Ratio Profile: Physiologic Approach to Estimating Appropriate Intravenous Fluid Rate to Manage Hyponatremia in the Syndrome of Inappropriate Antidiuresis

Abstract

A hyponatremic patient with the syndrome of inappropriate antidiuresis (SIAD) gets normal saline (NS), and the plasma sodium decreases, paradoxically. To explain, desalination is often invoked: if urine is more concentrated than NS, the fluid’s salts are excreted while some water is reabsorbed, exacerbating hyponatremia. But comparing concentrations can be deceiving. They should be converted to quantities because mass balance is key to unlocking the paradox. The [sodium] equation can legitimately be used to track all of the sodium, potassium, and water entering and leaving the body. Each input or output “module” can be counterbalanced by a chosen iv fluid so that the plasma sodium stays stable. This equipoise is expressed in terms of the iv fluid’s infusion rate, an easy calculation called the ratio profile. Knowing the infusion rate that maintains steady state, we can prescribe the iv fluid at a faster rate in order to raise the plasma sodium. Rates less than the ratio profile may risk a paradox, which essentially is caused by an iv fluid underdosing. Selecting an iv fluid that is more concentrated than urine is not enough to prevent paradoxes; even 3% saline can be underdosed. Drinking water adds to the ratio profile and is underestimated in its ability to provoke a paradox. In conclusion, the quantitative approach demystifies the paradoxical worsening of hyponatremia in SIAD and offers a prescriptive guide to keep the paradox from happening. The ratio profile method is objective and quickly deployable on rounds, where it may change patient management for the better.

Introduction

One patient with hyponatremia at a plasma sodium of 120 mEq/L is treated with normal saline (NS) at 50 ml/h, and the next day’s sodium goes up to 127 mEq/L—a good correction rate (1). Another patient with hyponatremia, also at a sodium of 120 mEq/L, is also treated with NS at 50 ml/h. The next day, the sodium drops to 117 mEq/L, and the team is confused by the paradoxical response (2). What is the difference between the two patients? Nephrologists blame the paradoxical worsening of hyponatremia on the syndrome of inappropriate antidiuresis (SIAD) (3,4), but does SIAD always lead to a fall in the plasma sodium if NS is given? Can NS be given at a fast enough rate to overcome the SIAD and actually raise the sodium? In turn, can SIAD overcome 3% saline, which is about three times as concentrated as NS, and cause a paradox? We will address these questions by discussing the fundamentals of sodium physiology, explaining how SIAD can lead to a paradox, knowing when a patient is at risk, and formulating a therapy to avoid the paradox.

Fundamentals

The plasma sodium concentration is determined by three parameters, primarily, according to the Edelman equation: total body exchangeable sodium, total body exchangeable potassium, and total body water (TBW) (5). Add up the first two parameters (quantities, usually in mEq) and divide by the third (a volume, usually in liters) to get the plasma [sodium]:

$${\left[\text{Na}\right]}_{\text{p}}=\frac{\stackrel{\text{Total Na}+\text{K amount}}{\overbrace{{\text{Na}}_{\text{TB}}+{\text{K}}_{\text{TB}}}}}{\text{TBW}}$$

a simplified formula that rounds the slope to 1 and y intercept to 0. To infer the total Na+K mass, we can compute [Na]_p×TBW. In a sense, the Edelman equation is describing the effect of osmosis to equilibrate the concentrations of the two major tonic cations: sodium in the extracellular fluid and potassium in the intracellular fluid, as weighted by the sizes of their respective compartments (6). The extracellular space is about one-third of TBW, whereas the intracellular space is about two-thirds of TBW. So, the intracellular [K] is not budged as much by a perturbation like the infusion of an intravenous (iv) fluid.

Normal Saline Must Raise Plasma Sodium

Can a more concentrated salt solution of 154 mEq/L lower a hyponatremic [sodium]? If an iv fluid is infused in isolation—no other inputs or outputs—the Adrogué–Madias formula can be used to predict the change in plasma sodium (7). Using the corrected Adrogué–Madias formula, we observe that the Δ[Na] must always be positive (8):

$$\text{Δ}\left[\text{Na}\right]=\frac{{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}-{\left[\text{Na}\right]}_{\text{p}}}{\text{TBW}+V}\cdot V$$

In the numerator, any isotonic iv fluid such as NS is more concentrated than any plasma sodium during hyponatremia, by definition, so the difference is positive. The denominator of TBW plus added iv fluid volume (V) is strictly positive. Multiply by V, which is also positive, and the overall Δ[Na] has to be positive. This sign implies that Δ[Na] must be an increase. If NS by itself can only raise the [sodium], then how does the paradox happen?

Kidney in Severe SIAD Must Lower Plasma Sodium

Besides iv fluids, the kidney also influences the plasma sodium. Water is retained in SIAD, so this volume expansion causes the kidney to excrete sodium and potassium in an inappropriate natriuresis. The arginine vasopressin-induced electrolyte-free water retention and the cationic solute loss begets hyponatremia. Producing urine that is more concentrated than plasma, i.e., [Na+K]_u>[Na]_p, can occur in SIAD, and then hyponatremia will get worse due to renal actions alone. When this renal effect outweighs the NS effect above, the plasma sodium falls, making it appear as if NS were exacerbating hyponatremia, when in fact it was due to the kidney.

Edelman Accounting Tool

How do we determine which effect will predominate, and what will the new plasma sodium be? These questions can be answered by the Edelman equation. The numerator tracks all of the cationic solutes, and the denominator tracks all of the water (6,9). The tracking version of the Edelman equation can be conceptualized as

$${\left[\text{Na}\right]}_{\text{new}}=\frac{\text{Initial Na}+\text{K amount}+\text{Na}+\text{K solutes gained by IV fluid}-\text{Na}+\text{K solutes lost by urine}}{\text{Initial TBW}+\text{Volume gained by IV fluid}-\text{Volume lost by urine}}$$

For example, if a liter of NS is given to a patient, then her numerator increases by 154 mEq, and the denominator increases by 1 L. Over the same time, if she produces 0.5 L of urine with a [Na+K]_u=100 mEq/L, then her numerator decreases by 50 mEq, and the denominator decreases by 0.5 L. Combine all of the input and output data with initial conditions, and the Edelman equation can predict, in theory, the new plasma sodium. Let us say that she started with a plasma sodium of 140 mEq/L and had a TBW of 42 L. Then, the equation would be calculated as:

$$\frac{\stackrel{\text{Initial Na}+\text{K amount}}{\overbrace{140\cdot 42}}+\stackrel{\text{Na}+\text{K in NS}}{\overbrace{154}}-\stackrel{\text{Na}+\text{K in urine}}{\overbrace{50}}}{\underset{\text{Initial TBW}}{\underbrace{42}}+\underset{\text{Volume of NS}}{\underbrace{1}}-\underset{\text{Volume of urine}}{\underbrace{0.5}}}=140.8$$ mEq/L.

The Δ[Na]_p is $+0.8$ mEq/L, which can be verified by the Barsoum–Levine Equation (10):

$$\text{Δ}{\left[\text{Na}\right]}_{\text{p}}=\frac{{\text{V}}_{\text{IVF}}\cdot {\left[\text{Na}+\text{K}\right]}_{\text{IVF}}-{\text{V}}_{\text{u}}\cdot {\left[\text{Na}+\text{K}\right]}_{\text{u}}-\text{ΔV}\cdot {\left[\text{Na}\right]}_0}{\text{TBW}+\text{ΔV}}=\frac{\stackrel{{\text{Vol}}_{\text{NS}}}{\overbrace{1}}\cdot \stackrel{{\left[\text{Na}+\text{K}\right]}_{\text{NS}}}{\overbrace{154}}-\stackrel{{\text{Vol}}_{\text{u}}}{\overbrace{0.5}}\cdot \stackrel{{\left[\text{Na}+\text{K}\right]}_{\text{u}}}{\overbrace{100}}-\stackrel{\text{ΔVol}}{\overbrace{0.5}}\cdot \stackrel{\text{Initial}\left[\text{Na}\right]}{\overbrace{140}}}{\underset{\text{Initial TBW}}{\underbrace{42}}+\underset{\text{ΔVol}}{\underbrace{0.5}}}=0.8.$$

All of the practical sodium equations exploit the principle that all gains and losses (iv fluids, oral intake, urine, stool, etc.) can be separated into their Na/K/H₂O components and sorted into their proper place in the Edelman fraction (6,8,9,11,12).

When the iv fluid and urine are continuous, then time is introduced into the Edelman equation. If a patient with SIAD gets NS at 30 ml/h (0.03 L/h) and concurrently has a urine [Na+K] of 140 mEq/L and a urine output of 3 L per day, then knowing (or at least estimating) a few initial conditions is sufficient to predict the plasma sodium. Say the patient starts with a [Na]_p of 120 mEq/L and a TBW of 42 L. After 24 hours, the Edelman accounting tool would yield:

$${\left[\text{Na}\right]}_{24\hspace{0.17em}\text{h}}=\frac{\stackrel{\text{Initial Na}+\text{K amount}}{\overbrace{120\cdot 42}}-\stackrel{\text{Na}+\text{K in}3\text{L of urine}}{\overbrace{140\cdot 3}}+\stackrel{\text{Na}+\text{K in NS at}30\text{mL/h for}24\text{h}}{\overbrace{154\cdot 0.03\cdot 24}}}{\underset{\text{Initial TBW}}{\underbrace{42}}-\underset{\text{Volume of urine}}{\underbrace{3}}+\underset{\text{Volume of NS}}{\underbrace{0.03\cdot 24}}}\approx 119$$ mEq/L

Notably, the [Na]_p is a little worse.

Desalination Argument Is Incomplete

Above, the plasma sodium decreased from 120 to 119 mEq/L when NS was given. This paradox is typically explained by desalination (13): the urine in SIAD is more concentrated than NS, so all of the Na in NS is dumped (desalinated), and some of the H₂O in NS is reabsorbed to bring its concentration up to the urine’s, thereby worsening the hyponatremia—except here, the urine was not more concentrated than NS. The [Na+K]_u was 140 mEq/L versus the NS’s 154 mEq/L. For this kidney to modify NS into urine, it would have to dilute the NS down to the urine’s concentration, resulting in the excretion of free water. Water loss should raise the plasma sodium, so why did it go down instead? The main reason is that the kidney ran out of NS to process and worked on the plasma instead (not to be taken literally). The NS rate of 30 ml/h is less than the urine output rate of 3000 ml/24 hours=125 ml/h, so the kidney has to find the Na/K/H₂O elsewhere. It modifies the [Na]_p at 120 mEq/L up to the [Na+K]_u at 140 mEq/L, reabsorbing water. This exceeds the earlier free water excretion and lowers the plasma sodium overall. Thus, simply comparing the urine concentration to the iv fluid concentration does not tell the whole story. The paradox can still occur, even if care is taken to use an iv fluid that is hypertonic to urine.

Intravenous Fluid Rate Is Important Too

Having laid the physiologic groundwork, we can apply our knowledge to avoid the paradox. If it is not sufficient to simply choose an iv fluid that is hypertonic to urine, then we need to “dose” the iv fluid properly. To find the right dose, we can solve for the infusion rate that is equipotent with the kidney, such that the plasma sodium remains stable. That equilibrium point becomes the benchmark for an iv fluid prescription. If it is infused at a lower rate, the iv fluid is underdosed, and it will permit the sodium to go down. If it is infused at a higher rate than the benchmark, the iv fluid should make the plasma sodium go up and prevent a paradox.

The Ratio Profile

The Edelman accounting tool would say that a chosen iv fluid and the urine would affect the plasma sodium over some period of time in the following way:

$${\left[\text{Na}\right]}_t=\frac{\stackrel{\text{Initial Na}+\text{K amount}}{\overbrace{{\left[\text{Na}\right]}_0\cdot \text{TBW}}}-\stackrel{\text{Na}+\text{K solutes lost by}\hspace{0.17em}\text{urine}}{\overbrace{{\left[\text{Na}+\text{K}\right]}_{\text{u}}\cdot {\text{Rate}}_{\text{u}}\cdot t}}+\stackrel{\text{Na}+\text{K solutes gained by IV fluid}}{\overbrace{{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}\cdot {\text{Rate}}_{\text{IVF}}\cdot t}}}{\underset{\text{Initial TBW}}{\underbrace{\text{TBW}}}-\underset{\text{Volume lost by urine}}{\underbrace{{\text{Rate}}_{\text{u}}\cdot t}}+\underset{\text{Volume gained by IV fluid}}{\underbrace{{\text{Rate}}_{\text{IVF}}\cdot t}}}$$

If the iv fluid exactly neutralizes the effect of the urine, then the future [Na]_t will remain equal to the initial [Na]₀. Equating [Na]_t and [Na]₀ gives:

$$\frac{{\left[\text{Na}\right]}_0\cdot \text{TBW}-{\left[\text{Na}+\text{K}\right]}_{\text{u}}\cdot {\text{Rate}}_{\text{u}}\cdot t+{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}\cdot {\text{Rate}}_{\text{IVF}}\cdot t}{\text{TBW}-{\text{Rate}}_{\text{u}}\cdot t+{\text{Rate}}_{\text{IVF}}\cdot t}={\left[\text{Na}\right]}_0$$ (1)

Solve for the benchmark Rate_IVF (see Supplemental Material for algebraic details):

$${\text{Rate}}_{\text{IVF}}={\text{Rate}}_{\text{u}}\cdot \frac{{\left[\text{Na}+\text{K}\right]}_{\text{u}}-{\left[\text{Na}\right]}_0}{{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}-{\left[\text{Na}\right]}_0}$$ (2)

Because the latter fraction is a ratio of two differences, each subtracting [Na]₀, we call it the ratio profile. Fortunately, Rate_IVF is independent of the patient’s TBW, which is estimated imprecisely.

Real-time [Sodium] Evolution

Because Equation (1) incorporates time, it shows how the paradox evolves. A patient with [Na]₀ of 120 mEq/L, TBW of 42 L, [Na+K]_u of 200 mEq/L, and urine output of 40 ml/h would have a ratio profile vis-à-vis NS of

$${\text{Rate}}_{\text{u}}\cdot \frac{{\left[\text{Na}+\text{K}\right]}_{\text{u}}-{\left[\text{Na}\right]}_0}{{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}-{\left[\text{Na}\right]}_0}=40\cdot \frac{200-120}{154-120}\approx 94$$ ml/h

Below this rate, NS would lead to a paradox, like at a to-keep-open rate of 10 ml/h. Over 24 hours, [Na]_t ends up at 118 mEq/L (Figure 1, red solid). But above 94 ml/h, NS would avert a paradox, say at 150 ml/h. Over 24 hours, [Na]_t ends up at 121 mEq/L (Figure 1, blue dash). On clinical timescales, the red solid and blue dash look linear, but they are curves (8). With urine being more concentrated than the iv fluid in this SIAD case, the kidney seems fully capable of desalinating NS. Yet, a large enough dose of NS did improve the sodium, so a paradox with isotonic fluids is not inevitable.

Figure 1.

Evolution of a syndrome of inappropriate antidiuresis paradox. Equation (1) is graphed with time as the independent variable (x axis) and plasma sodium as the dependent variable (y axis). Parameters: [Na]₀=120 mEq/L, TBW=42 L, [Na+K]_u=200 mEq/L, and Rate_u=40 ml/h. If normal saline (NS) is given at less than its ratio profile (underdose), then [Na]_t decreases paradoxically to 118 mEq/L at 24 hours (red solid). But if NS is given at more than its ratio profile, like at 150 ml/h, then [Na]_t increases to 121 mEq/L at 24 hours (blue dash). The equations being graphed are shown on the left, for open source verification.

Type of IV Fluid Matters

To raise the plasma sodium, an iv fluid has to be infused faster than Rate_IVF, but can any iv fluid be given? Intuitively, we cannot give 5% dextrose in water (D5W) at >Rate_IVF and expect the [Na]_p to go up. The iv fluid’s Na/K composition matters, and the [Na+K]_IVF threshold is evident from the denominator in Equation (2). The iv fluid needs to be hypertonic to the initial sodium, so that [Na+K]_IVF–[Na]₀ is positive. This implies that we only have to consider the iv fluids that are typically used to treat hyponatremia, which at the lower end of the tonicity spectrum would include NS, lactated Ringer’s, and Plasma-Lyte. Although isotonic, D5W does not qualify.

Generic Outputs

Equation (2) refers to urine, but it applies to outputs in general. If, say, the diarrheal [Na+K] and rate are known, then the ratio profile for the diarrhea component would be:

$${\text{Rate}}_{\text{IVF}}={\text{Rate}}_{\text{stool}}\cdot \frac{{\left[\text{Na}+\text{K}\right]}_{\text{stool}}-{\left[\text{Na}\right]}_0}{{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}-{\left[\text{Na}\right]}_0}$$

The same would go for insensible losses, although they are usually not considered. Such outputs can still be estimated using data from human studies (14,15). The generic output ratio profile is:

$${\text{Rate}}_{\text{IVF}}={\text{Rate}}_{\text{output}}\cdot \frac{{\left[\text{Na}+\text{K}\right]}_{\text{output}}-{\left[\text{Na}\right]}_0}{{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}-{\left[\text{Na}\right]}_0}$$ (3)

Generic Inputs

For inputs, their ratio profiles follow the same pattern, except the sign is negative. For example, if a medication is being given in 0.45% NaCl at 50 ml/h, how fast does NS need to be infused to neutralize the 1/2 NS and keep the patient’s [Na]_p steady?:

$${\text{Rate}}_{\text{NS}}=-{\text{Rate}}_{\frac{1}{2}\text{NS}}\cdot \frac{{\left[\text{Na}+\text{K}\right]}_{\frac{1}{2}\text{NS}}-{\left[\text{Na}\right]}_0}{{\left[\text{Na}+\text{K}\right]}_{\text{NS}}-{\left[\text{Na}\right]}_0}=-50\cdot \frac{77-120}{154-120}\approx 63.2\frac{\text{mL}}{\text{h}}$$

This means NS would have to be given at >63.2 ml/h to overcome 0.45% NaCl and raise the sodium. If 1/2 NS is this hard to neutralize, D5W is more formidable. With a [Na+K]_D5W of 0, it imparts full force to the numerator’s [Na]₀. Luckily, D5W is avoided during hyponatremia, but a secret source of D5W is as a carrier for medication drips, like iv antibiotics or pressors (16).

Modularity

Can individual outputs and inputs be combined into one big net ratio profile? Mathematically, each behaves like a separate module, and they are added or subtracted according to their signs. The net ratio profile can be expressed as:

$${\text{Rate}}_{\text{IVF}}=\frac{+\stackrel{\text{generic output module}}{\overbrace{{\text{Rate}}_{\text{output}\left(\text{s}\right)}\cdot \left({\left[\text{Na}+\text{K}\right]}_{\text{output}\left(\text{s}\right)}-{\left[\text{Na}\right]}_0\right)}}+\stackrel{\text{etc}.}{\overbrace{\cdots }}-\stackrel{\text{generic input module}}{\overbrace{{\text{Rate}}_{\text{input}\left(\text{s}\right)}\cdot \left({\left[\text{Na}+\text{K}\right]}_{\text{input}\left(\text{s}\right)}-{\left[\text{Na}\right]}_0\right)}}-\stackrel{\text{etc}.}{\overbrace{\cdots }}}{{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}-{\left[\text{Na}\right]}_0}$$ (4)

Include as many outputs and inputs as are known to get a more complete sense of the chosen iv fluid’s minimum infusion rate to avoid a paradox. As the ratio profile is to be used prospectively, there will be unknown modules that need to be predicted or estimated (8). In the Supplemental Material, all three patient cases demonstrate how to aggregate the modules into a net ratio profile.

Modifying Urine

Certain treatments for hyponatremia modify the urine. By opposing SIAD at the kidney level, they help to avoid the paradox. Urea acts as an osmotic diuretic and increases the water in urine relative to its Na/K (17–19). The vasopressin antagonists inhibit the vasopressin2 receptor and cause an aquaresis (20,21). Loop diuretics diminish the medullary gradient and reduce the driving force for water reabsorption (22). Any of them, urea/vasopressin antagonists/loop diuretics, should lower [Na+K]_u. If [Na+K]_u is indeed diluted by the treatment, then updated urinary data can be plugged into the ratio profile. When the [Na+K]_u drops below [Na]₀, Equation (3) becomes negative, but that negative is for the urine module. Other modules may yet be positive. At least the negative urine module will reduce the net ratio profile and make it easier to prescribe an iv fluid at greater than its Rate_IVF.

Potency of Drinking Water

An important oral input is drinking water (or other liquid). If a patient drinks 1200 ml of water in a day, and assuming the worst-case scenario where all of the water is absorbed in the intestine, then the ratio profile is:

$${\text{Rate}}_{\text{IVF}}=-{\text{Rate}}_{\text{water}}\cdot \frac{{\left[\text{Na}+\text{K}\right]}_{\text{water}}-{\left[\text{Na}\right]}_0}{{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}-{\left[\text{Na}\right]}_0}=-\frac{\text{1,200}}{24}\cdot \frac{0-120}{154-120}\approx 176.5\frac{\text{mL}}{\text{h}}$$

Amazingly, NS would have to be given at ≥176.5 ml/h to keep the [sodium] from dropping. Such high rates are not commonly ordered for NS, so the simple act of drinking water can make NS appear to have a paradoxical effect to worsen hyponatremia. Really, NS was just underdosed. The paradox does not even have to occur in the setting of SIAD. The kidney could be excreting a lot of free water, but that can be undone by drinking even more water, which is not hard to do. See Supplemental Material, cases 1 and 2, for clinical examples of water drinking contributing to the paradox.

Salt Tablets

Salt tablets are used to treat hyponatremia, but they are solid and do not have a volume rate like the other liquid outputs and inputs. Returning to the fundamental equation, we can define the salt tablet rate, Rate_salt, as the sodium quantity per time. The tablets are taken intermittently, e.g., 1 g three times a day, but it is all right to pretend that salt tablets are taken continuously as long as time cancels out:

$$\frac{{\left[\text{Na}\right]}_0\cdot \text{TBW}+{\text{Rate}}_{\text{salt}}\cdot t+{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}\cdot {\text{Rate}}_{\text{IVF}}\cdot t}{\text{TBW}+{\text{Rate}}_{\text{IVF}}\cdot t}={\left[\text{Na}\right]}_0$$

Solve for the benchmark Rate_IVF (see Supplemental Material for algebraic details):

$${\text{Rate}}_{\text{IVF}}=-\frac{{\text{Rate}}_{\text{salt}}}{{\left[\text{Na}+\text{K}\right]}_{\text{IVF}}-{\left[\text{Na}\right]}_0}$$ (5)

Like other inputs, salt tablets give the ratio profile a leading negative sign. If Rate_salt is in mEq/h, then dividing by [Na+K]_IVF–[Na]₀ in mEq/L will give Rate_IVF in liters per hour, which is easy to convert to milliliters per hour. If an iv fluid is hypertonic to the initial sodium, i.e., [Na+K]_IVF>[Na]₀—as it should be to treat hyponatremia—then the ratio profile will always be negative by inspection. The salt tablet module has to reduce the net ratio profile, lowering the Rate_IVF that needs to be exceeded. Case 3 in the Supplemental Material shows how to incorporate salt tablets into the ratio profile calculation.

Potency of 3% Saline

3% saline at 513 mEq/L is more concentrated than a urine [Na+K] can ever get. Common sense would say that 3% must raise the plasma sodium, like how drinking seawater makes one thirstier and more dehydrated because the human kidney can only make urine that is less salty than seawater, which is about 3.5% saline. What the kidney lacks in concentrating power can be compensated for by nonstop desalination. In addition, 3% saline can be handicapped by limiting its dose. At some point, the 3% will have been excreted by the kidney, and then the kidney can desalinate the plasma, resulting in a lower plasma sodium. In this example where [Na]₀=120 mEq/L, TBW=42 L, urine output is 3 L/day with [Na+K]_u=140 mEq/L, and 3% saline is infused at just 1 ml/h, we calculate that:

$$\frac{\stackrel{\text{Initial Na}+\text{K amount}}{\overbrace{120\cdot 42}}-\stackrel{\text{Na}+\text{K in}3\text{L/day of urine}}{\overbrace{140\cdot 3}}+\stackrel{\text{Na in 3\% saline at}1\text{mL/h for}24\text{h}}{\overbrace{513\cdot 0.001\cdot 24}}}{\underset{\text{Initial TBW}}{\underbrace{42}}-\underset{\text{Volume of urine}}{\underbrace{3}}+\underset{\text{Volume of 3\% saline}}{\underbrace{0.001\cdot 24}}}\approx 118.7\text{mEq/L}$$

Surprisingly, the 3% can be overcome by the kidney, even if urine [Na+K] stays well below 513 mEq/L. See case 3 in the Supplemental Material for a real example of 3% saline not being able to raise the sodium.

Limitations

The cases show that the ratio profile can predict an SIAD paradox in real life, but can the method be used to prevent a paradoxical worsening of hyponatremia? Cases will be hard to find because the ratio profile during SIAD tends to calculate a high infusion rate for isotonic fluids. In case 2, the rate for NS had to be >168.8 ml/h, which over the course of a day would add >4 L to a patient’s volume. That is just to keep the plasma sodium stable; even more NS has to be given to actually raise the sodium, and modestly at best (Figure 1). Besides fluid overload, other effects of saline overexpansion may include pulmonary edema, renal injury, and hyperchloremic acidosis (23,24). A more realistic case series to test the ratio profile’s utility would be to study 3% NaCl. Its infusion rate is going to be lower than that of NS and other isotonic fluids, so patients would be prescribed 3% at rates either higher or lower than the ratio profile. If lower rates consistently result in a paradox, and higher rates reliably avoid it, then the use of the ratio profile is supported. It would also teach us that conventional rates of 3% NaCl could still lead to desalination.

The ratio profile is only as good as its underlying mathematical model, and the Edelman equation represents our best understanding of sodium physiology (25). But all of the practical formulas based on Edelman, like Adrogué–Madias and Barsoum–Levine, do not always accurately predict the [sodium] after iv fluids (26–28). To be fair, some studies have shown the formulas to be reasonably accurate (9,29–31). Part of the inaccuracy may stem from using the Edelman equation in a simplified form. The original equation had a non-1 slope and a non-0 y intercept, believed to be due to a Gibbs–Donnan equilibrium and an osmotically inactive store of the total Na/K (32–34). The ratio profile can be derived from the original Edelman equation (35), but the slope and y intercept values are not agreed upon (36). Until better measurements are made, it may be prudent to denote the slope and y intercept as the symbols m and b. Also, there may be hidden variables that determine the plasma sodium, yet to be discovered.

Like all formulas, the ratio profile is only as valid as the numbers being entered. Clerical errors can be made, relevant input/output data may be omitted, and laboratory measurements have an inherent error of uncertainty. Most importantly, the numbers themselves change because SIAD is a dynamic, evolving disease. The urine [Na+K] was found to fluctuate hour to hour, but this study lacked urine output information, so the true variability of the Na/K/H₂O balance is not known (37). Still, it highlights the need to measure the urine and other parameters frequently, so the ratio profile or other dysnatremia tool can be updated. Although theoretically sound, the ratio profile needs to be clinically validated. Even then, there is no substitute for frequent monitoring of plasma sodium when correcting hyponatremia.

Conclusion

The paradox starts with a propensity toward hyponatremia, usually SIAD, that is countered by a weak therapeutic response. This underdose is more likely to happen with isotonic fluids. They are not all that hypertonic to the plasma sodium, even when hyponatremia is severe, and they are prescribed at standard rates that, unintentionally, are inadequate. Our quantitative tool, dubbed the ratio profile, calculates the rate that an iv fluid should be infused at in order to keep [sodium] steady. To raise the plasma sodium, we have to prescribe a rate greater than the ratio profile. 3% saline can be used, but it too can be overcome. Drinking water is an underappreciated reason for the paradox. The ratio profile is readily calculable on rounds, and we hope it will help doctors to avoid the paradoxical worsening of hyponatremia in SIAD when iv fluids are given.

Disclosures

R. Chiaramonte has ownership interest in BioPharma Credit/Pharmakon, ITB-MED/Zelarion, ProKidney, and Royalty Pharma; patents or royalties with Royalty Pharma; and an advisory or leadership role with ITB-MED/Zelarion. J. Shey reports being an employee of West Coast Kidney Institute and has ownership interest in Fresenius Kidney Care. The remaining author has nothing to disclose.

Funding

None.

Author Contributions

S. Chen was responsible for the investigation, project administration, supervision, and visualization, and wrote the original draft of the manuscript; S. Chen and R. Chiaramonte were responsible for the formal analysis and validation; S. Chen and J. Shey were responsible for the conceptualization; and all authors were responsible for the methodology and reviewed and edited the manuscript.

Supplemental Material

This article contains the following supplemental material online at http://kidney360.asnjournals.org/lookup/suppl/doi:10.34067/KID.0004882022/-/DCSupplemental.

Derive the ratio profile formula.

Derive the salt tablets module.

Case 1: Mild SIAD still led to paradox.

Case 2: No SIAD but paradox anyway.

Case 3: Severe SIAD desalinated 3% saline.