Digital Twin (DT) technology models and synchronizes physical systems through virtual counterparts. Existing mathematical formulations of DTs concentrate on physical-virtual synchronization and say little about how representations of the same system at different levels of detail should be related. This paper builds a mathematical foundation for Granular Digital Twins (GDTs) by combining Granular Computing with DT theory. We introduce information granules, the Granular Representation Mapping, the Granular Twin Space, and the Granular Digital Twin, and then construct a Granular Twin Algebra whose aggregation, refinement, and adaptive transformation operators act on an ordered family of granula... More >

Compartmental and tracer-kinetic models are widely used to describe radiopharmaceutical uptake in brain PET and SPECT, but in simple analytical models the fraction of transferred activity that reaches the brain is seldom isolated as a parameter of its own. This study defines a dimensionless Brain Uptake Constant, $\psi$, as the uptake fraction in a two-stage first-order model of cerebral radiopharmaceutical activity and examines its behaviour in a purely synthetic setting. Activity in the systemic pool, $A(t)$, is lost at a first-order rate $k$; cerebral activity, $A_c(t)$, receives a source term $\psi k A(t)$ and is lost at an effective decay rate $\lambda$. The closed-form solution was eva... More >

Glioblastoma (GBM) is a highly infiltrative primary brain tumor, and reaction-diffusion models are widely used to describe how its cells spread and proliferate. We derive the classical Fisher-KPP equation from mass conservation, Fickian diffusion, and logistic growth, and extend it to fractionated radiotherapy through a linear-quadratic survival update applied to the tumor-density field at each fraction. The Fisher-KPP equation is solved by finite differences on a three-dimensional voxel grid with 0.5-mm spacing; Geant4 is used only for Monte Carlo photon transport and dose calculation. The schedule of 60~Gy in 30 fractions from the clinical case reported by Roda et al. is reproduced, and th... More >

In this paper, we study the unsteady magnetohydrodynamic (MHD) flow and the heat and mass transfer of a Williamson tri-hybrid nanofluid (THNF) composed of water and Al$_2$O$_3$, TiO$_2$ and Cu nanoparticles over a stretching sheet. The model incorporates thermal radiation, activation energy, chemical reaction, a heat source and the bioconvection of motile gyrotactic microorganisms. An artificial neural network (ANN) is employed for the predictive modeling of such THNF flows under these combined effects. Similarity transformations reduce the governing nonlinear partial differential equations (PDEs) to coupled ordinary differential equations (ODEs), which are solved numerically with a boundary... More >

This study investigates the two-dimensional magnetohydrodynamic (MHD) slip flow of a tangent-hyperbolic (TGH) aluminum-alloy hybrid nanofluid over a stretching surface in a porous medium. The model incorporates magnetic effects, velocity and thermal slip, thermal radiation, heat generation, viscous dissipation, chemical reaction, and activation energy. The governing nonlinear partial differential equations are transformed into ordinary differential equations using similarity transformations and solved numerically in MATLAB using the bvp4c solver. The effects of key parameters on the velocity, temperature, and concentration profiles are analyzed, together with the skin-friction coefficient, N... More >