In creatinine kinetics, the glomerular filtration rate always moves the serum creatinine in the opposite direction

Abstract Introduction When the serum [creatinine] is changing, creatinine kinetics can still gauge the kidney function, and knowing the kinetic glomerular filtration rate (GFR) helps doctors take care of patients with renal failure. We wondered how the serum [creatinine] would respond if the kinetic GFR were tweaked. In every scenario, if the kinetic GFR decreased, the [creatinine] would increase, and vice versa. This opposing relationship was hypothesized to be universal. Methods Serum [creatinine] and kinetic GFR, along with other parameters, are described by a differential equation. We differentiated [creatinine] with respect to kinetic GFR to test if the two variables would change oppositely of each other, throughout the gamut of all allowable clinical values. To remove the discontinuities in the derivative, limits were solved. Results The derivative and its limits were comprehensively analyzed and proved to have a sign that is always negative, meaning that [creatinine] and kinetic GFR must indeed move in opposite directions. The derivative is bigger in absolute value at the higher end of the [creatinine] scale, where a small drop in the kinetic GFR can cause the [creatinine] to shoot upward, making acute kidney injury similar to chronic kidney disease in that regard. Conclusions All else being equal, a change in the kinetic GFR obligates the [creatinine] to change in the opposite direction. This does not negate the fact that an increasing [creatinine] can be compatible with a rising kinetic GFR, due to differences in how the time variable is treated.

is a decidedly qualitative art. Creatinine slope analysis became more quantitative when the science of creatinine kinetics was applied, a technique that has been called the kinetic glomerular filtration rate (GFR) (Chen, 2013(Chen, , 2018a(Chen, , 2018bChen & Chiaramonte, 2019). Now, the creatinine trajectory with all its slopes can be decoded into a kinetic GFR format that shows how the kidney function is evolving. Insight into the underlying kidney function by using the kinetic GFR has improved the diagnosis and treatment of patients who suffer from an acute kidney injury (AKI) (Bairy, 2020;Bairy et al., 2018;Dash et al., 2020;Dewitte et al., 2015;Endre et al., 2016;Khayat et al., 2019;Kwong et al., 2019;Latha et al., 2020;O'Sullivan & Doyle, 2017;Pianta et al., 2015;Yoshida et al., 2019). With creatinine kinetics being modeled mathematically, a question that naturally arises is how much does the [creatinine] change in response to a change in kinetic GFR? The answer can be found using derivatives.

| Kinetic GFR equation
Changes in the serum creatinine follow passively from changes in the GFR, so like cause and effect, the change in GFR precedes and actively drives the change in creatinine. The creatinine serves mostly as a surrogate to calculate the GFR. The [creatinine] is relatable to the GFR by a differential equation (Chen, 2018a;Chen & Chiaramonte, 2019). The rate of change in the creatinine amount is a function of the rate of creatinine coming in minus the rate of creatinine going out of its volume of distribution, which is total body water (TBW) (Chow, 1985;Edwards, 1959;Jones & Burnett, 1974;Pickering et al., 2013). Most of the gain comes from creatinine being generated by the muscles, the mass of which remains more or less the same in the short term, so that the rate of creatinine addition is usually taken to be a constant: Gen. Most of the creatinine loss occurs via the kidneys, so that the rate of creatinine excretion is GFR K ·[Cr] t , a product of the kinetic GFR and the serum [creatinine] at an instant in time. On the left side of the differential equation, the instantaneous rate of change in creatinine amount is d dt [Cr] t ⋅ V t , where [Cr] t is the serum [creatinine] at a given point in time, as before, and V t is the volume of distribution at the same point in time. Further, V t is allowed to vary at a steady rate to mimic the clinical reality that patients have many fluid inputs and outputs (I's/O's), like intravenous (IV) fluids and urine output. The net balance of the I's/O's divided by the time period over which they occur can be modeled as a constant rate of change in the creatinine's volume of distribution: ΔV Δt . Thus, volume as a function of time is V t = V 0 + ΔV Δt t, where V 0 is the initial volume at a time zero for each clinical time interval. The full differential equation can be written as: This first-order linear differential equation's solution, as previously published, is: where [Cr] 0 is the initial serum [creatinine] for each clinical time interval (Chen, 2018a;Chen & Chiaramonte, 2019). That is to say, the serum [creatinine] at a given time is equal to the initial [creatinine] plus a time-evolved portion of the spread between the initial [creatinine] and the [creatinine] reached at a new steady state if the kinetic GFR and volume change rate remained at those levels.
To show how the [creatinine] trajectory is shaped by the kinetic GFR and other variables, we can graph the [Cr] t for a typical occurrence of AKI. Let us say that a patient with a baseline [creatinine] of 1.0 mg/dL develops acute tubular injury, and his GFR drops to 10 mL/min and stays there. He receives copious amounts of IV fluids in an attempt to reverse the renal failure, so that the net of the I's/O's is +6 L in 24 h, or ΔV Δt = 0.25 L∕h. The creatinine generation rate is found by multiplying the baseline [creatinine] by its corresponding steady-state GFR: Gen = 1 mg∕dL × 90 mL∕min . His initial volume of distribution, V 0 , is taken to be TBW (Bjornsson, 1979), and a typical value is 42 L. The drop in GFR to a constant (average) value of 10 mL/min perturbs the steady state, so [creatinine] will change. We graphed the serum [creatinine] (y) versus kinetic GFR (x) to directly visualize the relationship between the two. If we imagine the tangent to this curve, it appears that the slope is consistently negative (Figure 1). We hypothesize that the tangential slope of [creatinine] versus kinetic GFR will always be negative.

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CHEN and CHIARAMONTE Next, the derivative of Putting it all together and using the product rule on Equation (2), we see that [Cr] t GFR K is:

| Limits
The derivative in Equation (3) has three special cases that threaten a division by zero: (1) ΔV Δt = 0, (2) GFR K = − ΔV Δt , and (3) GFR K = ΔV Δt = 0, which itself is a special case of (2). Each of these cases can be resolved by using limits. See Appendix for derivation details.
Of the three cases, this is the most important clinically, as a stable volume is common or at least is assumed when information on I's/O's is unavailable.

| Hypothesis plausibility
To see if our intuition is on the right track that [Cr]

| Rules for the proof
All of the variables have constraints on their values as imposed by clinical reality. The rules are: A. Creatinine must be positive: [Cr] t > 0 and [Cr] 0 > 0. B. Volume of distribution must be positive: V 0 > 0 and V 0 + ΔV Δt t > 0. C. Kinetic GFR is traditionally non-negative: GFR K ≥ 0. (3) as written in Equation (3). The kinetic GFR Equation (2) is graphed as GFR K on the x -axis and [Cr] t on the yaxis. The red curve slopes downward from left to right, and all of its tangent slopes appear to be negative. Two such tangent slopes at GFR K = 10 (orange dot) and GFR K = − 90 (green dot) were calculated by the The derivative slopes are negative, as expected, and the calculated lines (black dash for GFR K = 10 and purple dash for GFR K = − 90) do appear to be tangents. D. The creatinine generation rate must be positive: Gen > 0. E. Volume change rate can be positive or negative, but it cannot be so negative in magnitude that a patient would lose all of the volume of distribution in an allotted time In theory, a positive volume change rate has no upper limit on magnitude ΔV Δt < + ∞ .

| Proof of the hypothesis
The derivative, rearranged to emphasize the leading negative sign, is hypothesized to always be negative for clinically realistic values, that is, the expression in the curly brace must be positive.
The general strategy will be a proof by contradiction. We test whether the expression within the curly brace can be negative and prove that it cannot, in all six of the scenarios that are possible. Thus, the partial derivative is always negative (except when it is zero at t = 0, which is a trivial case). Due to its length, the complete proof is presented in the Appendix.

| Where is the derivative bigger?
An observation familiar to nephrologists is that a small change at the low end of the [creatinine] scale represents a big change in GFR, whereas a big change at the high end of the [creatinine] scale represents a small change in GFR. A typical anecdote is that an increase in the [creatinine] from 1.0 to 2.0 mg/dL is akin to a 50% decrease in the GFR. But the same absolute increase in the [creatinine] from 9.0 to 10.0 mg/dL is only a 10% drop in the GFR. The statements assume that each [creatinine] was more or less in a steady state, which models a slowly progressive chronic kidney disease (CKD). In CKD, the creatinine excretion rate is nearly equal to but is slightly less than the creatinine production rate, which is why the serum [creatinine] slowly increases over time. Since the creatinine production rate is relatively constant if the patient's muscle mass remains stable, then the clearance expression is more or less a constant divided by a [creatinine] that varies, i.e., a reciprocal function. On a graph of [Cr] versus GFR in CKD, at the low end of the [Cr] scale, a big loss of GFR (x-axis) is needed to raise the [Cr]  Do the slope lessons of CKD carry over into AKI? That is, does the AKI analog of Δ[Cr] ΔGFR , i.e., [Cr] t GFR K , follow the same pattern as before: (in magnitude)? Try a low initial [Cr] like 1.0 mg/dL. In steady state, that may correspond to a GFR of 90 mL/min if the creatinine generation rate were 90 mg/dL·mL/min. Let the kinetic GFR change from 90 mL/min and graph its effect on the [Cr] t , using Equation (2). For the other parameters of AKI, we kept t = 24 h and V 0 = 42 L and ΔV Δt = 0.25 L∕h. As seen in Figure 4, the tangent to the curve at GFR K = 90 mL∕min has a [Cr] t GFR K slope that is shallow ( − 0.009722536 mg∕dL per mL∕min). Now, try a high initial [Cr] like 9.0 mg/dL, which in steady state would correspond to a GFR of 10 mL/min. If we kept all other parameters the same, the tangent to this other (3) is graphed against GFR K as the independent variable. The red curve retains the miscellaneous clinical values as chosen in Figure 1. All of its yvalues curve at GFR K = 10 mL∕min has a [Cr] t GFR K slope that is steeper ( − 0.217521268 mg∕dL per mL∕min) ( Figure 4). This suggests that AKI recapitulates the behavior of CKD in terms of Δ[Cr] ΔGFR , albeit in a blunted way.

| Clinical significance
The consistently negative value for [Cr] t GFR K in the real world means that changes in kinetic GFR must always move in the opposite direction of changes in the serum [creatinine]. If the GFR goes up (even in the slightest), then more creatinine is excreted in that instant, and the [creatinine] must go down. Conversely, if the GFR goes down (even in the slightest), then less creatinine is excreted in that instant, and the [creatinine] must go up. Our proof of these assertions would affirm what most physicians intuitively sense about the relationship of GFR to the [creatinine]. Of course, this assumes that all of the other variables of creatinine kinetics like time and volume status remain constant, as is done when taking a partial derivative. All else being equal, a doctor can confidently know that the GFR can never change the [creatinine] in a parallel way. If at the 24-h mark the [creatinine] were hypothetically greater than the actual-a positive Δ [Cr]-, then the GFR would have to be lower-a negative ΔGFR K .

| Conundrum
One important lesson from applying the kinetic GFR to patient care was that an increasing [creatinine] does not always imply a decreasing GFR. Most of the time though, the two move in opposite directions, and this mental shortcut is used daily on the wards. However, there is a gray zone where some instances of a rising [creatinine] actually indicate an improving GFR. Let us say during an ΔGFR slopes in chronic kidney disease are also negative. In CKD, the [creatinine] (y-axis) is simply the Gen(in this case 90 mg/dL ⋅ mL/min) divided by the GFR (x-axis), a reciprocal function as shown by the red curve. At the low end of the [creatinine] scale, e.g., green dot ( ΔGFR slope is small in magnitude (−1/90, green dash line is tangent). At the high end of the [creatinine] scale, e.g., blue dot ([Cr] = 9 mg∕dL), the Δ[Cr] ΔGFR slope is much larger in magnitude (−0.9, blue dash line is tangent).

F I G U R E 4
To see if acute kidney injury might follow a similar pattern as chronic kidney disease, we visualized the steepness of the episode of AKI that a low GFR rises slightly. That would increase the creatinine excretion, but if that excretion remained less than the creatinine production, then the net positive balance would obligate the serum [creatinine] to rise still. It is not until the GFR increases by a sufficient amount to make the creatinine excretion equal to the creatinine production that the serum

| What if Gen is changing?
Differential Equation (1) is set up to assume that the creatinine generation rate, Gen, is constant. That may be true for the most part, but Gen could be decreased during a critical illness such as sepsis (Doi et al., 2009;Prowle et al., 2014). More than an academic exercise, this finding has important clinical ramifications. If less creatinine is being produced, then a [creatinine] trajectory may not rise as quickly, and that will mute the apparent severity of the GFR loss during AKI, for example. Or, it could make the [creatinine] trajectory fall more quickly during a renal recovery, making the GFR gain seem more robust than it really is. Accounting for a changing Gen can potentially improve the accuracy of the kinetic GFR calculations. Unfortunately, it is not known how Gen evolves over time in most of our critically ill patients. We can measure it to be decreased, but was the evolution a sudden drop, a gradual and linear drop, a logistic model drop, etc.? Until patient data are gathered, it may be acceptable to treat a changing Gen as a sudden drop to a new value that remains stably low throughout the critical illness. If the acute drop is completed within 24 h, then the kinetic GFR calculations are easy to adapt. Just use the reduced Gen, whatever it is estimated (or, better yet, measured) to be, in the kinetic spreadsheet at the onset of and for the duration of the critical illness. That said, a Gen drop that is linear, like how volume change is modeled, can be incorporated into our differential Equation (1) to yield a closed-form solution. However, its utility is limited to one or two rounds of calculation, since Gen cannot descend into negative values.

| Conclusion
Doctors need to estimate the kidney function in the nonsteady state to care for their patients who develop acute renal failure. The GFR affects most facets of diagnosis and therapy, and the most cost-effective way to estimate GFR at the bedside is to use the kinetic GFR equation (Endre et al., 2016;Khayat et al., 2019). Its math contains a lot of relationships that are waiting to be discovered.  [Cr] t with respect to kinetic GFR GFR K is: The following three cases of a division by zero represent removable discontinuities that can be resolved by using limits.
Case 1: ΔV Δt = 0. We will solve the limit piecewise, proceeding through the various components of To begin with, the exponential can be resolved as: The dominant term in the Taylor series for ln 1 + Now, let ΔV Δt go to zero, and the limit is seen to be: Another part of the derivative affected by ΔV . Again, using the dominant term in the Taylor series for ln , the limit becomes: The rest of the derivative in the limit as ΔV Δt approaches zero is straightforward: Although it is unlikely that the GFR K will equal the negative of the volume change rate, the limit is interesting to solve. Letting GFR K approach − ΔV Δt is equivalent to stating that GFR K = − ΔV Δt + h, and letting h approach zero. This substitution transforms [Cr] t GFR K into: We will solve the limit piecewise. First, find the Taylor expansion of the e 0 -like expression: Substitute in the e 0 -like Taylor expansion, and the first half of the limit becomes: Substitute in the e 0 -like Taylor expansion, and the second half of the limit becomes: Add the two halves of the limit together, and the problematic division by h → 0 is subtracted out: Now, let h go to zero, and the limit is seen to be: Case 3: GFR K = ΔV Δt = 0.
In the even more unlikely event that the kinetic GFR and the volume change rate are both zero, the dual limit approaching zero can be solved by at least two ways: (1). Equation (1) was the limit of [Cr] t GFR K as ΔV Δt → 0. Now, take the limit as GFR K → 0: The Taylor expansion of the e 0 -like term e The problematic division by GFR K → 0 is subtracted out: Now, let GFR K go to zero, and the limit is seen to be: (2). The same limit as above would be found if Equation (2)  Now, let ΔV Δt go to zero, and the limit is seen to be: The dual limits in Equations (3) and (4) are mathematically equivalent, which also validates the precursor limits in Equation (1): stable volume ΔV Δt = 0 and Equation (2): GFR K = − ΔV Δt . A fourth case of division by zero, V 0 + ΔV Δt t = 0, has a limit that is undefined, which is just as well since patients cannot have a zero volume.

Proof that [Cr] t GFR K is always negative
The derivative, rearranged to emphasize the leading negative sign, is hypothesized to always be negative for clinically realistic values, i.e., the expression in the curly brace must be positive.
The general strategy will be a proof by contradiction. We test whether the expression within the curly brace can be negative and prove that it cannot, in all six of the scenarios that are possible.
The curly brace's sum breaks up into two groups:

GROUP 1:
The ratio of initial-to-final volume Δt t must be positive, and a positive number raised to any real exponent is also positive. In , which makes its logarithm positive. Here, the product can take on either sign, it determines the overall sign of the three-term product in Group 1.

GROUP 2:
In Gen GFR K + ΔV Δt 2 , Gen is positive, and the square GFR K + ΔV Δt 2 is positive in the real numbers. [ Broadly, there are 3 ways to falsify the hypothesis by having a curly brace value that is negative. Referring to expression (5)-Group 1-and expression (6)-Group 2, the ways are: Analyze the more interesting bound on the exponent: 1 − ii. If ΔV Δt is positive, then Positive and the whole expression is positive, specifically that 0 < 1 − again.
In both cases, the 1 − was positive, but it needed to be negative. Therefore, Way #1 does not falsify the hypothesis.

As learned in 1, if
has to be positive. Together, they satisfy two out of the three conditions in 2. The 3rd condition, then, is key. . Substitute these in: (Note: ΔV Δt = − GFR K and ΔV Δt = GFR K = 0 are covered separately.)

Given that
Is this function always negative when x > 0 Differentiate the function: f � (x) = e x − a − 1 a . Because x > 0, then e x > 1. In contrast, because a > 1, then 0 < a − 1 a < 1. The subtraction yields an f � (x) that is positive, meaning that f (x) is increasing on x > 0. As x approaches zero from the right, f (x) must have a minimum of lim Thus, f (x) is always positive, but that contradicts f (x) = e x − a − 1 a ⋅ x − 1 needing to be negative, so Way #2 does not falsify the hypothesis.
In Group 2, . Substitute these in: Disregarding the trivial case of t = 0, term by term, the signs have to be − The negative sign out in front makes the entire derivative limit negative, which supports the hypothesis that [Cr] t GFR K