To explain the varied BOLD responses to progressive hypercapnia in patients with known cerebrovascular steno-occlusive disease.In the presence of cerebrovascular disease, the blood flow response to a ramp increase in the end-tidal partial pressure of CO₂ (P_ETCO₂) is distributed between regions according to competition for a limited blood supply, and thereby produces various patterns of flow response in different regions. These flow response patterns take on four distinct characteristics, and we hypothesised that these patterns result from underlying sigmoidal changes in flow resistance with P_ETCO₂. We used a simple model of two vascular beds (voxels) competing for a limited supply via a fixed arterial flow resistance (Figure 1), Vascular beds with resistances R1 and R2 are perfused via an arterial flow resistance (Rart) from mean arterial blood pressure (MAP). The pressure perfusing the two branches (Pbranch) establishes flows through each branch (F1 and F2), that sum to Ftotal. The vascular bed resistances respond to the P_ETCO₂, and were derived from BOLD estimates of blood flow. We monitored the BOLD response as a surrogate measure of cerebral blood flow in an 18 year old patient with Moyamoya disease, and scaled all voxel BOLD responses by the same factor to convert them to model flow. After establishing the key model parameter, the supply artery flow resistance (Rart), from the whole brain average BOLD response, we chose one voxel with a strong BOLD response as the reference vascular bed and then selected another voxel BOLD response to compete with it. With the known key model parameters, the BOLD responses for each branch were used to calculate the resistance responses, which were assumed to be sigmoidal and fitted accordingly. Using the fitted sigmoidal resistances we calculated the resulting flow patterns and compared them to the measured BOLD responses as a check of the model appropriateness. Figures 2 to 5 illustrate the use of the model, comparing resistance responses of voxels with the four types of patterns of BOLD responses with the resistance response of a chosen reference voxel in the example patient. In each figure: (A) The type map for an axial slice shows the position of the reference voxel (red cursors) and the chosen voxel (blue cursors). (B) Similarly, the cerebrovascular reactivity map (deltaBOLD/deltaP_ETCO₂) shows the positions of the chosen voxels. (C) The scaled BOLD responses to P_ETCO₂ for the chosen voxel (blue open circles) and the reference voxel (red open circles), with the light blue lines showing the type fitting of the chosen voxel. The BOLD response curves for the chosen voxel (blue line) and reference (red line) voxels were calculated from the fitted resistance responses. (D) The resistance responses calculated from the BOLD responses using the model for the chosen voxel (blue open circles) and the reference voxel (red open circles), and the fitted sigmoid curves (lines).This is the first report providing an explanation of the BOLD response patterns to a ramp P_ETCO₂ stimulus. We found that the model reproduced the four types of response patterns using sigmoidal resistance responses calculated from the measured BOLD responses. These individual examples suggested that the pattern types can be characterized by their underlying resistance responses according to their strength of response (amplitude) and the P_ETCO₂ at their maximum sensitivity (sigmoid midpoint) as follows: Type A: A strong response (highest amplitude) with a midpoint P_ETCO₂ representing the PCO₂ of healthy tissue. Type B: A response with a midpoint P_ETCO₂ lower than that of type A; responds as P_ETCO₂ rises but then reaches its maximum and is stolen from by regions still increasing. Type C: A weak response (lowest amplitude) stolen from at all ranges of P_ETCO₂. Type D: A response with a midpoint P_ETCO₂ higher than that of type A; does not respond until a higher P_ETCO₂ is reached and so is stolen from at first but then responds. The model explains the why the BOLD signal resulting from a ramp P_ETCO₂ challenge is not a simple linear or sigmoidal relationship. It provides a physiological explanation in terms of the differences in amplitude and midpoint of the sigmoidal resistance dependence on PCO₂ of a particular region compared to competing regions, increasing insight into pathophysiology.