where 𝐹 𝑡 is the time-dependent concentration of hyperpolarized [1-13 C]fumarate, 𝑀 𝑡 is the timedependent concentration of hyperpolarized [1-13 C]malate and [4- 13 C]malate, and 𝑘 is the forward reaction-rate constant. In this model, for simplicity we assume that both species relax with the same time constant 𝑇 , which is a combination of 𝑇 and the signal decay due to successive pulses. 𝑇 is known from the [1-13 C]fumarate measurement discussed above, and the rate constant k was treated as a fitting parameter. This model yielded k = 1.62 ± 0.01 x 10 -3 s -1 . Including a rate constant for the reverse reaction did not improve the quality of the fit. We note that the measured rate constant is higher than has been previously reported . This may be due to a difference in the experimental conditions; specifically, a higher cell concentration was used in this work.