A new generalised two-dimensional time and space variable-order fractional Bloch-Torrey equation is developed in this study. The variable-order Riesz fractional derivative and variable diffusion coefficient are introduced to simulate diffusion phenomena in heterogeneous, irregularly shaped biological tissues. The fractional Bloch-Torrey equation is discretised by the weighted and shifted Grünwald-Letnikov formula with respect to time and by finite volume method with respect to space. Additionally, to improve the accuracy of the numerical method for dealing with non-smooth solutions, some appropriate correction terms are introduced in the time approximation. Numerical examples on different irregular domains with various non-smooth solutions are explored to verify the effectiveness of the presented numerical scheme. Furthermore, we also solve the coupled variable-order fractional Bloch-Torrey equation on a human brain-like domain which is composed of white matter and grey matter. The solution behaviour of this model is compared with that of the constant-order fractional model, and the transverse magnetisation in magnetic resonance imaging on different biological micro-environments are graphically analysed. Results suggest that incorporation of the non-local property and spatial heterogeneity in the model by use of fractional operators can lead to a better capability for capturing the complexities of diffusion phenomena in biological tissues. This research may provide a basis for further research on the application of fractional calculus to clinical research and medical imaging.

本研究开发了一个新的广义二维时空变阶分数布洛赫-托雷方程。引入变阶 Riesz 分数阶导数和可变扩散系数来模拟异质、不规则形状生物组织中的扩散现象。分数 Bloch-Torrey 方程通过加权和移动的 Grünwald-Letnikov 公式关于时间离散，并通过有限体积方法关于空间离散。此外，为了提高数值方法处理非光滑解的精度，在时间近似中引入了一些适当的修正项。探索了具有各种非光滑解的不同不规则域的数值例子，以验证所提出数值方案的有效性。此外，我们还在由白质和灰质组成的类人脑域上求解耦合的可变阶分数 Bloch-Torrey 方程。将该模型的求解行为与恒阶分数模型的求解行为进行了比较，并对不同生物微环境下磁共振成像中的横向磁化强度进行了图形分析。结果表明，通过使用分数运算符将非局部属性和空间异质性纳入模型中，可以更好地捕捉生物组织中扩散现象的复杂性。该研究可为进一步研究分数阶微积分在临床研究和医学影像中的应用提供依据。