t 1 , P t 2 )) -1 = (s src (P t 1 )) -1 + (s obs (P t 2 )) -1

We can now define the reflection coefficients as:

R s,h = - -4 ξ L e -j (ξ L ) 3 12 G s,h (ξ L , X L ) (8)
with:

ξ L = -2m(Q R ) cos θ i and X L = 2k L(Q R , Q R ) cos 2 θ i (9)
with θ i the reflection angle respect to cylinder normal at Q R .

And the diffraction coefficients are expressed as:

T 1,2 s,h = -m(Q 1,3 ) 2 k e -jkt 1,2 G s,h (ξ d1,2 , X d1,2 ) (10)
with t 1 (resp. t 2 ) the length of the considered cylinder geodesic between Q 1 and Q 2 (resp. Q 3 and Q 4 ) and:

ξ d1,2 = m(Q 1,3 ) R c (Q 1,3 ) t 1,2 and X d1,2 = k L(Q 1,3 , Q 2,4 )(ξ d1,2 ) 2 2(m(Q 1,3 )) 2 (11)
Finally, a common function G s,h (ξ, X) appears in equations ( 8) and ( 10). This function is expressed as:

G s,h (ξ, X) = e -jπ/4 2ξ √ π [1 -F (X)] + P s,h (ξ) (12)
depending on the "Transition" function F (X) expressed as:

F (X) = 2j √ X e jX +∞ √ X e -jt 2 dt ( 13
)
and on the "Pekeris" function P s,h (ξ) expressed as:

P s,h (ξ) = e -jπ/4 √ π +∞ -∞ e -jξt V (t) -q s,h V (t) W 2 (t) -q s,h W 2 (t) dt (14)
Fock-type Airy functions V (t), V (t), W 2 (t) and W 2 (t) appearing in equation ( 14) could be expressed according to the well-known Airy functions [START_REF] Plouhinec | Etude et Extension de Modèles de Prédiction de la Propagation[END_REF]. The "Pekeris" function is the only term that takes into account the dielectric nature of the cylinder depending on polarization, thanks to q s,h parameter expressed as:

q s = -jm(P t)K and q h = -j m(P t) K

with P t = Q R for the reflected ray, P t = Q 1 (resp. Q 3 ) for ray 1 (resp. ray 2) and:

K = √ r = r -j[ r + 60λσ ] (16)
with r the complex permittivity of the cylinder and λ the wavelength. r , r and σ are the cylinder dielectric properties and are called, respectively, the relative permittivity real part, the relative permittivity imaginary part and the conductivity.

Measurement Campaign Description

The measurement campaign was carried out in the Satimo SG32 near field antenna mesurement chamber of IETR ("Electronics and Telecommunications Institute of Rennes") illustrated by Fig. 2 (a). The cylinder under test is a custom built phantom depicted by Fig. 2 (b). It is nearly a cylindrical plastic bottle filled with MSL1800 phantom liquid. The dielectric properties of this liquid are given in [START_REF] Mhedhbi | Requirements for UWB Antenna Electromagnetic Simulations[END_REF] for the [0.3 -3] GHz frequency range. The size of the phantom of radius R = 3.5 cm was chosen to represent a human arm.

The bottle was fitted with a screw in the bottom to fix it on the platform grooves or holes (cf. Fig. 3). The used Ultra Wide Band (UWB) printed circular dipole antenna [START_REF] Guéguen | A Low Cost UWB Printed Dipole Antenna with High Performances[END_REF] was placed in the centre of the measurement cartesian coordinates system illustrated in Fig. 3. Its azimuth radiation pattern (here in the xy plane) is omnidirectional in the [3 -6] GHz frequency range.

The measurements were made in the [0.8 -5.95] GHz frequency range for various antenna-phantom distances d varying from 3 cm to 13.5 cm [START_REF] Mhedhbi | D2.1 -On-Body Antennas Characterization and Exploitable Radiation Properties[END_REF]. The measurements were carried out in the spherical coordinates system ( r, θ, φ),

for θ ∈ [0 • ; 180 • [ and φ ∈ [0 • ; 360 • [ with a 2
• step, for the received farfield as well as for the "antenna without phantom" radiation pattern called F e (θ e Rx , φ e Rx ) (directly included in the presented model thanks to equation ( 2)) with receiver Rx in the far-field region.

Comparison Results

For this comparison, the cylinder axis is the z axis to be coherent with the measurements cartesian coordinate system. So, for the model, cylinder point A becomes the cartesian coordinates system origin (cf. Fig. 1), different from the measurements system origin (cf. Fig. 3). S has an ordinate y s = 0 and consequently an abscissa x s = d + R. Moreover, S height is z s = 7 cm. Finally, we have to say that circular cylinder UTD scattering is only valid if kR ≥ 2 [START_REF] Govaerts | Electromagnetic Wave Scattering by a Circular Cylinder of Arbitrary Radius[END_REF], i.e. f ≥ 2.73 GHz for the presented comparison: that is why only results for frequencies greater than 3 GHz are presented. First, results for θ = 90 • (2D configuration) are given, for f = 4, 5 GHz and d = 7 cm, by Figures 4 (a) and 4 (b) for polarizations θ and φ respectively. Because the used frequency is greater than 3 GHz, we used the dielectric properties given by [START_REF] Koutitas | Multiple Human Effects in Body Area Networks[END_REF] for the "Muscle" model, i.e. r = r = 48.2 We can make the same conclusions for all the other possible scenarios with a minimum usable frequency of 1.5 GHz instead of 2.73 GHz calculated above. Under this minimum frequency, the model results move away from the measurements particularly for the smallest distance (d = 3 cm) for which S seems too close to the cylinder regarding the used wavelength. Moreover, we have to notice that results for polarization φ are not crucial: because of the used antenna [START_REF] Guéguen | A Low Cost UWB Printed Dipole Antenna with High Performances[END_REF], the emitted electric field is necessarily weaker for polarization φ and close to noise level. So, we will now only focus on polarization θ.

Secondly, for θ = 90 • (3D configuration), we achieve the same conclusions as for 2D configuration when 4 rays are found for LOS and 2 rays are found for NLOS. So we only focus in this paper on the cases for which not all the rays are found. For instance, Fig. 5 gives the results for polarization θ with d = 13 cm and f = 3 GHz. For θ = 70 • (Fig. 5 (a)), diffracted rays are only found for φ ∈ [142 • ; 218 • ] but model results are quite close to the measurements: so we can say that, if the forgotted diffracted rays maybe exist for the measurements, they are so weak that they have no influence on the total received field. For θ = 64 • (Fig. 5 (b)), the model does not find diffracted rays. Reflected rays are not found for φ ∈ [108 • ; 252 • ] because of the finite length of the 3D cylinder. And the model predicts no field in the NLOS region, i.e. for φ ∈ [170 • ; 190 • ], while a received field is measured. This case highlights the fact that the rays diffracted by the bottom and top disks of the cylinder have to be taken into account if we consider a canonical scenario, especially for φ ∈ [108 • ; 252 • ] that is the φ range for which model results and measurements are quite different. For θ = 60 • (Fig. 5 (c)), S and P are always in LOS. The model only finds the direct ray because the raytracing finds a reflection point that is always off the cylinder. But we can notice a measurements oscillatory behavior implying interference between, at least, 2 rays: we notice, once again, that additive rays, diffracted by the cylinder disks, have to be taken into account for a canonical case. Finally, for θ = 6 • (Fig. 5 (d)), the model only predicts a direct ray but, this time, its results are quite close to the measurements: for this case, the influence of the "disk-diffracted" rays seems less problematic.

Conclusion

A 3D deterministic propagation model, based on ray-tracing and UTD "Scattering" case, has been described and validated for the particular case of a 3D single dielectric circular cylinder of finite length. The developed model originalities are its deterministic nature and its ability to take into account not only the cylinder dielectric nature but also any antenna pattern including NFC (Near-Field Chamber) measured patterns. We have also highlighted the fact that the top and bottom disks cylinder diffracted fields have an influence on the total received field for the presented canonical problem. However, this issue does not exist in a WBAN context. Now, the model can be improved for the more general case of an elliptic cylinder. But 2 other scenarios could also be interesting for WBAN: the UTD "Radiation" scenario (with the Source (resp. the Receiver) on the cylinder and the Receiver (resp. the Source) off the cylinder) and the UTD "Coupling" scenario (when both Source and Receiver are on the cylinder). These scenarios imply the computation of new integrals, different from the Pekeris function of equation [START_REF] Rusch | Observation-Point-Dependent Blocking Shadows on a Reflector Antenna[END_REF]. Then, we can also study the possibility of adapting the model for the scenario of several cylinders in order to really model the human body. After, we can imagine a dynamic model taking into account the human body motion. Then we could imagine a scenario with several persons... This model should also be optimized in term of computation time if it has to be implemented into more complex WBAN propagation models. For this, the model could take advantage of spherical harmonics [1] [18] for antennas pattern description.

Figure 1 :

 1 Figure 1: 3D Ray-tracing illustrations 4

Figure 2 :Figure 3 :

 23 Figure 2: Measurement Configuration

Figure 4 :

 4 Figure 4: Comparison for θ = 90 • , d = 7 cm and f = 4.5 GHz.

Figure 5 :

 5 Figure 5: Polarization θ with d = 13 cm and f = 3 GHz.

  θ e P t , φ e P t ) = F θ (θ e P t , φ e P t ) θ e P t + F φ (θ e P t , φ e P t ) φ e P t

the 2D rotation matrix from Q 2,4 local diffracted base[START_REF] Pathak | A Uniform GTD Analysis of the Diffraction of Electromagnetic Waves by a Smooth Convex Surface[END_REF] to (Q 2,4 , θ d1,2 P , φ d1,2 P ) with

Acknowledgment

This work has been carried out in the frame of the CORMORAN project, which is funded by the French National Research Agency (ANR) under the contract number ANR-11-INFR-010.