The effect of impermeable boundaries of arbitrary geometry on the apparent diffusion coefficient Astrid F. Frøhlich a, * , Sune N. Jespersen a , Leif Østergaard a , Valerij G. Kiselev b a Department of Neuroradiology, Center of Functionally Integrative Neuroscience (CFIN), University Hospital of Aarhus, Nørrebrogade 44, Building 30, 8000 Aarhus C, Denmark b Medical Physics, Department of Diagnostic Radiology, University Hospital Freiburg, Freiburg, Germany article info Article history: Received 23 April 2008 Revised 13 June 2008 Available online 24 June 2008 Keywords: NMR Diffusion Restricted diffusion Cumulant expansion Apparent diffusion coefficient Biological boundaries abstract The apparent diffusion coefficient (ADC) obtained from NMR measurements is modelled for diffusion in a compartment restricted by an impermeable boundary. For a given pulse sequence, the ADC can be deter- mined from the connected velocity autocorrelation function (the second-order velocity cumulant), which we show can be expressed as a double surface integral over the boundary, involving the probability for molecules to diffuse from one boundary point to another. There is no restriction on the geometry of the boundary. This result allows a fast calculation of the ADC for an arbitrary time course of the diffusion-sen- sitizing gradient. Explicit examples are given for diffusion within three basic geometries for different pulse sequences. The ADCs measured with the Stejskal–Tanner pulse sequence and a more realistic pulse sequence with slice selection gradient and eddy current compensation are found to yield almost identical results. The application of the results are discussed in relation to determination of the microscopic struc- ture of brain white matter. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction Diffusion-weighted MRI probes cellular structure of living tis- sue. However, the detailed relation between the biological microstructure and the acquired MR signal is extremely complex and the study of this relation remains an area of active research. Considerable insight into this field may be gleaned from the exploration of the structure of porous media by diffusion- weighted MRI [1–3]. In such media, an NMR visible fluid is con- fined in an impermeable NMR-invisible matrix. The physics underlying diffusion and the MRI signal is well understood in this context [4]. In particular, the short-time behavior of the apparent diffusion coefficient (ADC) can be related to the geometry of the matrix: D ¼ D 0 1  c  ffiffiffiffiffiffiffiffi D 0 T p S V   ; ð1Þ where S and V are the surface and volume of the pore space, D 0 is the diffusion coefficient in the bulk fluid, T is the diffusion time, and c is a constant depending on the pulse sequence [1–3]. By vir- tue of the central limit theorem, the diffusion takes its free form again for long times, with a reduced ADC. The reduction is de- scribed by the so-called tortuosity, k, defined through D ¼ D 0 =k 2 . The diffusion-weighted signal in biological tissue is more diffi- cult to analyze, for experimentally relevant diffusion times, since many of the simplifications applicable to porous materials do not apply. In most tissue types, NMR visible water is present inside and outside cells. Furthermore, the majority of biological mem- branes are penetrable for water molecules, and there are few com- partments with free diffusion. For example, the medium surrounding neuronal fibers consists of glia cells and an abundance of macro-molecules, and the diffusion in such a medium is hin- dered. This issue was considered in Ref. [5], where a short-time expression for the ADC was derived for molecules diffusing in a heterogeneous medium with restrictive boundaries in one-dimen- sion. No simplifying assumptions were made about the diffusion in the bulk medium. The results ruled out a simple interpretation of the diffusion-weighted signal in the spirit of Eq. (1). The dependence of the ADC on the measurement technique presents another problem for the interpretation of the diffusion- weighted signal. In the narrow pulse approximation, with its unambiguous definition of the diffusion time and the Fourier rela- tion between the spin displacement distribution and the NMR sig- nal, the ADC is simply related to the mean square displacement of individual spins. However, the narrow pulse approximation is rarely achievable with the gradient systems of clinical scanners, and for the more complicated pulse sequences employed in practice, the relation of the ADC to microscopic variables is less di- rect. For these reasons, it is important to develop methods for the calculation of the ADCs for different realistic pulse sequences. 1090-7807/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmr.2008.06.009 * Corresponding author. Fax: +45 89894400. E-mail address: astrid@pet.auh.dk (A.F. Frøhlich). Journal of Magnetic Resonance 194 (2008) 128–135 Contents lists available at ScienceDirect Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr