Advanced diffusion encoding methods in MRI 2.4.2 Average Propagator with Short Gradient Pulses 2.4.3 Arbitrary Gradient Waveforms and Multiple Propagators 2.4.4 Double Encoding with a Mixing Block 2.4.5 Approximations for Spin echo Attenuation 2.4.6 Spin echo as a Characteristic Functional of a Stochastic Process 2.4.7 Cross correlation Tensors and the Gaussian Approximation of the Cumulant Expansion 2.4.8 Multi Gaussian Diffusion in the Low b Regime 2.5 Frequency domain Analysis with the Gaussian Approximation of the Cumulant Expansion 2.5.1 Encoding and Diffusion Spectra 3.2.1 Haeberlen Convention for Tensor Size and Shape 3.2.2 Measures of Tensor Anisotropy 3.2.3 Positive (Semi)definiteness 3.3 Distributions of Axisymmetric Diffusion Tensors 3.3.1 Normalization 3.3.2 Low dimensional Projections 3.3.3 Statistical Descriptors 3.3.4 Component Binning 3.3.5 Ensemble averaging 3.4 Tensor valued Diffusion Encoding 3.4.1 Diffusion Weighting q Space Trajectories and b Tensor 3.4.2 Computing the Frobenius Inner Product 3.4.3 Qualitative Interpretation 3.5 Estimation of Statistical Descriptors 2.5.2 Directional Average Signal Attenuation due to Time dependent Diffusion in Multicompartment Systems 2.5.3 Mean Spectrum and Spectral Anisotropy 2.5.4 Signal Attenuation for Directionally Averaged Multi compartment Systems 2.5.5 Case of Axisymmetric Diffusion Spectra 2.5.6 Case of Gaussian Diffusion 2.5.7 Spectral Tuning of b Tensors 2.5.8 Effect of Spectral Anisotropy 2.5.9 Low frequency Expansion 2.6 Conclusions Acknowledgements References Chapter 3 Diffusion Anisotropy and Tensor valued Encoding 3.1 Introduction 3.2 Symmetric Tensors Cover Preface Contents Chapter 1 Translational Motion of Water in Biological Tissues A Brief Primer 1.1 Introduction 1.2 A Molecular Perspective on Water Diffusion in Biological Tissues 1.2.1 Biomembranes 1.2.2 Macromolecules 1.3 Experimental Data on Biomembrane Permeability and Macromolecular Obstruction 1.4 Diffusivity Restriction Anisotropy Exchange and Flow 1.5 Summary Acknowledgements References Chapter 2 Diffusion Encoding with General Gradient Waveforms 2.1 Introduction 2.2 Stochastic Processes 2.2.1 Probability Distributions 2.2.2 Markov Process 2.2.3 Moments and Cumulants 2.3 NMR Signal and Spin Phase 2.3.1 Phase of a Single Spin Contribution 2.3.2 Spin echo and the Effective Gradient Waveform 2.3.3 NMR Signal for an Ensemble of Spins 2.3.4 Non moving Spins 2.3.5 Spin echo Condition 2.3.6 Spin echo for Spins Undergoing Translational Motion 2.3.7 Coherent Motion Single Ensemble 2.3.8 Coherent Motion Multiple Sub ensembles 2.3.9 Spin Phase as a Function of Displacement or Velocity 2.4 Stochastic Motion 2.4.1 Spin Position as a Stationary Markov Process The medical MRI community is by far the largest user of diffusion NMR techniques and this book captures the current surge of methods and provides a primary source to aid adoption in this field