Penny J. Davies

Measurements and modelling of the compliance of human and porcine organs

Stress-strain data obtained from animal and human tissue have several applications including medical diagnosis, assisting in surgical instrument design and the production of realistic computer-based simulators for training in minimal access surgery. Such data may also be useful for corroborating mathematical models of tissue response. This paper presents data obtained from ex-vivo and in-vivo tissue indentation tests using a small indentor that is similar to instruments used in minimal access surgery. In addition, uniform stress tests provide basic material property data, via an exponential stress-strain law, to allow a finite element method to be used to predict the response for the non-uniform stresses produced by the small indentor. Data are obtained from harvested pig liver and spleen using a static compliance probe. Data for human liver are obtained from volunteer patients, undergoing minor open surgery, using a sterile hand-held compliance probe. All the results demonstrate highly non-linear stress-strain behaviour. Pig spleen is shown to be much more compliant than pig liver with mean elastic moduli of 0.11 and 4.0 MPa respectively. The right lobe of human liver had a mean elastic modulus of about 0.27 MPa. However, a single case of a diseased liver had a mean modulus of 0.74 MPa--nearly three times the stiffness. It was found that an exponential stress-strain law could accurately fit uniform stress test data and that subsequent finite element modelling for non-uniform stress around a small indentor matched measured force characteristics.

Stability and Convergence of Collocation Schemes for Retarded Potential Integral Equations

Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations. Solving such equations numerically is both complicated and computationally intensive, and numerical methods often prove to be unstable. Collocation schemes are easier to implement than full finite element formulations, but little appears to be known about their stability and convergence. Here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering, and use (spatial) Fourier and (temporal) Laplace transform techniques to demonstrate that such stable schemes are second order convergent.

Numerical stability and convergence of approximations of retarded potential integral equations

Stability and convergence results for approximations of retarded potential integral equations are given. Such integral approximations arise in time marching schemes for direct scattering problems, and are both extremely complicated and computationally intensive. They are often observed to have exponentially increasing unstable solutions. A stability analysis is performed which is shown to provide a way of predicting whether a given scheme is stable or not. The proposed test is supported by numerical experiments and requires much less computation than the scheme itself. It is also shown that if an approximation for this type of integral equation is both stable and consistent, then it is convergent.

A stability analysis of a time marching scheme for the general surface electric field integral equation

We present a stability analysis of a mixed finite element approximation of the time dependent electric field integral equation which is used to compute the current induced on a perfect conductor by a transient electric field. The analysis shows that the solution has fast growing unstable modes that are due to a combination of the finite element basis functions and the integral quadrature rule. These modes cannot be removed by standard time averaging methods, but they can be eliminated by averaging in space appropriately.

Mathematical Modelling for Keyhole Surgery Simulations: Spleen Capsule as an Elastic Membrabe

We show that the mechanical properties of the outer covering (or capsule) of pig spleen can be modelled as a nonlinear elastic membrane with an exponential stress-strain Law. Knowledge of the caspules elastic properties is important for the development of a virtual reality training package for keyhole surgery. Bccause capsule tissue is very fragile and tears when clamped and pulled it is not possible to obtain experimental data for which the usual uniform stretch data-fitting approach can be adoptcd.Instcad we perform experiments in which the capsule deforms non-uniformly and describe how numerical solutions of the corresponding boundary value problem can be fitted to the experimerimental data in order to validate the biomechanical model.

Convolution-in-Time Approximations of Time Domain Boundary Integral Equations

We present a new temporal approximation scheme for the boundary integral formulation of time-dependent scattering problems which can be combined with either collocation or Galerkin approximation in space. It uses the backward-in-time framework introduced in [P. J. Davies and D. B. Duncan, Convolution Spline Approximations of Volterra Integral Equations, www.mathstat.strath.ac.uk/research/reports/2012 (2012)] with new temporal basis functions which share some properties with radial basis function multiquadrics. We analyze the stability and convergence properties of the new scheme for associated Volterra integral equations and perform extensive numerical tests for scattering from flat polygonal plates and open and closed cubes and spheres, which demonstrate effectiveness of this approach.

The time domain integral equation for a straight thin-wire antenna with the reduced kernel is not well-posed

We show that the Pocklington integral equation for time-domain scattering from thin-wire antennas is not mathematically well-posed. This has considerable implications for numerical solution schemes. In particular, our argument explains the observed occurrence of rapidly oscillating errors in numerical solutions as the numerical grid sizes are reduced.

Puncture forces of solid organ surfaces

AbstractBackground: In this experimental study, we measured the force needed to puncture the liver (low elastin) and the spleen (high elastin). The surface displacement preceding puncture was also measured. These data are relevant to an understanding of surgical technique and are essential to the development of electronic surgical simulators. Methods: Controlled puncture experiments were performed on intact organs harvested from pigs and sheep, as well as on their surface capsules following removal and suspension at zero strain and at three increasing levels of prestrain. The biomechanical data were compared with information obtained from histological studies. Results: The spleen has a higher puncture force than the liver and suffers greater displacement before puncture (p < 0.05). Prestrain decreases displacement before puncture (p < 0.05) but has no effect on puncture force. Conclusion: The higher puncture force and displacement of spleen, as compared with liver, is probably due to its higher elastin content.

Numerical stability of collocation schemes for time domain boundary integral equations

Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations (RPIEs). Collocation schemes for RPIEs are often unstable, having errors which oscillate and grow exponentially with time. We describe how Fourier analysis can be used to analyse the stability of uniform grid schemes and to show that the instabilities are often very different from those observed in PDE approximations. We also present a new stable collocation scheme for a scalar RPIE, and show that it converges.

On the behaviour of time discretisations of the electric field integral equation

We derive a separation of variables solution for time-domain electromagnetic scattering from a perfectly conducting infinite flat plate. The time dependent part of the equations are then used as a model problem in order to study the effects of various time discretisations on the full scattering problem. We examine and explain how exponential and polynomial instabilities arise in the approximation schemes, and show that the time averaging which is often used in an attempt to stabilise solutions of the full problem acts to destabilise some of the schemes. Our results show that two of the time discretisations can produce good results when coupled with a space-exact approximation, and indicate that they will be useful when coupled with an accurate enough spatial approximation.