Direct multiple shooting and direct collocation perform similarly in biomechanical predictive simulations

Direct multiple shooting (DMS) and direct collocation (DC) are two common transcription methods for solving optimal control problems (OCP) in biomechanics and robotics. They have rarely been compared in terms of solution and speed. Through five examples of predictive simulations solved using five transcription methods and 100 initial guesses in the Bioptim software, we showed that not a single method outperformed systematically better. All methods converged to almost the same solution (cost, states, and controls) in all but one OCP, with several local minima being found in the latter. Nevertheless, DC based on fourth-order Legendre polynomials provided overall better results, especially in terms of dynamic consistency compared to DMS based on a fourth-order Runge-Kutta method. Furthermore, expressing the rigid-body constraints using inverse dynamics was usually faster than forward dynamics. DC with dynamics constraints based on inverse dynamics converged to better and less variable solutions. Consequently, we recommend starting with this transcription to solve OCPs but keep testing other methods.Comment: 19 pages, 4 figure

A new ghost cell/level set method for moving boundary problems:application to tumor growth

In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth

Computational study of flow–induced oscillation of a simplified soft palate

Two-dimensional numerical simulations are employed to study fluid structure interaction soft palate in the pharynx for uniform inhalation. We take a next step towards a better biomechanical system by modeling the motion of an inextensible flexible plate. The improved structural model discretized by a low order difference method permits us to simulate the two-dimensional motion of the flexible plate. The inspiratory airflow is described by the Navier–Stokes equations for compressible flow solved by a high order difference method. The explicitly coupled fluid re interaction model is based on the Arbitrary Lagrangian–Eulerian formulation

Lumped Parameter Model for a Self Powered Fontan Palliation of the Hypoplastic Left Heart Syndrome

Out of all newborn infants with congenital heart disease (CHD), 8% have a single functioning ventricle. The Fontan surgical procedure, where the superior and inferior venous returns are connected directly to the pulmonary arteries to allow the single functioning ventricle to perfuse the systemic circulation, has been around for decades, yet the patients who undergo this operation suffer from chronic illnesses and their survivability is less than 50% by adulthood [1-10]. Some suggest that the Fontan operation can be improved by implanting a synthetic pump. However, synthetic pumps also present some complications due to mechanical failure, risk of stroke, and risk of infection [23-30]. The purpose of this study is to numerically simulate the hemodynamics of a self-powered Fontan circulation aided by the reserve ventricular energy captured by the entrainment effect of an Injection Jet Shunt (IJS). A simulation is created to identify important physiological parameters caused be the IJS. By numerically approximating the solution using a lumped parameter model (LPM) of a single ventricle cardiovascular system, the physiological parameters can be approximated. Systemic flow, pulmonary flow, caval pressure and ratio between the pulmonary and systemic flows will be determined to verify whether the IJS is beneficial to the Fontan circulation

Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete LQR problem to find a locally stabilizing controller for a 40 DOF system with 6 constraints.Comment: 13 page

Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model

This work aims at identifying and quantifying uncertainties related to elastic and viscoelastic parameters, which characterize the arterial wall behavior, in one-dimensional modeling of the human arterial hemodynamics. The chosen uncertain parameters are modeled as random Gaussian-distributed variables, making stochastic the system of governing equations. The proposed methodology is initially validated on a model equation, presenting a thorough convergence study which confirms the spectral accuracy of the stochastic collocation method and the second-order accuracy of the IMEX finite volume scheme chosen to solve the mathematical model. Then, univariate and multivariate uncertain quantification analyses are applied to the a-FSI blood flow model, concerning baseline and patient-specific single-artery test cases. A different sensitivity is depicted when comparing the variability of flow rate and velocity waveforms to the variability of pressure and area, the latter ones resulting much more sensitive to the parametric uncertainties underlying the mechanical characterization of vessel walls. Simulations performed considering both the simple elastic and the more realistic viscoelastic constitutive law show that the great uncertainty of the viscosity parameter plays a major role in the prediction of pressure waveforms, enlarging the confidence interval of this variable. In-vivo recorded patient-specific pressure data falls within the confidence interval of the output obtained with the proposed methodology and expectations of the computed pressures are comparable to the recorded waveforms

Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete LQR problem to find a locally stabilizing controller for a 40 DOF system with 6 constraints.Comment: 13 page