The mean squared displacement can then be calculated from the second moment of this distribution:

This is the well-known Einstein equation of diffusion and this model is used to derive the NMR signal attenuation due to diffusion of molecules (4, 7).

Most modern diffusion imaging techniques employ variations of bipolar pulsed field gradient methods (8) to obtain diffusion sensitivity. In these sequences, two magnetic field gradients are applied sequentially to dephase and then rephase spins or protons. Any spins that have moved in the interval between the two gradients do not experience similar magnetic field as during the first gradient and thus they do not get rephased, resulting in a net loss of spin coherence and a quantifiable loss in the MRI signal (9) which can be related to the apparent diffusion coefficient or ADC. In the simplest case, with spin-echo MRI and considering for the moment only a single diffusion coefficient per voxel, it can be shown (starting from the Einstein formulation of diffusion) that the signal and the ADC are related as follows (10):
where S is the measured signal, S ₀ the signal in absence of the gradients, and D is the ADC. The b-factor in this equation is a function of the gradient strength G, duration δ, and time separation Δ between the gradient pulses. The b-factor must be calculated for each sequence, which can be non-trivial when the diffusion gradients interact with the imaging gradients in a complicated fashion. For the simple case where imaging gradients and the resultant cross terms in the b-factor calculations are ignored, , (10) where γ is a physical constant, the gyromagnetic ratio. D can be determined from the above equation if at least two images are acquired with different diffusion weightings (and thus different b-factors). It also becomes evident from this equation that any movement of spins during the diffusion time Δ will result in MRI signal loss. Thus diffusion MRI is sensitive to all motion of molecules during the sequence, and not just purely diffusive motion. Moreover, the observed diffusion coefficients in tissues reflect diffusion in several compartments—extracellular and intracellular, intranuclear, mitochondrial, etc. —which may all have different diffusivities if one could resolve them. It is for these reasons that the diffusion coefficient and tensor as measured with MRI are called the apparent diffusion coefficient (ADC) and apparent diffusion tensor (ADT), respectively.

The diffusivity of a substance is not always identical in all directions. For example, diffusion coefficients can be relatively large in one direction and small in another. Diffusion is more generally characterized by a second-order tensor, or a matrix of nine coefficients:

This implies that nine diffusivities would have to be determined to measure the tensor. However, the diffusion tensor is symmetric, meaning that , , and . Thus, only six diffusivities need to be determined. When the axes of the measurement system coincide exactly with the axes of the fiber or object being studied, the off-diagonal elements of the tensor (i.e., D _xy , D _xz , and D _yz ) become zero, and the measured diagonal diffusivites D _xx , D _yy , and D _zz are termed the principal diffusivities, often annotated D ₁₁, D ₂₂, and D ₃₃ or λ ₁, λ ₂ , and λ ₃. Of course, in the general case, the measurement and principal axes of diffusion do not coincide for most fibers. In such a case, these principal diffusivities, which are also called the eigenvalues of the diffusion tensor, along with the eigenvectors of the system (defining the directions of the principal diffusivities), can be calculated from the measured tensor by a simple mathematical transformation (see Note 1).

In an environment where the barriers to diffusion are identical in all directions, diffusion is considered isotropic. This is the case, for example, in a cup of water, where even if molecules are allowed to diffuse for relatively long times, they are unlikely to encounter a barrier to diffusion so that movement in one direction is not favored over any other. In such situations, the diffusion coefficients in all directions are equivalent (i.e., , etc.). For isotropic cases, measurement of a single scalar coefficient suffices to completely characterize the system. This condition is often not met in tissue measurements where cell membranes, packing, and other ordering in tissue structures promote directed water diffusion. For medical imaging applications, this is the case for example in white matter, muscle, cartilage, kidneys, tumors, and in the lens of the eye (11–19). In such settings, diffusion is said to be anisotropic, reflecting the non-equal diffusivities in various directions.

The coefficients along the x, y, and z-directions (D _xx , D _yy , and D _zz ) are simply the diagonal terms of the tensor and are the principal diffusivities of the system only when the laboratory coordinate frame coincides with that of the sample orientation. In the general case where the laboratory axes are different from the orientation of the sample, the principal diffusivities are found by determining the eigenvalues of the measured diffusion matrix (1, 15, 16). These principal diffusivities are independent of the relation between the coordinate system of the laboratory and the coordinate system of the sample (1, 16).

The tensor nature of diffusion must be taken into account in the mathematical framework for describing diffusion. Torrey derived the Bloch equations with diffusion terms and included the rudiments of the tensor analysis needed to consider anisotropic diffusion (20). In 1965, Stejskal extended this analysis to give a formal description of the effect of anisotropic diffusion on the NMR signal (7). Basser et al. further extended this treatment and measured the ADT by applying gradients in various directions (15, 16). Starting with the mathematical expressions for the description of magnetization in the presence of spin diffusion (20) they derived the expressions necessary to calculate the various terms of D (or ADT) from an MRI experiment. They showed that
where

Here, the indices i and j represent the measurement axes (1, 2, and 3 being x, y, and z, respectively). G _i and G _j thus represent the applied diffusion gradients in various directions. This equation bears obvious resemblance to the case for measuring a single diffusion coefficient. However, all six unique diffusivities must be accounted for, and thus this implies that at least seven images must be acquired to completely determine the seven variables in this equation (S ₀ and the six unique diffusivities). Obviously, the calculation of the b-matrix as per Eq. [6] is considerably more complex than that of b-factors for a uni-directional experiment and care must be taken to account for the interaction between the various diffusion and imaging gradients.