A cylinder fork model (CFM) was previously proposed [1] to simulate blood oxygenation level dependent (BOLD) contrast effects in functional magnetic resonance imaging (fMRI). The CFM reflects a more realistic model of the human cortical vasculature [2,3] compared to infinite long cylinders [4] by incorporating bifurcations and vessel size. The effect of diffusion was also investigated using the CFM where a clear dependence was observed between the relaxation times and the bifurcation angle at different diffusion scenarios [5]. This study further improves the CFM model by incorporating a rotational movement on the plane of the CFM relative to the magnetic field. Since tumorous tissue exhibits vessels with increased tortuosity and bifurcations with larger diameters compared to that of normal tissue [3], the rotational angle (ϕ–Fig. 1a) can reflect the tortuosity of vessels which adds an extra dimension for characterizing normal and tumor vessels, in addition to the bifurcation angle (β–Fig. 1b) and the vessel diameter. In this study, we quantified the CFM topology using ϕ and explored its effect on R2* at different β and vessel diameters. We also investigated the effects of diffusion and susceptibility (χ) on the R2* relationship with ϕ at different β and vessel diameters within a range of physiological values.Monte Carlo methods were used to quantify R2* for cylindrical fork perturbers at different rotational and bifurcation angles (Fig. 1de) with the magnetic field B₀ oriented in three different directions (Fig. 1c). We modeled the vasculature using a CFM composed of straight trunk (prior to bifurcation) of half cube length and the bifurcating segments. The cube is of 64 µm sides and incorporates cylinder fork segments (Fig. 1b shows 1, 3, 5, 7 and 9 segments at β=0°) with varying angles (ϕ=0°, 15°, 30°, 45°, 60°, 75°, 90° and β=0°, 45°) that were arranged close to symmetry without any overlapping vessels. For all the orientations and with multiple forks in the same cube, the cylinder trunks were set in a parallel orientation. The angle θ (Fig. 1a) between the trunk and the magnetic field was 90° for orientations 1 and 3 with the vessel lying in the xy-plane (Fig. 1c), while θ=0° for orientation 2. This model was converted into a cubic 128×128×128 matrix. Magnetic field perturbations were calculated using a forward 3D Fourier transform of the susceptibility distribution of the CFM [6]. The simulations were performed with a random walk of 40,000 protons using the same technique as described before [1] using the following parameters: true vessel diameters of 2.8, 5.6 and 8.6 µm; diffusion constant (D) values of 1×10^-9, 1×10^-10 and 1×10^-11 m²/s; χ values of 3×10^-8, 1×10^-7, and 4×10^-7 in cgs units. R2* was calculated by linear least-square fitting of log signal intensity versus volume fraction. Relaxation rates per volume fraction unit were used to remove the dependence on vessel lengths and emphasize the role of the ϕ. Analysis of variance (ANOVA) test was performed to check for significant effects of ϕ on R2* at different D values.R2* showed a clear dependence on ϕ in all orientations and β values for all diameters, D, and χ values. Orientation 1 showed a symmetrical profile of R2* around ϕ=45° more clearly pronounced at β=0° compared to β=45°, which is evident at the higher diameter and χ values (Figs. 2,3). A larger decrease of R2* with increasing ϕ at β=0° compared to β=45° was apparent with orientation 3 (Figs. 4,5). Orientation 2 (data not shown) showed a mirror effect of orientation 3 depicting an increase in R2* with increasing ϕ since ϕ=0° in one case corresponds to ϕ=90° for the other due to the orientation of the magnetic field (Fig. 1c). For both orientations 1 and 3 at β=0° and β=45°, the R2* gap between the lower two D values and the highest D value becomes larger with increasing diameter and χ values (for e.g. Figs. 2adg & 2def). This can reflect a physiological scenario where with larger diameters (tumorous tissue) and increased susceptibility (presence of contrast agent) can clearly delineate R2* in relation to ϕ as being either in the fast exchange regime (high D value) or in the intermediate-slow exchange regime (low D values). Since the water exchange differs between normal and tumor tissue [7], these quantifications can potentially aid in differentiating them. R2* measurements indicated a clear dependence on ϕ in all orientations. The change in R2* at larger values of β, χ, and diameter with respect to D can potentially be exploited to distinguish between normal and tumor vessels.