Bayesian analysis of random regression models using B-splines to model test-day milk yield of Holstein cattle in Brazil

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Random regression analyses using B-splines functions to model growth from birth to adult age in Canchim cattle

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Otimização de forma aplicando B-splines sob critério integral de tensões

This work proposes a computational methodology to solve problems of optimization in structural design. The application develops, implements and integrates methods for structural analysis, geometric modeling, design sensitivity analysis and optimization. So, the optimum design problem is particularized for plane stress case, with the objective to minimize the structural mass subject to a stress criterion. Notice that, these constraints must be evaluated at a series of discrete points, whose distribution should be dense enough in order to minimize the chance of any significant constraint violation between specified points. Therefore, the local stress constraints are transformed into a global stress measure reducing the computational cost in deriving the optimal shape design. The problem is approximated by Finite Element Method using Lagrangian triangular elements with six nodes, and use a automatic mesh generation with a mesh quality criterion of geometric element. The geometric modeling, i.e., the contour is defined by parametric curves of type B-splines, these curves hold suitable characteristics to implement the Shape Optimization Method, that uses the key points like design variables to determine the solution of minimum problem. A reliable tool for design sensitivity analysis is a prerequisite for performing interactive structural design, synthesis and optimization. General expressions for design sensitivity analysis are derived with respect to key points of B-splines. The method of design sensitivity analysis used is the adjoin approach and the analytical method. The formulation of the optimization problem applies the Augmented Lagrangian Method, which convert an optimization problem constrained problem in an unconstrained. The solution of the Augmented Lagrangian function is achieved by determining the analysis of sensitivity. Therefore, the optimization problem reduces to the solution of a sequence of problems with lateral limits constraints, which is solved by the Memoryless Quasi-Newton Method It is demonstrated by several examples that this new approach of analytical design sensitivity analysis of integrated shape design optimization with a global stress criterion purpose is computationally efficient

Bayesian analysis of random regression models using B-splines to model test-day milk yield of Holstein cattle in Brazil

The objective of this paper is to model variations in test-day milk yields of first lactations of Holstein cows by RR using B-spline functions and Bayesian inference in order to fit adequate and parsimonious models for the estimation of genetic parameters. They used 152,145 test day milk yield records from 7317 first lactations of Holstein cows. The model established in this study was additive, permanent environmental and residual random effects. In addition, contemporary group and linear and quadratic effects of the age of cow at calving were included as fixed effects. Authors modeled the average lactation curve of the population with a fourth-order orthogonal Legendre polynomial. They concluded that a cubic B-spline with seven random regression coefficients for both the additive genetic and permanent environment effects was to be the best according to residual mean square and residual variance estimates. Moreover they urged a lower order model (quadratic B-spline with seven random regression coefficients for both random effects) could be adopted because it yielded practically the same genetic parameter estimates with parsimony. (C) 2012 Elsevier B.V. All rights reserved.

Random regression analyses using B-spline functions to model growth of Nellore cattle

The objective of this study was to estimate (co)variance components using random regression on B-spline functions to weight records obtained from birth to adulthood. A total of 82 064 weight records of 8145 females obtained from the data bank of the Nellore Breeding Program (PMGRN/Nellore Brazil) which started in 1987, were used. The models included direct additive and maternal genetic effects and animal and maternal permanent environmental effects as random. Contemporary group and dam age at calving (linear and quadratic effect) were included as fixed effects, and orthogonal Legendre polynomials of age (cubic regression) were considered as random covariate. The random effects were modeled using B-spline functions considering linear, quadratic and cubic polynomials for each individual segment. Residual variances were grouped in five age classes. Direct additive genetic and animal permanent environmental effects were modeled using up to seven knots (six segments). A single segment with two knots at the end points of the curve was used for the estimation of maternal genetic and maternal permanent environmental effects. A total of 15 models were studied, with the number of parameters ranging from 17 to 81. The models that used B-splines were compared with multi-trait analyses with nine weight traits and to a random regression model that used orthogonal Legendre polynomials. A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and maternal permanent environmental effect, was the most appropriate and parsimonious model to describe the covariance structure of the data. Selection for higher weight, such as at young ages, should be performed taking into account an increase in mature cow weight. Particularly, this is important in most of Nellore beef cattle production systems, where the cow herd is maintained on range conditions. There is limited modification of the growth curve of Nellore cattle with respect to the aim of selecting them for rapid growth at young ages while maintaining constant adult weight.

Random regression analyses using B-spline functions to model growth of Nellore cattle

The objective of this study was to estimate (co)variance components using random regression on B-spline functions to weight records obtained from birth to adulthood. A total of 82 064 weight records of 8145 females obtained from the data bank of the Nellore Breeding Program (PMGRN/Nellore Brazil) which started in 1987, were used. The models included direct additive and maternal genetic effects and animal and maternal permanent environmental effects as random. Contemporary group and dam age at calving (linear and quadratic effect) were included as fixed effects, and orthogonal Legendre polynomials of age (cubic regression) were considered as random covariate. The random effects were modeled using B-spline functions considering linear, quadratic and cubic polynomials for each individual segment. Residual variances were grouped in five age classes. Direct additive genetic and animal permanent environmental effects were modeled using up to seven knots (six segments). A single segment with two knots at the end points of the curve was used for the estimation of maternal genetic and maternal permanent environmental effects. A total of 15 models were studied, with the number of parameters ranging from 17 to 81. The models that used B-splines were compared with multi-trait analyses with nine weight traits and to a random regression model that used orthogonal Legendre polynomials. A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and maternal permanent environmental effect, was the most appropriate and parsimonious model to describe the covariance structure of the data. Selection for higher weight, such as at young ages, should be performed taking into account an increase in mature cow weight. Particularly, this is important in most of Nellore beef cattle production systems, where the cow herd is maintained on range conditions. There is limited modification of the growth curve of Nellore cattle with respect to the aim of selecting them for rapid growth at young ages while maintaining constant adult weight.

Hardware accelerated ray cast of volume data and volume gradient for an optimized splines-based multi-resolution 2D-3D registration

This paper describes a method for DRR generation as well as for volume gradients projection using hardware accelerated 2D texture mapping and accumulation buffering and demonstrates its application in 2D-3D registration of X-ray fluoroscopy to CT images. The robustness of the present registration scheme are guaranteed by taking advantage of a coarse-to-fine processing of the volume/image pyramids based on cubic B-splines. A human cadaveric spine specimen together with its ground truth was used to compare the present scheme with a purely software-based scheme in three aspects: accuracy, speed, and capture ranges. Our experiments revealed an equivalent accuracy and capture ranges but with much shorter registration time with the present scheme. More specifically, the results showed 0.8 mm average target registration error, 55 second average execution time per registration, and 10 mm and 10° capture ranges for the present scheme when tested on a 3.0 GHz Pentium 4 computer.

Estudo de transições a dois fotões em iões com um e dois electrões

Neste trabalho foi efectuado um estudo teórico de todas as transições de emissão espontânea a dois fotões dos estados iniciais n = 2, 3 em iões hidrogenoides. Foram obtidos valores precisos para a taxa de emissão através da resolução da equação de Dirac e da utilização de uma base de B-splines para valores da carga nuclear Z até 92. Foram também calculadas as taxas de transição a dois fotões para os estados 21S e 23S0 em iões heliumoides na aproximação que os electrões não interactuam entre si. Os resultados obtidos estão de acordo com o esperado. Este trabalho é a continuação de um trabalho anterior na qual foi efectuado um estudo da transição a dois fotões 2s → 1s em sistemas hidrogenoides usando uma base de B-splines [1] e uma dedução de uma expressão teórica para a taxa de emissão a dois fotões em sistemas heliumoides [2].

Central and Cortical Gray Mater Segmentation of Magnetic Resonance Images of the Fetal Brain

Motivation. The study of human brain development in itsearly stage is today possible thanks to in vivo fetalmagnetic resonance imaging (MRI) techniques. Aquantitative analysis of fetal cortical surfacerepresents a new approach which can be used as a markerof the cerebral maturation (as gyration) and also forstudying central nervous system pathologies [1]. However,this quantitative approach is a major challenge forseveral reasons. First, movement of the fetus inside theamniotic cavity requires very fast MRI sequences tominimize motion artifacts, resulting in a poor spatialresolution and/or lower SNR. Second, due to the ongoingmyelination and cortical maturation, the appearance ofthe developing brain differs very much from thehomogenous tissue types found in adults. Third, due tolow resolution, fetal MR images considerably suffer ofpartial volume (PV) effect, sometimes in large areas.Today extensive efforts are made to deal with thereconstruction of high resolution 3D fetal volumes[2,3,4] to cope with intra-volume motion and low SNR.However, few studies exist related to the automatedsegmentation of MR fetal imaging. [5] and [6] work on thesegmentation of specific areas of the fetal brain such asposterior fossa, brainstem or germinal matrix. Firstattempt for automated brain tissue segmentation has beenpresented in [7] and in our previous work [8]. Bothmethods apply the Expectation-Maximization Markov RandomField (EM-MRF) framework but contrary to [7] we do notneed from any anatomical atlas prior. Data set &Methods. Prenatal MR imaging was performed with a 1-Tsystem (GE Medical Systems, Milwaukee) using single shotfast spin echo (ssFSE) sequences (TR 7000 ms, TE 180 ms,FOV 40 x 40 cm, slice thickness 5.4mm, in plane spatialresolution 1.09mm). Each fetus has 6 axial volumes(around 15 slices per volume), each of them acquired inabout 1 min. Each volume is shifted by 1 mm with respectto the previous one. Gestational age (GA) ranges from 29to 32 weeks. Mother is under sedation. Each volume ismanually segmented to extract fetal brain fromsurrounding maternal tissues. Then, in-homogeneityintensity correction is performed using [9] and linearintensity normalization is performed to have intensityvalues that range from 0 to 255. Note that due tointra-tissue variability of developing brain someintensity variability still remains. For each fetus, ahigh spatial resolution image of isotropic voxel size of1.09 mm is created applying [2] and using B-splines forthe scattered data interpolation [10] (see Fig. 1). Then,basal ganglia (BS) segmentation is performed on thissuper reconstructed volume. Active contour framework witha Level Set (LS) implementation is used. Our LS follows aslightly different formulation from well-known Chan-Vese[11] formulation. In our case, the LS evolves forcing themean of the inside of the curve to be the mean intensityof basal ganglia. Moreover, we add local spatial priorthrough a probabilistic map created by fitting anellipsoid onto the basal ganglia region. Some userinteraction is needed to set the mean intensity of BG(green dots in Fig. 2) and the initial fitting points forthe probabilistic prior map (blue points in Fig. 2). Oncebasal ganglia are removed from the image, brain tissuesegmentation is performed as described in [8]. Results.The case study presented here has 29 weeks of GA. Thehigh resolution reconstructed volume is presented in Fig.1. The steps of BG segmentation are shown in Fig. 2.Overlap in comparison with manual segmentation isquantified by the Dice similarity index (DSI) equal to0.829 (values above 0.7 are considered a very goodagreement). Such BG segmentation has been applied on 3other subjects ranging for 29 to 32 GA and the DSI hasbeen of 0.856, 0.794 and 0.785. Our segmentation of theinner (red and blue contours) and outer cortical surface(green contour) is presented in Fig. 3. Finally, torefine the results we include our WM segmentation in theFreesurfer software [12] and some manual corrections toobtain Fig.4. Discussion. Precise cortical surfaceextraction of fetal brain is needed for quantitativestudies of early human brain development. Our workcombines the well known statistical classificationframework with the active contour segmentation forcentral gray mater extraction. A main advantage of thepresented procedure for fetal brain surface extraction isthat we do not include any spatial prior coming fromanatomical atlases. The results presented here arepreliminary but promising. Our efforts are now in testingsuch approach on a wider range of gestational ages thatwe will include in the final version of this work andstudying as well its generalization to different scannersand different type of MRI sequences. References. [1]Guibaud, Prenatal Diagnosis 29(4) (2009). [2] Rousseau,Acad. Rad. 13(9), 2006, [3] Jiang, IEEE TMI 2007. [4]Warfield IADB, MICCAI 2009. [5] Claude, IEEE Trans. Bio.Eng. 51(4) (2004). [6] Habas, MICCAI (Pt. 1) 2008. [7]Bertelsen, ISMRM 2009 [8] Bach Cuadra, IADB, MICCAI 2009.[9] Styner, IEEE TMI 19(39 (2000). [10] Lee, IEEE Trans.Visual. And Comp. Graph. 3(3), 1997, [11] Chan, IEEETrans. Img. Proc, 10(2), 2001 [12] Freesurfer,http://surfer.nmr.mgh.harvard.edu.

A post-processing pipeline to reconstruct the developing fetal brain using low-resolution MRI

Introduction. Development of the fetal brain surfacewith concomitant gyrification is one of the majormaturational processes of the human brain. Firstdelineated by postmortem studies or by ultrasound, MRIhas recently become a powerful tool for studying in vivothe structural correlates of brain maturation. However,the quantitative measurement of fetal brain developmentis a major challenge because of the movement of the fetusinside the amniotic cavity, the poor spatial resolution,the partial volume effect and the changing appearance ofthe developing brain. Today extensive efforts are made todeal with the âeurooepost-acquisitionâeuro reconstruction ofhigh-resolution 3D fetal volumes based on severalacquisitions with lower resolution (Rousseau, F., 2006;Jiang, S., 2007). We here propose a framework devoted tothe segmentation of the basal ganglia, the gray-whitetissue segmentation, and in turn the 3D corticalreconstruction of the fetal brain. Method. Prenatal MRimaging was performed with a 1-T system (GE MedicalSystems, Milwaukee) using single shot fast spin echo(ssFSE) sequences in fetuses aged from 29 to 32gestational weeks (slice thickness 5.4mm, in planespatial resolution 1.09mm). For each fetus, 6 axialvolumes shifted by 1 mm were acquired (about 1 min pervolume). First, each volume is manually segmented toextract fetal brain from surrounding fetal and maternaltissues. Inhomogeneity intensity correction and linearintensity normalization are then performed. A highspatial resolution image of isotropic voxel size of 1.09mm is created for each fetus as previously published byothers (Rousseau, F., 2006). B-splines are used for thescattered data interpolation (Lee, 1997). Then, basalganglia segmentation is performed on this superreconstructed volume using active contour framework witha Level Set implementation (Bach Cuadra, M., 2010). Oncebasal ganglia are removed from the image, brain tissuesegmentation is performed (Bach Cuadra, M., 2009). Theresulting white matter image is then binarized andfurther given as an input in the Freesurfer software(http://surfer.nmr.mgh.harvard.edu/) to provide accuratethree-dimensional reconstructions of the fetal brain.Results. High-resolution images of the cerebral fetalbrain, as obtained from the low-resolution acquired MRI,are presented for 4 subjects of age ranging from 29 to 32GA. An example is depicted in Figure 1. Accuracy in theautomated basal ganglia segmentation is compared withmanual segmentation using measurement of Dice similarity(DSI), with values above 0.7 considering to be a verygood agreement. In our sample we observed DSI valuesbetween 0.785 and 0.856. We further show the results ofgray-white matter segmentation overlaid on thehigh-resolution gray-scale images. The results arevisually checked for accuracy using the same principlesas commonly accepted in adult neuroimaging. Preliminary3D cortical reconstructions of the fetal brain are shownin Figure 2. Conclusion. We hereby present a completepipeline for the automated extraction of accuratethree-dimensional cortical surface of the fetal brain.These results are preliminary but promising, with theultimate goal to provide âeurooemovieâeuro of the normal gyraldevelopment. In turn, a precise knowledge of the normalfetal brain development will allow the quantification ofsubtle and early but clinically relevant deviations.Moreover, a precise understanding of the gyraldevelopment process may help to build hypotheses tounderstand the pathogenesis of several neurodevelopmentalconditions in which gyrification have been shown to bealtered (e.g. schizophrenia, autismâeuro¦). References.Rousseau, F. (2006), 'Registration-Based Approach forReconstruction of High-Resolution In Utero Fetal MR Brainimages', IEEE Transactions on Medical Imaging, vol. 13,no. 9, pp. 1072-1081. Jiang, S. (2007), 'MRI of MovingSubjects Using Multislice Snapshot Images With VolumeReconstruction (SVR): Application to Fetal, Neonatal, andAdult Brain Studies', IEEE Transactions on MedicalImaging, vol. 26, no. 7, pp. 967-980. Lee, S. (1997),'Scattered data interpolation with multilevel B-splines',IEEE Transactions on Visualization and Computer Graphics,vol. 3, no. 3, pp. 228-244. Bach Cuadra, M. (2010),'Central and Cortical Gray Mater Segmentation of MagneticResonance Images of the Fetal Brain', ISMRM Conference.Bach Cuadra, M. (2009), 'Brain tissue segmentation offetal MR images', MICCAI.

Quantitative morphometry analysis of the fetal brain using clinical MR imaging

In vivo fetal magnetic resonance imaging provides aunique approach for the study of early human braindevelopment [1]. In utero cerebral morphometry couldpotentially be used as a marker of the cerebralmaturation and help to distinguish between normal andabnormal development in ambiguous situations. However,this quantitative approach is a major challenge becauseof the movement of the fetus inside the amniotic cavity,the poor spatial resolution provided by very fast MRIsequences and the partial volume effect. Extensiveefforts are made to deal with the reconstruction ofhigh-resolution 3D fetal volumes based on severalacquisitions with lower resolution [2,3,4]. Frameworkswere developed for the segmentation of specific regionsof the fetal brain such as posterior fossa, brainstem orgerminal matrix [5,6], or for the entire brain tissue[7,8], applying the Expectation-Maximization MarkovRandom Field (EM-MRF) framework. However, many of theseprevious works focused on the young fetus (i.e. before 24weeks) and use anatomical atlas priors to segment thedifferent tissue or regions. As most of the gyraldevelopment takes place after the 24th week, acomprehensive and clinically meaningful study of thefetal brain should not dismiss the third trimester ofgestation. To cope with the rapidly changing appearanceof the developing brain, some authors proposed a dynamicatlas [8]. To our opinion, this approach however faces arisk of circularity: each brain will be analyzed /deformed using the template of its biological age,potentially biasing the effective developmental delay.Here, we expand our previous work [9] to proposepost-processing pipeline without prior that allow acomprehensive set of morphometric measurement devoted toclinical application. Data set & Methods: Prenatal MRimaging was performed with a 1-T system (GE MedicalSystems, Milwaukee) using single shot fast spin echo(ssFSE) sequences (TR 7000 ms, TE 180 ms, FOV 40 x 40 cm,slice thickness 5.4mm, in plane spatial resolution1.09mm). For each fetus, 6 axial volumes shifted by 1 mmwere acquired under motherâeuro?s sedation (about 1min pervolume). First, each volume is segmentedsemi-automatically using region-growing algorithms toextract fetal brain from surrounding maternal tissues.Inhomogeneity intensity correction [10] and linearintensity normalization are then performed. Brain tissues(CSF, GM and WM) are then segmented based on thelow-resolution volumes as presented in [9]. Ahigh-resolution image with isotropic voxel size of 1.09mm is created as proposed in [2] and using B-splines forthe scattered data interpolation [11]. Basal gangliasegmentation is performed using a levet setimplementation on the high-resolution volume [12]. Theresulting white matter image is then binarized and givenas an input in FreeSurfer software(http://surfer.nmr.mgh.harvard.edu) to providetopologically accurate three-dimensional reconstructionsof the fetal brain according to the local intensitygradient. References: [1] Guibaud, Prenatal Diagnosis29(4) (2009). [2] Rousseau, Acad. Rad. 13(9), 2006. [3]Jiang, IEEE TMI 2007. [4] Warfield IADB, MICCAI 2009. [5]Claude, IEEE Trans. Bio. Eng. 51(4) 2004. [6] Habas,MICCAI 2008. [7] Bertelsen, ISMRM 2009. [8] Habas,Neuroimage 53(2) 2010. [9] Bach Cuadra, IADB, MICCAI2009. [10] Styner, IEEE TMI 19(39 (2000). [11] Lee, IEEETrans. Visual. And Comp. Graph. 3(3), 1997. [12] BachCuadra, ISMRM 2010.

3D Surface Modeling From Point Clouds

The goal of this thesis is to implement software for creating 3D models from point clouds. Point clouds are acquired with stereo cameras, monocular systems or laser scanners. The created 3D models are triangular models or NURBS (Non-Uniform Rational B-Splines) models. Triangular models are constructed from selected areas from the point clouds and resulted triangular models are translated into a set of quads. The quads are further translated into an estimated grid structure and used for NURBS surface approximation. Finally, we have a set of NURBS surfaces which represent the whole model. The problem wasn’t so easy to solve. The selected triangular surface reconstruction algorithm did not deal well with noise in point clouds. To handle this problem, a clustering method is introduced for simplificating the model and removing noise. As we had better results with the smaller point clouds produced by clustering, we used points in clusters to better estimate the grids for NURBS models. The overall results were good when the point cloud did not have much noise. The point clouds with small amount of error had good results as the triangular model was solid. NURBS surface reconstruction performed well on solid models.

Neurofuzzy network model construction using Bézier-Bernstein polynomial functions

Neurofuzzy modelling systems combine fuzzy logic with quantitative artificial neural networks via a concept of fuzzification by using a fuzzy membership function usually based on B-splines and algebraic operators for inference, etc. The paper introduces a neurofuzzy model construction algorithm using Bezier-Bernstein polynomial functions as basis functions. The new network maintains most of the properties of the B-spline expansion based neurofuzzy system, such as the non-negativity of the basis functions, and unity of support but with the additional advantages of structural parsimony and Delaunay input space partitioning, avoiding the inherent computational problems of lattice networks. This new modelling network is based on the idea that an input vector can be mapped into barycentric co-ordinates with respect to a set of predetermined knots as vertices of a polygon (a set of tiled Delaunay triangles) over the input space. The network is expressed as the Bezier-Bernstein polynomial function of barycentric co-ordinates of the input vector. An inverse de Casteljau procedure using backpropagation is developed to obtain the input vector's barycentric co-ordinates that form the basis functions. Extension of the Bezier-Bernstein neurofuzzy algorithm to n-dimensional inputs is discussed followed by numerical examples to demonstrate the effectiveness of this new data based modelling approach.

Otimização de pré-formas e matrizes em problemas bidimensionais de forjamento

Este trabalho apresenta uma sistemática para realizar a otimização numérica de pré-formas e de matrizes em problemas de forjamento axissimétricos e em estado plano de deformações. Para este fim, desenvolveu-se um código computacional composto basicamente de três módulos: módulo de pré-processamento, módulo de análise e módulo de otimização. Cada um destes foi elaborado acrescentando rotinas em programas comerciais ou acadêmicos disponíveis no GMAp e no CEMACOM. Um programa gerenciador foi desenvolvido para controlar os módulos citados no processo de otimização. A abordagem proposta apresenta uma nova função objetivo a minimizar, a qual está baseada em uma operação booleana XOR (exclusive or) sobre os dois polígonos planos que representam a geometria desejada para o componente e a obtida na simulação, respectivamente. Esta abordagem visa eliminar possíveis problemas geométricos associados com as funções objetivo comumente utilizadas em pesquisas correlatas. O trabalho emprega análise de sensibilidade numérica, via método das diferenças finitas. As dificuldades associadas a esta técnica são estudadas e dois pontos são identificados como limitadores da abordagem para problemas de conformação mecânica (grandes deformações elastoplásticas com contato friccional): baixa eficiência e contaminação dos gradientes na presença de remalhamentos. Um novo procedimento de diferenças finitas é desenvolvido, o qual elimina as dificuldades citadas, possibilitando a sua aplicação em problemas quaisquer, com características competitivas com as da abordagem analítica Malhas não estruturadas são tratadas mediante suavizações Laplacianas, mantendo as suas topologias. No caso de otimização de pré-formas, o contorno do componente a otimizar é parametrizado por B-Splines cujos pontos de controle são adotados como variáveis de projeto. Por outro lado, no caso de otimização de matrizes, a parametrização é realizada em termos de segmentos de reta e arcos de circunferências. As variáveis de projeto adotadas são, então, as coordenadas das extremidades das retas, os raios e centros dos arcos, etc. A sistemática é fechada pela aplicação dos algoritmos de programação matemática de Krister Svanberg (Método das Assíntotas Móveis Globalmente Convergente) e de Klaus Schittkowski (Programação Quadrática Sequencial – NLPQLP). Resultados numéricos são apresentados mostrando a evolução das implementações adotadas e o ganho de eficiência obtido.