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.　©　2021　Elsevier　Ltd