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The cerebral metabolic rate of oxygen (CMRO2) is an important indicator of brain function and pathology. Knowledge about its magnitude is also required for proper interpretation of the blood oxygenation level dependent (BOLD) signal measured with functional MRI (fMRI). The ability to measure CMRO2 with high spatial and temporal accuracy is thus highly desired. Traditionally the estimation of CMRO2 has been pursued with somewhat indirect approaches combining several different types of measurements with mathematical modeling of the underlying physiological processes. Given the numerous assumptions involved, questions have thus been raised about the accuracy of the resulting CMRO2 estimates. The recent ability to measure the level of oxygen (pO2) in cortex with high spatial resolution in in vivo conditions has provided a more direct way for estimating CMRO2. CMRO2 and pO2 are related via the Poisson partial differential equation. Assuming a constant CMRO2 and cylindrical symmetry around the blood vessel providing the oxygen, the so-called Krogh-Erlang formula relating the spatial pO2 profile to a constant CMRO2 value can be derived. This Krogh-Erlang formula has previously been used to estimate the average CMRO2 close to cortical blood vessels based on pO2 measurements in rats. Here we introduce a new method, the Laplace method, to provide spatial maps of CMRO2 based on the same measured pO2 profiles. The method has two key steps: First the measured pO2 profiles are spatially smoothed to reduce effects of spatial noise in the measurements. Next, the Laplace operator (a double spatial derivative) in two spatial dimensions is applied on the smoothed pO2 profiles to obtain spatially resolved CMRO2 estimates. The smoothing introduces a bias, and a balance must be found where the effects of the noise are sufficiently reduced without introducing too much bias. In this model-based study we explore this balance in situations where the ground truth, that is, spatial profile of CMRO2 is preset and thus known, and the corresponding pO2 profiles are found by solving the Poisson equation, either numerically or by taking advantage of the Krogh-Erlang formula. MATLAB code for using the Laplace method is provided.
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