Functional MRI (fMRI) using the blood oxygenation level dependent (BOLD) signal is a common technique in the study of brain function. The BOLD signal is sensitive to the complex interaction of physiological changes including cerebral blood flow (CBF), cerebral blood volume (CBV), and cerebral oxygen metabolism (CMRO2). A primary goal of quantitative fMRI methods is to combine BOLD imaging with other measurements (such as CBF measured with arterial spin labeling) to derive information about CMRO2. This requires an accurate mathematical model to relate the BOLD signal to the physiological and hemodynamic changes; the most commonly used of these is the Davis model. Here, we propose a new nonlinear model that is straightforward and shows heuristic value in clearly relating the BOLD signal to blood flow, blood volume and the blood flow-oxygen metabolism coupling ratio. The model was tested for accuracy against a more detailed model adapted for magnetic fields of 1.5, 3 and 7T. The mathematical form of the heuristic model suggests a new ratio method for comparing combined BOLD and CBF data from two different stimulus responses to determine whether CBF and CMRO2 coupling differs. The method does not require a calibration experiment or knowledge of parameter values as long as the exponential parameter describing the CBF-CBV relationship remains constant between stimuli. The method was found to work well for 1.5 and 3T but is prone to systematic error at 7T. If more specific information regarding changes in CMRO2 is required, then with accuracy similar to that of the Davis model, the heuristic model can be applied to calibrated BOLD data at 1.5T, 3T and 7T. Both models work well over a reasonable range of blood flow and oxygen metabolism changes but are less accurate when applied to a simulated caffeine experiment in which CBF decreases and CMRO2 increases.