Identification of a model of sound transmission in the human knee: vibroarthrographic signals as a diagnostic tool

Blind System Identification " 107 8.1 .Analytical Methods 107 8.1.1Multiple Blind System Identification .107 vii 8.1.2Orthonormal Filters 8.1.3Fast MBSI Algorithm 8.2 Simulations 8.2.1 Ideal Case 8.2.2Low Pass Filter Simulations 8.3 Experimental Results 8.4 Discussion and Future Work 8.4.1 Future Work 9 Conclusion 9.1 Contributions 9.2 Future Work Bibliography vii' List of Tables 4.1 Correlation Coefficients between trials.4.2 Correlation coefficients between accelerometers.5.1 %Variance Accounted For values of trials.5.2 Magnitude and Frequency Characteristics of Transfer Functions . .6.1 % Variance Accounted For values in cross-validation data of nonlinear kernel models 6.2 % Variance Accounted For values for Second order Volterra Series models in cross-validation and in-sample data with varying memory length .6.3 % Variance Accounted For values for block structured cascade models: Accelerometer 1 7.1 % Variance Accounted For values for input estimated using frequency deconvolution method with various S value cutoff levels.7.2 % Variance Accounted For values for input estimated using frequency deconvolution method with various SNR levels in output data 7.3 % Variance Accounted For values for input estimated using time invariant deconvolution method, using experimental data 7.4 % Variance ' Account ed For values for output signal resulting from the reconvolution of the estimated input with the identified IRF 8.1 % Variance Acco'unted For in the low pass filter simulations with var-bus amounts of additive noise.118 ix List of Figures 2.1 Anatomy of human knee 9 2.2 Ligaments in the human knee 3.1 Block diagram of a "black box" model 3.2 Diagram of a Wiener-Bose model 3.3 The effect of the delay parameter on Laguerre filters, 3.4 Block diagram of an LNL cascade structure.4.1 Measurement of joint angle S 4.2 Position of accelerometers on skin over patella.4.3 Schematic of experimental setup .4.4 Raw data collected from accelerometer 1, in trial 2. 4.5 Output data from accelerometer 1 in trials 1 and 2 4.6 Output data from accelerometer 1 in trials 2 and 6. 5.1 Block diagram representing the analyzed system.5.2 Impulse responses, accelerpmeter 1 5.3 Impulse responses, accelerometer 2 5.4 Impulse responses, accelerometer 3 5.5 Output error, trial 1, accelerometer 1 5.6 Transfer functions estimated at 90 degrees 5.7 Coherence squared functions.5.8 Magnitude characteristics at various joint angles.5.9 The effects of compression force 5.10 Cross spectral confidence, accelerometer 1 6.1.Second order FOA Volterra kernels, trial 2, accelerometer 1 6.2 Second order LET Volterra kernels, trial 2, accelerometer 1 6.3 Comparison of the 2nd order kernel diagonal with the 1st order kernel, trial 1 .6.4 Comparison of the 2nd order kernel diagonal with the 1st order kernel, trial 2 77 6.5 Second order FOA Volterra kernels, trial 6, accel.1. 78 6.6 Predicted FOA output, trial 6, accelerometer 1. 6.7 Second order FOA Volterra kernels, trial 8, accel. 1 79 6.8 Wiener cascade model, trial 1, accel. 1 82 6.9 Hammerstein cascade model trial 1 accel..1.83 6.10 LNL cascade model, trial 1, accel. 1. 84 7.1 Actual and predicted input for simulation system 93 7.2 Actual and predicted input sequence to simulated system with noise.94 7.3 Filter bank for simulation system 95 7.4 Time Varying System used for deconvolution simulations 95 7.5 Results of time varying deconvolution simulation, varying S cutoffs. .96 7.6 Results for time varying deconvolution simulation 96 7.7 Power spectra of significant error 97 7.8 Actual and predicted input for stationary knee vibration signal: Trial 9. 99 7.9 Comparison of reconvolved output and measured output with trial from 90 degrees.7.10 Significant frequencies of error for deconvolution of trial 9. 7.11 Input estimated for clinically acquired VAG signal.7.12 Comparison of the reconvolution of the estimated input with the measured output signal from 30 to 90 degrees.8.1 Blind System Identification Block Structure . . .8.2 The effect of a and L on the final IRF.All IRFs are calculated using coefficients of 1 for each of the basis functions in the filter bank. . . .8.3 Singular values identified by FBI algorithm in ideal case simulation . 8.4The results of the FBI algorithm applied to the ideal case simulation.8.5 The preliminary results of the FBI algorithrfi applied to the low pass filters simulation.8.6 Singular values of the Laguerre basis Hessian in low pass filter simulation 8.7 A block structure representation of the structure of a multiple channel system with common elements.8.8 The final results of the FBI algorithm applied to the low pass filters simulation., 8.9 The preliminary results of the FBI algorithm applied to experimental data collected at a 45 degree joint angle.' 119 8.10 The final results of the FBI algorithm applied to experimental data collected at a 45 dree joint angle 120 xi the principles of physics, biochemistry and physiology.System identification, on the other hand, requires little or no a priori knowledge the system, instead is estimated using inputs and outputs measured from a specially designed experiment. Experimental DesignThe need for both a measured output and input signal poses a significant challenge for studying VAG transmission, where the input signal to the system is unknown and impossible to measure non-invasively.With the help of an orthopedic surgeon an experimental protocol has been developed with an actuator implanted in the patello-femoral joint of a cadaver knee to acquire these signals.A band-pass filtered Gaussian voltage signal was injected into the actuator, causing it to vibrate, and the resulting vibrations were measured on the outer surface of the patella. Linear System IdentificationTime and frequency domain methods were used to estimate linear time-invariant systems: using an impulse response function (IRF) in the time domain and using spectra in the frequency domain.The IRFs give clues to the behaviour of a system indicating delays and the memory length of the responses (i.e., how long.after an input that the output begins and how long the dynamics last).Spectra, on the otherhand,.can indicate filtering characteristics of the system.By examining the magnitude and phase responses of a system's spectrum, zeros and poles of the systemS can be identified.Finally, estimating the spectrum allows examination of the coherence squared function of the system, a measure of how "linear" and/or noise-free it is.'Specific details of the components are discussed in Sec.4.3

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