[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 (GFRK ), 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 GFRK . If GFRK is below a calculable value, then the curve's minimum vanishes, letting it descend to - ∞ and not allowing the derivative to ever be positive. If GFRK lies between a lower and a higher calculable value, then the curve's maximum vanishes, letting the derivative diverge to + ∞ , though the clinical scenario is unrealistic. If GFRK 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 GFRK ). 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 GFRK that excretes creatinine faster than its volume of distribution declines.

INTRODUCTION

We previously showed that changes in the glomerular filtration rate (GFR) must always drive the serum creatinine in the opposite direction (Chen & Chiaramonte, 2021). If the GFR were to decrease, then the creatinine would have to increase, and vice versa. Though these statements seem obvious, they were only recently proved by differentiating the creatinine with respect to kinetic GFR (Chen, 2013) and finding that this partial derivative's sign is always negative for any possible set of real‐world values. It does not matter how extreme the variables are or how they are combined. The perpetual negative sign assures doctors that a rise in creatinine (positive creatinine) had to come about from a drop in kinetic GFR (negative GFRK) and vice versa, with all else being constant—particularly time.

Is creatinine also related to the volume change rate in an ever‐opposing way? If more volume is being gained (positive volume rate), the [creatinine] is further diluted and is negative. If more volume is being lost (negative volume rate), the [creatinine] is further concentrated and is positive. With the signs being opposite in our thought experiments, it would appear that the partial derivative of with respect to volume change rate is always negative, same as for kinetic GFR (Chen & Chiaramonte, 2021). But could unsuspected factors alter the sign? What if a negative volume rate concentrates the [creatinine] even further, but that enables the kinetic GFR to excrete more creatinine? Would the creatinine quantity decline faster than the volume of distribution, making the creatinine concentration fraction lower in value—a negative ? If so, the partial derivative would be positive, with volume rate and decreasing over the same time frame. The potential for a positive sign brings up a clinical paradox. Sometimes, being more aggressive with the volume removal may improve the serum creatinine (Hegde, 2020). Or, giving even more volume may increase the creatinine. These paradoxes can occur in septic patients on continuous dialysis, for purely mathematical reasons to be shown that need not involve the messiness of real life, which is more complex than the derivative model that assumes that only volume and [creatinine] can change.

Creatinine kinetics

The differential equation that underpins the kinetic GFR states that the rate of change in the creatinine mass is equal to the creatinine input rate minus the creatinine output rate (Chen, 2018a, 2018b; Chen & Chiaramonte, 2019). Further, the creatinine mass at any given time is the current [creatinine] times the volume of distribution, typically taken to be total body water (TBW) (Bjornsson, 1979; Chow, 1985; Edwards, 1959; Jones & Burnett, 1974; Pickering et al., 2013). To account for the concentrating and diluting effects on the [creatinine], its volume of distribution can be modeled to change at a constant rate: , where is the volume as a function of time, is the initial volume, is the (average) volume change rate, and is time. The creatinine input rate is primarily determined by the muscle mass, which tends to be fairly stable so that the creatinine generation rate is usually thought of as a constant: . The creatinine output rate is mostly determined by the kidney such that the excretion rate is equal to the kinetic GFR times the ambient [creatinine]: , where is the [creatinine] at a particular time. Thus, the differential equation is:

This first‐order linear differential equation's solution, as previously published, is (Chen, 2018a):

In other words, 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.

MATERIALS AND METHODS

Derivative of [creatinine] with respect to volume change rate

From Equation (2), we can deduce how the serum creatinine would change if one other variable were tweaked, and the partial derivative is suited to this task. Previously, the one other variable was kinetic GFR (Chen & Chiaramonte, 2021), but now the one other variable will be the volume change rate. The derivative of with respect to quantifies their relationship at every instant, allowing a comprehensive assessment of the sign. If the sign can be positive, then may change in the same direction as .

In Equation (2), the derivative of with respect to is:

To calculate the derivative of the exponential, let and use logarithms.

Differentiate with the product rule:

Next, in Equation (2), find the derivative of with respect to :

Putting Equations (3) and (4) together in using the product rule on Equation (2), we find that is:

Note: Unit conversions are not shown, but the final units could be , for example. The main conversion factors are and . The converts in L/h to ml/min: , and the converts in ml/min to L/h: .

Calculator and concept map

To follow the calculations in the Results, please download a spreadsheet we created to calculate the main equations in the manuscript. You can use the spreadsheet to explore your own scenarios and questions. For a map of the concepts being presented, the final algorithm in Section 3.8 may help with understanding when the first derivative in Equation (5) can be positive. First, and will be varied (Section 3.5), as their ratio is a principal determinant of positivity. Later, will also be varied (Section 3.6), as values above a calculable reference point can allow the first derivative to be positive in situations that are clinically encountered.

RESULTS

First derivative behavior and sign

To gauge the behavior and sign of the partial first derivative, we graphed (‐axis) vs. (‐axis) for an acute kidney injury (AKI): steady state GFR of 100 ml/min corresponding to an initial of 1.0 mg/dl that increases over the next 24 hours when the suddenly drops to 20 ml/min in a patient with a TBW (volume of distribution) of 42 L. According to the thought experiments, the sign of should be negative (below ) throughout the gamut of values (Figure 1). As approaches an extreme that would deplete all of the TBW by the 24‐h mark (in this case), the goes to (Figure 1). A volume of zero is nonsensical, so if needs to be , then must be . At the far right of the graph, as , the value of approaches zero from below, that is, (Figure 1). So far, seems to always be negative.

FIGURE 1

Example first derivative of [creatinine] with respect to volume change rate. Equation (5) is graphed with as an independent variable (‐axis) and as the dependent variable (‐axis). The other variables are mg/dLmL/min, L, h, mL/min, and mg/dL. From left to right, the starts at where approaches its leftmost value of . The curve rises quickly but remains negative. As increases, the flattens out as it approaches zero asymptotically. The first derivative is always negative in this example of acute kidney injury

Despite the thought experiment, can become positive under certain conditions? To find out, we varied the parameters and found conditions that work: creatinine generation rate of 60 mg/dl ml/min and of 8.0 mg/dl that decreases over the next 24 h when the suddenly increases to 100 ml/min (e.g., by renal replacement therapy) in a patient with a TBW of 42 L. The stays positive for most of the negative values and for even a few positive values (Figure 2, red line, 3 gray dots). Further, decreasing keeps positive for a wider range of s and makes the peak at a higher positive level (Figure 2, green curve). On the other hand, increasing the lowers the values, until a large enough make the persistently negative for all values (Figure 2, blue & black curves). Alternatively, the curves can be moved up or down by varying the (Figure 3). In general, is more likely to be positive if is small and is high; it helps if is larger and is negative. This family of curves has an absolute maximum. If we can find the curve whose maximum lies tangent to , that represents the border between a first derivative being perpetually negative versus potentially positive. In Figure 2, the (blue) curve comes closest to touching . Its maximum is , but we can place the peak right at 0.

FIGURE 2

Graphs of first derivative curves when varying only the . Equation (5) is graphed like before. Other variables are L, h, mL/min, and mg/dL. The is varied between 10 and 100 mg/dLmL/min. The smallest yields the highest curve (green). As the increases, the curves move downward, until yields the lowest curve (black). Above , the curves are wholly below the ‐axis, meaning that all of their values are negative. But, one other curve (red) is partially above the ‐axis, meaning that some of its values are positive. A positive is promoted by a that is on the smaller side

FIGURE 3

Graphs of first derivative curves when varying only the . Like Figure 2, the graphs of the first derivative (‐axis) vs. (‐axis) shift up or down depending on the initial [creatinine]. The fixed variables are L, h, mL/min, mg/dLmL/min, while the increases from 2 to 5 to 7 to –10 mg/dL. All of the curves are anchored to the same leftmost and point. From there, they take different paths with the bottommost curve arising from (black) and the uppermost one arising from (red). Some curves stay completely below the ‐axis, so their first derivatives are always negative. Some curves rise above the ‐axis for short stretches, after the gets to about 7 (blue), so their first derivatives are positive at times. A positive is fostered by a that is on the larger side

First derivative's peak

To calculate the peak of the vs. curve, we differentiated with respect to and then set this second derivative equal to zero. Without showing the differentiation steps, we calculated the second derivative to be:

We set and solved for by Newton's method or the secant method. At one root, our example first derivative (Section 3.1, second paragraph) attains its maximum and is positive. On either side of the root, the values are decreasing. At the other root, our first derivative has a relative minimum. At the left extreme, truncates the plot (Figure 2), because becomes a complex number: in Equation (5), once turns negative, then is a negative base raised to a non‐integer power. At the right extreme, makes approach zero in the limit. Overall, solving yields a single maximum for , one that happens to be absolute, and a single minimum for , one that is relative.

Making the first derivative's peak tangent to the ‐axis

Setting the second derivative equal to zero optimizes the first derivative, but the first derivative's absolute maximum is not necessarily zero. To find a curve whose maximum is tangent to , we devised a way to make both the first derivative and the second derivative equal to zero at the same time. In doing so, we find the transition to being potentially positive (the curve came close in Figure 2). To solve the simultaneous equations, and , we used algebraic substitution.

In the first and second derivatives [Equations (5) and (6)], only two variables can be explicitly solved for, namely and . Set the first derivative equal to zero and solve for :

Substitute this , arising from , in place of the from . After a lot of algebra, the key to the simultaneous equations is to solve:

In Equation (8), we supply values for , , and and then calculate using a root‐finding method. At that , the peak of the vs. curve will touch the ‐axis from below. However, Equation (8) does not contain either or . To find those values, we refer back to the first derivative equaling zero. When , Equation (7) yields . We just have to supply a value for and be sure to use the newly calculated , not the patient's actual . Or, if is known, as measured by the laboratory, then a rearrangement of yields :

Testing if the peak is tangent to the ‐axis

Equation (8) reveals how at its maximum can equal zero. From Figure 2, plug ml/min, L, and h into Equation (8). Use a root‐finding method to determine that … L/h. Figure 2 had a uniform of 8 mg/dL. Plug that into Equation (9) to find that … This is the value, not , that places the ’s absolute maximum on the ‐axis, exactly. Alternatively, plug into Equation (7) to find that a … would have also placed the curve's peak on the ‐axis. Any combination, really, of and would work as long as the ratio is … ml/min (in this case). Broadly, the ratio is a fixed attribute for a set of , , and inputs that allows the first and second derivatives to equal zero simultaneously.

and effects: lifting the peak into positive territory

Now that the peak can be positioned at the ‐axis, how can the be lifted above the ‐axis? The , , and are initial data, and is the independent variable on the graph. That leaves only and to be manipulated. Using the fixed ratio as a benchmark, we find that lower ratios shift the curve partially into positive territory, in keeping with the observation that smaller s and/or bigger s promote being positive. In practice, one can calculate by Equation (7), for example, and then ask if the patient's actual is larger, which lets be positive at times. Or, one can calculate the benchmark by Equation (9) and then ask if the patient's actual is smaller, which also permits to be positive.

effect: variant way for to be positive

In Figures 2 and 3, the stereotypical shape of the vs. curve, from left to right, is that rises from a negative value to peak at an absolute maximum which can be positive, then falls to a relative minimum (mentioned in Section 3.2) that is negative, and then asymptotically increases toward . The curve is shifted vertically, more or less, by varying the or . Well, the curve is shifted horizontally, mostly, by varying the . A higher pulls the curve rightward, and a lower pushes it leftward. Also, imagine that the left end of the curve is tethered to an invisible wall at but has the ability to slide up or down that wall. Then, a right shift would stretch the curve, flattening it out, and a left shift would compress the curve, bunching it up against the wall in an orderly way by making it bend and stack in layers (with no thickness). Can the be lowered sufficiently to left‐shift the absolute maximum until it is located at the leftmost , that is, ? Going further, can the left shift continue until the relative minimum is then pressed up against the leftmost wall? If so, these max/min at the leftmost would correspond to a second derivative equaling zero at two roots, one for the max and one for the min.

As is reduced, the curve acts like a rope being pushed leftward against a wall, based on tracking the maximum and minimum points and the sliding along the wall. In response to the push, the endpoint at the leftmost moves down, the maximum moves up, and the minimum moves down, like how a rope could fold to be more compact (Figure 4a). In addition to the vertical motions, the max/min points move horizontally to the left. Once the is lowered to ~58.34 (in this example), the bend at the maximum is very sharp and the maximum is left‐shifted all the way to (Figure 4b). As the is lowered some more, the minimum continues to move down and left but the absolute maximum is transitioned into the left endpoint of the curve sliding up the wall, on its way to (Figure 4c). In this way, certain s can enforce a positive .

FIGURE 4

Decreasing pushes the first derivative curve leftward along the ‐axis. Variables in common are mg/dLmL/min, mg/dL, L, and h. (a) As the decreases from 90 (red) to 80 (blue) to 70 (green) mL/min, the curve looks like it is being pushed to the left and is bending in the process. The maximum moves steadily up, the minimum moves down, and both of them move to the left. Also, the left endpoint slides down a virtual wall at the leftmost . (b) When decreases to 58.34 , the maximum has been pushed to the leftmost , and only a short tail to the left of the maximum is decreasing before it gets truncated at the wall (inset). (c) As decreases below 58.34, the maximum vanishes (blue) and transitions into a left tail that blows up to (red). (d) With no more maximum, the minimum is the sole critical point, and it continues to move down and left as the decreases further from 57 (red) to 50 (orange) mL/min. When the drops to 36.67 , the minimum has been pushed to the leftmost , and the left tail still diverges to (purple). (e) When decreases below 36.67, the minimum vanishes (purple) and transitions into a left tail that plunges to (red). From here, the first derivative is always negative

effect: keeping negative

As is further reduced, with no more sliding down the wall for now, the relative minimum becomes an absolute minimum (Figure 4c). As the reduction keeps pushing the curve/rope to the left against a wall, the bend gets sharper and the minimum moves even more to the left and down. When the gets down to ~36.67 (in this example), the minimum is left‐shifted all the way to the leftmost (Figure 4d), like the maximum was earlier. As is lowered past ~36.67, the absolute minimum is transitioned into the left endpoint of the curve sliding down the wall, on its way to (Figure 4e). After this transition, the will always be negative.

For details on how the kinetic GFR can alter the shape of the first derivative curve and help determine whether can be positive, please see the Appendix.

Algorithm to determine if can be positive

If all variables have allowable values , one way to detect potential positivity of is to compile the lessons above into an algorithm. If the roots are in a “permissive” order of , permitting to be positive, then:

For (bottom domain), the first derivative will always be negative. See A1.3.

For (middle domain), the first derivative has an absolute minimum and the left‐sided tail can diverge to at the leftmost , with exceptions (see A1.2).

For (top domain), the first derivative has an absolute maximum, which can be positive (see Section 3.2).

Calculate by a root‐finding method the at which the vs. curve is tangent to the ‐axis at its absolute maximum, that is, solve Equation (8) for (see Section 3.3).

Plug that and a known into Equation (7) to calculate a benchmark (Section 3.5).

If the patient's is greater than the benchmark , then the absolute maximum lies above the ‐axis and can be positive.

If the patient's is less than the benchmark , then the absolute maximum lies below the ‐axis and is always negative.

Alternatively, plug the from step c., i. and a known into Equation (9) to calculate a benchmark (see Section 3.5).

If the patient's is less than the benchmark , then the absolute maximum lies above the ‐axis and can be positive.

If the patient's is greater than the benchmark , then the absolute maximum lies below the ‐axis and is always negative.

To know if is positive at the patient's actual , not the calculated above, plug all of the patient's variables into Equation (5) and note the sign. One can also find the spread of values that yield a positive by a root‐finding method. Vary the initial guess to find both roots.

DISCUSSION

Positive paradox possible?

The positive paradox is fostered by the combination of a low creatinine generation rate and a high initial creatinine. The two conditions are not mutually exclusive but they are at odds with one another, making the combination rare but not impossible. For a low to be paired with a high , renal failure probably had to be sustained for a while. To permit to be positive, the has to be at least and preferably . The relatively high is going to decrease the [creatinine] over time . Though it is decreased overall, can decrease less due to a volume rate increase? Then the would be comparatively increased. Or, can decrease more due to a volume rate decrease? Then the would be comparatively decreased. Either scenario is compatible with a that is positive in sign. But what kind of patient fits the criteria of low , high , and a relatively high ? One plausible patient may have suffered sepsis that temporarily reduced the (Doi et al., 2009; Prowle et al., 2014). Sepsis may have also caused kidney failure, so the [creatinine] went fairly high. Doctors then initiated continuous renal replacement therapy (CRRT) that provided a greater than . ( here is not used in the literal sense of clearance done by the glomerulus. Rather, it is used in the broader sense of clearance done by any means, including extracorporeal).

Paradox by the numbers

The abstract math may be easier to grasp if we put some concrete numbers on it. Suppose that a septic patient now has a mg/dl ml/min. He develops acute tubular necrosis and the creatinine rises to 8 mg/dl. CRRT is started, and the total ml/min. The combination of conditions seems ripe for a positive , so the algorithm is consulted. First, the falls into the top domain, since it is , assuming his volume (TBW) is 42 L and the time interval is going to be 24 h. The top domain implies that the vs. curve will have an absolute maximum. To know where the maximum is tangent to the ‐axis, Equation (8) is solved by a root‐finding method to yield a … Plug that and the into Equation (9) to calculate a benchmark of 78.42… (Alternatively, plug that and the into Equation (7) to calculate a benchmark of 4.08…) The patient's of 40 is less than the benchmark , so the absolute maximum lies above the ‐axis. (Alternatively, the patient's of 8 is greater than the benchmark , and again the absolute maximum lies above the ‐axis). If the maximum is positive, then stays positive over a spread of values. The sign says that changes in move in the same direction as changes in .

Effect size

Say that the CRRT ultrafiltration (UF)—volume removal—rate is turned up from 100 to 300 ml/h, that is, the goes from to L/h, making the negative. At those two s, the first derivative is positive ( to mg/dl per L/h). That forces the to be negative. By Equation (2), the goes from 0.982… to 0.978… mg/dl, a decrease that represents a negative as advertised. Certainly, the change in [creatinine] is small, as predicted by the small . Importantly, the positive sign assures the nephrologist that turning up the UF rate will actually improve the next day's [creatinine]. One might posit that the improvement is due to the higher UF rate increasing convective clearance (Tandukar & Palevsky, 2019), but the math disproves that by holding the constant. Besides, turning up the UF rate will worsen the next day's [creatinine] if the is negative, so convective clearance does not always match with the [creatinine] trajectory.

Come‐from‐behind win: getting to a lower [creatinine]

By itself, volume loss should concentrate and thereby increase the [creatinine]. Somehow, this concentration effect is overridden by a creatinine‐lowering effect. In 24 h, L/h got to a lower [creatinine] than L/h. Having L/h would seem like a handicap, because removing more volume concentrates the [creatinine] and resists the that is trying to lower the [creatinine]. Thus, the (Figure 5, blue curve) has a higher [creatinine] than the (Figure 5, red curve) at almost all time points. After about 5.7 h, however, the blue curve starts to catch up to the red curve (Figure 5), which is peculiar as the two s have not changed. Apparently, concentrating the [creatinine] can be advantageous when the higher interacts with the steady to excrete more creatinine mass. That lowers the total creatinine (numerator) faster than its volume (denominator), such that the creatinine concentration starts to decline more quickly. The blue curve catches up to the red curve at ~22 h (Figure 5). Then, the blue curve barely edges out the red curve at the 24‐h mark (Figure 5, see inset), meaning that the higher UF rate came from behind to get to a lower [creatinine]. Despite the concentration disadvantage for most of the race, the higher UF rate's latent factor that slowly predominated was a synergy between the and the to boost creatinine excretion.

FIGURE 5

Positive derivative paradox viewed as evolution of [creatinine]. This paradox can happen in a septic patient with kidney failure on continuous dialysis. Raise the ultrafiltration rate from to L/h, which is a negative , and if the first derivative is positive, then tomorrow's [creatinine] will be further decreased, which is a negative . The fixed variables are L, mg/dLmL/min, mg/dL, and mL/min. Then, Equation (2) is graphed as (‐axis) versus time (‐axis). Seen as the evolution of , the red curve shows the effect of a baseline L/h, while the blue curve shows the effect of a L/h. Predictably, both [creatinine] curves decrease over time due to the relatively high of 80 mL/min. But the blue curve declines more slowly, because its greater volume removal will concentrate the more. As time goes by, the blue curve catches up to the red curve at about 22 h. After that, blue surpasses red and gets to a lower at the 24‐h mark (see inset), consistent with the first derivative being positive

Volume gain can increase the [creatinine]

In the same clinical example, the stays positive briefly into the positive zone. If volume is given , could that increase the [creatinine]? Yes. If the UF is turned off and CRRT is used to give volume, let us say that increases from to 80 ml/h. The is certainly positive. The remains positive. That forces to be positive too. In a race between and L/h, the [creatinine] at 24 h is 0.982… vs. 0.983… mg/dl, respectively. Counterintuitively, giving volume resulted in a higher [creatinine] than continuing the UF. The explanation is similar to before. The baseline UF rate , by virtue of the concentration effect, lags behind in lowering the . Meanwhile, volume gain is diluting the and helping the . Because the UF has a higher that is subjected to a relatively high for most of the race, more creatinine is excreted that eventually lowers the [creatinine] further versus a gain of volume, even with the latter's dilution effect advantage. So, the creatinine‐lowering effect that overcomes the volume effect is facilitated by a higher , which explains why the should be to get a positive .

Reality check

What if the is in the middle domain (see Section 3.8, b.)? That gives the curve an absolute minimum, and the tail to the left can be positive, maybe even going to . Unfortunately, obtaining a positive first derivative this way is clinically unrealistic. The is usually so negative that it would dry up nearly all of the TBW within an allotted time, killing the patient. Realistically, all of the positive first derivatives in medicine come from a being in the top domain of .

Big picture

A positive paradox may not happen all that often, but it is a real mathematical phenomenon that can occur under the right circumstances, especially in septic patients who have become quite azotemic and are being initiated on CRRT, a not uncommon scenario. In those cases, clinicians may want to pay attention to the CRRT volume settings. Turning up the UF rate, that is, decreasing the , can lower the a little more. On the other hand, turning down the UF rate (or giving volume), that is, increasing the , can raise the a little. This counterintuitive improvement or worsening of [creatinine] is marginal at best and pales in comparison to the overall effect that CRRT exerts on the [creatinine] trajectory. In addition, the paradox goes unnoticed because one patient cannot experience two separate rates to yield two s for comparison.

Most patients will not be at risk for a positive paradox. The combination of low , high , and high‐ish is rare. The is in many instances of AKI, so those patients are protected from a paradox and likely will behave as expected in response to fluids or diuresis. Milder cases of AKI can have a that lies between and , which does permit a paradox but only under a ludicrous rate of volume loss that is clinically unrealistic. If the is high‐ish enough to be , the paradox, if it occurs, alters in a negligible way. Finally, more than just the changes in clinical practice, so if the paradox seems to occur, it may be due to the other variables changing and confounding the picture. With all that said, we think the possibility of a positive is intellectually enlightening, and it differs markedly from the that was proved to always be negative (Chen & Chiaramonte, 2021).

CONFLICT OF INTEREST

The authors have no conflicts of interest.

AUTHOR CONTRIBUTION

Sheldon Chen: conception and design of study, mathematical derivations and equation graphs, interpretation of data, writing and revision of manuscript, and final approval of the manuscript. Robert Chiaramonte: confirmation of mathematical derivations and equation graphs, interpretation of data, revision, and final approval of the manuscript.