What is the input signal that causes insulin to be released from a beta cell?

The kidney filters potentially toxic substances in the blood, and thus “clears” the blood of those substances. this clearance function is dependent upon and proportional to the diffusion gradient of the substance across filtering capillaries, i.e. if the concentration of the substance is doubled, twice as much will be cleared from each ml of blood that is filtered. suppose that the body produces a constant amount of a substance x per unit of time. the kidneys eliminate substance x at a rate directly proportional to the concentration of the substance and the volume of blood cleared each minute (c): elimination = c × [x], where [x] is the steady-state concentration of substance x. imagine an individual with an initial concentration of x equal to [x]0 who develops kidney disease. her baseline clearance c0 drops to one half of the original (½c0). what is the new steady state concentration of x? (for simplicity, assume that substance x is 100% filtered by the kidney).