Professor Robert Stavins

Michel Damian
email: michel.damian@upmf-grenoble.fr

dodgy arguments

Keywords: Cap-and-trade, Command and control, Lead in gasoline, SO 2 emissions, California AB32, Climate Change JEL classification: H23, Q54, Q58

The Only Feasible Way of Cutting Emissions" is to set up a market for tradable permits. We review and criticize his mains arguments. Our purpose here is not to deny the possibility of carbon trading, but to call for a realistic assessment of the deployment of cap-

1.Introduction

On 1 June 2014 Robert Stavins, the Albert Pratt Professor of Business and Government at Harvard University published an opinion piece in the New York Times entitled "The Only Feasible Way of Cutting Emissions" [START_REF] Stavins | The Only Feasible Way of Cutting Emissions[END_REF]. By that he means the only effective way of reducing carbon-dioxide emissions -which involves setting targets, cutting costs and deploying low-carbon technologies -is to set up a market for tradable permits. He introduces and concludes his piece with these words: "A national carbonpricing regime is the only feasible way for the United States to reach its goal of reducing emissions to 83 percent below their 2005 level by 2050. [...] it becomes clear that cap and trade has emerged as the preferred approach for meaningful action to address climate change."

Stavins is one of the leading protagonists in the economics of the environment and global warming, having worked on carbon-pricing and permit markets for almost four decades. In 1988 he headed "Project 88: Harnessing Market Forces to Protect the Environment". This emblematic report, which enjoyed bipartisan support from both Democrats and Republicans, focused on market incentives as an alternative to regulatory public policies. About 50 experts, from universities, government, private business and environmental organizations, took part in the project. Its message was that the environment could be protected in other ways than by regulation, through market incentives such as tradable emissions permits. To the best of our knowledge this report launched the cap-and-trade concept -a system for capping and trading emissions permits -to combat climate change. It included a recommendation for "international trading in greenhouse gases" (Stavins, 1988, p. 10). Which is well and good.

But there is a sizable problem with Professor Stavins' recent article in the New York Times, key points in his analysis being open to question. Our purpose here is not to deny the possibility of carbon trading, but rather to raise another issue. We shall restrict ourselves to just a few comments.

The cap-and-trade system, Stavins explains, "has a history of successful adoption and implementation over three decades". Let us take a closer look at the examples and evidence he provides in support of this claim.

2.The phasedown of lead in gasoline

He starts by asserting that the trading programme introduced by the Environmental Protection Agency in 1982 to phase out leaded petrol "produced a more rapid elimination of leaded gasoline from the marketplace than had been anticipated, and at a savings of some $250 million per year". The literature confirms the fact that the trading programme, compared with regulation, lowered the cost of achieving the targets for phasing out leaded fuel [START_REF] Newell | The U.S. Experience with the Phasedown of Lead in Gasoline. Discussion Paper, Resources for the Future[END_REF]. But we need to look a little closer at what actually happened when this trading scheme was set up, only operating for five years, from late-1982 to the end of 1987. Of all the sales and purchases of permits, respectively 67% and 70% were made between refineries, as part of internal trades or compensations, not as part of trading on a market between firms (Harrison, 1999, p. 30). So this was not so much a market as a source of flexibility for controlling sources of pollution within the same firm.

It is also important to see this market in the context of regulation spanning two decades, in other words the standards for lead content gradually imposed by the EPA starting in the mid-1970s. As a result of this regulatory policy the lead content of fuel in the United States had already been cut by 80% by the start of the 1980s, prior to the introduction of lead credits. The latter system was launched, in particular, to lower the cost of achieving the target standards for small refiners, which were struggling to meet increasingly stringent requirements for lead reduction. In 1988 the EPA went back to imposing the same standard of performance on all refineries. Lead, as an additive in fuel, was virtually eliminated by the beginning of the 1990s. It was finally banned altogether in 1996 [START_REF] Newell | The U.S. Experience with the Phasedown of Lead in Gasoline. Discussion Paper, Resources for the Future[END_REF].

Stavins makes no mention of the US attempt -which was rapidly shelved -to set up a scheme for trading permits for ozone-depleting substances. Between 1989 and 1995 there were 561 trades; in 1992, the busiest year on this market, there were 160 trades within firms, with only 20 international trades in 1992-95. Alternative technologies, covered by the Montreal Protocol, were quickly deployed, killing the market. "Reductions appear to have been the result of cost-saving due to technological change and the development of substitutes, rather than cost-saving due to trading." (Harrison, 1999, p. 33).

Nor does Stavins say anything -obviously space is limited in an opinion piece published in the mainstream press -about the Regional Clean Air Incentives Market (RECLAIM), a trading programme launched in California in the late-1980s primarily targeting nitrogen oxide emissions. In the past he has been positive and enthusiastic in his appreciation of this scheme, unlike other authors (Fowlie et al., 2009, p. 3). Meredith Fowlie et al. conclude their very detailed analysis of NOx trading in California as follows: "Our results indicate that emissions at RECLAIM facilities fell approximately 20 percent, on average, relative to the control facilities. However, during the period of great permit price volatility, several facilities did not comply with the regulation. When these facilities are included in our analysis, we find that RECLAIM did not reduce emissions relative to command and control during this volatile time." (Fowlie et al., 2009, p. 35;authors' emphasis).

3.The SO 2 emissions trading scheme

Stavins then pulls out a joker, citing the US sulphur dioxide allowance trading system, which served as a model for the Kyoto Protocol in 1997 and the European Union Emissions Trading Scheme. He sweeps the board with a single sentence, containing one striking, round number. "Congress enacted a law proposed by President George H. W. Bush to use cap and trade to cut sulphur dioxide emissions (and consequently acid rain) by half, at a savings of $1 billion per year." But to understand and debate this substantial saving rather more detail is required.

The US SO 2 allowance trading system, which was started in 1980, was the first scheme for tradable permits to be deployed on such a large scale. Many authors maintain that it was an economic, environmental and technological success, but this is open to question [START_REF] Damian | La politique climatique change enfin de paradigme[END_REF][START_REF] Damian | Repenser l'économie du changement climatique[END_REF]. On the economic front the market purportedly reduced emissions by more than initially projected and at half the cost entailed by a regulatory policy (Ellerman et al., 2000). But in fact the trading system itself had little impact. Between 1985 and 1993 about 80% of the cuts in SO 2 emissions were achieved by supplying coal-fired power plants with coal with a lower sulphur content (which had become more competitive thanks to lower transport costs obtained by deregulating the rail system, from the mid-1970s onwards) (Ellerman, Montero, 1996, pp. 13 and 15). As a result the price of a tonne of SO 2 stayed very low for a long time, simply because use of lowsulphur coal made it pointless to resort to the trading system. So the claim that this was an 'economic' success should perhaps be taken with a pinch of salt. Although the cost of reducing emissions was lower than expected -perhaps as much as 30 times lower (Burtraw, 2012, p.8) -credit for this achievement was not exclusively due to the trading system.

As to the claim that it was an environmental success, the EU used a regulatory method, not market mechanisms, as the basis for reducing SO 2 emissions. Between 1980 and 2010 overall SO 2 emissions dropped by 86% in EU-15 countries and by 67% in the US. Over the same period per capita emissions were cut by a factor of more than 12 in EU-15 countries, but by less than 5 in the US (in 2010 SO 2 emissions amounted to 6kg per person in EU-15 countries, compared with 24kg in the US). Emissions in relation to gross domestic product were roughly equivalent on both sides of the Atlantic in 1980; in 2010 they amounted to 259kg per million dollars of GDP in EU-15 countries, as against 656kg in the US.

Finally at a technological level, several technologies (desulfurization filters) were available and long established, well before the cap-and-trade programme. All that was needed was to improve them and promote their widespread use. Yet two decades after the launch of the SO 2 trading system, one-third of US coal-fired power plants are still not fitted with a desulfurization system. In contrast Germany rejected the idea of a trading system and achieved large-scale take-up of desulfurization solutions. There can be no doubt about the conclusion. "Traditional economists' view is that command-andcontrol instruments are less efficient than economic instruments (e.g., taxes and tradable permits) […] However, our analysis suggests that the German command-and-control approach was most likely the best choice in terms of efficiency. […] The implementation of an SO 2 emissions trading scheme in the United States has not led to the development of technologies that could have been used to achieve the ambitious German reduction aims at lower costs." (Wätzold, 2004, pp. 37 and 38).

So there is nothing very compelling about the success of the US SO 2 allowance trading system, neither economically, nor environmentally and less still technologically. The alleged success is more an illusion. Furthermore the US is not the only place where an SO 2 trading system has run into problems. In China too, despite the full support of the US administration and international organizations, the attempt to set up a market for permits along these lines proved a failure [START_REF] Shin | China's failure of policy innovation: the case of sulphur dioxide emission trading[END_REF].

In the two examples cited above Stavins pulled out of his sleeve savings worth millions and billions of dollars, thanks to cap-and-trade systems. It all seems very appealing. But he is very sparing in his attention to the difficulties assessing the ex ante and ex post costs and benefits of policies based on regulation versus market incentives [START_REF] Harrington | On the Accuracy of Regulatory Cost Estimates[END_REF][START_REF] Schelling | Incentives for Environmental Protection[END_REF].

But there is a more important concern with the SO 2 trading system. It is now moribund, an empty shell with permit prices devoid of any economic significance. This is what happened. Coal-fired power plants do not only emit SO 2 but also NOx, various carcinogens and mercury which is highly toxic. Under George W. Bush the administration proposed stricter limitations on emissions, by lowering the number of SO 2 permits available and setting up a cap-and-trade system for mercury. Many American states had already introduced strict standards limiting mercury emissions. Environmental organizations argued that a toxin such as mercury could not be treated as a marketable commodity and that it should be strictly regulated [START_REF] Rabe | Environmental Policy and the Bush Era: The Collision Between the Administrative Presidency and State Experimentation[END_REF][START_REF] Bellas | Are Tradable Permits for Mercury Worthwhile ?[END_REF]. In 2008 US courts ruled in favour of the opponents of the cap-and-trade scheme: emission-reduction policies should be based on norms and standards (using the best available technologies, which generate the least pollution). As a direct consequence of establishing norms for mercury, SO 2 emissions also dropped, the two forms of pollution being associated. The new regulations should cut the latter emissions by a factor of three by 2015, compared with 2010. As a result the market value of SO 2 is currently close to zero (Burtraw, 2012, p. 8).

There is much to be learnt from recent regulations, and their effectiveness, with regard to the economics of the environment (and global warming). Dallas Burtraw, a fine analyst, particularly of US cap-and-trade systems, has made a substantial contribution to this process in "The Institutional Blind Spot in Environmental Economics", a work of great intellectual honesty in which he acknowledges past errors and blind spots, including his own: "With adoption of the sulfur dioxide program, many economists (including me) clamored initially that other regulations under the [Clean Air] Act were unnecessary, inefficient, and raised costs; but ultimately they delivered substantial public health and economic benefits" (Burtraw, 2012, p. 9).

4.California's cap-and-trade system

In his article in the press Stavins goes on to cite the two cap-and-trade regimes operational in the US: the first and oldest operating in the north-east; the other in California. In 2008 10 east-coast states, now only nine (Connecticut, Delaware, Maine, Maryland, Massachusetts, New Hampshire, New York, Rhode Island and Vermont; New Jersey, yielding to Republican pressure, pulled out in May 2011), set up the Regional Greenhouse Gas Initiative. Its modest objective was to reduce CO 2 emissions in power generation by 10% by 2018. The permits almost all found buyers on the auctions organized by the various states. The funds collected, amounting to more than $1 billion, were mainly directed towards measures promoting energy efficiency and renewables. This is indeed a success, from which lessons may be learned: the price of permits is low, about $4 to $5 per tonne of CO 2 , even after the recent reduction in the volume of authorized emissions. Electricity prices have only slightly increased, by only 1%. So the cost is acceptable for consumers, who have made greater savings on their electricity bill thanks to the energy-efficiency schemes funded by the electricity utilities [START_REF] Pool | The Proof Is in the Pudding: Regional Greenhouse Gas Initiative Shows Pollution Pricing Works[END_REF][START_REF] Navarro | Reaping a Bonus From Cap-and-Trade[END_REF].

Stavins sums up the Californian initiative in one, powerful sentence. "California's ambitious cap-and-trade system will soon cover 85% of the state's economy and reduce carbon dioxide emissions to their 1990 levels by 2020" (our emphasis). Readers of the New York Times, unless they are very well informed, will be impressed. All on its own -or so Stavins would have us believe -California's cap-and-trade system will bring CO 2 emissions back down to their 1990 level by 2020. This would indeed be quite an achievement, but unfortunately it is not the case and what Stavins has written is downright misleading. As we shall explain.

In 2006 California passed the Global Warming Solutions Act (also known as Assembly Bill 32), which aims to reduce the state's greenhouse gas emissions to their 1990 level by 2020. Measures include a tradable permit system which was launched, after a great deal of legal wrangling, in 2013. It is the largest cap-and-trade system in the US. It could represent about 20% -it all depends on the terms of its operational deploymentof efforts to cut emissions by 2020. "The vast majority of reductions -about 80% -to achieve California's emission reduction mandate are expected to come from policy instruments other than carbon pricing. The California cap-and-trade program is tasked with inducing about 20% of the reductions, 34.4 MMTCO2E [million metric tonnes of CO 2 equivalent] out of 174 total" (Hanemann, Bush, 2009, p. 6). So the vast majority of the expected gains will be achieved through public policies, consisting of a bundle of forty regulatory actions, according to Michael Hanemann: reduction in emissions from motor vehicles; greater share of renewables in electricity generation; energy savings; reforestation; and even smaller-scale measures, "such as regulating the handling of wet manure at animal facilities. […] Implicit in this approach was an emphasis on 'cap' rather than 'cap and trade'. […] it was a natural extension of the regulatory approach traditionally used with air pollution and energy efficiency in California." (Hanemann, 2008, p. 119). With very ambitious public policies it might even be possible to achieve the targets "without engaging in cap and trade." (Bushnell, Peterman, Wolfram, 2008, p. 190).

What is the point in exaggerating the true situation, as Stavins has done in the New York Times? Particularly as the future of cap-and-trade is likely to be settled in California, home to the world's 12th largest economy. Not only has this system been hotly disputed in the courts but it has proved difficult to harmonize with a complex bundle of regulatory measures. The EU may well learn from what is happening in California. "Cap and trade may add little to the many energy efficiency regulations California has long had in place. […] For Europe, California has lessons about what can be achieved by regulators. […] The Californian stress on, and success with, energy regulation deserves Europe's attention at a time of European nervousness concerning the inefficacy of the ETS." (Buchan, 2010, pp. 30 and 33).

Stavins devotes two paragraphs to the EU carbon trading system. We shall not comment on this, it not being to our purpose, except to note that he concludes that the system "is now being expanded". Really? Let us hope that his words will allay the worries of Eurocrats nursing the ETS. They might even remind them of those happy times when Christian de Perthuis (then a member of the climate change economics research unit at France's Caisse des Dépôts) could write, hailing the launch of the European system: "This new creature can look forward to a fine future" (de Perthuis, 2006).

5.Conclusion

In his article in the New York Times Stavins has overstated his case in favour of a solution based exclusively on cap-and-trade, for he is perfectly familiar with the economic literature on we have drawn. His apology, which includes several glaring omissions, does nothing to boost his credibility. What is needed now is an honourable, realistic defence of the deployment of cap-and-trade systems and their limitations.


Pierre-Yves Rohan

Ahmad Bijar

Georges Bettega

Pascal Perrier

Yohan Payan

Predicting the consequences of tongue cancer surgery: design of a 3D patient-specific biomechanical model and evaluation

Introduction

The preoperative prediction of the consequences of tongue cancer surgery on tongue mobility is a topic of paramount interest for maxillo-facial surgeons. Loss of tongue mobility is associated with an impairment of basic functions such as speech articulation and deglutition which can, in some cases, induce a substantial decrease in the quality of life of patients [START_REF] Schliephake | Speech, deglutition and life quality after intraoral tumour resection: A prospective study[END_REF]. Significant variability has been observed across patients in the recovery of these functions, which are linked with tumor location and reconstruction techniques, but also, strongly, with patient-specific characteristics, such as vocal-tract morphology and idiosyncratic motor control strategies and which call for the development of tools to predict the functional consequences of tongue cancer surgery.

Recent advances in our group on the development of a three-dimensional (3D) finite-element biomechanical model of the oral cavity [START_REF] Gérard | 3D biomechanical tongue modeling to study speech production[END_REF][START_REF] Buchaillard | A biomechanical model of cardinal vowel production: muscle activations and the impact of gravity on tongue positioning[END_REF] and on the design of automatic adaptation of finite-element meshes to subject-specific morphology [START_REF] Bucki | A fast and robust patient specific finite element mesh registration technique: application to 60 clinical cases[END_REF] constitute encouraging steps towards the development of pre-operative planning tongue surgery systems. In particular, Buchaillard et al. [START_REF] Buchaillard | Simulations of the consequences of tongue surgery on tongue mobility: implications for speech production in post-surgery conditions[END_REF] showed the potential of such models to qualitatively predict the impact of oral cancer ablative surgery and reconstruction on tongue mobility. Building upon this work, our objective is to perform a clinical evaluation of the proposed biomechanical model by comparing between the acoustic signal predicted by our model after tongue cancer surgery and the acoustic data collected in post-surgery conditions. In the present work, we focus on the design of a 3D patient-specific biomechanical model.

Materials and methods

Materials

Pre-operative Magnetic Resonance Images are acquired on the head of a 50 year old patient undergoing tongue cancer using a 3-T clinical scanner and pixel resolution 1 x 1 x 1 mm 3 . Acoustic data of the patient in both pre-surgery and post-surgery conditions were also collected following the procedure developped by Acher et al. [START_REF] Acher | Speech production after glossectomy: Methodological aspects[END_REF].

Methods

An improved version of the 3D Finite Element (FE) model of the oral cavity developed in ANSYS at the laboratory [START_REF] Buchaillard | A biomechanical model of cardinal vowel production: muscle activations and the impact of gravity on tongue positioning[END_REF] is used. It consists of a soft deformable tongue and the surrounding hard tissues with which the tongue interacts (jaw, teeth, palate and hyoid bone). The improvements relate, primarily, with the tongue mesh as described in [START_REF] Rohan | Finite element modelling of nearly incompressible materials and volumetric locking: a case study[END_REF]. The identification of each muscle group in the tongue mesh is performed automatically. Muscle activation is modelled using the FE formulation of the Hill muscle model proposed by Blemker et al. [START_REF] Blemker | A 3D model of muscle reveals the causes of nonuniform strains in the biceps brachii[END_REF][START_REF] Nazari | The distributed lambda (λ ) model (DLM): a 3-d, finiteelement muscle model based on feldman's λ model; assessment of orofacial gestures[END_REF] that includes both the active stress stiffening effect and the passive transversal isotropy of muscles. Passive tissues (inactive muscles) are modelled using a 5-parameter Mooney-Rivlin model.

The patient-specific model is automatically set up using pre-operative MR data coming from a patient undergoing tongue cancer surgery. A volume image registration-based approach is used, in which an atlas FE mesh is deformed using a 3D transform derived from patient-to-atlas volume (MR) image registration [START_REF] Tong | Generation of patient-specific Finite-Element mesh from 3D medical images[END_REF]. The volume image registration includes rigid/affine registration (as global alignment) and non-rigid registration (as local alignment). The global transformation removes the scale and position differences between the atlas and patient (MR) image volumes. Accordingly, the local transformation compensates the inter-patient morphological differences. Finally, the obtained 3D deformation field (derived from registration) is used to deform the tongue atlas mesh to fit the patient's tongue morphology. As the deformed mesh may possess irregular or low quality elements, an automatic mesh repair algorithm is used [START_REF] Bucki | A fast and robust patient specific finite element mesh registration technique: application to 60 clinical cases[END_REF].

Tongue cancer surgery was accounted for in the model by modifying the biomechanical properties of the tongue tissues that were excised and reconstructed during the surgical procedure. This was done in collaboration with a maxillo-facial surgeon. First, the zone that was resected was identified and the corresponding elements in the biomechanical model were removed. In our case, the hemi-glossectomy performed by the surgeon consisted in resecting partially or totally the styloglossus, the tranversalis, the hyoglossus and the genio-glossus muscles. Then, the tongue model was reconstructed with flaps having passive elastic properties. In practice, the surgeon used an anterolateral thigh cutaneaous flap. The flap is totally inactive during tongue activation.

The response of the tongue to the activation of muscles, responsible for the production of cardinal vowels /i/, /a/, /u/, after tongue surgery, is simulated. The deformed tongue surface is then used as input for the acoustic model of sound synthesis which, in turn, is used to compute the predicted spectral characteristics of vowels in post-surgery condition.

Results

The patient-specific tongue geometry and mesh reconstructed from the Pre-operative Magnetic Resonance Images is represented in figure 1 below.

Perspectives

Comparison between the acoustic signal predicted by our model and the acoustic data collected in postsurgery conditions (6 months after surgery) is currently being performed and will serve as basis for the evaluation of the prediction of the impact of the surgery.

Conclusion

A patient-specific biomechanical model is set up in this study to predict the impact of surgery on tongue mobility. Evaluation of the model is done by comparing the simulated acoustic signal with the acoustic data collected in both pre-surgery and post-surgery conditions. Further work is required to (i) develop appropriate numerical strategies that allow to accurately handle and account for the resection and reconstruction procedure and (ii) reduce computation time.

Figure 1 :

 1 Figure 1: (a) Pre-operative MR data of the patient undergoing tongue cancer surgery superimposed with the 3D patient-specific tongue mesh set-up. (b) Side, (c) isometric and (d) front views respectively of the 3D FE tongue mesh.

* yohan.payan@imag.fr

Acknowledgments

This work was supported by French state funds managed by the Grenoble University AGIR projects (CLAM) and by the ANR within the Investissements d'Avenir programme (Labex CAMI) under reference ANR-11-LABX-0004.


Baptiste Pierrat
email: baptiste.pierrat@gmail.com

Jérôme Molimard

Laurent Navarro

Stéphane Avril

Paul Calmels

Evaluation of the mechanical efficiency of knee braces based on computational modelling

Keywords: knee braces, knee orthoses, efficiency, discomfort tolerance, finite element

Introduction

The knee is the largest joint in the body and supports high loads, up to several times the body weight. It is vulnerable to injury during sport activities and to degenerative conditions such as arthrosis. Various syndromes are associated with an increased knee laxity, leading to a functional instability (i.e. a 'wobbly' feeling). The anterior cruciate ligament (ACL) is the main postero-anterior (P-A) stabilizing structure, preventing the knee from an anterior drawer movement, for instance when walking upstairs [START_REF] Vergis | Sagittal plane translations of the knee in anterior cruciate deficient subjects and controls[END_REF]). The ACL is involved in 24% of all knee injuries and 59% of ligamentous injuries [START_REF] Bollen | Epidemiology of knee injuries: diagnosis and triage[END_REF]. In the United States,the annual incidence in the general population is approximately 1 in 3500 with 100,000 ACL reconstructions performed each year [START_REF] Gordon | Anterior cruciate ligament injuries[END_REF][START_REF] Miyasaka | The incidence of knee ligament injuries in the general population[END_REF]. These conditions are a huge burden on individuals and healthcare systems.

Knee braces or orthoses are commonly prescribed by physicians and medical practitioners for pathologies involving knee pain/laxity. This choice is related to their claimed mechanical effects but rely on very few assessments, from biomechanical August 21, 2013 18:19 Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 2 studies to therapeutic trials [START_REF] Thoumie | Orthèses de genou. Première partie : Evaluation des propriétés physiologiques à partir d'une revue de la littérature[END_REF][START_REF] Thoumie | Orthèses de genou. Evaluation de l'efficacité clinique à partir d'une revue de la littérature[END_REF]. Numerous action mechanisms have been proposed and investigated such as: proprioceptive improvements [START_REF] Barrack | Proprioception in the anterior cruciate deficient knee[END_REF][START_REF] Corrigan | Proprioception in the cruciate deficient knee[END_REF][START_REF] Mcnair | Knee Bracing: Effects on Proprioception[END_REF][START_REF] Birmingham | Knee bracing for medial compartment osteoarthritis: effects on proprioception and postural control[END_REF][START_REF] Thijs | Does bracing influence brain activity during knee movement: an fMRI study[END_REF], strain decrease on ligaments [START_REF] Beynnon | The Effect of Functional Knee Bracing on the Anterior Cruciate Ligament in the Weightbearing and Nonweightbearing Knee[END_REF][START_REF] Beynnon | Anterior cruciate ligament strain in-vivo: A review of previous work[END_REF][START_REF] Fleming | The Influence of Functional Knee Bracing on the Anterior Cruciate Ligament Strain Biomechanics in Weightbearing and Nonweightbearing Knees[END_REF][START_REF] Hinterwimmer | Influence of a mono-centric knee brace on the tension of the collateral ligaments in knee joints after sectioning of the anterior cruciate ligament -an in vitro study[END_REF], neuromuscular control enhancement [START_REF] Osternig | Effects of prophylactic knee bracing on lower extremity joint position and muscle activation during running[END_REF][START_REF] Ramsey | Electromyographic and biomechanic analysis of anterior cruciate ligament deficiency and functional knee bracing[END_REF][START_REF] Théoret | Study on three-dimensional kinematics and electromyography of ACL deficient knee participants wearing a functional knee brace during running[END_REF] and joint stiffness increase [START_REF] Lunsford | Response of eight knee orthoses to valgus, varus and axial rotation loads[END_REF]. Other studies aimed to justify the use of knee orthoses in medical practice. These studies were reviewed by [START_REF] Paluska | Knee braces: current evidence and clinical recommendations for their use[END_REF]; [START_REF] Thoumie | Orthèses de genou. Première partie : Evaluation des propriétés physiologiques à partir d'une revue de la littérature[END_REF][START_REF] Thoumie | Orthèses de genou. Evaluation de l'efficacité clinique à partir d'une revue de la littérature[END_REF]; [START_REF] Genty | Place des orthèses en pathologie ligamentaire du genou. Revue de la littérature[END_REF]; [START_REF] Beaudreuil | Clinical practice guidelines for rest orthosis, knee sleeves, and unloading knee braces in knee osteoarthritis[END_REF]. The following conclusions can be drawn from the literature:

(1) Mechanical/physiological effects have been emphasized, but these mechanisms have been poorly characterized.

(2) Only a few high-level clinical studies exist, and the effectiveness of bracing versus no bracing on improving joint stability or reducing pain has not been conclusively demonstrated in practice.

Possible explanations of (1) having no perceptible effect on (2) are that mechanical action levels are too low, or that patients do not comply to the orthopedic treatment and do not wear enough the device due to comfort issues. What is more, subjective evaluations of patients highlight a large demand for these products; therefore, their efficiency is still widely discussed among medical experts.

As a consequence of these uncertainties, medical practitioners and manufacturers still lack a simple evaluation tool for knee orthoses. A French committee of experts highlighted this problem [START_REF] Ribinik | Evaluation des orthèses de genou et de cheville en pathologie de l'appareil locomoteur. Avis d'experts[END_REF]) and stated that orthoses must be evaluated by taking both the mechanisms of action and the desired therapeutic effects into account. Mechanical actions of knee orthoses have been evaluated using experimental devices either on cadaveric knees [START_REF] France | In vitro assessment of prophylactic knee brace function[END_REF] or on surrogate limbs [START_REF] Paulos | The biomechanics of lateral knee bracing[END_REF][START_REF] France | The biomechanics of lateral knee bracing[END_REF][START_REF] Cawley | Comparison of rehabilitative knee braces[END_REF][START_REF] Lunsford | Response of eight knee orthoses to valgus, varus and axial rotation loads[END_REF]). However, the cadaveric knee method leads to unreliable results because of substantial scatter (anatomical, physiological and methodological variances); as for the surrogate method, phantoms consisted in mechanical parts mimicking the joint, thigh and leg on which a specific kinematics was simulated (drawer, pivot shift, lateral impact...), and instrumented with tensiometers in order to quantify the mechanical effects of a brace. These devices were poorly representative of a real human limb (skin and soft tissue behaviour). What is more, tests were conducted on very specific braces and do not allow to understand bracing mechanisms in general. Besides, these tests are far from accurate in reproducing the dynamical conditions of real movements (running, walking...). Finite Element (FE) analysis is a powerful tool when it comes to complex mechanical simulations and would definitely help to understand force transfer mechanisms and unveil the influence of different brace characteristics. To our knowledge, there is no published computational analysis of a brace/limb system, although it would answer most of the above concerns.

An original FE model approach has been developed and is presented in this paper. This model was built in agreement and cooperation with medical practitioners and orthotic manufacturers, in a tentative of linking design problems, brace ability to prevent a given pathology (ACL deficiency) and brace discomfort tolerance. Studies dealing with passive motion of the knee using Robotic Testing Systems [START_REF] Woo | Biomechanics of the ACL: Measurements of in situ force in the ACL and knee kinematics[END_REF][START_REF] Lujan | Effect of ACL deficiency on MCL strains and joint kinematics[END_REF]) reported that removing the ACL greatly increases anterior tibial translation and reduces internal tibia rotation under an anterior load; knee braces prescribed for this condition should compensate this laxity and stiffen the joint in this direction. An efficiency evaluation index reflecting this objective is

needed.

As there is a huge variety of orthoses on the market, the focus was placed on manufactured knee braces, in opposition to individualized, custom-made orthotic devices. They are usually made of synthetic fabrics and may incorporate bilateral hinges and bars, straps, silicone anti-slipping pads and patella hole. A typical design of an usual brace is depicted in Figure 1. Such braces are prescribed either for prophylactic or functional purposes [START_REF] Thoumie | Orthèses de genou. Première partie : Evaluation des propriétés physiologiques à partir d'une revue de la littérature[END_REF].

This paper is organized as follows. First, we will present the development of the FE model of a braced knee including the geometry, mesh, constitutive parameters and boundary conditions (BCs). Then we will explain how this tool has been used to assess the brace effectiveness against a drawer movement, using a design of experiment approach in order to account for various brace designs. Finally we will propose an optimization criterion to maximize the brace effectiveness while limiting discomfort tolerance issues.

Methods

Finite element model of the braced knee

The model was developed under Abaqus ® v6.10-2. It was built as a generic tool to understand the force transfer mechanisms between the rigid parts of the knee brace (hinged bars) and the joint (bones). The aim was to investigate how this force transfer is altered by the deformation of the brace fabric and the patient soft tissues on the one hand, and sliding phenomena at the interfaces on the other. To this end, some geometrical and mechanical parameters were considered as factors and their influence on the global response were characterized.

The deformable limb

2.1.1.1 Geometry. 3D geometry of the human lower limb was obtained from a whole body PET-CT (Positron Emission Tomography -Computed Tomography) scan. This scan comes from an online DICOM sample image sets [START_REF] Melanix | DICOM sample image sets[END_REF]. The DICOM data states that the subject is a 34 year old female. The lower body consisted of about 500 slices of thickness 2 mm, and size 512x512 pixels with 0.98 mm/pixel resolution. A limb was cropped and segmented using the software ImageJ (Rasband 2011). Segmentation was performed by thresholding, resulting in one material identified as soft tissues after removal of bones areas and skin, as depicted in Figure 2.

A fine 3D mesh of the segmented stack was imported in Rhinoceros ® v4.0 for surface reconstruction. Finally, surfaces were imported in Abaqus ® in order to control mesh generation from this software. Skin was generated by offsetting the external boundary of the soft tissue part, resulting in a separate layer of thickness 1 mm [START_REF] Evans | Measuring the mechanical properties of human skin in vivo using digital image correlation and finite element modelling[END_REF]. The thigh was separated from the leg in order to get two separate parts as seen in Figure 3. This last step was done in order to avoid any internal knee stiffness and in a concern of modelling knee kinematics without convergence problems due to the high deformation of elements in the centre knee area. Patella was modelled as a separate shell part. Finally, the limb was scaled in order to reach the dimensions of a median French male limb (Institut Français du Textile et de l'Habillement 2006): circumference at the knee: 38 cm; +15 cm above the knee: 49.3 cm; -15 cm below the knee: 36.2 cm. The resulting parts can be seen in the exploded view of the assembly in Figure 3. It is noteworthy that the limb is not fully extended, it is slightly bent with an angle of 20°.

2.1.1.2 Mesh. Soft tissues of the thigh and leg were respectively meshed with 37572 and 22627 reduced linear hexahedral elements of type C3D8R (Abaqus 2010) using a custom meshing algorithm written in Python ® allowing a finer mesh at the skin interface and a coarser mesh around the bones. During this step the bone geometry was simplified. The patella was meshed with shell elements: 714 reduced linear quadrilateral elements of type S4R and 26 reduced linear triangular elements of type S3R (Abaqus 2010). The skin was meshed with 22648 linear hexahedral elements of type C3D8 (Abaqus 2010) using an offset algorithm, with two elements in the thickness. The resulting meshes are depicted in Figure 3.

2.1.1.3 Materials. The soft tissue material was defined as homogeneous, isotropic, quasi-incompressible and hyper-elastic. A Neo-Hookean strain energy function was used [START_REF] Linder-Ganz | Assessment of mechanical conditions in subdermal tissues during sitting: a combined experimental-MRI and finite element approach[END_REF][START_REF] Avril | ASME; Mixed Experimental and Numerical Approach for Characterizing the Biomechanical Response of the Human Leg Under Elastic Compression[END_REF][START_REF] Dubuis | Identification of the material parameters of soft tissues in the compressed leg[END_REF]. This function may be written:

W = G 2 (I 1 -3) + K 2 (J -1) 2 (1)
where G and K are the material parameters, I 1 = T r(F.F t ) is the first deviatoric strain invariant, J = det(F) the volume ratio, F the deformation gradient, F = J -1/3 F the deviatoric part of the deformation gradient and T r the trace of a matrix. The constitutive properties represent the homogenized properties of muscles, fat, tendons and fascias. Values for G have already been identified for the leg for both passive muscles (G = 5 kPa) by [START_REF] Dubuis | Identification of the material parameters of soft tissues in the compressed leg[END_REF] and contracted muscles (G = 400 kPa) by [START_REF] Iivarinen | Experimental and computational analysis of soft tissue stiffness in forearm using a manual indentation device[END_REF]. Soft tissue stiffness was considered as a parameter in the model to account for passive/active muscles. K was set such as K = 100G in order to have a quasi-incompressible material (corresponding to an initial Poisson's ratio of 0.49). Bones were modeled as rigid bodies by fixing the surface nodes. The skin material was defined as homogeneous, isotropic, quasiincompressible and hyper-elastic. An Ogden strain energy function was used [START_REF] Evans | Measuring the mechanical properties of human skin in vivo using digital image correlation and finite element modelling[END_REF][START_REF] Flynn | Mechanical characterisation of in vivo human skin using a 3D force-sensitive micro-robot and finite element analysis[END_REF]. This function may be written:

W = 2µ α 2 (λ α 1 + λ α 2 + λ α 3 ) + K 2 (J -1) 2 (2)
where α, µ and K are the constitutive parameters, λ i = J -1/3 λ i are the deviatoric principal stretches, λ i the principal stretches, J = det(F) the volume ratio and F the deformation gradient. Values of α and µ have been identified by [START_REF] Evans | Measuring the mechanical properties of human skin in vivo using digital image correlation and finite element modelling[END_REF] on the forearm. µ was set to 130 Pa, α to 26 and K to 6.5 kPa. [START_REF] Evans | Measuring the mechanical properties of human skin in vivo using digital image correlation and finite element modelling[END_REF] also identified an initial strain of 0.2, so a corresponding pre-stress of 1.35 kPa was applied in circumferential and longitudinal directions of the skin at the start of the analysis.

The orthosis

2.1.2.1 Geometry. Geometry of the orthosis was designed from determination of the mechanically important features of usual existing braces. The identified features, as depicted in Figure 1, are:

-three metal bars on each side. August 21, 2013 18:19 Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 6 -An articulated system between bars consisting of two hinges on each side with a blocking feature to prevent hyper-extension. -A polymeric textile with identified orientations.

-Fitting straps made of a different, stiffer textile.

The orthosis was generated as a slightly conical cylinder. Different regions were defined on the part: brace fabric, rigid bars and straps (Figure 3), each of which was assigned different mechanical properties. Rigid bars were connected using hinge connectors (Abaqus 2010) with a blocking feature, allowing them to pivot with the joint but blocking them in hyper-extension. An assumption was made that a patella hole, which is a circular opening above the patella area, as seen in Figure 1, has not a significant mechanical effect on the brace stiffness, hence the choice not to model it. The resulting brace model is reported in Figure 3. The size (cylinder circumference) and length (cylinder height) of the brace were considered as parameters in the model.

2.1.2.2 Mesh. The orthosis was meshed with reduced linear quadrilateral shell elements of type S4R (Abaqus 2010). The number of elements varied from 14790 to 43690 depending on the length of the brace.

2.1.2.3 Material. Mechanical behaviour of fabric has been already successfully modelled using shell elements [START_REF] Yu | Drape Simulation of Woven Fabrics by Using Explicit Dynamic Analysis[END_REF]. The material was defined as homogeneous, orthotropic and linear elastic. The constitutive equations, written in vectorial form, are:

    N 11 N 22 N 12     =     E1 1-ν12ν21 ν21E1 1-ν12ν21 0 ν12E2 1-ν12ν21 E2 1-ν12ν21 0 0 0 G 12         11 22 2 12     (3)
and

  M 11 M 22 M 12   =   F 1 µ 2 F 1 0 µ 1 F 2 F 2 0 0 0 τ 12     κ 11 κ 22 2κ 12   (4)
where N ij and M ij are the tensions and bending moments of the fabric, ij and κ ij the strains and bending strains, E i the tensile rigidities, G 12 the shear rigidity, ν ij the Poisson's ratios, F i the bending rigidities, τ 12 the torsional rigidity and µ i parameters analogous to Poisson's ratios. Subscripts 1 and 2 represent the longitudinal and circumferential directions of the brace cylinder and the directions along and across the straps respectively. Tensile rigidities, shear rigidity and Poisson's ratios were obtained from unidirectional and off-axis tensile tests on an Instron ® machine at speeds of 50 mm/min on 40×20 mm fabric samples. The linear elasticity assumption was judged reasonable from tensile tests for strains ≤40%. Bending rigidities were measured using a KES-F (Kawabata Evaluation System for Fabrics) device [START_REF] Yu | Drape Simulation of Woven Fabrics by Using Explicit Dynamic Analysis[END_REF][START_REF] Wu | Mechanical properties of fabric materials for draping simulation[END_REF]. Samples were taken from four commercially available orthoses and their straps. Fabric stiffness was considered as a parameter in the model. In order to reduce the number of parameters and based on measured properties, only E 1 varied and the other properties were derived as reported in Table 1. The strap properties were consistent among braces and are provided in Table 1. August 21, 2013 18:19 Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 7

Table 1. Mechanical properties of brace and strap fabrics.

Property The bars were modelled as rigid, considering the fact that they are usually made of 2 mm thick aluminium.

E 1 = E 2 (N/m) G 12 (N/m) ν 12 = ν 21 F 1 = F 2 (N.m) τ 12 (N.m) µ 1 = µ 2 Brace [5000; 20000] E 1 /3 0 f (E 1 ) * F 1 /3 0 Straps 26400 9100 0.45 10 -3 0.5 × 10 -3 0 * f (E 1 ) = 2.37 × 10 -4 log(E 1 ) -1.415 × 10 -3 (from mechanical

Interfaces

A basic Coulomb friction model was used for the orthosis/skin and skin/soft tissues interactions in which contact pressure is linearly related to the equivalent shear stress with a constant friction coefficient µ. Values of µ for different fabric/skin systems are available in the literature, averaging 0.7 for Spenco ® [START_REF] Sanders | Material properties of commonly-used interface materials and their static coefficients of friction with skin and socks[END_REF], or ranging from 0.3 (Teflon ® ) to 0.43 (cotton and polyester) [START_REF] Gerhardt | Skin-textile friction and skin elasticity in young and aged persons[END_REF].

Concerning the skin/soft tissues contact, the choice of modelling skin as a separate layer comes from a preliminary study, in which the effect of in-plane pulling of the skin on an area 10 cm above the knee was observed by ultrasound (Aixplorer ® system with auto time gain compensation mode). A 10 N traction was performed using duct tape and displacements of the skin and underlying structures were observed in the sagittal plane (4(a)). The PIV (Particle Image Velocimetry) ImageJ plugin (Rasband 2011;[START_REF] Tseng | Etude d'architecture multicellulaire avec le microenvironnement contrôlé[END_REF]) was used on 1166×666 pixels images with successive interrogation window sizes of 160×160;100×100;70×70 pixels to compute the displacement field (4(b)), showing a very narrow gliding plane between the skin/fat and underlying structures. From the work of [START_REF] Guimberteau | Introduction à la connaissance du glissement des structures sous-cutanées humaines[END_REF], who described this interface, it is probable that modelling a separate layer is a good approximation of this mechanical behaviour, as long as slipping magnitudes are relatively small (∼1 cm in the case of these observations). No data was found in the literature for friction coefficient measurements of this interface, but a value of 0.1 was chosen from measurements of this coefficient between tendons and the structures over which they slide [START_REF] Albin | Vivo Estimation of the Coefficient of Friction between Extrinsic Flexor Tendons and Surrounding Structures in the Carpal Tunnel[END_REF]. Skin was attached to soft tissues at the top and bottom of the limb.

No contact was defined between the thigh and leg.

Analysis steps and BCs

A quasi-static analysis was performed using the Explicit solver (Abaqus 2010) in order to solve significant discontinuities (fabric creases, contacts). Time scale and material density were carefully chosen to prevent dynamic effects (kinetic energy August 21, 2013 18:19 Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 8

Initial strap tightening

Initial strain along strap to account for strap tightening, as described in Section 2.1.4 5-15 % * * * Discrete factor. * * Continuous factor.

was much inferior to external work). The simulation consisted in three steps:

Step 1: a displacement field was applied to the brace to enlarge it and make it fit at the right place around the joint. The pre-stress was applied to the skin.

Step 2: contacts were activated, previously applied displacements were released in order to let the brace compress the limb and reach the mechanical equilibrium; bones were fixed; the straps were pre-stressed to simulate a real fitting (this pre-stress was considered as a parameter).

Step 3: a joint kinematics was imposed on the tibia/fibula, in this case a 1-DOF (Degree Of Freedom) P-A drawer of magnitude 20 mm; femur and patella were fixed.

Design of experiment approach

In order to study the influence of several mechanical and geometrical parameters of the brace/limb system, a design of experiment technique was used. It consists in selecting relevant parameters, choosing a plausible domain for each of them, using a sampling method to pick the different experimental points, run the analyses, choose one or several output data, extract the outputs from the analyses and build one or several response surfaces from these output data.

Parameter screening and domain selection

The basic ideas, terminology and techniques of design of experiments was explained by [START_REF] Goupy | Introduction to Design of Experiments with JMP Examples[END_REF]. Six parameters, or factors, were selected. Definition of these factors and the selected domains are reported in Table 2.

Soft tissue stiffness was used as a 2-level factor to reproduce the passive/contracted stiffness of the limb. As a consequence, it involves the following approximations: the contracted limb has the same geometry than the passive limb and the stiffness increase is homogeneous and isotropic. This factor is preponderant compared to inter-individual stiffness variability as there is a factor 80 between passive and active muscles. The brace circumference domain is justified as the indicated brace size for the limb (manufacturer's size table) ± 1 size. Brace length is usually consistent between brace manufacturers but no scientific argument supports this choice. It corresponds to the intermediate value of our 3-level factor (34 cm). The lower level is supposed to be the minimum length to tighten the straps above and below the joint (17 cm), and the upper level is 51 cm. Fabric stiffness was slightly extrapolated from mechanical characterization: usual braces are made with fabrics of stiffness usually comprised between 0.5 and 1.5 kN/m. August 21, 2013 18:19 Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 9

The orthosis/skin friction coefficient ranged from 0.1 to 1 (see Section 2.1.3). This factor domain would need to be supported by tribological characterization of different fabric/skin interfaces with and without anti-sliding features. Finally, initial strap tightening ranged between 5 and 10 %. The lower value of its domain is just enough to contact the skin without applying a significant pressure while the higher value roughly corresponds to highly tightened straps on a subject with a similar brace.

Sampling method

The use of Latin hypercube for efficient sampling has been described by [START_REF] Olsson | On Latin hypercube sampling for structural reliability analysis[END_REF]. This sampling method corresponds to the generalization of Latin squares in multiple dimensions. It allows variance reduction by partitioning the input factor space into equal probability disjoint sets. This method was first introduced in 1979 by [START_REF] Mckay | Comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF]. It was chosen in a concern of finding a good balance between total computational time and study domain coverage. The principle of Latin hypercubes is to partition a space of n factors into m n parts, each factor being divided into m equally probable intervals. Then, m samples are taken in the n dimensions space so that only one division of each factor contains one sample. Stratified Latin hypercube is a special type of Latin hypercube in which discrete factors are allowed (Matlab 2011). The choice of the stratified Latin hypercube was done because two of the six factors are discrete. Variables were centred around 0 and scaled to a [-1; 1] interval, resulting in coded variables. There are several methods used to construct the Latin hypercube, we chose the minimizing of the Root Mean Square (RMS) variation of the cumulative distribution function in order to obtain a smooth distribution. This sampling resulted in a factor matrix X of n = 6 columns and m = 100 rows with an average shortest distance between points of 0.72 ± 0.23.

Responses

Two different responses were studied: the stabilizing effect of the brace along the P-A movement and the pressures on the skin. The relevance of these indexes will be discussed in Section 4.1.3. Calculations were performed with Matlab ® . 2.2.3.1 Joint stiffening. The response of the limb only was computed thanks to a simulation without bracing and subtracted from the braced limb responses in order to get rid of the skin stiffness contribution. A typical response of the reaction force of the tibia along the P-A direction as a function of the applied P-A drawer magnitude is depicted in Figure 5. The analysis yielded minor damping effects due to the Explicit solver. These oscillations were filtered by a quadratic regression. In all cases, responses were nicely fitted by this kind of polynomial. Finally, the sagittal-plane shear stiffness of the orthosis was calculated as the slope of the 2nd order polynomial at a displacement of 5 mm (see Figure 5). [START_REF] Eagar | A method for quantifying the anterior load-displacement behavior of the human knee in both the low and high stiffness regions[END_REF] showed that the typical load-displacement curve of a knee exhibits a low stiffness region followed by a high stiffness region. This transition appeared at a displacement of ∼5 mm and is caused by the tension of passive structures, especially the ACL. The k index is related to this behaviour. This index was calculated for the 100 simulations resulting in a response vector k = {k 1 , k 2 , ..., k 100 }.

2.2.3.2 Pressure distribution. The pressure field applied by the brace on the skin elements was extracted and the mean pressure was computed. This discomfort tolerance index was calculated for the 100 simulations resulting in a response vector p = {p 1 , p 2 , ..., p 100 }. Although the pressure field was not uniform, the mean pressure was chosen over the maximum pressure because a global index was judged more reliable than local maxima, which might be influenced by element size and numerical singularities.

Response surfaces

6-dimensional response surfaces f and g respectively express stiffness and discomfort tolerance indexes k and p as a function of the factor matrix X:

k = f (X) + k p = g(X) + p (5)
k and p are the residuals due to the lack of fit, depending on the fitting accuracy of chosen functions f and g.

Different functions f and g were fitted on the simulated data. A first degree polynomial response (6 linear coefficients + 1 constant term) was primarily fitted with a multi-dimensional linear regression method. Each polynomial coefficient is the partial derivative of the response with respect to a factor, thus it represents the global influence of this factor on the studied response. Then a second degree polynomial response with full interactions (6 linear coefficients + 21 quadratic coefficients + 1 constant term) was fitted to study the level of interactions between factors. Finally, an interpolation was performed thanks to a sum of radial basis functions (RBFs), as described in Matlab (2011). This function matches values of all data points and interpolates between them, hence residuals are equal to zero. A multiquadratic base was used. The obtained fit functions f rbf and g rbf were used to perform a parameter optimization.

Different statistical methods were used to assess the fitting accuracy. The Root Mean Squared Error (RMSE) and the Predicted Residual Sum of Squares Root Mean Square Prediction Error (PRESS RMSE) were calculated for the linear and quadratic regressions. A Fisher's test was applied on the quantity defined as the ratio of Mean Square of Regression (MSR) to Mean Square Error (MSE) in order to test the overall model significance. A Student's test was applied to each coefficients in the model to test their individual significance.

Design optimization

Thanks to both index response surfaces interpolated with RBFs, it is possible to compute a brace design for which parameters would be optimised to maximize the stiffness index while minimizing the discomfort tolerance index. With this in mind, a tolerance threshold was set to various values to investigate the optimal brace design corresponding to a discomfort tolerance level. A genetic algorithm (Matlab 2011) was used in Matlab ® with enough runs to ensure a global minimization. The August 21, 2013 18:19 Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 11 fitness function to maximize was set to f rbf and a non-linear constraint function was chosen such as g rbf ≤ p t with p t a given mean pressure tolerance threshold.

Results

Exploratory FE results

A single FE analysis completed in about 6 hours on 8 CPUs at 2.4 Ghz. All 100 simulations were completed successfully. The mechanical equilibrium was checked by observing energy quantities to verify that dynamic effects had dampened out.

Because of the various factor values that were simulated, no general results can be given. Instead two extreme cases were selected: simulations A and B yield the highest and lowest stiffness indexes respectively. The characteristics of these simulations are reported in Table 3.

Reaction forces

The response curves of both these simulations are reported in Figure 6. Not only did simulation A exhibit a higher stiffness index, it also had a high reaction force at zero displacement. This means that this brace alters the natural resting position of the joint. This behaviour was observed for some high k index simulations. What is more, a reaction force drop can be noticed at 15 mm for this brace. A deeper analysis of the results showed a stick-and-slip behaviour: a mechanical discontinuity (balance between normal and tangential forces) suddenly allowed the brace to slip where it previously adhered to the skin, leading to a sudden mechanical efficiency loss.

Strains

Strain results in the fabric are reported in Table 4. For simulation A, the fabric strain was quite high in the circumferential direction after the fitting step due to the small initial circumference of the brace. However the fabric deformed mainly in the longitudinal direction during the drawer step. Some creases appeared on the brace fabric in the popliteal area (behind the joint) for simulation B due to the buckling of shell elements under compressive forces. These creases are usually observed on real braces in this region. A higher compression of soft tissues observable on real braces at the brace/limb interfaces and under the straps was also observed in these FE results. Skin did not deform much during the fitting step: compressive logarithmic strains of 20-30 % were located underneath the brace for simulation A and 10-20 % in the patella area for simulation B.

Contact pressure

A very inhomogeneous pressure distribution was observed for both simulations, as depicted in Figure 7. Concerning simulation A, very high local pressures were located underneath the straps and the rigid bars. Such high values (mean pressure of 23.5 kPa and local values up to 398 kPa) would probably lead to serious discomfort. This is why the brace from simulation A, although very efficient for preventing a drawer, is not an acceptable design. Here lies the interest of the parametric study and the optimization of a effective brace design which does not apply too high pressure. Simulation B exhibited lower pressures. Maximum local pressures were located on the patella, the tibia area and below the rigid bars, but their magnitudes were considerably lower than simulation A (mean pressure of 0.79 kPa and local values up to 8.61 kPa). Large areas were even not in contact with the brace due to its low tightening.

Interface sliding

The skin/soft tissues interface exhibited substantial sliding in our simulations. Sliding magnitudes after the drawer step reached about 1.5 cm all around the thigh and leg under the braced area for tight, short braces on contracted limbs. The brace also moved relative to the skin for such braces with a magnitude of ∼2 cm in the same area, but this sliding was much more localized in the popliteal area within the thigh/leg interface area and the brace adhered everywhere else.

Design of experiments and optimization results

The mean (± st. dev.) k value for all the simulations was 3.18 (± 2.22) N/mm. Maximum and minimum values were 10.9 and 0.58 N/mm. p values averaged 6.63 (± 4.45) kPa with maximum and minimum values of 23.5 and 0.72 kPa. In all cases, the overall model significance was higher than 0.9999.

Linear regression

The linear regression with 7 terms exhibited a RMSE of 1.25 N/mm and a PRESS RMSE of 1.31 N/mm for the stiffness index, a RMSE of 2.09 kPa and a PRESS RMSE of 2.20 kPa for the discomfort tolerance index.

The computed polynomial coefficients for both responses are represented in response. For example, by increasing the fabric stiffness of 1 in the coded space (i.e. 0.75 N/m), the brace reaction to a drawer is expected to increase of 950 N/mm and the mean pressure of 1.91 kPa. Surprisingly, the brace length was found to be the most influential factor for joint stiffening, and in favour of the small brace.

The most influential parameter for the p-response was the strap tightening. The friction coefficient had a relatively low influence on both responses. Comparing the influence of each coefficient on both responses, it was found that increasing the k-index of 1 N/mm lead to an increase of the p-index of about 2 kPa. This finding is valid when changing the soft tissue stiffness, the brace size and the fabric stiffness. Increasing the friction coefficient allowed to increase the k-index without significantly increasing the mean pressure, but only to a small extent. However, the brace length and strap tightening were found to be key parameters because they gave strong leverage on an index but not much on the other.

Quadratic regression

Concerning the quadratic regression, the terms which turned out to be insignificant (p<0.95) were removed. For the stiffness index, 15 terms were retained. The regression had a RMSE of 0.58 N/mm and a PRESS RMSE of 0.64 N/mm. For the discomfort tolerance index, 20 terms were retained, resulting in a RMSE of 0.59 kPa and a PRESS RMSE of 0.70 kPa. The computed coefficients are represented in Figure 9. Colour intensities show the influence of each term, therefore the interaction levels between parameters can be easily visualized. This revealed that some terms interact strongly. For the k-response, soft tissue stiffness with brace length and strap tightening with brace length had the higher interaction coefficients. The quadratic term associated with brace length was also high, indicating a strongly non-linear response. For the p-response, soft tissue stiffness with strap tightening and strap tightening with brace length had the higher interaction coefficients.

Brace optimization

The exploitation of these results naturally leads to an optimization of the brace design in order to maximize the k index without exceeding a certain pressure threshold. As the soft tissue stiffness is a patient-related parameter, the design was separated between two half-planes, resulting in two responses for each index: first an optimised brace for a passive limb, then an optimised brace for a contracted limb. Finally, it was hypothesized that a compromise could be reached for a brace that applies low pressures during muscle rest and a maximized joint stabilization during muscle contraction, for instance during the stance phase of gait. The results of these optimizations are reported in Table 5.

It is noteworthy that the indicated brace size (33 cm) for this limb in the manufacturer's size table was found to be close to optimal values. Fabric stiffness was found to be a parameter that should be maximized. Finally, strap tightening appeared to be a very efficient parameter to navigate between different pressure thresholds. August 21, 2013 18:19 Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 15 7.5 8.5 9.4 5.5 6.3 6.9 7.9 8.9 9.9 * Threshold not to be exceeded. * * Maximized index.

Discussion

Methodological justifications

Finite element model

The proposed finite element model of the deformable limb is not patient-specific, it is a generic model representative of a median male lower limb. Some studies in the literature highlight the importance of patient-specific geometry and mechanical properties for biomechanical studies. For example, [START_REF] Dubuis | Biomechanics of soft tissues of human leg under elastic compressionEcole Nationale Supérieure des Mines de Saint-Etienne[END_REF] showed a significant inter-subject variability of pressure levels applied by elastic compression. Even if it has not been demonstrated, it is highly probable that several patient-specific factors influence the mechanical response of the brace-limb system such as mechanical properties of the different limb constituents (skin, soft tissues) as well as the thickness of adipose tissue or the geometry of the limb itself. The same remark is valid for the finite element brace. It is mechanically representative of usual commercially available braces but some specificities are missing. For instance, most braces have a patella opening with a silicon ring. In view of simplification, this feature was not implemented in this model. It is designed to enhance the flexural behaviour of the brace during knee bending. It requires further investigation to confirm that a patella opening has a limited mechanical effect on both indexes. Nevertheless, the purpose of this work is not to compute the actual response of a particular bracelimb system, but to understand the general mechanisms governing force transfers. In that way, the developed generic model is representative of a median limb with a general brace and is perfectly suited for exploratory biomechanical investigation.

Concerning the interfaces, the model allows skin sliding on underlying tissues and brace sliding on skin. The latter phenomenon leads to brace migration, it is problematic for patients and well known in clinical practice (rehabilitation and sport). Industrials have tried to limit this sliding by attaching adhesive silicon pads at some places inside the brace. These pads were not modelled as such, but changing the global orthosis/skin friction coefficient accounted for this feature. This study shows that they have a limited impact on both responses, but still it appeared that it allowed to gain little efficiency without altering the mean pressure (Figure 8). However these pads are essentially meant to prevent sliding during knee bending and in dynamic conditions [START_REF] Van Leerdam | Chicago, IL: The Genux, A New Knee Brace with an Innovative Non-Slip System[END_REF]. As for the skin/soft tissues interface, it must be emphasized that there is a strong lack of mechanical data in the literature. This interface would probably exhibit a more complex behaviour than a basic isotropic quasi-frictionless Coulomb friction model, but our model is a good approximation for low sliding magnitudes, which is the case in most areas.

It was chosen not to model the intra-articular joint elements (cartilage, liga- ments...) because this would be computationally expensive and the goal was to study the effect of braces on the joint stiffening and not on inner structures. This is why a simple translation was applied as a BC to approximate the motion of a real ACL-deficient joint instability. This kinematics may also be subject to discussion. [START_REF] Christel | The contribution of each anterior cruciate ligament bundle to the Lachman test: a cadaver investigation[END_REF] investigated the actual kinematics during an anterior translation caused by the Lachman test; they showed that a complete removal of the ACL resulted in almost pure tibial translation. [START_REF] Lujan | Effect of ACL deficiency on MCL strains and joint kinematics[END_REF] suggested that the ACL encourages internal rotation by "unwinding" during anterior translation. The choice of constraining the DOFs and allowing only anterior tibial translation was made to reproduce a injured kinematics; it will thus be possible to compare previous and future results with this study, as it is a standard test in the medical field to assess knee stability after an ACL rupture (cadaveric studies, Lachman test, arthrometers such as the KT-1000, etc...). What is more, it has the advantage of being numerically much more stable.

Finally, another strong assumption of the model is the quasi-static analysis. Knee instabilities are often characterized by a rapid motion of one segment with respect to the other. The dynamic response of the brace-limb system may be slightly different from the quasi-static response. The textile used for braces often includes a high ratio of synthetic fibres, which exhibit a viscoelatic behaviour, as do body tissues. However, the factor domains cover the eventual stiffness increase of materials for rapid solicitation. Dynamic forces induced by acceleration have almost no effect on the brace because of its low mass. The quasi-static approach is then an appropriate approximation.

Design of experiment

The method of design of experiment is a solid approach to investigate the influence of each factor, however the results are dependent of the choice to include or not a factor, and of the chosen domains. The number of investigated factors was first dictated by the maximum number of simulations that could be performed in a reasonable amount of time. Several factors were omitted either in a concern of simplification (influence of a patella opening, of brace misalignment, of fabric anisotropy and of patient specific factors) or because they were thought to have low mechanical impact on the studied responses (effect of helical straps, strap stiffness, and of the articulation system linking bars).

All these factors proved to interact strongly for both responses, as seen in Figure 9. This means that the linear regression is only a rough approximation of the responses, as shown by the RMSEs. However it is easy to interpret, as each coefficient value is the overall effect of the corresponding factor. The quadratic regression is a highly accurate response, the RMSEs are relatively low. As some parameters are aliased, it is more complicated to physically explain each polynomial factor. Concerning the interpolation by radial basis functions, it passes through all test points so the RMSE is zero. But this does not mean that it would predict the responses of a new simulation without error, because the distance between test points is quite large compared to the size of the domain. This is especially true at the borders of the domain where the functions are extrapolated. This is why a validation test was performed to predict the response of two optimised braces from Table 5. The first is the brace for a passive limb with a pressure threshold of 4 kPa, with a predicted k value of 3.21 N/mm. A FE analysis of this parameter set yielded a p value of 4.12 kPa and a k value of 3.27 N/mm (errors of 3 and 2 %). The second is the brace for an active limb with a pressure threshold of 4 kPa, with a predicted k value of 6.90 N/mm. The FE analysis of this parameter set computed a p value of 3.97 kPa and a k value of 6.02 N/mm (errors of 1 and 13 %). August 21, 2013 18:19 Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 17

Stiffness and discomfort tolerance indexes

Both indexes were chosen as the primary aim of a knee brace: increasing joint stability while being relatively comfortable.

The k index is a good indicator of the additional stiffness brought by the brace at low anterior displacement. This implies the hypothesis that the stiffness response of a braced joint under a drawer is the sum of the inherent joint stiffness with the additional brace stiffness, as the stiffness of two springs in parallel is the sum of each individual stiffness. This index does not account for the initial force, which was not negligible for some braces, as seen in Figure 6 and could help maintaining knee stability by replacing the ACL initial tension, as highlighted by [START_REF] Solomonow | Sensory -Motor control of ligaments and associated neuromuscular disorders[END_REF].

The idea behind the p index is to quantify discomfort as a global tolerance level. However, discomfort tolerance is a very subjective sensation and depends not only on the mean pressure but also on local cutaneous solicitations (pinching, denting), on the solicited area, fabric type, etc... The literature is very scarce in comfort/tolerance assessment. As orthotic devices are typically worn for a few hours, global discomfort tolerance might be estimated by avoiding pressures above the ischemic level, i.e. the level at which capillaries are unable to irrigate tissues. The value of 4.3 kPa from the work of [START_REF] Landis | Micro-injection studies of capillary blood pressure in human skin[END_REF] is traditionally used.

Result outcomes

The main effects governing brace efficiency and their effect on joint stabilization are described here. Some practical learnings for manufacturers and physicians are also given.

Force transfer mechanisms

Two main force transfer mechanisms have been identified: the sleeve compression effect and the force transmission efficiency from the bars to the straps.

The first effect depends on the pressure applied by the brace on the skin due to elastic compression. The brace stiffening efficiency greatly depends on these compressive forces. In fact, the brace is secured to the limb (no slip) at a point if:

F t ≤ µ F n (6)
with F t and F n the tangential and normal forces and µ the friction coefficient of the interface. Compressive forces are thus related to the resistive forces preventing the drawer. The Laplace law for elastic compression says that F n is proportional to the tension in the fabric along the circumferential direction [START_REF] Dubuis | Biomechanics of soft tissues of human leg under elastic compressionEcole Nationale Supérieure des Mines de Saint-Etienne[END_REF]:

P = F n a = T r (7)
with P the pressure applied by the brace, T the tension in the circumferential direction of the fabric, r the curvature radius in the same direction and a the area.

F n is then maximized by increasing fabric stiffness and decreasing the initial brace circumference. However, it also increases the mean pressure, this explains why these two factors had the same relative effect on k and p (Figure 8). This compression effect also limits skin sliding on soft tissues, but to a lesser extent because of the lower friction coefficient.

The second effect is the efficiency of force transfer from the bars to the straps. It can be explained by considering the attach between straps and rigid bars as anchoring points of the brace on the limb, as depicted in Figure 10. The stabilizing effect is affected by the movement of bars relative to the limb segments (loosening). As the strap fabric is much stiffer than the brace fabric, the latter one deforms easily. The ability to tilt the hinge link depends on the strength of anchors and the lever arm, i.e. on the strap tightening and the brace length. If the straps are attached to the bars far away from the joint, there is a long lever arm and tilting the hinge link is easy; in contrary, a small lever arm makes it more resistant to tilting. As a matter of fact, the brace length might be misleading, a long brace with anchoring points (straps) close to the joint would probably result in the same improvement. This effect is essential for designing an optimised brace because it allows to increase the force transfer without necessarily increasing the mean pressure, unlike the previous effect. This effect explains the major interaction between strap tightening and brace length for the k index (Section 3.2.2).

Mechanical stiffening level

It has been found that conventional braces have a joint stiffening effect and this effect has been quantified and optimised. [START_REF] Liu | Comparison of functional knee braces for control of anterior tibial displacement[END_REF] experimentally tested 10 braces on a surrogate leg using the same kinematics. From their data, an average k-index of 11.3 N/mm (min: 5.1, max: 18.3) has been computed. These values are high, and it is probable that the testing support has a high influence because a rigid leg was used, and the stiffening effect may be overestimated. This study proves that the mechanical realism of the testing support is important and the use of FE modelling allowed to obtain more reliable responses.

It is hard to tell if the computed stabilization levels are high enough to efficiently reduce an actual drawer laxity. [START_REF] Eagar | A method for quantifying the anterior load-displacement behavior of the human knee in both the low and high stiffness regions[END_REF] performed the same P-A test on 7 cadaveric knees, with and without the ACL. Results of this study are shown in Figure 11 and clearly illustrate the high laxity of an ACL-deficient knee. The response of a brace with a k index of 8 N/mm has been plotted and added to the ACL-deficient knee in order to predict the response of a braced injured knee: it compensates the injury and is comparable to the healthy knee for low displacements.

However, joint stability also comes from muscle activation. [START_REF] Wojtys | Edward M. Wojtys[END_REF] used an arthrometer to measure sagittal-plane shear stiffness of passive and ac- tive knees. Passive knees exhibited a mean stiffness of 18.7 N/mm (men) and 19.3 N/mm (women) while active limbs resulted in stiffer joints: 70.9 N/mm (men) and 40.7 N/mm (women). A good order of magnitude of in-vivo forces in the knee was given by [START_REF] Kutzner | Loading of the knee joint during activities of daily living measured in vivo in five subjects[END_REF], who measured anterior forces as high as 30 % bodyweight (240 N for 80 kg) when descending stairs. The passive structures only need to bear a small fraction of this force when muscles are contracted. That is, the stabilizing effect of an optimised brace could be enough to compensate for a deficient ACL if the muscles are recruited as the main active stabilizers. The sufficiency of the stiffening effect of the brace remains a difficult question because the ACL is not only a passive stabilizing mechanical structure, but also a major sensory organ capable of stimulating the active recruitment of the musculature to participate in maintaining joint stability [START_REF] Solomonow | Sensory -Motor control of ligaments and associated neuromuscular disorders[END_REF]). An optimised brace could compensate the mechanical deficiency at low forces/displacements, but it is not sure at which extent it can compensate the lack of sensory feedback by proprioceptive effect. If it turns out that braces can effectively trigger muscle contraction thanks to this effect, the joint stiffening induced by muscle contraction would be much higher that what is brought by the brace. Several studies show that muscle activation is modified by such devices [START_REF] Osternig | Effects of prophylactic knee bracing on lower extremity joint position and muscle activation during running[END_REF][START_REF] Ramsey | Electromyographic and biomechanic analysis of anterior cruciate ligament deficiency and functional knee bracing[END_REF][START_REF] Théoret | Study on three-dimensional kinematics and electromyography of ACL deficient knee participants wearing a functional knee brace during running[END_REF], but the level of actions are not known and the mechanisms not fully understood.

Finally, in order to restore the ACL function, a brace should not only restrain anterior tibial translation but also restore internal tibial rotation during the drawer [START_REF] Lujan | Effect of ACL deficiency on MCL strains and joint kinematics[END_REF]). The simulated 1-DOF kinematics does not allow the tibia to rotate, but this effect can be investigated by looking at the internal tibial moment during the motion. This moment was found to be very low (-0.1 to 0.3 N.m depending on the case), probably due to the symmetric design of the brace.

Brace design and fitting recommendations

As for now, most investigated design parameters are chosen arbitrarily or empirically. This study suggests a few guidelines to design more efficient and comfortable braces. These braces should be seen as prototypes because they were optimised only for two purposes, which are drawer prevention and discomfort tolerance; they August 21, 2013 18:19 Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 20

might not be suitable for unusual limb geometries, high flexion angles... Knowing this, the brace size from the manufacturer's size chart was found to be adapted to this limb. Anti-sliding pads systems are probably more efficient to limit dynamic sliding rather than to improve sagittal-plane shear stiffness. Besides, it is recommended to use stiffer fabric than usually found for the brace body. Further investigations are needed to determine whether a stiffness of 2 kN/m is convenient for knee bending. Finally, the most important recommendation is to attach the straps close to the joint, in order to keep a short lever arm between the centre of the joint and the rigid anchoring points. This can be done by designing shorter braces, which could reduce manufacturing costs. A non-symmetric design (of the body or straps) could help restoring the internal tibial rotation during the drawer.

Practitioners should insist on the fact that a correct strap tightening is very important when prescribing a brace to a patient. If the straps are too loose, they do not play their role of anchoring points effectively. It they are too tight, the applied pressure is too high and the discomfort tolerance is increased.

Conclusion

An original FE model of a braced deformable limb has been developed and used to assess the influence of various design factors of usual knee braces on the ability to prevent a drawer. Two efficiency indexes have been proposed: the mechanical stabilization (stiffening) of the joint and the pressure applied onto the skin. A parametric optimization resulted in brace designs yielding a stiffening effect high enough to compensate for the structural role of the ACL at low displacement. Two main force transfer mechanisms have been identified: a sleeve compression effect and a load transfer to the side bars through the straps. The latter is governed by strap tightening and brace length, which were found to be key factors to improve brace designs. This study proposes a new methodology to assess the biomechanical efficiency of knee orthoses and provides substantive guidance to manufacturers and practitioners. The model and findings will be validated in future studies, first through experimental measurements with an instrumented limb simulator, then through a clinical trial using an arthrometer. The characterization of commercially available braces will lead to an objective ranking based on their stiffness index. Finally, the role of orthoses on inner structures (muscles, ligaments) needs to be investigated to understand how other mechanisms such as proprioceptive action, localized structural unloading or muscle recruitment participate in the brace effect.

Figure 1 .

 1 Figure 1. Usual mass-produced knee brace with product characteristics.

  Pet-CT scan slices from the thigh to the leg (left-right). (b) Corresponding images after segmentation.

Figure 2 .

 2 Figure 2. Illustration of the segmentation process.

Figure 3 .

 3 Figure 3. Exploded view of the different meshed parts constituting the deformable limb (thigh, leg, patella, skin layer) and the brace with a detail on the hinged rigid bars only.

Figure 4 .

 4 Figure 4. (a) Reference image in the sagittal plane as observed by ultrasound. Scale is in cm. (b) Correlated displacement in the area of interest after a 10 N pull of the skin. The white area correspond to a place where the correlation process failed. Scale of the arrows is 1/5.

Figure 5 .

 5 Figure 5. Reaction force of the tibia along the P-A direction vs. applied P-A drawer magnitude to determine k.

Figure 6 .

 6 Figure 6. Reaction force of the tibia along the P-A direction vs. applied P-A drawer: joint stabilizing effect of simulations A and B (maximum and minimum stiffness indexes). The response of the limb only has been subtracted.

  Figure 8. Each coefficient is representative of the influence of a factor on the overall

Figure 7 .

 7 Figure7. Deformed brace mesh after the drawer step (shape before drawer in overlay) and pressure map on the surface of the skin for both simulations (log. scale).

Figure 8 .

 8 Figure 8. Coefficient values for the linear regression on both responses in the coded space. s stands for significance.

Figure 9 .

 9 Figure 9. Coefficient values for the quadratic regression on both responses in the coded space (graphic representation of the interaction matrices). s stands for significance.

Figure 10 .

 10 Figure 10. Mechanism explaining the advantage of small braces to prevent drawer.

Figure 11 .

 11 Figure 11. Quantification of the additional stiffness brought by an optimal brace (k = 8 N/mm) to an ACL-deficient cadaveric knee under a P-A displacement. Data kindly provided by Eagar et al. (2001).

Table 2 .

 2 Selected model factors with their domains.Brace lengthBrace cylinder height, as described in Section 2.1.2.1 17|34|51 cm *

	Factor name	Description	Domain
	Soft tissue stiffness	Neo-Hookean parameter G for passive/contracted soft tissues of the limb, as described in Section 2.1.1.3	5|400 kPa *
	Brace size	Brace cylinder circumference at the knee, as described in Section 2.1.2.1	28-38 cm * *
	Fabric stiffness	Single parameter to account for brace fabric stiffness, as described in Section 2.1.2.3	0.5-2 kN/m * *
	Brace/skin friction coefficient	Orthosis/skin Coulomb friction coefficient, as de-scribed in Section 2.1.3	0.1-1 * *

Table 3 .

 3 Characteristics of simulations A and B.

		Simulation A	Simulation B
	k index value (N/mm)	10.9	0.58
	p index value (kPa)	23.5	0.79
	Soft tissue stiffness (kPa)	400	5
	Brace circumference (cm)	30.1	37.7
	Brace height (cm)	17	51
	Fabric stiffness (kN/m)	2.00	0.96
	Friction coefficient (-)	0.26	0.84
	Initial strap strain (%)

Table 4 .

 4 Fabric strain results for simulations A and B.

	August

Table 5 .

 5 Characteristics of optimised braces.

			Passive limb			Active limb			Compromise
	Passive p value (kPa)	3 *	4 *	5 *	-	-	-	3 *	4 *	5 *
	Passive k value (N/mm)	2.83 * *	3.21 * *	3.54 * *	-	-	-	2.79	3.17	3.49
	Active p value (kPa)	-	-	-	3 *	4 *	5 *	8.34	9.53	10.59
	Active k value (N/mm)	-	-	-	6.42 * *	6.90 * *	7.41 * *	6.82 * *	9.22 * *	11.65 * *
	Brace circumference (cm)	32.9	31.9	31.1	33.5	34.0	33.8	33.7	32.7	31.9
	Brace height (cm)	17	17	17	17	17	17	17	17	17
	Fabric stiffness (kN/m)	1.98	1.96	1.96	2	2	2	2	2	2
	Friction coefficient (-)	0.48	0.45	0.42	0.93	0.76	0.70	0.60	0.58	0.56
	Initial strap strain (%)

Acknowledgements

This work was funded in part by the ANRT (Association Nationale de la Recherche et de la Technologie) and the following orthotic manufacturers: Thuasne ® , Gibaud ® and Lohmann-Rauscher ® .

 August 21, 2013 18:19Computer Methods in Biomechanics and Biomedical Engineering pier-rat˙final 22