روش تجزیه و تحلیل حساسیت و کاربرد آن در مدل سازی حرکت غیرسخت مبتنی بر فیزیک

Parameters used in physical models for nonrigid and articulated motion analysis are often not known with high precision. It has been recognized that commonly used assumptions about the parameters may have adverse effect on modeling quality. In this paper, we present an efficient sensitivity analysis method to assess the impact of those assumptions by examining the model's spatial response to parameter perturbation. Numerical experiments with a synthetic model and skin tissues show that: (1) normalized sensitivity distribution can help determine the relative importance of different parameters; (2) dimensional sensitivity is useful in the assessment of a particular parameter assumption; and (3) models are more sensitive at the locations of property discontinuity (heterogeneity). The formulation of the proposed sensitivity analysis method is general and can be applied to assessment of other types of assumptions, such as those related to nonlinearity and anisotropy.

During the past two decades, physics-based modeling has emerged as one of the most important techniques for nonrigid and articulated motion analysis [18]. Its popularity is evidenced by the increasing number of publications each year as well as the diversity of the fields in which the papers appeared. For example, physical model has found applications in: nonrigid and articulated motion tracking [21], [22], [15] and [34], realistic facial animation [19] and [17], surgery simulation and operation planning [6], [3] and [7], elastic medical image registration [4], [36] and [14], dynamic cloud simulation using satellite images [33], as well as shape representation and recognition [25] and [24], to name a few. More comprehensive survey and in-depth discussions on physics-based nonrigid motion analysis can be found in [1] and [20]. In comparison to geometrical and mass-spring models, physical models that are based on continuum mechanics have high computational complexity. As a result, various assumptions are often made to simplify the model and its parameters. For example, a commonly used assumption is that the material properties of an object are isotropic and homogeneous. However, results from large amount of biomechanical tests [12] indicate that the mechanical behavior of many biological materials, especially soft tissues, cannot be accurately described by such a simplified model. Certain types of muscles (such as skeletal muscle) are characterized by strong anisotropic behaviors. More importantly, the property heterogeneity of several orders of magnitude is also common in human organs [9]. Recently, measuring elastic property of abnormalities caused by the pathological processes has been utilized for early cancer detection [11] and [10]. Recognizing the inadequacy of simplified physical models, researchers have started to investigate to what degree the various assumptions, especially those about the material properties and the boundary conditions, may affect the model's performance by means of sensitivity analysis. For example, Alterovitz et al. [2] studied the influence of both the physician-controlled parameters and the intrinsic material parameters on the accuracy of needle insertion simulation. In the study of model-based breast cancer diagnosis, Tanner et al. [28] compared the results of biomechanical models with different settings of boundary condition and material properties. However, those studies were done on the case-by-case basis using ad hoc comparison methods, and therefore the conclusions cannot be readily generalized to other domains. Moreover, the experiments were performed based on the assumption that the model has homogeneous material properties, which implies that the solution of a Dirichlet type problem could be independent of the internal property variation, and hence the subsequent sensitivity analysis results may not be valid. Another limitation of their methods is that the sensitivity data is incapable of providing a complete picture of the model's spatial response to the parameter variation on each individual point of the model. In this paper, we propose a local gradient based computation method that can be used to conduct a systematic and comprehensive sensitivity analysis of any motion model. Specifically, physics-based nonrigid motion modeling will benefit from such a sensitivity analysis in the following aspects: 1. The proposed method allows us to compare the relative importance of different parameters using the normalized sensitivity data and quantify the impact of various assumptions on model's performance using the dimensional sensitivity data; 2. The algorithm is designed based on the adjoint state method, which significantly reduces the computational cost and is suited for handling large scale finite element models; 3. The sensitivity contour map enables us to identify the vulnerable areas of a model that are most affected by a poor assumption, so that further improvement can be made; 4. The parameter value can be obtained by either the direct measurement [12] or the indirect inference [23]. But those acquisition procedures are time-consuming and expensive. It would be economic to first conduct a sensitivity analysis to identify the primary parameters and then to concentrate our effort on the acquisition of those parameters.

Model-based nonrigid and articulated motion analysis requires careful design and calibration of the model being used, whether it is a geometrical model or a physical model. Various assumptions about the model and its parameters must be thoroughly evaluated. Ad hoc sensitivity analysis method is impractical for this task, especially when dealing with large scale problems. We propose a systematic sensitivity analysis method that is capable of making a quantitative and reasoned diagnosis of the model's performance related to our assumptions about the parameters. The proposed method is formulated using the first-order local gradient information. The adjoint state method is employed to reduce the computational cost of large scale numerical models. A two-step procedure is recommended: (1) the normalized sensitivity is suited for identifying the most significant parameters; (2) the dimensional sensitivity can be used to assess and improve our assumption about a particular parameter. Based on the experiments with synthetic models, burn scars, and faces, several observations can be made: (1) the models are more sensitive to the change of the Young's modulus than to the change of the Poisson's ratio; (2) the models are most sensitive at the property discontinuities (heterogeneity); (3) sensitivity map is informative about the location/geometry of the property abnormalities. It should be stressed that the experiments reported here were performed using a linear elastic model. The system response of a nonlinear model to the parameter variation is more complex. In future investigations, we will apply the proposed method to address the following issues: (1) How sensitive an elastic model is to an assumed isotropic property? (2) What is the sensitivity distribution of an elastic model under various loading conditions (boundary conditions)? (3) What is the sensitivity response of a nonlinear model? We will study cases that are both geometrically nonlinear and materially nonlinear. Our primary interest is to model the nonrigid motion of skin tissue to facilitate medical diagnosis and face recognition. Like many other soft tissues, skin has nonlinear stress–strain curves undergoing large deformation. To model this nonlinear behavior, Cauchy stress tensor and infinitesimal strain tensor have to be replaced by second Piola-Krichoff stress tensor and Green–Lagrange strain tensor, respectively. The constitutive equation of skin can be approximated as either hyperelastic (assuming pseudo elastic) or truly viscoelastic. For example, Tong and Fung [12] proposed a pseudo strain energy function of the form equation(35) View the MathML sourcew=12(a1E112+a2E222+2a3E122+2a4E11E22)+12Cexp(b1E112+b2E222+2b3E122+2b4E11E22+c1E113+c2E223+c3E112E22+c4E11E222) Turn MathJax on where ai,bi,ci are constants to be determined by experiments and Eij are Green–Lagrange strain components. The second Piola-Kirchoff stress tensor can be determined by differentiating the strain energy function with respect to the Green–Lagrange strain components. The tangent stiffness can be determined as the second derivative of the strain energy function. If skin is to be modeled as hyperelastic (neglecting loading and unloading effects), the path-independent nature of the problem allows us to conduct sensitivity analysis using the same method developed for linear elastic deformation. However, if a viscoelastic model is adopted, tow potential problems may arise due to its path-dependent nature. First, the objective function may not be differentiable which would result in both theoretical and numerical difficulties. More importantly, the approach that computes the sensitivity values outside the finite element model becomes very expensive, because the current sensitivity value will depend upon both the deformation history and the sensitivity history. Therefore, it is reasonable to study the sensitivity response of hyperelastic model first and then to examine more complex viscoelastic cases. Another important issue in sensitivity analysis is that the initial parameter values assigned to the model should be based on the data reported in literature, especially those collected from the biomechanical tensile tests with real soft tissues [12] and [9]. Using those data will ensure that the sensitivity values are in a reasonable range and the interpretation is both physically and biologically sound.