960 resultados para Non-constant coefficient diffusion equations


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The concept of conditional stability constant is extended to the competitive binding of small molecules to heterogeneous surfaces or macromolecules via the introduction of the conditional affinity spectrum (CAS). The CAS describes the distribution of effective binding energies experienced by one complexing agent at a fixed concentration of the rest. We show that, when the multicomponent system can be described in terms of an underlying affinity spectrum [integral equation (IE) approach], the system can always be characterized by means of a CAS. The thermodynamic properties of the CAS and its dependence on the concentration of the rest of components are discussed. In the context of metal/proton competition, analytical expressions for the mean (conditional average affinity) and the variance (conditional heterogeneity) of the CAS as functions of pH are reported and their physical interpretation discussed. Furthermore, we show that the dependence of the CAS variance on pH allows for the analytical determination of the correlation coefficient between the binding energies of the metal and the proton. Nonideal competitive adsorption isotherm and Frumkin isotherms are used to illustrate the results of this work. Finally, the possibility of using CAS when the IE approach does not apply (for instance, when multidentate binding is present) is explored. © 2006 American Institute of Physics.

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MR imaging is currently regarded as a pivotal technique for the assessment of a variety of musculoskeletal conditions. Diffusion-weighted MR imaging (DWI) is a relatively recent sequence that provides information on the degree of cellularity of lesions. Apparent diffusion coefficient (ADC) value provides information on the movement of water molecules outside the cells. The literature contains many studies that have evaluated the role of DWI in musculoskeletal diseases. However, to date they yielded conflicting results on the use and the diagnostic capabilities of DWI in the area of musculoskeletal diseases. However, many of them have showed that DWI is a useful technique for the evaluation of the extent of the disease in a subset of musculoskeletal cancers. In terms of tissue characterization, DWI may be an adjunct to the more conventional MR imaging techniques but should be interpreted along with the signal of the lesion as observed on conventional sequences, especially in musculoskeletal cancers. Regarding the monitoring of response to therapy in cancer or inflammatory disease, the use of ADC value may represent a more reliable additional tool but must be compared to the initial ADC value of the lesions along with the knowledge of the actual therapy.

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Forest management for groundwater protection is a cheap solution for a vital question, which is implemented for decades all over the world. The main challenge is to insure a constant adequate forest management to preserve the service provided. In Lombok Island, the problem is the lack of implementation of the public regulation in the forest area. Therefore payments for environmental services (PES) are used as an alternative in this weak institutional environment. The results of the field research show that, surprisingly, the "famous" Lombok PES case is not a PES at all, even if there are some payments. This research has however happy ends because other "forest for water" PES have been identified in the field. In addition, the legal review identified a way to solve the lack of legal base for PES implementation. Thus, the PES examples that we identified could be spread all over Indonesia without conflicting other regulations (fiscal, local finance, forest, etc.) and circumventing the forest administrations.

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Työn päätavoitteena oli selvittää hinnan ja kilpailutilanteen vaikutusta matkaviestinnän diffuusioon. Työn empiirinen osuus tarkasteli matkapuhelinliittymien hinnan vaikutusta liittymien diffuusioon sekä sitä, miten alan kilpailu on vaikuttanut matkaviestinnän hintatasoon. Työssä analysoitiin myös matkaviestinnän kilpailutilannetta Suomen markkinoilla. Tutkimuksen empiirinen aineisto kerättiin toissijaisista lähteistä, esimerkiksi EMC-tietokannasta. Tutkimus oli luonteeltaan kvantitatiivinen.Empiirisessä osassa käytetyt mallit oli muodostettu aikaisempien tutkimuksien perusteella. Regressioanalyysiä käytettiin arvioitaessa hinnan vaikutusta diffuusionopeuteen ja mahdollisten omaksujien määrään. Regressioanalyysissä sovellettiin ei-lineaarista mallia.Tutkimustulokset osoittivat, että tasaisesti laskevilla matkapuhelinliittymien sekä matkapuhelimien hinnoilla ei ole merkittävää vaikutusta matkaviestinnän diffuusioon. Myöskään kilpailutilanne ei ole vaikuttanut paljon matkaviestinnän yleiseen hintatasoon. Työn tulosten perusteella voitiin antaa myös muutamia toimenpide-ehdotuksia jatkotutkimuksia varten.

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The main subject of this master's thesis was predicting diffusion of innovations. The prediction was done in a special case: product has been available in some countries, and based on its diffusion in those countries the prediction is done for other countries. The prediction was based on finding similar countries with Self-Organizing Map~(SOM), using parameters of countries. Parameters included various economical and social key figures. SOM was optimised for different products using two different methods: (a) by adding diffusion information of products to the country parameters, and (b) by weighting the country parameters based on their importance for the diffusion of different products. A novel method using Differential Evolution (DE) was developed to solve the latter, highly non-linear optimisation problem. Results were fairly good. The prediction method seems to be on a solid theoretical foundation. The results based on country data were good. Instead, optimisation for different products did not generally offer clear benefit, but in some cases the improvement was clearly noticeable. The weights found for the parameters of the countries with the developed SOM optimisation method were interesting, and most of them could be explained by properties of the products.

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In this paper we explore the use of non-linear transformations in order to improve the performance of an entropy based voice activity detector (VAD). The idea of using a non-linear transformation comes from some previous work done in speech linear prediction (LPC) field based in source separation techniques, where the score function was added into the classical equations in order to take into account the real distribution of the signal. We explore the possibility of estimating the entropy of frames after calculating its score function, instead of using original frames. We observe that if signal is clean, estimated entropy is essentially the same; but if signal is noisy transformed frames (with score function) are able to give different entropy if the frame is voiced against unvoiced ones. Experimental results show that this fact permits to detect voice activity under high noise, where simple entropy method fails.

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By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.

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A series of poly(butylene terephthalate) copolyesters containing 5-tert-butyl isophthalate units up to 50%-mole, as well as the homopolyester entirely made of these units, were prepared by polycondensation from the melt. The microstructure of the copolymers was determined by NMR to be at random for the whole range of compositions. The effect exerted by the 5-tert-butyl isophthalate units on thermal, tensile and gas transport properties was evaluated. Both Tm and crystallinity as well as the mechanical moduli were found to decrease steadily with copolymerization whereas Tg increased and the polyesters became more brittle. Permeability and solubility sligthly increased also with the content in substituted units whereas the diffusion coefficient remained practically constant. For the homopolyester poly(5-tert-butyl isophthalate), all these properties were found to deviate significantly from the general trend displayed by copolyesters suggesting that a different chain mode of packing in the amorphous phase is likely adopted in this case.

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Yksivuotisten kasvien (non-wood) kuitua verrataan usein lehtipuukuituihin. Käytetyimpiä non-wood kasveja ovat vehnän olki, bambu, järviruoko ja bagassi. Non-wood massan erottaa puumassasta kuitenkin korkea silikaattipitoisuus sekä parenkyymisolupitoisuus, joka antaa massalle korkean hienoainepitoisuuden. Tämä yhdessä korkean hemiselluloosapitoisuuden ja lyhyen kuidun pituuden kanssa heikentävät voimakkaasti non-wood massan vedenpoisto-ominaisuuksia. Non-wood kuidulla voidaan korvata lehtipuumassaa hienopapereissa. Non-wood kuitu antaa paperille hyvän opasiteetin, korkean valonsirontakertoimen sekä sileän painopinnan. Massaan lisättävä pitkäkuituinen havupuumassa parantaa ajetta¬vuutta paperikoneella ja helpottaa massan vedenpoistoa. Non-wood massan vedenpoistoa voidaan tehostaa esimerkiksi poistamalla osa hienoaineesta, käyttämällä non-wood massalle sopivaa keittotapaa sekä käyttämällä märkä¬puristuksessa pitkänippityyppistä puristinratkaisua. Myös non-wood kuidun kuivaaminen parantaa vedenpoistoa. Tässä tutkimuksessa kirjallisuusosassa keskityttiin yleisimpiin paperin valmistuksessa käytettäviin non-wood kuidun lähteisiin, märkäpuristuksen teoriaan ja tapoihin tehostaa vedenpoistoa. Kokeellisessa osassa tutkittiin vehnänolkimassan käyttäytymistä märkäpuristuksessa erilaisten ominaisuuksien pohjalta. Tutkimuksen kohteena oli non-wood massan keittotapa (hapan/alkali), hienoainepitoisuus, silikaattipitoisuus sekä kuivattu/kuivaamaton kuitu. Vertailun vuoksi tutkimuksessa oli mukana myös yksi järviruokomassa. Tuloksista huomataan, että non-wood massan vedenpoistoon vaikuttaa hienoainepitoisuus, kuidun kuivaus sekä massan valmistustapa. Järviruokomassan veden¬poisto on tehokkaampaa kuin vehnänolkimassan paremman kuitu¬koostumuksensa takia. Jos hienopaperimassassa korvataan lehtipuumassaa non-wood kuidulla maksimissaan 40 %, massan vedenpoistoa voidaan hyvin arvioida erilaisten suotautuvuusmittojen, kuten freeneksen, vedenpidätyskyvyn ja suotautumisajan, avulla.

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Intravoxel incoherent motion (IVIM) MRI is a method to extract microvascular blood flow information out of diffusion-weighted images acquired at multiple b-values. We hypothesized that IVIM can identify the muscles selectively involved in a specific task, by measuring changes in activity-induced local muscular perfusion after exercise. We tested this hypothesis using a widely used clinical maneuver, the lift-off test, which is known to assess specifically the subscapularis muscle functional integrity. Twelve shoulders from six healthy male volunteers were imaged at 3 T, at rest, as well as after a lift-off test hold against resistance for 30 s, 1 and 2 min respectively, in three independent sessions. IVIM parameters, consisting of perfusion fraction (f), diffusion coefficient (D), pseudo-diffusion coefficient D* and blood flow-related fD*, were estimated within outlined muscles of the rotator cuff and the deltoid bundles. The mean values at rest and after the lift-off tests were compared in each muscle using a one-way ANOVA. A statistically significant increase in fD* was measured in the subscapularis, after a lift-off test of any duration, as well as in D. A fD* increase was the most marked (30 s, +103%; 1 min, +130%; 2 min, +156%) and was gradual with the duration of the test (in 10(-3) mm(2) /s: rest, 1.41 ± 0.50; 30 s, 2.86 ± 1.17; 1 min, 3.23 ± 1.22; 2 min, 3.60 ± 1.21). A significant increase in fD* and D was also visible in the posterior bundle of the deltoid. No significant change was consistently visible in the other investigated muscles of the rotator cuff and the other bundles of the deltoid. In conclusion, IVIM fD* allows the demonstration of a task-related microvascular perfusion increase after a specific task and suggests a direct relationship between microvascular perfusion and the duration of the effort. It is a promising method to investigate non-invasively skeletal muscle physiology and clinical perfusion-related muscular disorders.

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Understanding and quantifying seismic energy dissipation, which manifests itself in terms of velocity dispersion and attenuation, in fluid-saturated porous rocks is of considerable interest, since it offers the perspective of extracting information with regard to the elastic and hydraulic rock properties. There is increasing evidence to suggest that wave-induced fluid flow, or simply WIFF, is the dominant underlying physical mechanism governing these phenomena throughout the seismic, sonic, and ultrasonic frequency ranges. This mechanism, which can prevail at the microscopic, mesoscopic, and macroscopic scale ranges, operates through viscous energy dissipation in response to fluid pressure gradients and inertial effects induced by the passing wavefield. In the first part of this thesis, we present an analysis of broad-band multi-frequency sonic log data from a borehole penetrating water-saturated unconsolidated glacio-fluvial sediments. An inherent complication arising in the interpretation of the observed P-wave attenuation and velocity dispersion is, however, that the relative importance of WIFF at the various scales is unknown and difficult to unravel. An important generic result of our work is that the levels of attenuation and velocity dispersion due to the presence of mesoscopic heterogeneities in water-saturated unconsolidated clastic sediments are expected to be largely negligible. Conversely, WIFF at the macroscopic scale allows for explaining most of the considered data while refinements provided by including WIFF at the microscopic scale in the analysis are locally meaningful. Using a Monte-Carlo-type inversion approach, we compare the capability of the different models describing WIFF at the macroscopic and microscopic scales with regard to their ability to constrain the dry frame elastic moduli and the permeability as well as their local probability distribution. In the second part of this thesis, we explore the issue of determining the size of a representative elementary volume (REV) arising in the numerical upscaling procedures of effective seismic velocity dispersion and attenuation of heterogeneous media. To this end, we focus on a set of idealized synthetic rock samples characterized by the presence of layers, fractures or patchy saturation in the mesocopic scale range. These scenarios are highly pertinent because they tend to be associated with very high levels of velocity dispersion and attenuation caused by WIFF in the mesoscopic scale range. The problem of determining the REV size for generic heterogeneous rocks is extremely complex and entirely unexplored in the given context. In this pilot study, we have therefore focused on periodic media, which assures the inherent self- similarity of the considered samples regardless of their size and thus simplifies the problem to a systematic analysis of the dependence of the REV size on the applied boundary conditions in the numerical simulations. Our results demonstrate that boundary condition effects are absent for layered media and negligible in the presence of patchy saturation, thus resulting in minimum REV sizes. Conversely, strong boundary condition effects arise in the presence of a periodic distribution of finite-length fractures, thus leading to large REV sizes. In the third part of the thesis, we propose a novel effective poroelastic model for periodic media characterized by mesoscopic layering, which accounts for WIFF at both the macroscopic and mesoscopic scales as well as for the anisotropy associated with the layering. Correspondingly, this model correctly predicts the existence of the fast and slow P-waves as well as quasi and pure S-waves for any direction of wave propagation as long as the corresponding wavelengths are much larger than the layer thicknesses. The primary motivation for this work is that, for formations of intermediate to high permeability, such as, for example, unconsolidated sediments, clean sandstones, or fractured rocks, these two WIFF mechanisms may prevail at similar frequencies. This scenario, which can be expected rather common, cannot be accounted for by existing models for layered porous media. Comparisons of analytical solutions of the P- and S-wave phase velocities and inverse quality factors for wave propagation perpendicular to the layering with those obtained from numerical simulations based on a ID finite-element solution of the poroelastic equations of motion show very good agreement as long as the assumption of long wavelengths remains valid. A limitation of the proposed model is its inability to account for inertial effects in mesoscopic WIFF when both WIFF mechanisms prevail at similar frequencies. Our results do, however, also indicate that the associated error is likely to be relatively small, as, even at frequencies at which both inertial and scattering effects are expected to be at play, the proposed model provides a solution that is remarkably close to its numerical benchmark. -- Comprendre et pouvoir quantifier la dissipation d'énergie sismique qui se traduit par la dispersion et l'atténuation des vitesses dans les roches poreuses et saturées en fluide est un intérêt primordial pour obtenir des informations à propos des propriétés élastique et hydraulique des roches en question. De plus en plus d'études montrent que le déplacement relatif du fluide par rapport au solide induit par le passage de l'onde (wave induced fluid flow en anglais, dont on gardera ici l'abréviation largement utilisée, WIFF), représente le principal mécanisme physique qui régit ces phénomènes, pour la gamme des fréquences sismiques, sonique et jusqu'à l'ultrasonique. Ce mécanisme, qui prédomine aux échelles microscopique, mésoscopique et macroscopique, est lié à la dissipation d'énergie visqueuse résultant des gradients de pression de fluide et des effets inertiels induits par le passage du champ d'onde. Dans la première partie de cette thèse, nous présentons une analyse de données de diagraphie acoustique à large bande et multifréquences, issues d'un forage réalisé dans des sédiments glaciaux-fluviaux, non-consolidés et saturés en eau. La difficulté inhérente à l'interprétation de l'atténuation et de la dispersion des vitesses des ondes P observées, est que l'importance des WIFF aux différentes échelles est inconnue et difficile à quantifier. Notre étude montre que l'on peut négliger le taux d'atténuation et de dispersion des vitesses dû à la présence d'hétérogénéités à l'échelle mésoscopique dans des sédiments clastiques, non- consolidés et saturés en eau. A l'inverse, les WIFF à l'échelle macroscopique expliquent la plupart des données, tandis que les précisions apportées par les WIFF à l'échelle microscopique sont localement significatives. En utilisant une méthode d'inversion du type Monte-Carlo, nous avons comparé, pour les deux modèles WIFF aux échelles macroscopique et microscopique, leur capacité à contraindre les modules élastiques de la matrice sèche et la perméabilité ainsi que leur distribution de probabilité locale. Dans une seconde partie de cette thèse, nous cherchons une solution pour déterminer la dimension d'un volume élémentaire représentatif (noté VER). Cette problématique se pose dans les procédures numériques de changement d'échelle pour déterminer l'atténuation effective et la dispersion effective de la vitesse sismique dans un milieu hétérogène. Pour ce faire, nous nous concentrons sur un ensemble d'échantillons de roches synthétiques idéalisés incluant des strates, des fissures, ou une saturation partielle à l'échelle mésoscopique. Ces scénarios sont hautement pertinents, car ils sont associés à un taux très élevé d'atténuation et de dispersion des vitesses causé par les WIFF à l'échelle mésoscopique. L'enjeu de déterminer la dimension d'un VER pour une roche hétérogène est très complexe et encore inexploré dans le contexte actuel. Dans cette étude-pilote, nous nous focalisons sur des milieux périodiques, qui assurent l'autosimilarité des échantillons considérés indépendamment de leur taille. Ainsi, nous simplifions le problème à une analyse systématique de la dépendance de la dimension des VER aux conditions aux limites appliquées. Nos résultats indiquent que les effets des conditions aux limites sont absents pour un milieu stratifié, et négligeables pour un milieu à saturation partielle : cela résultant à des dimensions petites des VER. Au contraire, de forts effets des conditions aux limites apparaissent dans les milieux présentant une distribution périodique de fissures de taille finie : cela conduisant à de grandes dimensions des VER. Dans la troisième partie de cette thèse, nous proposons un nouveau modèle poro- élastique effectif, pour les milieux périodiques caractérisés par une stratification mésoscopique, qui prendra en compte les WIFF à la fois aux échelles mésoscopique et macroscopique, ainsi que l'anisotropie associée à ces strates. Ce modèle prédit alors avec exactitude l'existence des ondes P rapides et lentes ainsi que les quasis et pures ondes S, pour toutes les directions de propagation de l'onde, tant que la longueur d'onde correspondante est bien plus grande que l'épaisseur de la strate. L'intérêt principal de ce travail est que, pour les formations à perméabilité moyenne à élevée, comme, par exemple, les sédiments non- consolidés, les grès ou encore les roches fissurées, ces deux mécanismes d'WIFF peuvent avoir lieu à des fréquences similaires. Or, ce scénario, qui est assez commun, n'est pas décrit par les modèles existants pour les milieux poreux stratifiés. Les comparaisons des solutions analytiques des vitesses des ondes P et S et de l'atténuation de la propagation des ondes perpendiculaires à la stratification, avec les solutions obtenues à partir de simulations numériques en éléments finis, fondées sur une solution obtenue en 1D des équations poro- élastiques, montrent un très bon accord, tant que l'hypothèse des grandes longueurs d'onde reste valable. Il y a cependant une limitation de ce modèle qui est liée à son incapacité à prendre en compte les effets inertiels dans les WIFF mésoscopiques quand les deux mécanismes d'WIFF prédominent à des fréquences similaires. Néanmoins, nos résultats montrent aussi que l'erreur associée est relativement faible, même à des fréquences à laquelle sont attendus les deux effets d'inertie et de diffusion, indiquant que le modèle proposé fournit une solution qui est remarquablement proche de sa référence numérique.

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Focal epilepsy is increasingly recognized as the result of an altered brain network, both on the structural and functional levels and the characterization of these widespread brain alterations is crucial for our understanding of the clinical manifestation of seizure and cognitive deficits as well as for the management of candidates to epilepsy surgery. Tractography based on Diffusion Tensor Imaging allows non-invasive mapping of white matter tracts in vivo. Recently, diffusion spectrum imaging (DSI), based on an increased number of diffusion directions and intensities, has improved the sensitivity of tractography, notably with respect to the problem of fiber crossing and recent developments allow acquisition times compatible with clinical application. We used DSI and parcellation of the gray matter in regions of interest to build whole-brain connectivity matrices describing the mutual connections between cortical and subcortical regions in patients with focal epilepsy and healthy controls. In addition, the high angular and radial resolution of DSI allowed us to evaluate also some of the biophysical compartment models, to better understand the cause of the changes in diffusion anisotropy. Global connectivity, hub architecture and regional connectivity patterns were altered in TLE patients and showed different characteristics in RTLE vs LTLE with stronger abnormalities in RTLE. The microstructural analysis suggested that disturbed axonal density contributed more than fiber orientation to the connectivity changes affecting the temporal lobes whereas fiber orientation changes were more involved in extratemporal lobe changes. Our study provides further structural evidence that RTLE and LTLE are not symmetrical entities and DSI-based imaging could help investigate the microstructural correlate of these imaging abnormalities.

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The authors report a case where a quantitative assessment of the apparent diffusion coefficient (ADC) of liver metastasis in a patient undergoing chemotherapy has shown to be an effective early marker for predicting therapeutic response, anticipating changes in tumor size. A lesion with lower initial ADC value and early increase in such value in the course of the treatment tends to present a better therapeutic response.

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La dermatite irritative est décrite comme une réaction réversible, non immunologique caractérisée par des lésions d'aspect très variable, allant de la simple rougeur jusqu'à la formation de bulles voire d'une nécrose, accompagnée de prurit ou d'une sensation de brûlure suite à I' application d'une substance chimique. Le teste de prédiction d'irritation cutanée est traditionnellement depuis les années 1940 le Test de Draize. Ce test consiste en l'application d'une substance chimique sur une peau rasée de lapin pendant 4h et de regarder à 24h si des signes cliniques d'irritations sont présents. Cette méthode critiquable autant d'un point de vue éthique que qualitative reste actuellement le teste le plus utilisé. Depuis le début des années 2000 de nouvelles méthodes in vitro se sont développées tel que le model d'épiderme humain recombiné (RHE). Il s agit d'une multicouche de kératinocyte bien différencié obtenu depuis une culture de don d'ovocyte. Cependant cette méthode en plus d'être très couteuse n'obtient au mieux que 76% de résultat similaire comparé au test in vivo humain. Il existe donc la nécessité de développer une nouvelle méthode in vitro qui simulerait encore mieux la réalité anatomique et physiologique retrouvée in vivo. Notre objectif a été de développer cette nouvelle méthode in vitro. Pour cela nous avons travaillé avec de la peau humaine directement prélevée après une abdominoplastie. Celle ci après préparation avec un dermatome, un couteau dont la lame est réglable pour découper l'épaisseur souhaitée de peau, est montée dans un système de diffusion cellulaire. La couche cornée est alors exposée de manière optimale à 1 ml de la substance chimique testée pendant 4h. L'échantillon de peau est alors fixé dans du formaldéhyde pour permettre la préparation de lames standards d'hématoxyline et éosine. L'irritation est alors investiguée selon des critères histopathologiques de spongioses, de nécroses et de vacuolisations cellulaires. Les résultats de ce.tte première batterie de testes sont plus que prometteurs. En effet, comparé au résultat in vivo, nous obtenons 100% de concordance pour les 4 même substances testes irritantes ou non irritantes, ce qui est supérieur au model d épiderme humain recombiné (76%). De plus le coefficient de variation entre les 3 différentes séries est inférieur à 0.1 ce qui montre une bonne reproductibilité dans un même laboratoire. Dans le futur cette méthode va devoir être testée avec un plus grand nombre de substances chimiques et sa reproductibilité évaluée dans différents laboratoires. Mais cette première evaluation, très encourageante, ouvre des pistes précieuses pour l'avenir des tests irritatifs.

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The chemistry of gold dissolution in alkaline cyanide solution has continually received attention and new rate equations expressing the gold leaching are still developed. The effect of leaching parameters on gold gold cyanidation is studied in this work in order to optimize the leaching process. A gold leaching model, based on the well-known shrinking-core model, is presented in this work. It is proposed that the reaction takes place at the reacting particle surface which is continuously reduced as the reaction proceeds. The model parameters are estimated by comparing experimental data and simulations. The experimental data used in this work was obtained from Ling et al. (1996) and de Andrade Lima and Hodouin (2005). Two different rate equations, where the unreacted amount of gold is considered in one equation, are investigated. In this work, it is presented that the reaction at the surface is the rate controlling step since there is no internal diffusion limitation. The model considering the effect of non-reacting gold shows that the reaction orders are consistent with the experimental observations reported by Ling et al. (1996) and de Andrade Lima and Hodouin (2005). However, it should be noted that the model obtained in this work is based on assumptions of no side reactions, no solid-liquid mass transfer resistances and no effect from temperature.