Title:

Functional MRI entropy measurements of brain complexity

The analysis of complex systems has attracted great interest in recent years due to its potential application in medicine. Biological signals are decidedly nonlinear (Bertolaccini, Bussolati & Padovini 1978) and usually exhibit complex behaviour with nonlinear dynamic properties. Examples of these signals are electroencephalogram (EEG) signals, which measure the electrical activity of the brain and the blood oxygen level dependent (BOLD) signal of the brain from functional magnetic resonance imaging (fMRI) acquisition. The changes and differences in the signals produced by biological or physiological systems are mostly undetected by conventional linear signal processing statistics such as mean, standard deviation, variance, Fourier transforms and the like. Nonlinear approaches may be better approach than conventional linear methods in characterizing the complex nature of biological or physiological signals and revealing subtle and important insights into biological processes. The literature study on the application of the nonlinear dynamics theory to analyze physiological signals shows that the complexity of a system's output is an indication of the capacity to adapt to perturbation (Goldberger 1996); (Goldberger, Peng & Lipsitz 2002); (Vaillancourt, Newell 2002). Therefore, the characterization and analysis of the brain's output in terms of its complexity may reveal a better understanding of an individual's health, robustness and adaptive capacity in terms of brain ageing, diseases and invivo effect of drugs. Lipsitz (Lipsitz 2004) has suggested that some systems loose complexity in their output with ageing and disease. Vaillancourt and Newell have characterised two types of systems which have either a fixed point or an oscillatory attractor. An attractor is the state to which a system 'wants' to return to after perturbation (Vaillancourt and Newell, 2002). In a fixedpoint attractor system they proposed that the complexity of the output reduces with age and disease, while in the oscillatory attractor system the opposite is the case. The study of nonlinear dynamics and concepts of complexity can give the opportunities to develop new approaches that are needed to understand and control the complex system in biology and medicine. The historical developments of the concepts of complexity has centered on measuring regularity using various types of entropy measures. Entropy measures the randomness and predictability of a stochastic process and in general increases with greater randomness. Regularity statistics such as approximate entropy (ApEn} and sample entropy (SampEn) are measures that have evolved from the historical developments, adaptations and interdisciplinary applications of entropy. These types of entropy measures can be used to characterize and analyse the different aspects of complex physiological signals brought about by ageing, diseases and the invivo effect of drug. These entropy measures have been applied to analyse 1 dimensional (ID) EEG data of heart rate, nerve activity, arterial pressure and respiratory signals. The main challenge of this project was how to pioneer the implementation of these measures to 4 dimensional (4D) MRI data such as BOLD signal, which has a very poor temporal resolution, with the intension of producing whole brain entropy maps. The aim of this project was to develop a robust and computationally less intensive image processing algorithm for the measurement of entropy with fMRI data. Also, to evaluate and test the fMRI entropy codes by measuring the complexity differences in brain fMRI data and investigating its association with a range of phenomena, such as cognitive performance, normal ageing, anaesthesia, dyslexia and schizophrenia. The implementation of these measures of entropy involved a careful selection of parameters such as the tolerance value, r, number of time points N, the pattern length, m and the time delay, t that are appropriate for fMRI data. Recent studies and my investigation (Appendix A and B) have shown that the recommended range of r (0.1 ~ r~0.2) in the literature is not appropriate for fMRI data (Lu et al. 2008). I found that the appropriate r in an fMRI context was 2 to 3 times that recommended for other modalities. The effect of the number of time points, N on r shows that the appropriate r changed with the number of time points. As a result of these it is necessary to derive an appropriate value of r for each fMRI study. The fMRI entropy codes were developed on a MA TLAB and C platform, the codes were tested on fMRI data acquired while examining resting state normal ageing, dyslexia, cognitive performance, Propofol anaesthesia and Schizophrenia. The results obtained from these analyses show that individuals who were older, had dyslexia, who has lower than expected cognitive performance and who were under the influence of anaesthesia has lower entropy in particular regions than their respective control groups, This is consistent with the GoldbergerlLipsitz model for robustness (Goldberger 1996, Goldberger, Peng & Lipsitz 2002, Goldberger et al. 2002, Goldberger 1997) where complexity decreases with age and disease. The results obtained from applying SampEn to fMRI data of schizophrenia patients indicate an opposite effect. This result is however consistent with other findings in schizophrenia which have consistently demonstrated complex behavioural and physiological output in schizophrenics. This result is also consistent with the second postulate of Vaillancourt and Newell which says that complexity increases with age and diseases (Vaillancourt, Newell 2002) in systems governed by an oscillatory attractor. The results do however suggest that, in this context, that too much complexity is not a good thing and represents abnormal function. This may be a result of impairments in the feedback mechanism of the dopamine system, which is responsible for keeping the system (s) stable. This approach of measuring entropy in fMRI data may find applications in many diverse pathologic and nonpathologic areas of the medical field.
