Estimating in vivo potassium distribution and fluxes with stable potassium isotopes

Abstract

Extracellular potassium (K⁺) homeostasis is achieved by a concerted effort of multiple organs and tissues. A limitation in studies of K⁺ homeostasis is inadequate techniques to quantify K⁺ fluxes into and out of organs and tissues in vivo. The goal of the present study was to test the feasibility of a novel approach to estimate K⁺ distribution and fluxes in vivo using stable K⁺ isotopes. ⁴¹K was infused as KCl into rats consuming control or K⁺-deficient chow (n = 4 each), ⁴¹K-to-³⁹K ratios in plasma and red blood cells (RBCs) were measured by inductively coupled plasma mass spectrometry, and results were subjected to compartmental modeling. The plasma ⁴¹K/³⁹K increased during ⁴¹K infusion and decreased upon infusion cessation, without altering plasma total K⁺ concentration ([K⁺], i.e., ⁴¹K + ³⁹K). The time course of changes was analyzed with a two-compartmental model of K⁺ distribution and elimination. Model parameters, representing transport into and out of the intracellular pool and renal excretion, were identified in each rat, accurately predicting decreased renal K⁺ excretion in rats fed K⁺-deficient vs. control diet (P < 0.05). To estimate rate constants of K⁺ transport into and out of RBCs, ⁴¹K/³⁹K were subjected to a simple model, indicating no effects of the K⁺-deficient diet. The findings support the feasibility of the novel stable isotope approach to quantify K⁺ fluxes in vivo and sets a foundation for experimental protocols using more complex models to identify heterogeneous intracellular K⁺ pools and to answer questions pertaining to K⁺ homeostatic mechanisms in vivo.

INTRODUCTION

Potassium (K⁺) homeostasis is critical for normal cardiovascular and neuromuscular function (1, 2), as the transmembrane K⁺ gradient is a major component of membrane potential that determines the excitability and contractility of cardiac and skeletal muscle and nervous tissue. K⁺ is transported into cells by the sodium pump (i.e., Na-K-ATPase), generating and maintaining the transmembrane gradient. Because of the ubiquitous sodium pump, K⁺ is primarily distributed intracellularly, resulting in a small extracellular K⁺ pool accounting for only 2% of the total body K⁺ (3). A small extracellular pool presents a significant homeostatic challenge; compared with other major electrolytes, K⁺ has a higher ratio of dietary intake to extracellular pool size (i.e., turnover) (4). Extracellular K⁺ concentration ([K⁺]) can be substantially increased by large influx from a K⁺-rich meal or release from exercising muscles. To meet these challenges, the K⁺ homeostatic system adapts quickly and efficiently to altered K⁺ intake as described below.

Extracellular K⁺ homeostasis is maintained by renal and extrarenal mechanisms. The kidneys have a remarkable capacity to regulate K⁺ excretion to match K⁺ intake and play the major role in maintaining chronic K⁺ balance (5–8). In addition, extrarenal tissues (mainly skeletal muscle, the major K⁺ store) provide K⁺ buffering capacity by shifting K⁺ between the extracellular (ECF) and intracellular (ICF) fluids (9, 10). In previous studies, we demonstrated that both renal K⁺ excretion and K⁺ transport into extrarenal tissues are regulated by dietary K⁺ intake (11–14). We also established that gut sensing of K⁺ intake (“gut factor”) plays a major role in renal and extrarenal regulation (15, 16). These findings indicate that cross talk likely exists between the gut and renal and/or extrarenal tissues during K⁺ intake (i.e., the postprandial or absorptive state) to regulate and maintain ECF K⁺ homeostasis. Cross talk may also exist in the postabsorptive state between renal and extrarenal tissues to balance K⁺ fluxes into (from extrarenal tissues) and out of (via renal excretion) the ECF K⁺ pool (17).

Despite the marked progress made in understanding ECF K⁺ homeostasis (17), many important issues remain unaddressed. For example, although cross talk is apparent between organs and tissues, the underlying mechanisms of communication are elusive. In addition, molecular mechanisms regulating the K⁺ shift from the ECF into the ICF during K⁺ intake (via insulin) or exercise (via epinephrine) are not completely understood; although Na-K-ATPase stimulation, which increases cellular K⁺ uptake, may be the major mechanism for this shift, it is unknown whether K⁺ channels or cotransporters, which determine K⁺ efflux and thus affect net K⁺ shift, play a role (14). Finally, it is often difficult to assess relative contributions of renal versus extrarenal effects of an agent, e.g., aldosterone (19–23), to ECF K⁺ regulation or whether a pathophysiological resetting of ECF K⁺ is due to a lesion in muscle or kidney. Improvements in our understanding of K⁺ homeostatic mechanisms can facilitate development of new strategies for treating hyperkalemic or hypokalemic states (24).

A main limitation in studies of ECF K⁺ homeostasis is the inability to quantify K⁺ fluxes in vivo, particularly those into and out of the ICF. Past experimental work studying K⁺ fluxes relied on the use of ⁸⁶Rb, a radioactive surrogate tracer for K⁺ movements (10, 25). This approach has significant drawbacks, such as radioactivity exposure and the short half-life of ⁸⁶Rb, which has limited its use to in vitro studies. The stable isotopes of potassium, ⁴¹K and ³⁹K, which constitute 6.7302% and 93.2561% of naturally occurring K⁺, respectively, offer an alternative to ⁸⁶Rb to characterize potassium fluxes directly without the drawbacks. Recent advances in multicollector inductively coupled plasma mass spectrometry (MC-ICP-MS) now permit reliable and precise measurements of natural variability in the ⁴¹K-to-³⁹K ratio (⁴¹K/³⁹K) at the ∼0.01% (or 0.1‰) level. These measurements have been used by Higgins and colleagues (26, 27) and others (28) to document natural variations in the ⁴¹K/³⁹K in a wide range of geological samples. This state-of-the-art technology provides the opportunity to estimate K⁺ distribution and fluxes in vivo based on stable isotope ratios, that is, without using radioactive tracers. In the present study, we examine the impact of ⁴¹K infusion and provide evidence for significant alterations of plasma ⁴¹K/³⁹K, which can be utilized to estimate K⁺ distribution and fluxes in vivo with compartmental model analysis. Our findings establish the feasibility of exploiting the ⁴¹K-to-³⁹K stable isotope ratio to quantify K⁺ fluxes in vivo. The novel approach provides a new tool to address key gaps in understanding K⁺ homeostatic mechanisms and (patho)physiology in vivo.

MATERIALS AND METHODS

Animals and Catheterization

All procedures involving animals were approved by the Institutional Animal Care and Use Committee at the University of Southern California. Male Wistar rats weighing 280–300 g (∼9 wk old) were obtained from Envigo and housed under controlled temperature (22 ± 2°C) and lighting (12-h light: 6:00 AM–6:00 PM; 12-h dark: 6:00 PM–6:00 AM) with free access to water and standard rat chow. Rats were maintained on either normal 1% K⁺ diet (TD 08267; Envigo) or K⁺-deficient diet (TD 88239; Envigo) for 5 days before the experiment. Rats were placed in individual cages with wire rack floors and accommodated to tail restraints for 3 or 4 days before the experiment, required to protect tail blood vessel catheters during the experiments (11, 12). The animals could move about the cage and were allowed unrestricted access to food and water. A tail vein catheter for intravenous infusion and a tail artery catheter for blood sampling were placed under local anesthesia with lidocaine the morning of the experiment (∼7:00 AM) (11, 12).

⁴¹K Infusion

Rats were studied in a conscious state after a 6-h fast. After basal blood samples were obtained from the tail artery catheter at ∼1:00 PM, ⁴¹K (ISOFLEX USA, San Francisco, CA; enriched to 96.5% of total K; as KCl) was infused intravenously (0.5 mg ⁴¹K dissolved in 1 mL normal saline; 12.2 mEq/L) for 1 h through the tail vein catheter, and blood samples were collected before (t = −30 and −10 min; basal), during (t = 5, 15, 40, and 60 min), and after (t = 65, 75, 100, 120, and 180 min) the ⁴¹K infusion (t = 0–60 min). Blood samples were rapidly spun for 1 min and separated into plasma and red blood cells (RBCs), which were then analyzed for ⁴¹K/³⁹K. In addition, urine passed was collected from the bottom of cages, as previously described (15, 16), for 3-h periods immediately before and during the ⁴¹K infusion. All samples were frozen and stored at −20°C until analysis for ⁴¹K/³⁹K. In addition, total plasma and urine K⁺ levels were determined by flame photometry as previously reported (11, 12).

Ion Chromatography and Isotope Ratio Mass Spectrometry for ⁴¹K/³⁹K Determination

K⁺ was purified from plasma, urine, and RBC samples for isotopic analyses with an automated high-pressure ion chromatography (IC) system, as previously described (26, 27). The accuracy of our chromatographic methods was verified by purifying and analyzing external standards [NIST SRM3141a (⁴¹K/³⁹K = ∼0.0722) and SRM70b (⁴¹K/³⁹K = ∼0.0721)] alongside unknown samples. Purified aliquots of K⁺ were analyzed in 2% HNO₃ for their isotopic compositions on a Thermo Scientific Neptune Plus multicollector inductively coupled plasma mass spectrometer (MC-ICP-MS) at Princeton University, with previously published methods (27). The external reproducibility of our protocols (chromatography and mass spectrometry) as determined through replicate measurements of international standards is ∼0.015%.

Compartmental Analysis

Changes in ⁴¹K/³⁹K from the baseline ratios (Δ⁴¹K/³⁹K) were analyzed by using a two-compartmental (2-C) model that comprises the ECF and ICF K⁺ pools, K⁺ fluxes between the compartments, and renal K⁺ excretion (Fig. 1A). We view this as a “minimal model” of K⁺ kinetics in vivo, employing a single, homogeneous ICF pool, which can be expanded to a more complex model with two or more heterogeneous ICF pools (see below). The 2-C model is represented by the following differential equations:

$$\text{d}{y}_{\text{1}}\left(t\right)/\text{d}t=-\left(k_{\text{21}}+k_{0\text{1}}\right)\times y_{\text{1}}\left(t\right)+k_{\text{12}}\times y_{\text{2}}\left(t\right)\times {\text{K}}_{\text{ICF}}/{\text{K}}_{\text{ECF}}+\text{ Inf}\left(t\right)/{\text{K}}_{\text{ECF}}$$

$$\text{d}{y}_{\text{2}}\left(t\right)/\text{d}t=-k_{\text{12}}\times y_{\text{2}}\left(t\right)+k_{\text{21}}\times y_{\text{1}}\left(t\right)\times {\text{K}}_{\text{ECF}}/{\text{K}}_{\text{ICF}}$$

where y₁(t) and y₂(t) represent Δ⁴¹K/³⁹K in the ECF and the ICF, respectively, K_ECF and K_ICF represent the amounts of K⁺ in the ECF and ICF compartments, respectively, and k₀₁, k₂₁, and k₁₂ are the rate constants for renal K⁺ excretion and K⁺ transport into and out of the ICF, respectively. Inf(t) is equal to the ⁴¹K infusion rate from 0 to 60 min and becomes 0 after 60 min. Model parameters (k₀₁, k₁₂, k₂₁, and K_ECF) were identified in individual animals from plasma Δ⁴¹K/³⁹K profiles with a Levenberg–Marquardt nonlinear algorithm on MATLAB (source code available at https://doi.org/10.6084/m9.figshare.17001139.v1). K_ICF was calculated to be K_ECF × k_21/k_12.

Figure 1.

A: 2-compartmental model of K⁺ distribution and fluxes in vivo. K_ECF is the amount of K⁺ in the extracellular fluid (ECF), and k₀₁, k₂₁, and k₁₂ are rate constants for renal excretion and transport into and out of the intracellular fluid (ICF), respectively. B: a single-compartmental model of red blood cell (RBC) K⁺ turnover in which k_in and k_out are first-order rate constants for K⁺ influx and efflux, respectively. Δ⁴¹K/³⁹K, increment in ⁴¹K-to-³⁹K ratio.

In addition, we used a simple model to describe changes in Δ⁴¹K/³⁹K in RBCs (Fig. 1B) that assumes a single compartment for RBC ICF with discrete rate constants for K⁺ influx (k_in) and efflux (k_out) and is represented by the following differential equation:

$$\text{d}y\left(t\right)/\text{d}t=-k_{\text{out}}\times y\left(t\right)+k_{\text{in}}\times \text{P}\left(t\right)$$

where y(t) represents Δ⁴¹K/³⁹K in RBCs, P(t) is Δ⁴¹K/³⁹K in plasma, and k_in and k_out are the rate constants for K⁺ transport into and out of the RBC ICF. With the plasma Δ⁴¹K/³⁹K profile as the input, the model produces dynamics of RBC Δ⁴¹K/³⁹K. Model parameters (k_in and k_out) that best fit the data were identified by a Levenberg–Marquardt nonlinear algorithm on MATLAB. A model is considered theoretically identifiable if model parameters can be identified from simulated data (without noise) to be the same as those used in simulation. In a real situation, when unknown model parameters are identified from data (with noise) collected in experiments, a model may be considered uniquely identifiable if parameter identification with different initial estimates converges to the same set of parameter values.

Model Estimation and Direct Measurement of Renal K⁺ Excretion

The whole body 2-C modeling provides an estimate of renal K⁺ excretion, which is calculated as k₀₁ × K_ECF. This estimate, based on the plasma ⁴¹K/³⁹K profile, was compared with directly measured values as a way of validating the model assumptions. Rate of renal K⁺ excretion was directly measured by determining the volume and K⁺ concentration of urine collected from the bottom of cage before and during the experiment, as previously described (15, 16).

Monte Carlo Simulation

Computer simulation is an effective method to optimize protocols that saves time and animal experimentation. We performed a Monte Carlo simulation on MATLAB to test the effects of experimental and sampling protocols on the identification of whole body models of K⁺ distribution and elimination. First, the plasma Δ⁴¹K/³⁹K profile was simulated under given experimental and sampling protocols using a select model (and its assigned parameter values). Subsequently, random noises at different levels were added to the simulated data, and these noise-added simulated data were analyzed to identify model parameters. We repeated this process 500 times at each noise level to estimate variations in model parameters identified from noise-added simulated data. This variation is expected to increase as noise levels increase, and the slope of this relationship (i.e., the sensitivity of parameter identification to noise) reflects the robustness of parameter identification with each experimental/sampling protocol. Simulation was performed with various experimental/sampling protocols to evaluate the impact of different protocols on the robustness of parameter identification. In this simulation study, we arbitrarily defined “unidentifiability” as the probability of estimated parameters being more than three times different from the true values.

Statistical Analysis

All data are expressed as means ± SE The significance of differences in the mean value was assessed on ProStat (Poly Software International, Inc.) by Student’s t tests. P values were adjusted for multiple comparisons by the Benjamini–Hochberg method. A P value <0.05 was considered statistically significant.

RESULTS

Effects of ⁴¹K Infusion on ⁴¹K/³⁹K in Plasma and RBC

Total K⁺ concentration ([K⁺]) was unchanged in plasma by the ⁴¹K infusion (Fig. 2A). In control rats (Fig. 2B), plasma ⁴¹K/³⁹K, expressed as change from the baseline (Δ⁴¹K/³⁹K in ‰), increased during the 1-h ⁴¹K infusion period. This increase was very rapid during the first 5 min and then slowed progressively through the remaining 1 h of ⁴¹K infusion, reaching ∼1.7‰ at 60 min of infusion. Upon cessation of the ⁴¹K infusion, Δ⁴¹K/³⁹K decreased rapidly for 5 min to ∼0.9‰ and then slowly for the subsequent 2 h to ∼0.5‰. In RBCs, the ⁴¹K/³⁹K (Fig. 2C) progressed rather linearly for the duration of the infusion and beyond until ∼90 min and remained steady through 3 h. Despite the continuous increase, Δ⁴¹K/³⁹K was significantly lower in RBCs than in plasma at the end of the experiment (0.14 ± 0.01‰ vs. 0.51 ± 0.07‰, P < 0.002).

Figure 2.

Effects of ⁴¹K infusion on plasma K⁺ concentration ([K⁺]) (A) and on ⁴¹K-to-³⁹K ratio (⁴¹K/³⁹K) in plasma (B) and red blood cells (RBCs) (C) in rats fed control and K⁺-deficient diets. Box indicates ⁴¹K infusion period. Isotope ratios are expressed as increments (Δ) from basal values. Area under the curve was significantly different between the 2 diet groups for plasma (P < 0.02, 2-tailed t test) but not RBC (P > 0.05) Δ⁴¹K/³⁹K. Data are means ± SE. n = 4 for each group.

Previous studies showed that K⁺-deficient diet causes adaptations of the kidneys and skeletal muscle to conserve ECF K⁺ by reducing K⁺ excretion and cellular K⁺ uptake, respectively (11–13). These adaptations occur rapidly, within a day (29). To test whether such changes are detectable with this stable isotope approach, rats were maintained on either normal or K⁺-deficient diet for 5 days before the experiment. We found no differences in food intake, body weight, and plasma Na⁺ concentration between the control and K⁺-deficient diet groups (Table 1). In contrast, plasma concentration of total K⁺ was lower by ∼24% in in rats fed K⁺-deficient versus control diet (P < 0.01), as in previous studies (11). The time course of plasma Δ⁴¹K/³⁹K (Fig. 2B) was significantly different between the two diet groups; although the overall pattern of change was similar, the rise in ⁴¹K/³⁹K during the ⁴¹K infusion was greater in rats fed K⁺-deficient diet compared with the control group, resulting in higher profiles of plasma Δ⁴¹K/³⁹K throughout the experiment (P < 0.02; see Fig. 2). In contrast, K⁺-deficient diet did not alter the time course of Δ⁴¹K/³⁹K in RBCs; thus, the curves were identical between the two groups (Fig. 2C).

Table 1.

Body weight, food intake, and plasma Na⁺ and K⁺ concentrations in rats fed control and K⁺-deficient diets

	Control Diet	K+-Deficient Diet
Body weight, g	322 ± 4	324 ± 4
Food intake, g/day	28.7 ± 0.9	26.7 ± 1.0
plasma [Na+], mEq/L	139 ± 2	139 ± 0.4
plasma [K+], mEq/L	4.1 ± 0.2	3.1 ± 0.1*
Δ41K/39K, ‰#	0.004 ± 0.005	−0.059 ± 0.012*

Data are means ± SE (n = 4 for each). [K⁺], K⁺ concentration; [Na⁺], Na⁺ concentration; Δ⁴¹K/³⁹K, change in ⁴¹K-to-³⁹K ratio. #Expressed as deviations from the K⁺ standard NIST SRM3141a (see materials and methods). *P < 0.01 (2-tailed t test).

2-C Modeling of Whole Body K⁺ Distribution and Fluxes

The time course of change in plasma Δ⁴¹K/³⁹K was analyzed in each rat with a 2-C model of K⁺ distribution and fluxes (Fig. 1), which includes the ECF (compartment 1) and the ICF (compartment 2) K⁺ pools, K⁺ fluxes between the pools, and renal K⁺ excretion. We assume that ECF instantly equilibrates with plasma and plasma Δ⁴¹K/³⁹K faithfully reflects the ratio in ECF. This is an assumption often used in kinetic modeling of many small molecules and electrolytes, supported by the findings that interstitial (or ECF) concentrations are much closer to plasma than intracellular concentrations. We consider this a minimal model of K⁺ distribution in vivo, employing a single homogeneous ICF pool. Despite the minimal complexity, the model fit the data quite well, resulting in residuals (i.e., differences between model prediction and observed data) not statistically different from zero (Fig. 3). Model parameters were uniquely identified in every animal (Fig. 4), and variations [i.e., fractional standard deviations (FSDs)] in estimated parameter values were <77% for all parameters in both groups (Table 2). ECF K⁺ pool (i.e., K_ECF) was estimated to be lower by 34% in rats fed K⁺-deficient versus control diet. Although this did not gain statistical significance because of the small number of animals used, this difference was consistent with the 24% lower plasma K⁺ concentration in these animals (P < 0.01; Table 1 and Fig. 2A). The size of the ICF K⁺ pool (i.e., K_ICF) and the rate constants k₀₁ (excretion), k₂₁ (ECF→ICF flux), and k₁₂ (ICF→ECF flux) were not significantly different between the groups.

Figure 3.

A and B: 2-compartmental (2-C) model fit to plasma increment in ⁴¹K-to-³⁹K ratio (Δ⁴¹K/³⁹K) in rats fed control (A) and K⁺- deficient (B) diets performed on MATLAB. Curves represent the averages of model fits in individual rats, and shaded areas represent 95% confidence intervals (i.e., mean ± 2 SE). Boxes indicate ⁴¹K infusion period. C and D: residuals (differences between observed and model-estimated values), expressed as % of estimated values, are shown for rats fed control (C) and K⁺-deficient (D) diets. None of the residuals is statistically different from 0 (2-tailed t test; P values were adjusted for multiple comparisons by the Benjamini–Hochberg method). Data are means ± SE. n = 4 for each group.

Figure 4.

Two-compartmental (2-C) model fit to plasma increment in ⁴¹K-to-³⁹K ratio (Δ⁴¹K/³⁹K) in individual rats (0–7) fed control (left) and K⁺-deficient (right) diets. Boxes indicate ⁴¹K infusion period.

Table 2.

Model parameters identified by fitting the 2-C model to the plasma profile of Δ⁴¹K/³⁹K in each rat

	Control Diet	K+-Deficient Diet	% Change
k 01	0.053 ± 0.017	0.050 ± 0.018	↓6%
k 21	0.235 ± 0.090	0.227 ± 0.061	↓3%
k 12	0.022 ± 0.003	0.016 ± 0.002	↓28%
KECF	1.10 ± 0.30	0.73 ± 0.25	↓34%
KICF	8.79 ± 1.21	8.54 ± 1.53	↓3%
KECF × k01	2.71 ± 0.49	1.57 ± 0.31*	↓42%
Urinary excretion#	0.067 ± 0.005	0.002 ± 0.001**	↓96%

Data are means ± SE (n = 4 for each). Four parameters {rate constants k₀₁ (renal K⁺ excretion) and k₂₁ and k₁₂ [K⁺ transport into and out of intracellular fluid (ICF)] and K_ECF [amount of K⁺ in extracellular fluid (ECF)]} were identified by fitting the model to the data, and K_ICF (amount of K⁺ in ICF) was calculated as K_ECF × (k₂₁/k₁₂) and renal excretion as K_ECF × k₀₁. Units are min⁻¹ for k₀₁, k₂₁, and k₁₂; mEq for K_ECF and K_ICF; and mEq/h for urinary excretion. Δ⁴¹K/³⁹K, change in ⁴¹K-to-³⁹K ratio. #Directly measured from urine collected from the bottom of cage. *P < 0.05, **P < 0.0001 vs. control (1-tailed t test).

Renal K⁺ Excretion: Model Estimated vs. Directly Measured

The results of the 2-C modeling also predicted that the rate of renal K⁺ excretion, calculated as K_ECF × k₀₁, was significantly lower in rats fed K⁺-deficient diet versus control rats (P < 0.05; Table 2), consistent with K⁺ conservation in rats fed K⁺-deficient diet (11–14). We compared these estimates with urinary K⁺ excretion rates directly determined by measuring total K⁺ (³⁹K + ⁴¹K) in the urine passed during the experimental period. Table 2 shows that urinary K⁺ excretion was indeed lower in rats fed K⁺-deficient diet, consistent with the differences predicted by the modeling. However, these data also showed that model estimates of K⁺ excretion were much greater than those directly measured from the urine, suggesting that the ⁴¹K disappearance from the ECF K⁺ pool may include fluxes other than renal K⁺ excretion (see discussion).

Modeling of Δ⁴¹K/³⁹K in RBCs

We applied a simple model to predict Δ⁴¹K/³⁹K in RBCs (Fig. 1B). This model assumes a single compartment for the RBC ICF with two rate constants for K⁺ influx (k_in) and efflux (k_out). Plasma bathes RBCs, and plasma Δ⁴¹K/³⁹K provokes Δ⁴¹K/³⁹K in RBCs. With plasma Δ⁴¹K/³⁹K as the input, the model identified the rate constants in each rat that produced dynamics of Δ⁴¹K/³⁹K best fitting the data (Fig. 5). k_in and k_out were estimated to be 0.0015 ± 0.0003 and 0.0045 ± 0.0026 min⁻¹, respectively, for the control group versus 0.0012 ± 0.0002 and 0.0033 ± 0.0021 min⁻¹, respectively, for the K⁺-deficient diet group. The estimated values for k_in and k_out were not statistically different between the two groups (P > 0.05).

Figure 5.

Model fit to red blood cell (RBC) increment in ⁴¹K-to-³⁹K ratio (Δ⁴¹K/³⁹K) in rats fed control (A) and K⁺- deficient (B) diets. Box indicates ⁴¹K infusion period. Data are means ± SE. n = 4 for each group.

3-C Modeling of Whole Body K⁺ Distribution and Fluxes

Although the 2-C model was able to fit the data, we noted that the time course of Δ⁴¹K/³⁹K predicted for the ICF pool was much faster than that observed in RBCs (Fig. 6). This may indicate a limitation of the 2-C model assuming a single ICF K⁺ pool; the ICF compartment in this minimal model may represent ICF K⁺ pools in fast equilibrium with the ECF but not those in slow equilibrium, as in RBCs. Therefore, the model may require an expansion to include at least two ICF compartments, representing not only tissues in fast equilibrium with the ECF (“fast pool”) but also those in slow equilibrium with the ECF (“slow pool”). However, including too many ICF pools would make it difficult to identify and validate model parameters; there is a trade-off between the model complexity and identifiability. A three-compartmental (3-C) model with two ICF compartments (Fig. 7) offers a representation of heterogeneous ICF K⁺ pools with a complexity allowing model identification. Although the 3-C model is theoretically identifiable, not all six parameters of the 3-C model were uniquely identified with the present data sets (data not shown). Identifying such a complex model may require data sets more enriched than the present ones, which may be attained by increasing the number of samplings, modifying the ⁴¹K infusion mode, optimizing the sampling schedule, and/or extending the observation period. To test this idea, we performed a Monte Carlo simulation to examine the effects of experimental protocols on the identification of the 3-C model’s parameters (see below).

Figure 6.

Change in ⁴¹K-to-³⁹K ratio (Δ⁴¹K/³⁹K) in plasma and red blood cells (RBCs) during and after ⁴¹K infusion (indicated by box), compared with the dynamics predicted by the 2-compartmental (2-C) model for the extracellular fluid (ECF) or intracellular fluid (ICF), in rats fed control (A) and K⁺-deficient (B) diets. Data are means ± SE. n = 4 for each group.

Figure 7.

An expansion of the 2-compartmental (2-C) model (Fig. 1) to a 3-compartmental (3-C) model of K⁺ distribution and fluxes in vivo to include a “slow” intracellular fluid (ICF) K⁺ pool. The 3-C model has 6 parameters: K_ECF, the amount of K⁺ in the extracellular fluid (ECF), and 5 rate constants [i.e., k₀₁ (renal K⁺ excretion), k₂₁ and k₁₂ (K⁺ transport into and out of fast ICF pool), and k₃₁ and k₁₃ (K⁺ transport into and out of slow ICF pool)]. Figure was created with BioRender.com.

Effects of Experimental Protocols on the Identification of the 3-C Model

The plasma Δ⁴¹K/³⁹K profile was simulated with the 3-C model. Parameter values used in this simulation were those identified by the 2-C and RBC modeling [i.e., K_ECF = 1.1 mEq (2-C model), k₁₂ = 0.022 min⁻¹ (2-C model), and k₁₃ = 0.0045 min⁻¹ (RBC model, k_out)]. In addition, k₀₁ (= 0.0015 min⁻¹) was estimated from measured renal excretion, and k₂₁ and k₃₁ were adjusted (0.195 and 0.040 min⁻¹, respectively) to produce plasma and slow pool Δ⁴¹K/³⁹K profiles similar to those observed in plasma and RBCs, respectively. We then added random noises at different levels (0–5%) to the simulated data, and the resulting noise-added data were analyzed to identify model parameters (see materials and methods).

Analyzing simulated data without noise resulted in estimated parameters identical to the original values (i.e., those used in simulation), confirming that the 3-C model is identifiable. Adding random noise to simulated data resulted in parameter values varying around the original values, and this variation increased as noise levels increased. Simulation results showed that the ECF’s (i.e., K_ECF) and fast pool’s (i.e., k₂₁ and k₁₂) model parameters were robustly identifiable even with the present sampling/experimental protocol. In contrast, the slow pool’s (i.e., k₃₁ and k₁₃) parameters were getting more difficult to accurately estimate as noise levels increased (Fig. 8). “Unidentifiability,” arbitrarily defined as the probability of identified parameters being more than three times different from the original values, increased significantly as noise levels increased, especially for k₁₃, the rate constant of the return flux from the slow pool. Both experiment duration and frequency of sampling (see Fig. 8) impacted the unidentifiability of k₁₃, with the former having more impact than the latter. These effects were additive to improve the identification of the slow pool parameters.

Figure 8.

Effects of increased number of samplings, experiment duration, or both compared with the current sampling schedule on identification of the slow pool’s parameters of the 3-compartmental (3-C) model [i.e., k₃₁ (K⁺ transport in; A) and k₁₃ (K⁺ transport out; B)], assessed by a Monte Carlo simulation (see materials and methods). In this simulation, all parameters of the 3-C model but k₀₁ (renal K⁺ excretion), which was fixed, were estimated from noise-added simulated data. The number of samplings and the experiment’s duration were increased 2-fold (i.e., from 10 to 20 and from 3 h to 6 h, respectively) without altering the ⁴¹K infusion period and rate. With the doubling of experiment duration and/or sampling number, sampling schedules were adjusted; both uniform and nonuniform (empirical) sampling schedules were tested, and the data presented here are with an empirical sampling schedule, which showed better performance than uniform sampling schedules. “Unidentifiability” was arbitrarily defined as the probability of estimated parameters being >3 times different from the true values.

To understand whether increasing experimental duration would improve the estimation of the return flux from the slow pool, we performed further computer simulations. The plasma ⁴¹K/³⁹K over time was simulated with the 3-C model (Fig. 7). The 2-C model parameters were estimated using the data simulated under the present sampling schedule (up to 180 min) (Fig. 9); the 2-C model fit the simulated data quite well up to 180 min. When the 2-C model was simulated beyond 180 min, the curve diverged from that simulated with the 3-C model. This difference may arise from the return flux of ⁴¹K from the slow pool, which occurs in the 3-C, but not the 2-C, model simulation; the difference increases significantly over time when the simulation is extended beyond 180 min (see discussion). Thus, additional data beyond 180 min would provide evidence for the slow pool and facilitate estimation of the slow pool parameters.

Figure 9.

The time course of plasma increment in ⁴¹K-to-³⁹K ratio (Δ⁴¹K/³⁹K) simulated using the 3-compartmental (3-C) or the 2-compartmental (2-C) model. The 3-C model was simulated using the parameters discussed in the text. The 2-C model parameters were estimated from the 3-C model-simulated data up to 180 min, i.e., under the present sampling schedule. Estimated 2-C model parameters were k₀₁ (renal K⁺ excretion) = 0.019 min⁻¹, k₂₁ [transport into intracellular fluid (ICF) pool] = 0.219 min⁻¹, k₁₂ (transport out of ICF pool) = 0.0218 min⁻¹, and K_ECF [amount of K⁺ in extracellular fluid (ECF)] = 1.09 mEq. The time courses were superimposed for 180 min but diverged subsequently, reflecting the return flux from the “slow” K⁺ pool in the simulation with the 3-C model (Fig. 7; see text).

DISCUSSION

The present study demonstrates that an intravenous infusion of a tracer amount of ⁴¹K in rats increased the ⁴¹K-to-³⁹K ratio (⁴¹K/³⁹K) in plasma and RBCs in a predictable manner without significantly increasing total (⁴¹K + ³⁹K) plasma K⁺ concentration. This approach is possible because of the extremely accurate and precise measurement of ⁴¹K/³⁹K (at the ∼0.01% or 0.1‰ level) afforded by inductively coupled plasma mass spectrometry (ICP-MS). In addition, the measured changes in ⁴¹K/³⁹K were amenable to analysis by compartmental models, permitting the estimation of model variables representing K⁺ distribution and fluxes in vivo. Furthermore, plasma ⁴¹K/³⁹K responses to ⁴¹K infusion were significantly different in rats fed K⁺-deficient diet versus control diet, and our analysis of these data indicated decreased renal K⁺ excretion in rats fed K⁺-deficient chow, as expected from previous studies (13, 29) and confirmed by directly measuring urinary K⁺ excreted and collected during these experiments. Thus, the present study introduces a new approach to quantify in vivo K⁺ distribution and fluxes, employing stable K⁺ isotopes, which can be used to identify mechanisms in (patho)physiological regulation of K⁺ homeostasis.

One limitation of the present study is that the employed sample size (n = 4) was too small to detect some expected effects of a low-K⁺ diet. However, the focus of the present study was to test the feasibility of the new approach rather than to investigate low-K⁺ diet effects, and the present data support that analyzing stable K⁺ isotope ratios after ⁴¹K infusion provides estimates of in vivo K⁺ distribution and fluxes. The results also indicate that although a 2-C model fits the data, it may not provide the complexity to characterize heterogeneous intracellular K⁺ pools, and a more complex (e.g., 3-C) model is required to accurately estimate rate constants (see below). Furthermore, simulation results suggest that experimental protocols can be optimized for identification of such a complex model by increasing experimental duration and/or sampling frequency. Thus, the present study provides convincing data to support the feasibility of the new approach and set a foundation for future studies to improve and optimize the model and experimental protocols to be used in studies of K⁺ homeostatic mechanisms in vivo.

Changes in plasma ⁴¹K/³⁹K induced by the ⁴¹K infusion were well described by the 2-C model of K⁺ distribution and fluxes. However, this single-ICF K⁺ pool model is a minimal model that exhibits limitations in accurately predicting certain aspects of K⁺ distribution and fluxes. For example, the 2-C model appears to underestimate the ICF K⁺ pool’s size; the model-estimated size of the ICF K⁺ pool was 8.8 mEq in control rats, which is only half of the expected 16.8 mEq K⁺ pool for 300-g rats [= 0.3 kg body wt × 0.4 L/kg body wt (ICF volume) × 140 mEq/L (ICF K⁺ concentration)]. One possibility is that not all ICF K⁺ is available for exchange with ECF ⁴¹K within the 3-h experimental period. Skeletal muscle, a major tissue for ICF K⁺ store, is highly structured and compartmentalized, and ICF K⁺ pools rapidly exchangeable with ECF K⁺ may be a fraction of its entire K⁺ pool. Another possibility is that the analysis using the 2-C model with a single ICF pool may detect ICF K⁺ pools in tissues in fast equilibrium with the ECF (fast pools) but not those in slow equilibrium with the ECF (slow pools), such as in RBCs. To support this idea, 2-C model analysis of 3-C model-simulated data (Fig. 9) showed that the estimated ICF pool size (10.9 mEq) was substantially smaller than the ICF pool size of the 3-C model (18.6 mEq), the sum of the fast and slow pools. Fast K⁺ pools may be readily detectable by analyzing plasma ⁴¹K/³⁹K, as a significant fraction of ⁴¹K transported into fast ICF pools later reappears in the ECF, impacting plasma ⁴¹K/³⁹K. In contrast, slow K⁺ pools may be difficult to detect, as ⁴¹K accumulation (or increases in ⁴¹K/³⁹K) in these pools would be small, as seen in RBCs, because of slow exchanges and/or large pools of ICF K⁺, and, as a result, these pools impact plasma ⁴¹K/³⁹K much less. Since ⁴¹K reappearance from these pools is difficult to detect over short time intervals, the ⁴¹K flux into the slow pool should be indistinguishable from irreversible loss from the ECF (e.g., renal excretion; see below).

Another limitation of the 2-C model was evident in model-estimated renal K⁺ excretion. Specifically, although the 2-C modeling, based on plasma ⁴¹K/³⁹K, accurately predicted the reduction in renal K⁺ excretion in rats fed K⁺-deficient diet, the model overestimated the real values of urinary K⁺ measured in collected urine from both control diet- and K⁺-deficient diet-fed rats (Table 2). These data suggest that the irreversible K⁺ loss from the ECF, quantified as renal K⁺ excretion in the model, may include fluxes into slow turnover K⁺ pools, as discussed above, in addition to renal K⁺ excretion, explaining the overestimation of renal K⁺ excretion with the 2-C modeling.

The findings suggest that a next step is to expand beyond a 2-C model to a more complex model that has at least two ICF compartments, representing tissues in fast equilibrium with the ECF (i.e., fast pools) and those in slow equilibrium (i.e., slow pools). Similar 3-C models have been widely used in whole body kinetic studies of many substrates and hormones (30, 31). The proposed 3-C model (Fig. 7) should not only better describe the dynamics of plasma ⁴¹K/³⁹K but also provide a more realistic representation of heterogeneous ICF K⁺ pools. However, not all six parameters of the 3-C model were uniquely identified with the data sets in the present study. As the 3-C model is theoretically identifiable, this finding suggests that the present experimental and sampling protocols are not optimal for identifying the 3-C model. Simulation studies showed that model parameters for the slow K⁺ pool (i.e., k₃₁ and k₁₃), particularly k₁₃, were difficult to identify with the present data sets. Increasing the number of samplings helped improve the identification of the slow pool parameters, but extending the observation period had a greater impact (Fig. 8). This makes sense intuitively, as ⁴¹K/³⁹K rises at a slower rate in the slow pool; thus the impact of ⁴¹K accumulation in slow K⁺ pools on plasma ⁴¹K/³⁹K would become more impactful with time. To support this idea, our simulation (Fig. 9) showed that the time courses of plasma ⁴¹K/³⁹K predicted by the 2-C and 3-C models diverged as the observation period was extended, reflecting the return flux from the slow pool. Thus, additional data from an extended period would help detect the flux from the slow pool. We are currently performing simulation using the 3-C model in a systematic way (i.e., with many combinations of experimental durations, sampling frequencies, and ⁴¹K infusion modes) to identify an experimental protocol that is both optimal for 3-C model identification and practical for experimentation. Once such a protocol is identified, we will need to validate it in animal experiments. Thus, the present study provides a foundation for the next streps to achieve the goal of accurate estimation of K⁺ distribution and fluxes in vivo.

We also introduce a novel approach for assessing K⁺ transport (or turnover) in RBCs. Measurements of time course of changes in the ⁴¹K/³⁹K in RBCs during and after ⁴¹K infusion can directly estimate K⁺ transport activities into and out of RBCs (k_in and k_out; Fig. 1B and Fig. 5). However, this approach does not allow the estimation of the RBC pool size. Pool sizes can be estimated in compartmental analysis if the input or flux into the pool (or the system) is known, which is the case in the whole body modeling. The RBC pool is one of many K⁺ pools into which the infused ⁴¹K goes, and there is no way to estimate the ⁴¹K flux into the RBC pool, making the RBC pool size indeterminate based on modeling. If RBCs are isolated and incubated in vitro, the RBC modeling should be able to estimate the size of the RBC pool, together with its turnover, as in this case ⁴¹K flux into the RBC pool can be estimated by analyzing the changes in the medium. Previous studies reported that Na-K-ATPase activity in RBCs is altered with obesity, hypertension, and diabetes (32–34). The present study may provide a new method for estimating Na-K-ATPase activity in RBCs in vivo, using k_in. Parameter identification in the RBC modeling was not as robust as that of the 2-C whole body modeling despite a simpler structure of the model with fewer parameters. The parameter identification algorithm often stopped at “local minimum,” and finding true solutions required multiple trials with different initial estimates of model parameters. One reason may be that, compared with the plasma data, changes in the ⁴¹K/³⁹K in RBCs were rather monotonous, revealing less of its dynamics (Fig. 2). Also, the signals (Δ⁴¹K/³⁹K) were much lower, causing low signal-to-noise (or high noise-to-signal) levels, making model identification less robust. Future studies are warranted to optimize experimental protocols for parameter identification and address the above issues with more confidence.

Δ⁴¹K/³⁹K could be measured in individual tissues at the end of the experiments (or certain times after ⁴¹K infusion) to reveal relative net K⁺ transport activities across tissues and classify these tissues into fast versus slow K⁺ pools. This approach would benefit determination of tissue-specific responses to stimuli (e.g., insulin or epinephrine). Analogous to the 2-deoxyglucose method for estimating glucose uptake by individual tissues in vivo (35, 36), K⁺ transport into individual tissues can be estimated as Δ⁴¹K/³⁹K normalized to the area under the plasma Δ⁴¹K/³⁹K profile. We predict that the liver with a high blood flow and a “leaky” endothelial wall will be a major fast pool of ICF K⁺ whereas skeletal muscle, the major body K⁺ store, will be a slow K⁺ pool, as it receives a low blood flow in resting states. As skeletal muscle is the major site of ECF K⁺ regulation, this may be another reason to explore the 3-C model to characterize the slow pool and its parameters.

In the present study, we assume that ⁴¹K behaves identically to ³⁹K (i.e., no isotope effects) to serve as a tracer for all K⁺ (i.e., ⁴¹K and ³⁹K). Previous studies (27, 37) reported variations in ⁴¹K/³⁹K in biological samples, suggesting isotope effects in biological processes. However, these variations were small, ranging <0.1‰, suggesting that possible isotope effects are small and ⁴¹K can be used to estimate K⁺ fluxes in vivo.

In summary, we explored the idea of determining K⁺ distributions and fluxes in vivo using stable K⁺ isotopes. We demonstrated that an intravenous infusion of tracer amounts of ⁴¹K in rats increased the ⁴¹K/³⁹K in plasma without altering total K⁺ concentration, because of the high sensitivity of the measurement of the ⁴¹K/³⁹K by MC-ICP-MS. We also demonstrated that the dynamics of the provoked changes in ⁴¹K/³⁹K could be analyzed by using a 2-C model to estimate K⁺ distribution and fluxes. These findings support the feasibility of the novel stable isotope approach to quantify K⁺ fluxes in vivo. Further studies can be designed to develop, and experimentally validate, an experimental protocol that allows the estimation of parameters for a more complex model (e.g., 3-C model) with heterogeneous ICF K⁺ pools. Ultimately, this new cutting-edge approach has the potential to answer important questions and thus fill gaps in our understanding of K⁺ homeostasis mechanisms in vivo.

SUPPLEMENTAL DATA

Supplemental Data: https://doi.org/10.6084/m9.figshare.17001139.v1.

GRANTS

This work was supported by an Innovative Pilot Grant from the University Kidney Research Organization at USC to J.H.Y., J.H., and A.A.M. This study was also supported by National Institute of Diabetes and Digestive and Kidney Diseases Grant R56 DK123780 (to A. A. M).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

J.H.Y., Y.T.O., A.A.M., and J.H. conceived and designed research; Y.T.O. and S.G. performed experiments; J.H.Y., Y.T.O., S.G., and J.H. analyzed data; J.H.Y., Y.T.O., S.G., A.A.M., and J.H. interpreted results of experiments; J.H.Y. prepared figures; J.H.Y. drafted manuscript; J.H.Y., Y.T.O., S.G., A.A.M., and J.H. edited and revised manuscript; J.H.Y., Y.T.O., S.G., A.A.M., and J.H. approved final version of manuscript.