2599 cubic metre [44]. The frequency resolution of MRE data gained can be entered into various models to obtain two, three, or four independent constitutive parameters [48]. The relatively simple Voight model has been used in the characterisation of liver fibrosis because it fits the viscoelas- tic parameters of soft materials well [49]. Here, the stress– strain relationship is: σ ¼ðμ ζδtÞ" with σ the strain, ε the stress, μ the shear modulus, ζ the shear viscosity (attenuation within the medium) and δt the time- derivative operator. For a homogeneous isotropic viscoelas- tic medium, the complex modulus can then be related to elasticity (μ) and viscosity (η) using: (4) GM ðωÞ ¼ μ þ i ωη (5) This results in six equations, three for the imaginary part and three for the real part, to estimate the two unknown viscoelastic parameters μ and η [48, 49]. A novel fast approach in MRE is to use broadband frequency encoding to acquire superposed tissue oscillations at multiple frequencies rather than one frequency at a time [48, 50]. This method is based on the increases in wave propagation speed and the damping of shear waves in soft tissue with the harmonic driving frequency as a result of the dispersion (i.e. frequency dependence) of elastic waves in viscous media. Several studies, both single frequency MRE [44] and multifrequency MRE [51], have shown that in high-grade liver fibrosis viscosity and/or elastic moduli are significantly elevated compared with the results in healthy volunteers. In a recent large study of 96 patients suspected of having chronic liver disease the elasticity was found to be significantly elevated at histopathology METAVIR scores for fibrosis higher than F1 [52]. Optimal cutoff values of elasticity were 2.5, 3.1 and 4.3 kPa for fibrosis scores greater than F1, F2 and F3, respectively [52]. Systematic liver stiffness increases with increased fibrosis stage of chronic liver disease were also shown by others who examined 50 patients and concluded that the predicted sensitivity and specificity for detecting all grades of liver fibrosis was 98–99% [53]. In a third study (of 44 patients) MRE of liver tumours revealed significantly greater mean shear stiffness in malignant tumours (10.1 kPa) compared with fibrotic liver (5.9 kPa), benign tumours (2.7 kPa) and normal liver (2.3 kPa) [54]. This is illustrated by Fig. 4. A cutoff value of 5 kPa differentiated benign tumours from malignant liver tumours. The masses examined ranged from 1.4 to 11 cm in diameter, but increasing tumour size, a factor expected to lead to decreases in stiffness as necrosis-associated water and fat contents increase, was not taken into account in this preliminary study [54]. It is concluded that MRE appears to be well suited to the noninvasive differentiation of fibrotic liver and malignant liver tumours from normal liver and benign lesions. According to a recent compre- hensive review comparing different imaging modalities in the diagnosis of hepatic fibrosis and cirrhosis, to date only MRE has been able to stage fibrosis or diagnose mild disease while ultrasonic elastography and DWI appear next most promising [55]. Fig. 4 Patient with hepatic ad- enoma. a T2-weighted MR image shows hyperintense 8-cm adenoma (arrow) in the right lobe of the liver. b Gadolinium- enhanced 3D spoiled gradient- recalled echo MR image shows intense arterial phase enhance- ment (arrow). Washout was evident in the portal venous phase (not shown). c Axial MR elastographic wave image shows good illumination of the tumour (circle). Waves in the tumour have a slightly longer length than those in the sur- rounding normal liver paren- chyma. d Elastogram with a region of interest corresponding to the tumour shows a shear stiffness value of the tumour of 3.1 kPa and of the surrounding liver of 2.4 kPa (reproduced with permission from Venkatesh et al. [54])