Damion Robin A, Radjenovic Aleksandra, Ingham Eileen, Jin Zhongmin, Ries Michael E
School of Physics and Astronomy, University of Leeds, Leeds, West Yorkshire, United Kingdom ; Institute of Medical and Biological Engineering, School of Mechanical Engineering, University of Leeds, Leeds, West Yorkshire, United Kingdom.
PLoS One. 2013 Nov 11;8(11):e78798. doi: 10.1371/journal.pone.0078798. eCollection 2013.
We develop a curvilinear invariant set of the diffusion tensor which may be applied to Diffusion Tensor Imaging measurements on tissues and porous media. This new set is an alternative to the more common invariants such as fractional anisotropy and the diffusion mode. The alternative invariant set possesses a different structure to the other known invariant sets; the second and third members of the curvilinear set measure the degree of orthotropy and oblateness/prolateness, respectively. The proposed advantage of these invariants is that they may work well in situations of low diffusion anisotropy and isotropy, as is often observed in tissues such as cartilage. We also explore the other orthogonal invariant sets in terms of their geometry in relation to eigenvalue space; a cylindrical set, a spherical set (including fractional anisotropy and the mode), and a log-Euclidean set. These three sets have a common structure. The first invariant measures the magnitude of the diffusion, the second and third invariants capture aspects of the anisotropy; the magnitude of the anisotropy and the shape of the diffusion ellipsoid (the manner in which the anisotropy is realised). We also show a simple method to prove the orthogonality of the invariants within a set.
我们开发了一种扩散张量的曲线不变集，可应用于对组织和多孔介质的扩散张量成像测量。这个新的不变集是分数各向异性和扩散模式等更常见不变量的替代方案。该替代不变集具有与其他已知不变集不同的结构；曲线集的第二个和第三个成员分别测量正交性程度和扁率/长率。这些不变量的一个优点是，正如在软骨等组织中经常观察到的那样，它们在低扩散各向异性和各向同性的情况下可能表现良好。我们还根据它们与特征值空间相关的几何结构探索了其他正交不变集；一个柱形集、一个球形集（包括分数各向异性和模式）和一个对数欧几里得集。这三个集具有共同的结构。第一个不变量测量扩散的大小，第二个和第三个不变量捕捉各向异性的方面；各向异性的大小和扩散椭球体的形状（各向异性实现的方式）。我们还展示了一种简单的方法来证明一个集合内不变量的正交性。
