[Creatinine] can change in an unexpected direction due to the volume change rate that interacts with kinetic GFR: Potentially positive paradox

[Creatinine] was proved to change in the opposite direction of the kinetic GFR (GFR(K)), but does the [creatinine] also change in the opposite direction of the volume rate? If volume is administered and the [creatinine] actually goes up, then the two changes move in the same direction and their ratio is positive, paradoxically. The equation that describes [creatinine] as a function of time was differentiated with respect to the volume rate. This partial first derivative has a global maximum that can be positive under definable conditions. Knowing what makes the maximum positive informs when the derivative will be positive over some continuous domain of volume rate inputs. The first derivative versus volume rate curve has a maximum and a minimum point depending on the GFR(K). If GFR(K) is below a calculable value, then the curve's minimum vanishes, letting it descend to -infinity and not allowing the derivative to ever be positive. If GFR(K) lies between a lower and a higher calculable value, then the curve's maximum vanishes, letting the derivative diverge to +infinity, though the clinical scenario is unrealistic. If GFR(K) is above the higher calculable value, then the curve's absolute maximum can become positive by decreasing the creatinine generation rate or increasing the initial [creatinine]. The derivative is potentially positive under these clinically realizable circumstances. The combination of parameters above can align in septic patients (low creatinine generation rate) with kidney failure (high initial [creatinine]) who are put on continuous dialysis (high GFR(K)). If a first derivative is positive, removing more volume can improve the [creatinine] and, dismayingly, giving more volume can worsen the [creatinine]. This paradox is explained by a covert interplay between the ambient [creatinine] and GFR(K) that excretes creatinine faster than its volume of distribution declines.