Uncertainty Management in Computational Simulations of Medical Devices1

Abstract

Certification of the designs of medical devices requires decisions to be made based on uncertainty. This is true whether the designer is basing the decision on information derived from empirical tests or computational simulations. Design of experiments (DoE) and sensitivity analysis (SA) can be used in both empirical testing and computational simulation to evaluate uncertainty. DoE and SA can be prohibitively expensive for empirical testing if more than just a few parameters are tested. And testing at the statistical limits of the parameters is seldom possible. Because computational simulations are often orders of magnitude less costly than empirical tests, they offer the possibility of fully examining the design space out to all conceivable limits. This provides the data needed to better understand the effect of uncertainty and the probability that a design will meet its desired function over its intended lifetime. Computational simulation allows for risk based designs with increased safety while lowering the number of redesigns required in the design cycle, lowering post market warranty exposure, and lowering the likelihood of recalls.Uncertainty in the design of devices comes in various forms such as physical uncertainty, model uncertainty, and statistical uncertainty.Physical uncertainty is defined by the loads, dimensions, and material properties that always have some uncertainty, or inherent randomness, associated with them. For example, the loads may depend on environmental effects. Since the environmental effects are random, the loads are represented as random variables. The variations in the manufacturing process and quality control introduce uncertainty in the dimensions and material properties.There are several sources of model uncertainty. One source of uncertainty is derived from neglecting some important random variables. Another is the fact that mathematical models of the structure always require some simplifying assumptions and approximations. A third factor is the adjustment of analysis approximations to be on the conservative side. The cumulative effect of these assumptions produces uncertainty in the predicted response. Further, risk analysis methods used to compute the probability of failure are also approximate and introduce more uncertainty in the reliability estimate.There is always statistical uncertainty, even if the available test data are adequate, there is error in the estimation of the random variable distribution parameters such as mean value and standard deviation.Managing uncertainty comes at a price—more testing to better define the variability in input parameters, higher fidelity analyses at a finer scale to limit the uncertainty in the physics, etc. Variability in each input parameter does not affect the uncertainty in the system response equally. Nor does every data refinement reduce the uncertainty in the system response. This paper presents a computational methodology that estimates the sensitivity of uncertainty in input variables and the sensitivity of modeling approximations to the risk of failure of the device. In the current age of large multidisciplinary virtual simulation, this is useful in determining how to minimize overall uncertainty in analytical predictions. In addition, the methodology can be used to optimize for the best use of computational and testing resources to arrive at most robust predictions. Although the methodology was developed for the U.S. Air Force Airframe Digital Twin Program, it is consistent with the Food and Drug Administration's Center for Devices and Radiological Health vision of a faster, better medical device development and evaluation substantially augmented by computational simulation. It is applicable to any system or component with high cost of failure. An example application to a medical device will be shown.Uncertainties arising from physical variability, information uncertainty, and model and measurement errors are evaluated. Physical uncertainties are represented in the analysis by defining them as random variables. The distribution information of the random variables (e.g., mean value, standard deviation, and density function) is obtained from the sample data. This information is then used in risk analysis methods. Physical uncertainty primarily manifests itself as input error and output measurement error. Measurement error in the input variables is propagated to the prediction of the output. When the relationship between input and output is given by a physical or empirical model, then the error in the prediction of the output due to the measurement error in the input variables may be approximated using SA. The measurement error in the output variable is a separate error component, whereas the measurement error in the input variables is compounded through propagation in the prediction model.Model uncertainty is estimated in terms of both solution approximation error due to numerical discretization in the solution of mathematical equations and model form error due to the selection of particular model form. Approximation errors are quantified through SA and numerical experimentation such as mesh refinement. Model form error is quantified by comparison of surrogate model prediction with high fidelity model prediction, and the knowledge of approximation errors.Statistical uncertainty is defined by a confidence interval on each parameter to represent this error. In Bayesian statistics, the uncertainty in the distribution parameters is represented by treating the distribution parameters themselves as random variables. This Bayesian representation facilitates easier treatment of errors in the distribution parameters while computing their effect on the performance.The uncertainty is managed using VEXTEC's virtual life management® (vlm®) software to propagate the uncertainty through the design analysis. The device geometry, loads, and materials data (both deterministic and random) along with the computational models and their error are used to create a virtual twin® (vt®) of the device. The vt is a digital representation of the device with all of the associated uncertainties explicitly defined.The vt was created of a defibrillator lead device (shown in Fig. 1) design that included a manufacturing process model, a structural analysis model, and a material fatigue model. The vt was used to estimate the sensitivity of uncertainty in input variables and the sensitivity of modeling approximations to the final output.The flow of the design analysis is shown in Fig. 2 in which a manufacturing process model provides residual stress input to the structural model which provide stresses to the material failure model. The coil (or filar) winding, x, can have uncertainty, ε, which produces a residual stress output, ypredi, from the process model with uncertainty, ɛinpi. These are input to the structural model. The structural model also has a modeling uncertainty (from finite element analysis discretization) ɛmodel1, along with other random variables. Likewise for the material model, obtaining a final output that includes uncertainties in material inclusion size and density is done by obtaining the corrected output at each level (discipline) sequentially. Uncertainty propagation in unidirectional coupled levels is analogous to the propagation of input error. That is, corrected output from the previous level is used as the input to the next model (ypredi in Fig. 2), and the output of the next then includes the combination of the errors in the previous level.Figure 2 shows the cumulative probability distribution function of the cycles to failure by considering all uncertainties compared with actual test results which are shown as solid circles in the figure. What is apparent from the simulation results is that the fatigue durability results are highly sensitive to uncertainty in residual stress which comes from the manufacturing process model. As such residual stress values are difficult to estimate from process models, since there is so much variability in the winding forces of the defibrillator lead filars. What VEXTEC's uncertainty propagation process demonstrates is by using computational virtual simulation one can develop robust methods to estimate such parameters with confidence. Simulations were performed for three different residual stress levels with statistical mean values at 50 ksi, 45 ksi, and 40 ksi (Fig. 3). Results show that a residual stress of 40 ksi captures the actual test results accurately. The Food and Drug Administration (FDA) has recognized vlm and vt, for the above presented example, at a mature stage of development. They have accepted VEXTEC into the recently announced Medical Device Development Tool pilot program, citing vlm's the potential for major efficiencies to be gained in medical device development and evaluation time.Results show that the proposed methodology can effectively quantify the errors and evaluate their importance according to their contribution to the uncertainty in the final output. What is insightful from the above analysis of the design is that, with limited resources and budget, one needs to concentrate on the material modeling discipline. The vt is useful in determining how to minimize overall uncertainty in analytical predictions. In addition, the vt can be used to optimize for the best use of computational and testing resources to arrive at the most robust predictions.