Martin L. Tanaka

Hybrid Mathematical Model of Glioma Progression

Objectives:  Gliomas are an important form of brain cancer, with high mortality rate. Mathematical models are often used to understand and predict their behaviour. However, using current modeling techniques one must choose between simulating individual cell behaviour and modeling tumours of clinically significant size.

Separatrices and basins of stability from time series data: an application to biodynamics

An approach is presented for identifying separatrices in state space generated from noisy time series data sets which are representative of those generated from experiments. We demonstrate how these separatrices can be found using Lagrangian coherent structures (LCSs), ridges in the state space distribution of the maximum finite-time Lyapunov exponent. As opposed to the current approach which requires a vector field in the state space at each instant of time, this method can be performed using only trajectories reconstructed from time series. As such, this paper forms a bridge connecting methods for evaluating time series data with methods used to evaluate LCSs in vector fields. The methods are applied to a problem in musculoskeletal biomechanics, considered as an exemplar of a class of experimental systems that contain separatrices. In this case, the separatrix reveals a basin of stability for a balancing task, outside of which is a zone of failure. We demonstrate that LCSs calculated from only trajectory data, which samples only portions of the state space, align well with LCSs found using a known vector field. In general, we believe this method provides a fruitful approach for extracting information from noisy experimental data regarding boundaries between qualitatively different kinds of behavior.

Detecting dynamical boundaries from kinematic data in biomechanics

Ridges in the state space distribution of finite-time Lyapunov exponents can be used to locate dynamical boundaries. We describe a method for obtaining dynamical boundaries using only trajectories reconstructed from time series, expanding on the current approach which requires a vector field in the phase space. We analyze problems in musculoskeletal biomechanics, considered as exemplars of a class of experimental systems that contain separatrix features. Particular focus is given to postural control and balance, considering both models and experimental data. Our success in determining the boundary between recovery and failure in human balance activities suggests this approach will provide new robust stability measures, as well as measures of fall risk, that currently are not available and may have benefits for the analysis and prevention of low back pain and falls leading to injury, both of which affect a significant portion of the population.

Comparison of Biomechanical Properties of Native Menisci and Bacterial Cellulose Implant

The menisci are crescent shaped fibrocartilaginous structures in the knee that may become damaged due to traumatic injury or degeneration resulting in pain and a loss of joint function. The goal of this study is to evaluate the mechanical properties of bacterial cellulose (BC) produced by Gluconacetobacter xylinus as a meniscus implants and compare it to native menisci from pigs, sheep, and human. The modulus of BC was varied by controlling water content and tested at four different stiffness values. The modulus of BC ranged from 2.2 MPa for native hydrogel (1% cellulose) to 242 MPa for BC with 30% cellulose. SEM showed a much denser network as the cellulose content increased. Suture retention tests gave a load to break of 20 N and 30 N for 10% and 20% BC, respectively. This study shows promising results for the potential use of BC as a meniscus implant.

Coupled mathematical model of tumorigenesis and angiogenesis in vascular tumours

Objectives:  Mathematical models are useful for studying vascular and avascular tumours, because these allow for more logical experimental design and provide valuable insights into the underlying mechanisms of their growth and development. The processes of avascular tumour growth and the development of capillary networks through tumour‐induced angiogenesis have already been extensively investigated, albeit separately. Despite the clinical significance of vascular tumours, few studies have combined these approaches to develop a single comprehensive growth and development model.

An empirical method for estimating thermal system parameters based on operating data in smart grids

An experimental methodology was developed for online system identification of a thermal system or heated space. In this setting, the intelligent controller detects system parameters during normal operation and adapts its performance accordingly. The ultimate goal is to demonstrate that load leveling with demand side management can be used to reduce peak power consumption while maintaining residential room temperatures at a comfortable level. A prototype enclosure was built and equipped with a heater and thermal measuring equipment. Data was collected during a 17 hour temperature regulation experiment using a bang-bang controller similar to those commonly used for residential heating control. First and second order mathematical models were developed for thermal system identification. The mathematical models utilized the collected temperature data to estimate the net thermal resistance and capacitance using system identification techniques. Results showed the second order model to match the real system characteristics reasonably well. It was found that even for a small prototype enclosure, the estimated thermal parameters showed quite large values of thermal capacitance which can be a great asset for demand side management and control applications in a smart grid. The system identification method developed here is an important step toward the development of intelligent controllers.

An Investigation of Parametric Load Leveling Control Methodologies for Resistive Heaters in Smart Grids

The main goal in this study is to demonstrate that load levelling with demand side management in smart grids can be achieved to reduce peak power consumption while maintaining residential room temperatures at a comfortable level. A prototype enclosure was built and equipped with a heater and thermal measuring equipment. Data was collected during a 17 hour temperature regulation experiment using a traditional on-off (bang-bang) controller similar to those commonly used for residential heating control. A second order mathematical model was utilized to estimate the net thermal resistances and capacitances using system identification techniques at two different temperature set points. The enclosure system was used to determine if peak power could be reduced by slowly varying loads utilizing a different type of controller. Two different linear control techniques (using K-Factor and PI approaches) and the associated power electronics circuitry were implemented and tuned. Both controller systems successfully leveled the load and reduced the peak power demand.

First metatarsophalangeal arthrodesis: a biomechanical comparison of three fixation constructs.

First metatarsophalangeal joint arthrodesis is utilized in the treatment of severe arthritis and hallux valgus. Successful fusion relies on limiting motion at the fusion site and may be achieved through numerous methods. Use of locking plates has recently generated considerable interest, but whether they provide any biomechanical advantage over other available constructs is unclear. Utilizing cyclic loading intended to mimic early weight bearing, the stiffness of three fixation methods for first metatarsophalangeal arthrodesis was compared using Sawbones. The one-third tubular plate completed 1.8 and 2.4 times more cycles before failure than the X-type locking plate or crossed screws, respectively. No difference was detected in cycles to failure between the X-type locking plate and crossed screws. One-third tubular plate mean stiffness was 49% greater than crossed screws at all cycles and greater than X-type locking plate by an average of 25%, beginning at cycle 50.

Determining the Threshold of Stability During Unstable Sitting

Diverse parameters have been developed to quantify spinal stability or system robustness. Kinematic variability (KV) has commonly been used based on the assumption that more robust systems will be able to more effectively reduce system variability. Some of these KV parameters include displacement, standard deviation, RMS area and path velocity of the center of mass (COM) or center of pressure (COP). In addition, stability diffusion analysis and Lyapunov exponents have been used to quantify stability, where lower diffusion or divergence rates indicate a more stable system.© 2009 ASME

Using topological equivalence to discover stable control parameters in biodynamic systems.

In order to better understand the mechanisms that contribute to low back pain, researchers have developed mathematical models and simulations. A mathematical model including neuromuscular feedback control is developed for a person balancing on an unstable sitting apparatus, the wobble chair. When the application of a direct method fails to discover appropriate controller gain parameters for the wobble chair, we show how topological equivalence can be used to indirectly identify appropriate parameter values. The solution is found by first transforming the wobble chair into the Acrobot, another member of the same family of topologically equivalent dynamical systems. After finding appropriate gain parameters for the Acrobot, a continuous transformation is performed to convert the Acrobot back to the wobble chair, during which the gain parameters are adjusted to maintain stability. Thus, we demonstrate how topological equivalence can be used to indirectly solve a problem that was difficult to solve directly.