Fig. 1A shows APs (membrane potential, V${}_{\text{m}}$) obtained after a simulated period of 10 min of stimulation at 50 min${}^{-1}$ under control conditions (‘baseline’, blue trace) and at 10, 20, and 50% hypernatremia (green, orange, and purple traces, respectively). Such a 10-minute period is sufficient to obtain steady-state or quasi-steady-state behavior at each level of hypernatremia and each stimulation rate tested. Fig. 1B–D shows the associated intracellular Na${}^{+}$, K${}^{+}$, and Ca${}^{\text{2+}}$ concentrations (denoted by [Na${}^{+}$]${}_{\text{i}}$, [K${}^{+}$]${}_{\text{i}}$, and [Ca${}^{\text{2+}}$]${}_{\text{i}}$, respectively). The cell shrinkage that is caused by the hyperosmosis of the extracellular solution results in increased levels of both [Na${}^{+}$]${}_{\text{i}}$ and [K${}^{+}$]${}_{\text{i}}$ (Fig. 1B,C). The diastolic resting level of [Ca${}^{\text{2+}}$]${}_{\text{i}}$ is hardly affected by the hypernatremia, but there is an increase in its systolic peak value with increasing levels of hypernatremia (Fig. 1D).

Fig. 1.

Effects of hypernatremia on the electrical activity of a single human ventricular cardiomyocyte (BPS2020 model) at a beating rate of 50 bpm. (A) Membrane potential (V${}_{\text{m}}$). (B) Intracellular Na${}^{+}$ concentration ([Na${}^{+}$]${}_{\text{i}}$). (C) Intracellular K${}^{+}$ concentration ([K${}^{+}$]${}_{\text{i}}$). (D) Intracellular Ca${}^{\text{2+}}$ concentration ([Ca${}^{\text{2+}}$]${}_{\text{i}}$). (E) Na${}^{+}$–K${}^{+}$ pump current (I${}_{\text{NaK}}$). (F) Na${}^{+}$–Ca${}^{\text{2+}}$ exchange current (I${}_{\text{NaCa}}$). (G) Transient outward K${}^{+}$ current (I${}_{\text{to}}$). (H) Rapid delayed rectifier K${}^{+}$ current (I${}_{\text{Kr}}$). (I) Slow delayed rectifier K${}^{+}$ current (I${}_{\text{Ks}}$). (J) Inward rectifier K${}^{+}$ current (I${}_{\text{K1}}$). (K) L-type Ca${}^{\text{2+}}$current (I${}_{\text{CaL}}$). (L) Fast Na${}^{+}$ current (I${}_{\text{Na}}$). The vertical arrow indicates the increase in I${}_{\text{Na}}$ amplitude with increasing levels of hypernatremia. (M) Late Na${}^{+}$ current (I${}_{\text{NaL}}$). (N) Background Na${}^{+}$ current (I${}_{\text{Nab}}$). Note the differences in current scales. The inset to (L) shows the threshold stimulus current (I${}_{\text{th}}$) at the different levels of hypernatremia. BPS2020, Bartolucci-Passini-Severi model as published in 2020; bpm, beats per minute.

The activity of the Na${}^{+}$–K${}^{+}$ pump is, on the one hand, enhanced by the increase in [Na${}^{+}$]${}_{\text{i}}$, but on the other hand, reduced by the increase in [K${}^{+}$]${}_{\text{i}}$ as well as by the cell shrinkage per se (as observed experimentally, as described in the Introduction section, and represented in the model as set out in the Materials and Methods section). The net effect is an increase in I${}_{\text{NaK}}$ (Fig. 1E), which is, however, insufficient to keep [Na${}^{+}$]${}_{\text{i}}$ at its baseline level. The activity of the Na${}^{+}$–Ca${}^{\text{2+}}$ exchanger is not only enhanced by the increase in [Ca${}^{\text{2+}}$]${}_{\text{i}}$ but also by the cell shrinkage per se (as observed experimentally, as described in the Introduction section, and represented in the model as set out in the Materials and Methods section). The net effect is an increase in I${}_{\text{NaCa}}$ (Fig. 1F). The time course of the transient outward current (I${}_{\text{to}}$; Fig. 1G), which is a K${}^{+}$ current, is only slightly dependent on the level of hypernatremia. This is because the activation of I${}_{\text{to}}$ is largely determined by the AP upstroke and early repolarization phases, which do not show a marked change (Fig. 1A). Its amplitude increases with increasing hypernatremia, among other things, due to the increase in its driving force by the hyperpolarization of the K${}^{+}$ equilibrium potential (E${}_{\text{K}}$) as a result of the increase in [K${}^{+}$]${}_{\text{i}}$ (Fig. 1C). The driving force of I${}_{\text{Kr}}$ and I${}_{\text{Ks}}$ is also increased. Yet, both currents show a decreased amplitude with increasing hypernatremia (Fig. 1H,I). This is largely due to the cell shrinkage-induced decrease in their fully activated conductance (as observed experimentally, as described in the Introduction section, and represented in the model as set out in the Materials and Methods section). I${}_{\text{K1}}$ increases with increasing hypernatremia (Fig. 1J), entirely due to its voltage dependence and the hypernatremia-induced hyperpolarization of V${}_{\text{m}}$ (Fig. 1A).

I${}_{\text{CaL}}$ shows a complex dependence on intracellular and extracellular ion concentrations and voltage. The net effect of the hypernatremia is a decrease in its amplitude (Fig. 1K). The fast Na${}^{+}$ current (I${}_{\text{Na}}$), on the other hand, shows an increase with increasing hypernatremia (Fig. 1L), which is largely due to the reduction in its steady-state inactivation due to the hyperpolarization of the resting membrane potential between consecutive APs (Fig. 1A). This increase per se would result in a faster activation of neighboring cells and an associated increase in conduction velocity. However, this is counteracted by reduced excitability, as reflected by the increase in threshold stimulus current (I${}_{\text{th}}$; Fig. 1L, inset). The late I${}_{\text{Na}}$ (I${}_{\text{NaL}}$) also shows an increase with increasing hypernatremia (Fig. 1M), which is also largely due to the reduction of its steady-state inactivation due to the hyperpolarization of the resting membrane potential between consecutive APs (Fig. 1A). The model cell has several other inward and outward currents, in addition to those shown in Fig. 1E–M. These include the background Na${}^{+}$, K${}^{+}$, and Ca${}^{\text{2+}}$ currents (I${}_{\text{Nab}}$, I${}_{\text{Kb}}$, and I${}_{\text{Cab}}$, respectively) and the sarcolemmal Ca${}^{\text{2+}}$ pump current (I${}_{\text{pCa}}$). As illustrated in Fig. 1N for I${}_{\text{Nab}}$, these other currents also depend on the level of hypernatremia through their dependence on ion concentrations and voltage. However, as illustrated in Fig. 1N, these currents are so small that they hardly contribute to the net membrane current.

The net effect of the hypernatremia-induced changes in ion concentrations (Fig. 1B–D) and membrane currents (Fig. 1E–N) is hyperpolarization and prolongation of the AP (Fig. 1A). The hyperpolarization amounts to 2.5, 4.8, and 10.3 mV under conditions of mild, severe, and extreme hypernatremia, respectively, whereas the AP duration (APD) at 90% repolarization (APD${}_{\text{90}}$) is increased by 7, 14, and 46%, respectively. As a direct effect of the increase in I${}_{\text{Na}}$ amplitude (Fig. 1L), the maximum AP upstroke velocity ((dV${}_{\text{m}}$/dt)${}_{\text{max}}$) is increased by 11, 18, and 23% under conditions of mild, severe, and extreme hypernatremia, respectively. At the same time, I${}_{\text{th}}$ is increased by 12, 25, and 55%, respectively (Fig. 1L, inset).

We repeated our simulations with the BPS2020 model at higher stimulation rates of 75 and 100 min${}^{-1}$. The results obtained at these two rates (Figs. 2,3) are qualitatively similar to those obtained at 50 min${}^{-1}$ (Fig. 1). Rate-dependent quantitative differences with respect to ion concentrations include a higher level of [Na${}^{+}$]${}_{\text{i}}$ (Figs. 2B,3B), higher peak amplitude of [Ca${}^{\text{2+}}$]${}_{\text{i}}$ (Figs. 2D,3D), and a higher activity of the Na${}^{+}$–K${}^{+}$ pump (Figs. 2E,3E). Rate-dependent quantitative differences concerning individual membrane currents include a decrease in I${}_{\text{to}}$ due to the smaller amount of time available between consecutive APs for its relatively slow recovery from inactivation and an increase in I${}_{\text{Ks}}$ due to the smaller amount of time available between consecutive APs for its relatively slow deactivation. The hypernatremia-induced AP hyperpolarization and prolongation observed at 50 bpm (Fig. 1A) were also examined at 75 bpm (Fig. 2A) and 100 bpm (Fig. 3A). The same holds for the hypernatremia-induced increase in I${}_{\text{Na}}$ amplitude (Figs. 2L,3L) and the associated increase in (dV${}_{\text{m}}$/dt)${}_{\text{max}}$, and the hypernatremia-induced increase in I${}_{\text{th}}$ (Fig. 2L, inset; Fig. 3L, inset).

Fig. 2.

Effects of hypernatremia on the electrical activity of a single human ventricular cardiomyocyte (BPS2020 model) at a beating rate of 75 bpm. (A) V${}_{\text{m}}$. (B) [Na${}^{+}$]${}_{\text{i}}$. (C) [K${}^{+}$]${}_{\text{i}}$. (D) [Ca${}^{\text{2+}}$]${}_{\text{i}}$. (E) I${}_{\text{NaK}}$. (F) I${}_{\text{NaCa}}$. (G) I${}_{\text{to}}$. (H) I${}_{\text{Kr}}$. (I) I${}_{\text{Ks}}$. (J) I${}_{\text{K1}}$. (K) I${}_{\text{CaL}}$. (L) I${}_{\text{Na}}$. The vertical arrow indicates the increase in I${}_{\text{Na}}$ amplitude with increasing levels of hypernatremia. (M) I${}_{\text{NaL}}$. (N) I${}_{\text{Nab}}$. Axis scales are identical to those in Fig. 1. The inset to (L) shows I${}_{\text{th}}$ at the different levels of hypernatremia. BPS2020, Bartolucci-Passini-Severi model as published in 2020; bpm, beats per minute.

Fig. 3.

Effects of hypernatremia on the electrical activity of a single human ventricular cardiomyocyte (BPS2020 model) at a beating rate of 100 bpm. (A) V${}_{\text{m}}$. (B) [Na${}^{+}$]${}_{\text{i}}$. (C) [K${}^{+}$]${}_{\text{i}}$. (D) [Ca${}^{\text{2+}}$]${}_{\text{i}}$. (E) I${}_{\text{NaK}}$. (F) I${}_{\text{NaCa}}$. (G) I${}_{\text{to}}$. (H) I${}_{\text{Kr}}$. (I) I${}_{\text{Ks}}$. (J) I${}_{\text{K1}}$. (K) I${}_{\text{CaL}}$. (L) I${}_{\text{Na}}$. The vertical arrow indicates the increase in I${}_{\text{Na}}$ amplitude with increasing levels of hypernatremia. (M) I${}_{\text{NaL}}$. (N) I${}_{\text{Nab}}$. Axis scales are identical to those in Figs. 1,2. The inset to (L) shows I${}_{\text{th}}$ at the different levels of hypernatremia. BPS2020, Bartolucci-Passini-Severi model as published in 2020; bpm, beats per minute.

As the Introduction mentions, the ToR–ORd and BPS2020 models can be considered major updates of the O’Hara et al. [30] “ORd” model. Since the ToR–ORd and BPS2020 models were developed largely independently and along different lines, meaning the simulation results obtained with the two models are not a priori highly similar, we repeated the above simulations using the ToR–ORd model. This model has default values for the extracellular Na${}^{+}$, K${}^{+}$, and Ca${}^{\text{2+}}$ concentrations (denoted by [Na${}^{+}$]${}_{\text{e}}$, [K${}^{+}$]${}_{\text{e}}$, and [Ca${}^{\text{2+}}$]${}_{\text{e}}$, respectively) of 140, 5.0, and 1.8 mmol/L, respectively, as opposed to 144, 5.4, and 2.7 mmol/L, respectively, in the BPS2020 model. Furthermore, unlike the BPS2020 model, it includes a Cl${}^{-}$ membrane current (I${}_{\text{Cl}}$). However, the intracellular Cl${}^{-}$ concentration cannot change dynamically, like [Na${}^{+}$]${}_{\text{i}}$, [K${}^{+}$]${}_{\text{i}}$, and [Ca${}^{\text{2+}}$]${}_{\text{i}}$ can. The equations describing the time or voltage dependence of individual membrane currents may differ between the two models. Moreover, parameters in these equations, such as the fully activated conductance of a specific current, may vary between the two models so that specific currents can have larger or smaller amplitudes in the ToR–ORd model than in the BPS2020 model and, thus, play a more or less important role in the ToR–ORd than in the BPS2020 model.

Fig. 4 shows the results obtained using the ToR–ORd model at a stimulation rate of 50 min${}^{-1}$. The format and time scale are identical to those in Figs. 1,2,3. However, not all of the ordinate scales are identical. Yet, without comparing the ordinate scales, it is immediately clear that the APs from the ToR–ORd and BPS2020 models differ in the presence of a notch. When comparing the ordinate scales of Fig. 4G and Fig. 1G, it is clear that I${}_{\text{to}}$ is approximately five times as large in the ToR–ORd model as in the BPS2020 model, giving way to a faster early repolarization and associated AP notch. At the same time, a comparison of the ordinate scales in Fig. 4I and Fig. 1I identifies that I${}_{\text{Ks}}$ is almost one order of magnitude smaller in the ToR–ORd model than in the BPS2020 model, which is important for our simulations because I${}_{\text{Ks}}$ is one of the currents that is reduced by cell shrinkage per se. Other remarkable differences are the [Na${}^{+}$]${}_{\text{i}}$ level, which reaches 23.3 mmol/L in the ToR–ORd model vs. 12.3 mmol/L in the BPS2020 model (Fig. 4B vs. Fig. 1B), and the absence of an increase in the amplitude of I${}_{\text{K1}}$ with an increase in the level of hypernatremia (Fig. 4J vs. Fig. 1J).

Fig. 4.

Effects of hypernatremia on the electrical activity of a single human ventricular cardiomyocyte (ToR–ORd model) at a beating rate of 50 bpm. (A) V${}_{\text{m}}$. (B) [Na${}^{+}$]${}_{\text{i}}$. (C) [K${}^{+}$]${}_{\text{i}}$. (D) [Ca${}^{\text{2+}}$]${}_{\text{i}}$. (E) I${}_{\text{NaK}}$. (F) I${}_{\text{NaCa}}$. (G) I${}_{\text{to}}$. (H) I${}_{\text{Kr}}$. (I) I${}_{\text{Ks}}$. (J) I${}_{\text{K1}}$. (K) I${}_{\text{CaL}}$. (L) I${}_{\text{Na}}$. The vertical arrow indicates the increase in I${}_{\text{Na}}$ amplitude with increasing levels of hypernatremia. (M) I${}_{\text{NaL}}$. (N) Chloride current (I${}_{\text{Cl}}$). Note that the ordinate scales are not identical to those in Figs. 1,2,3. The inset to (L) shows I${}_{\text{th}}$ at the different levels of hypernatremia. ToR–ORd, Tomek–Rodriguez model following the O’Hara–Rudy dynamic model; bpm, beats per minute.

Despite the remarkable differences in [Na${}^{+}$]${}_{\text{i}}$ level and some of the individual membrane currents, the net effects of the hypernatremia on the AP configuration are, at least qualitatively, quite similar to those observed using the BPS2020 model. The hypernatremia-induced AP hyperpolarization amounts to 2.3, 4.4, and 10.1 mV under conditions of mild, severe, and extreme hypernatremia, respectively, vs. values of 2.5, 4.8, and 10.3 mV in the BPS2020 model. The APD${}_{\text{90}}$ was increased by 6, 12, and 30% under mild, severe, and extreme hypernatremia conditions, whereas this prolongation amounted to 7, 14, and 46%, respectively, in the BPS2020 model. The (dV${}_{\text{m}}$/dt)${}_{\text{max}}$ was increased by 7, 11, and 12% under conditions of mild, severe, and extreme hypernatremia, respectively, whereas this increase amounted to 11, 18, and 23%, respectively, in the BPS2020 model. I${}_{\text{th}}$ showed an increase of 7, 15, and 34%, respectively, which was 12, 25, and 55%, respectively, in the BPS2020 model.

We repeated our simulations using the ToR–ORd model at higher stimulation rates of 75 and 100 min${}^{-1}$. The results obtained at these two rates (Figs. 5,6) are qualitatively similar to those obtained at 50 min${}^{-1}$ (Fig. 4). Rate-dependent quantitative differences with respect to ion concentrations include a somewhat higher level of [Na${}^{+}$]${}_{\text{i}}$ (Figs. 4B,5B,6B), the substantially higher peak amplitude of [Ca${}^{\text{2+}}$]${}_{\text{i}}$ (Figs. 4D,5D,6D), and a somewhat higher activity by the Na${}^{+}$–K${}^{+}$ pump (Figs. 4E,5E,6E). Rate-dependent quantitative differences with respect to individual membrane currents include a slight decrease in I${}_{\text{to}}$, as observed with the BPS2020 model. However, the rate-dependent increase in I${}_{\text{Ks}}$ is now only marginal (Figs. 4I,5I,6I). As mentioned, there is no increase in I${}_{\text{K1}}$ with increasing hypernatremia (Figs. 4J,5J,6J), in contrast to the BPS2020 model. This is due to differences in the current-voltage relationship of this current between the two models. The hypernatremia-induced AP hyperpolarization and prolongation observed at 50 bpm (Fig. 4A) were also examined at 75 bpm (Fig. 5A) and 100 bpm (Fig. 6A). The same holds for the hypernatremia-induced increase in I${}_{\text{Na}}$ amplitude (Figs. 5L,6L) and the associated increase in (dV${}_{\text{m}}$/dt)${}_{\text{max}}$, and the hypernatremia-induced increase in I${}_{\text{th}}$ (Fig. 5L, inset; Fig. 6L, inset).

Fig. 5.

Effects of hypernatremia on the electrical activity of a single human ventricular cardiomyocyte (ToR–ORd model) at a beating rate of 75 bpm. (A) V${}_{\text{m}}$. (B) [Na${}^{+}$]${}_{\text{i}}$. (C) [K${}^{+}$]${}_{\text{i}}$. (D) [Ca${}^{\text{2+}}$]${}_{\text{i}}$. (E) I${}_{\text{NaK}}$. (F) I${}_{\text{NaCa}}$. (G) I${}_{\text{to}}$. (H) I${}_{\text{Kr}}$. (I) I${}_{\text{Ks}}$. (J) I${}_{\text{K1}}$. (K) I${}_{\text{CaL}}$. (L) I${}_{\text{Na}}$. The vertical arrow indicates the increase in I${}_{\text{Na}}$ amplitude with increasing levels of hypernatremia. (M) I${}_{\text{NaL}}$. (N) I${}_{\text{Cl}}$. Axis scales are identical to those in Fig. 4. The inset to (L) shows I${}_{\text{th}}$ at the different levels of hypernatremia. ToR–ORd, Tomek–Rodriguez model following the O’Hara–Rudy dynamic model; bpm, beats per minute.

Fig. 6.

Effects of hypernatremia on the electrical activity of a single human ventricular cardiomyocyte (ToR–ORd model) at a beating rate of 100 bpm. (A) V${}_{\text{m}}$. (B) [Na${}^{+}$]${}_{\text{i}}$. (C) [K${}^{+}$]${}_{\text{i}}$. (D) [Ca${}^{\text{2+}}$]${}_{\text{i}}$. (E) I${}_{\text{NaK}}$. (F) I${}_{\text{NaCa}}$. (G) I${}_{\text{to}}$. (H) I${}_{\text{Kr}}$. (I) I${}_{\text{Ks}}$. (J) I${}_{\text{K1}}$. (K) I${}_{\text{CaL}}$. (L) I${}_{\text{Na}}$. The vertical arrow indicates the increase in I${}_{\text{Na}}$ amplitude with increasing levels of hypernatremia. (M) I${}_{\text{NaL}}$. (N) I${}_{\text{Cl}}$. Axis scales are identical to those in Figs. 4,5. The inset to (L) shows I${}_{\text{th}}$ at the different levels of hypernatremia. ToR–ORd, Tomek–Rodriguez model following the O’Hara–Rudy dynamic model; bpm, beats per minute.

The effects of hypernatremia in the two models of a single human ventricular cardiomyocyte are summarized in Figs. 7,8.

Fig. 7.

Effects of hypernatremia on the sodium and potassium equilibrium potentials and the peak intracellular calcium concentration in the BPS2020 and ToR–ORd models of a single human ventricular cardiomyocyte. (A) K${}^{+}$ equilibrium potential (E${}_{\text{K}}$) as a function of hypernatremia at stimulation rates of, from left to right, 50, 75, and 100 min${}^{-1}$. (B) Na${}^{+}$ equilibrium potential (E${}_{\text{Na}}$) as a function of hypernatremia at 50, 75, and 100 min${}^{-1}$ stimulation rates. (C) The peak amplitude of [Ca${}^{\text{2+}}$]${}_{\text{i}}$ as a function of hypernatremia at 50, 75, and 100 min${}^{-1}$ stimulation rates. Data from the BPS2020 and ToR–ORd models are shown by filled blue circles and filled purple squares, respectively. ToR–ORd, Tomek–Rodriguez model following the O’Hara–Rudy dynamic model; BPS2020, Bartolucci-Passini-Severi model as published in 2020; bpm, beats per minute.

Fig. 8.

Action potential (AP) parameters and threshold stimulus current of the BPS2020 and ToR–ORd models of a single human ventricular cardiomyocyte. (A) Resting membrane potential (RMP) as a function of hypernatremia at left to right, 50, 75, and 100 min${}^{-1}$ stimulation rates. (B) AP amplitude (APA) as a function of hypernatremia at stimulation rates of 50, 75, and 100 min${}^{-1}$. (C) AP duration at 50% repolarization (APD${}_{\text{50}}$) as a function of hypernatremia at stimulation rates of 50, 75, and 100 min${}^{-1}$. (D) AP duration at 90% repolarization (APD${}_{\text{90}}$) as a function of hypernatremia at stimulation rates of 50, 75, and 100 min${}^{-1}$. (E) Maximum AP upstroke velocity ((dV${}_{\text{m}}$/dt)${}_{\text{max}}$) as a function of hypernatremia at stimulation rates of 50, 75, and 100 min${}^{-1}$. (F) I${}_{\text{th}}$ as a function of hypernatremia at stimulation rates of 50, 75, and 100 min${}^{-1}$. Data from the BPS2020 and ToR–ORd models are shown by filled blue circles and filled purple squares, respectively. ToR–ORd, Tomek–Rodriguez model following the O’Hara–Rudy dynamic model; BPS2020, Bartolucci-Passini-Severi model as published in 2020; bpm, beats per minute.

In both models, the K${}^{+}$ and Na${}^{+}$ equilibrium potentials (E${}_{\text{K}}$ and E${}_{\text{Na}}$, respectively), as computed from the extracellular and intracellular K${}^{+}$ and Na${}^{+}$ concentrations, show a hyperpolarization, with a similar dependence on the level of hypernatremia at each of the stimulation rates tested (Fig. 7A,B). The hyperpolarization of E${}_{\text{K}}$ underlies the hypernatremia-induced hyperpolarization of the RMP in the two models. The hyperpolarization of E${}_{\text{K}}$ and E${}_{\text{Na}}$ is associated with changes in the driving force of individual membrane currents, which should be considered when studying the effects of hypernatremia on these currents, as performed in our simulations. Both models show a substantial increase in the peak amplitude of [Ca${}^{\text{2+}}$]${}_{\text{i}}$ with increasing levels of hypernatremia (Fig. 7C). At each of the beating rates tested, the two models show a highly similar dependence of this [Ca${}^{\text{2+}}$]${}_{\text{i}}$ peak amplitude on the level of hypernatremia.

Fig. 8 shows how the AP parameters and I${}_{\text{th}}$ of the BPS2020 and ToR–ORd model cardiomyocytes depend on the level of hypernatremia at each of the stimulation rates tested. As already noted, the resting membrane potential (RMP) shows a hyperpolarization that increases with increasing hypernatremia (Fig. 8A). This hyperpolarization is largely responsible for the observed increase in AP amplitude (APA; Fig. 8B). The hypernatremia-induced AP prolongation does not only translate into a hypernatremia-dependent increase in APD${}_{\text{90}}$ (Fig. 8D), as already noted in Sections 3.1 and 3.2, but also into a hypernatremia-dependent increase in the APD at 50% repolarization (APD${}_{50}$; Fig. 8C).

Fig. 8E demonstrates how (dV${}_{\text{m}}$/dt)${}_{\text{max}}$ increases with increasing hypernatremia, reflecting the hypernatremia-induced increase in I${}_{\text{Na}}$ amplitude. This increase was only small between 20 and 50% of the hypernatremia, particularly in the case of the ToR–ORd model, whereas I${}_{\text{th}}$ shows an almost linear dependence on the level of hypernatremia over the entire range of the hypernatremia tested. Consequently, AP conduction will be impaired at 50% hypernatremia as compared to lower levels of hypernatremia, provided that this high level of hypernatremia does not affect AP conduction in other ways, e.g., by cell shrinkage-induced structural perturbations of the nanodomains at the intercalated disks which are involved in cardiac conduction, and because of the localization of Na${}^{+}$ channels in the intercalated disks [35, 36].

As noted in the Introduction, there is an apparent discrepancy in the experimental data on the effects of acute exposure to a hyperosmotic solution on I${}_{\text{CaL}}$ (Table 1). Therefore, we repeated some of our simulations using I${}_{\text{CaL}}$ scaling factors other than the factor of 1.00, which was used in the simulations presented in Figs. 1,2,3,4,5,6,7,8. We selected the extreme case of 50% hypernatremia, where the effects are the most pronounced, and simulated both a decrease in I${}_{\text{CaL}}$ with a scaling factor of 0.72 and an increase with a scaling factor of 1.45. These scaling factors were derived from the experimental data of Ogura et al. [20] and Luo et al. [27], respectively (Table 1), disregarding our argument in the Introduction that the decrease in I${}_{\text{CaL}}$ amplitude in the study of Ogura et al. [20] was due to a rapid increase in [Ca${}^{\text{2+}}$]${}_{\text{i}}$ rather than a direct functional effect of the hyperosmosis on the I${}_{\text{CaL}}$ channels.

The results of our simulations are shown in Fig. 9, focusing on the effects on I${}_{\text{CaL}}$ and the associated effects on [Ca${}^{\text{2+}}$]${}_{\text{i}}$ and I${}_{\text{NaCa}}$. In both models, increasing the I${}_{\text{CaL}}$ scaling factor from 1.00 to 1.45 resulted in an increase in the inward peak of I${}_{\text{CaL}}$ (Fig. 9C,G), an increase in peak [Ca${}^{\text{2+}}$]${}_{\text{i}}$ (Fig. 9B,F), and an increase in the activity of the Na${}^{+}$–Ca${}^{\text{2+}}$ exchanger (Fig. 9D,H). As might be anticipated, decreasing the I${}_{\text{CaL}}$ scaling factor from 1.00 to 0.72 had the opposite effects. Interestingly, decreasing the I${}_{\text{CaL}}$ scaling factor from 1.00 to 0.72 still results in a substantial increase in peak [Ca${}^{\text{2+}}$]${}_{\text{i}}$ as compared to baseline in the BPS2020 model (Fig. 9B), but not in the ToR–ORd model, where there is a small decrease (Fig. 9F). Thus, there would be a positive effect on the contractile apparatus according to the BPS2020 model and a small negative effect according to the ToR–ORd model. Apparently, the ‘crossover’ from a positive to a negative effect has a different degree of decrease in the I${}_{\text{CaL}}$ scaling factor in the two models. In this regard, it should be noted that an increase in [Ca${}^{\text{2+}}$]${}_{\text{i}}$ compared to baseline, despite a decrease in the I${}_{\text{CaL}}$ scaling factor to 0.72, is largely due to the reduction in cell volume associated with the hypernatremia. Thus, even a substantially reduced amount of Ca${}^{\text{2+}}$ ions entering the cell or released from the sarcoplasmic reticulum can still increase [Ca${}^{\text{2+}}$]${}_{\text{i}}$, as in the BPS2020 model, or cause only a small decrease, as in the ToR–ORd model.

Fig. 9.

Effects of 50% hypernatremia and different scaling factors for I${}_{\text{𝐂𝐚𝐋}}$ on the electrical activity of the BPS2020 and ToR–ORd models for a single human ventricular cardiomyocyte at a beating rate of 75 bpm. (A) V${}_{\text{m}}$, (B) [Ca${}^{\text{2+}}$]${}_{\text{i}}$, (C) I${}_{\text{CaL}}$, and (D) I${}_{\text{NaCa}}$ in the BPS2020 model. (E) V${}_{\text{m}}$, (F) [Ca${}^{\text{2+}}$]${}_{\text{i}}$, (G) I${}_{\text{CaL}}$, and (H) I${}_{\text{NaCa}}$ in the ToR–ORd model. Note the differences in the ordinate scales. ToR–ORd, Tomek–Rodriguez model following the O’Hara–Rudy dynamic model; BPS2020, Bartolucci-Passini-Severi model as published in 2020; bpm, beats per minute.

When simulating hypernatremia, we incorporated both the hypernatremia-induced cell shrinkage and the hypernatremia-induced changes in I${}_{\text{Kr}}$, I${}_{\text{Ks}}$, I${}_{\text{NaCa}}$, and I${}_{\text{NaK}}$ using the scaling factors listed in Table 2. To test the effects of cell shrinkage and ion current scaling per se, we simulated two hypothetical cases of hypernatremia, one in the absence of cell shrinkage and one in the absence of ion current scaling. As in the simulations presented in Fig. 9, we selected the extreme 50% level of hypernatremia, where the effects are the most pronounced. The results of our simulations are shown in Fig. 10, focusing on [Na${}^{+}$]${}_{\text{i}}$, [K${}^{+}$]${}_{\text{i}}$, and [Ca${}^{\text{2+}}$]${}_{\text{i}}$, and the associated I${}_{\text{NaK}}$ and I${}_{\text{NaCa}}$.

Fig. 10.

Effects of 50% hypernatremia in the presence and absence of ion current scaling and the presence and absence of cell shrinkage on the electrical activity of the BPS2020 and ToR–ORd models for a single human ventricular cardiomyocyte at a beating rate of 75 bpm. (A) V${}_{\text{m}}$, (B) [Ca${}^{\text{2+}}$]${}_{\text{i}}$, (C) [K${}^{+}$]${}_{\text{i}}$, (D) [Na${}^{+}$]${}_{\text{i}}$, (E) I${}_{\text{NaK}}$, and (F) I${}_{\text{NaCa}}$ in the BPS2020 model. (G) V${}_{\text{m}}$, (H) [Ca${}^{\text{2+}}$]${}_{\text{i}}$, (I) [K${}^{+}$]${}_{\text{i}}$, (J) [Na${}^{+}$]${}_{\text{i}}$, (K) I${}_{\text{NaK}}$, and (L) I${}_{\text{NaCa}}$ in the ToR–ORd model. Note the differences in the ordinate scales. Note also that the green and orange traces coincidentally overlap almost completely in both (D) and (J). ToR–ORd, Tomek–Rodriguez model following the O’Hara–Rudy dynamic model; BPS2020, Bartolucci-Passini-Severi model as published in 2020; bpm, beats per minute.

Comparison of the APs obtained under baseline conditions and upon 50% hypernatremia without ion current scaling with those obtained during 50% hypernatremia with and without cell shrinkage (Fig. 10A,G) reveals that the hypernatremia-induced AP prolongation is largely determined by the ion current scaling, which includes scaling factors of 0.6, 0.5, and 0.333 for each of the repolarizing currents I${}_{\text{Kr}}$, I${}_{\text{Ks}}$, and I${}_{\text{NaK}}$, respectively (Table 2). Similarly, a comparison of the intracellular ion concentrations obtained under the different conditions (Fig. 10B–D,H–J) shows that the changes in [Ca${}^{\text{2+}}$]${}_{\text{i}}$, [K${}^{+}$]${}_{\text{i}}$, and [Na${}^{+}$]${}_{\text{i}}$ are largely, but certainly not completely, determined by the cell shrinkage. In particular, in the case of [Na${}^{+}$]${}_{\text{i}}$ (Fig. 10D,J), the ion current scaling has a strong effect. For example, in the ToR–ORd model, the hypernatremia induces an increase in [Na${}^{+}$]${}_{\text{i}}$ from its baseline value of 12.6 mmol/L to 24.7 mmol/L in the presence of the ion current scaling, which is a substantially smaller increase to 17.7 mmol/L in the absence of the ion current scaling (Fig. 10J). This effect is not highly surprising, given the hypernatremia-induced decrease in I${}_{\text{NaK}}$ and increase in I${}_{\text{NaCa}}$ (Table 2), which correspond to a reduced activity of the Na${}^{+}$–K${}^{+}$ pump and an enhanced activity of the Na${}^{+}$–Ca${}^{\text{2+}}$ exchanger, respectively, both of which tend to increase [Na${}^{+}$]${}_{\text{i}}$, in the presence of the ion current scaling. The hypernatremia-induced decrease in I${}_{\text{NaK}}$ and increase in I${}_{\text{NaCa}}$ are not immediately apparent from a direct comparison of the I${}_{\text{NaK}}$ traces in Fig. 10E,K and the I${}_{\text{NaCa}}$ traces in Fig. 10F,L. However, it should be noted that I${}_{\text{NaK}}$ and I${}_{\text{NaCa}}$ depend on the intracellular ion concentrations, which reached different levels under the four conditions tested.

Owing to the many effects of acute hypernatremia and the associated cell shrinkage and changes in intracellular ion concentrations on individual membrane currents, it is difficult to predict, if not qualitatively, then at least quantitatively, how acute hypernatremia will affect the ventricular AP. This is where comprehensive computer models of the ventricular cardiomyocyte come into play. With such models, it is possible to determine and understand the effects of different levels of acute hypernatremia on the individual membrane currents and their net effects on the ventricular AP. In the present study, we used two different comprehensive computer models of an isolated human ventricular cardiomyocyte to assess the effects of mild to extreme hypernatremia on the electrophysiology of such a cardiomyocyte. We observed a hyperpolarization of the RMP, a prolongation of the AP, an increase in (dV${}_{\text{m}}$/dt)${}_{\text{max}}$, and an increase in I${}_{\text{th}}$ at all levels of hypernatremia. The magnitude of these effects increased with increasing levels of hypernatremia.

Experimental data on the cardiac effects of acute hypernatremia at the cellular level are scarce. What we do know from the work of Bou-Abboud and Nattel [37] is that canine Purkinje fibers show small but statistically highly significant increases in their APD${}_{50}$ (+12.0%) and APD${}_{\text{95}}$ (+5.4%) and in their (dV${}_{\text{m}}$/dt)${}_{\text{max}}$ (+4.7%) when [Na${}^{+}$]${}_{\text{e}}$ is increased from 141 to 161 mmol/L (14% hypernatremia). Our simulation results at 10–20% hypernatremia, albeit for human ventricular cardiomyocytes rather than canine Purkinje fibers, correlate well with these experimental observations. More experimental data have been obtained on the effects of hyperosmotic extracellular solutions on the AP of cardiac myocytes at the cellular level, although these were obtained with sucrose-induced hyperosmosis rather than hypernatremia. From the 1997 study by Ogura et al. [20], we know that guinea pig ventricular cardiomyocytes, when stimulated at 1 Hz, show an increase in APD${}_{\text{90}}$ of 10 $\pm$ 3% (mean $\pm$ SEM, n = 4) and 11 $\pm$ 2% (n = 5) in 20% and 50% hyperosmotic solutions, respectively. In a more recent study, also using guinea pig ventricular cardiomyocytes stimulated at 1 Hz, Ogura et al. [22] observed a 16.7 $\pm$ 2.4% increase in APD${}_{\text{90}}$ (mean $\pm$ SEM, n = 5) in a 50% hyperosmotic solution as well as a 6.6 $\pm$ 0.2 mV hyperpolarization of the RMP. Qualitatively, such an increase in [K${}^{+}$]${}_{\text{i}}$ is supported by the –2.8 $\pm$ 0.3 and –7.3 $\pm$ 0.7 mV (mean $\pm$ SEM, n = 9) hyperpolarizing shifts in the I${}_{\text{K1}}$ reversal potential for 30% and 80% hyperosmotic solutions [23]. Although obtained with guinea pig rather than human ventricular cardiomyocytes and with a sucrose-induced hyperosmotic extracellular solution rather than a hypernatremic one, these experimental data on AP prolongation and RMP hyperpolarization correlate well with our simulation results. One would expect hyperosmosis and hypoosmosis to have opposite effects on APD, and this is indeed the case. Both Groh et al. [38] and Kocic et al. [39] found a decrease in APD${}_{\text{90}}$ in guinea pig ventricular cardiomyocytes upon hypoosmosis.

Ogura et al. [22] attributed the hyperpolarization of the RMP to an increase in [K${}^{+}$]${}_{\text{i}}$ in the osmotically shrunken cardiomyocytes, as also proposed by Missan et al. [23]. This is supported by our simulation results, which show an increase in both models. The increase in [K${}^{+}$]${}_{\text{i}}$ is even quantitatively very similar in the two models, which is, however, less the case for the absolute increase in [Na${}^{+}$]${}_{\text{i}}$. In this regard, we have to note that there is already a substantial difference in [Na${}^{+}$]${}_{\text{i}}$ between the two models at baseline, in contrast to the baseline level of [K${}^{+}$]${}_{\text{i}}$. For example, at 75 bpm, [Na${}^{+}$]${}_{\text{i}}$ is 12.6 mmol/L in the ToR–ORd model and 6.6 mmol/L in the BPS2020 model. These values increase to 24.7 and 12.8 mmol/L, respectively, with 50% hypernatremia. Thus, a near doubling of [Na${}^{+}$]${}_{\text{i}}$ can be observed in both models.

As mentioned in the Introduction, data on the direct effects of acute hypernatremia on individual membrane currents of cardiac myocytes are lacking. Data on changes in individual membrane currents of single cardiac myocytes that are acutely exposed to hyperosmotic solutions induced by the addition of sucrose or mannitol rather than hypernatremia are not equivocal, particularly in the case of I${}_{\text{CaL}}$. This is even more the case when data on hypoosmosis are considered. One would expect hyperosmosis and hypoosmosis to have opposite effects on individual ion currents, yet this is not always the case. From the data summarized in Table 1, one would expect an increase in both I${}_{\text{Kr}}$ and I${}_{\text{Ks}}$ upon hypoosmosis, while I${}_{\text{K1}}$ is unaffected. Groh et al. [38] reported no significant changes in I${}_{\text{Kr}}$ or I${}_{\text{K1}}$ in guinea pig ventricular cardiomyocytes exposed to a 20% hypoosmotic solution, whereas I${}_{\text{Ks}}$ increased (as expected) by $\approx$40%. Kocic et al. [39] also observed an increase in I${}_{\text{Ks}}$. The application of a 70% hypoosmolar bath solution by Sasaki et al. [26] increased I${}_{\text{Kr}}$ (as expected), whereas no noticeable change in I${}_{\text{K1}}$ was observed, which was in agreement with their observations with a 130% hyperosmolar solution. Furthermore, Rees et al. [40] provided evidence that cell swelling enhanced I${}_{\text{Ks}}$ (as expected) while inhibiting (rather than increasing) I${}_{\text{Kr}}$. Thus, these data on I${}_{\text{Ks}}$ and I${}_{\text{K1}}$ in response to hypoosmosis are entirely consistent with the data in response to hyperosmosis. However, the data on I${}_{\text{Kr}}$ are not: I${}_{\text{Kr}}$ is either increased, decreased, or unaffected by hypoosmosis.

From the data summarized in Table 1, one would also expect a decrease in I${}_{\text{NaCa}}$ and an increase in I${}_{\text{NaK}}$ in response to hypoosmosis. In about half of their cells, Sasaki et al. [26] observed a large increase in I${}_{\text{NaK}}$ upon superfusion with a hypoosmolar bath solution, consistent with their observations with a 130% hyperosmolar solution. Whalley et al. [25] also observed stimulation of the Na${}^{+}$–K${}^{+}$ pump during exposure to hypoosmolar solutions, consistent with their observations with hyperosmolar solutions. Wright et al. [29] showed a decrease in I${}_{\text{NaCa}}$ in response to 1.3-fold hypoosmotic treatment, consistent with their observations with hyperosmolar solutions. Thus, all of these data on I${}_{\text{NaCa}}$ and I${}_{\text{NaK}}$ in response to hypoosmosis are consistent with the data in response to hyperosmosis.

There are also some data on I${}_{\text{CaL}}$ in response to hypoosmosis. Groh et al. [38] reported no significant changes in I${}_{\text{CaL}}$ (measured with 10 mmol/L EGTA in the pipette solution) in guinea pig ventricular cardiomyocytes exposed to a 20% hypoosmotic solution. Similarly, Sasaki et al. [26] observed no apparent change in I${}_{\text{CaL}}$ (measured with 5.0 mmol/L EGTA or 10 mmol/L BAPTA) upon application of a 70% hypoosmolar bath solution, in agreement with their observations with a 130% hyperosmolar solution. Thus, these data on I${}_{\text{CaL}}$ in response to hypoosmosis support the hyperosmosis data of Ogura et al. [20] and Sasaki et al. [26] rather than those of Luo et al. [27].

Unfortunately, experimental data on the effects of hyperosmosis, let alone hypernatremia, on individual membrane currents are limited to the currents listed in Tables 1,2. Most importantly, there are no data on the effects of hyperosmosis on I${}_{\text{to}}$ and I${}_{\text{Na}}$. If such data had been available, we would have incorporated these into our models. Experimental data on the effects of hyperosmosis on intercellular coupling are also lacking, making conduction studies in strand or tissue models uncertain.

There are not many systematic studies of the effects of acute hypernatremia on the ECG. In a study of seven anesthetized dogs, Eliakim et al. [41] regularly observed bradycardia, a decrease in P wave amplitude, an increase in QRS complex amplitude, and QT prolongation after intravenous administration of hypertonic saline to induce acute hypernatremia. Gibson et al. [42] studied ECG changes in 14 anesthetized dogs upon intravenous infusion of sodium chloride (increase in serum Na${}^{+}$ levels of 28–56%, averaging 41%) and consistently observed an increase in the rate-corrected QT interval (3–26%, averaging 11%) and a decrease in the amplitude of the P wave and the QRS complex, without QRS prolongation, and with variable minor changes in rate and negligible changes in PR interval. Importantly, these ECG changes occurred with only a 5–15% increase in serum Na${}^{+}$ levels, and only one dog presented an arrhythmia, which was transient and occurred at the start of the infusion. In a study of 20 rabbits subjected to extreme acute hypernatremia, electrocardiographic tracings remained normal until respiratory arrest, “except in a few instances in which peaked T waves were observed in the terminal stages” [43]. Overall, these in vivo data from laboratory animals are inconsistent, except perhaps for increased QT prolongation. The inconsistencies may be related, at least in part, to the level of hypernatremia and a reduction in P and QRS amplitude due to alterations in blood conductivity with a short-circuiting effect on myocardial potentials, as discussed by Gibson et al. [42], although this cannot fully explain the conflicting data. An increase in QT prolongation is consistent not only with the increase in APD that was observed in vitro (see Section 4.1) and that we observed in silico (Fig. 8C,D) but also with clinical and experimental ECG data in case reports of the effects of (often extreme) acute hypernatremia, including a prolonged rate-corrected Q-U interval in a 12-year-old girl [44], QT prolongation in an 11-year-old girl [9], and QT prolongation in a 29-year-old woman [8].

Hypernatremia may also lead to left ventricular (LV) contractile dysfunction. In a prospective cohort study of subarachnoid hemorrhage patients, Fisher et al. [6] observed that hypernatremia was an independent predictor of a reduced LV ejection fraction of 50%. In line with this observation, King et al. [45] demonstrated that elevating [Na${}^{+}$]${}_{\text{e}}$ from 145 to 155 mmol/L in Langendorff-perfused isolated rat heart preparations decreased LV developed pressure (LVdP). Fisher et al. [6] hypothesized that the elevated [Na${}^{+}$]${}_{\text{e}}$ causes more Ca${}^{\text{2+}}$ to exit the cell via the sarcolemmal Na${}^{+}$–Ca${}^{\text{2+}}$ exchanger, which then results in reduced levels of [Ca${}^{\text{2+}}$]${}_{\text{i}}$ available for cardiac myocyte contraction, thus causing a negative inotropic effect. On the other hand, hypernatremia ranging from 163 to 218 mmol/L in 10 anesthetized dogs resulted in statistically significant increases in cardiac output, heart rate, and maximum rate of rise in LVdP in the study by Goodyer et al. [46].

The increase in the systolic peak amplitude of [Ca${}^{\text{2+}}$]${}_{\text{i}}$ that we found in our simulations is consistent with the findings of Goodyer et al. [46], especially when we compare the peak amplitude of [Ca${}^{\text{2+}}$]${}_{\text{i}}$ obtained at 20–50% hypernatremia and a beating rate of 75–100 bpm with the baseline peak [Ca${}^{\text{2+}}$]${}_{\text{i}}$ at 50 bpm to account for the increase in heart rate that accompanies the hypernatremia. At 10% hypernatremia, which best reflects mild hypernatremia, our simulations still predict an increase in peak [Ca${}^{\text{2+}}$]${}_{\text{i}}$, although less pronounced, which seems to contradict the observations of Fisher et al. [6] and King et al. [45] that mild hypernatremia impairs LVdP. This may indicate a shortcoming in our models; however, it may also point to a negative effect on the contractile apparatus that requires a higher increase in peak [Ca${}^{\text{2+}}$]${}_{\text{i}}$, as obtained at 20–50% hypernatremia, to be compensated. Notably, the hypothesis of Fisher et al. [6] regarding a hypernatremia-induced reduction in the level of [Ca${}^{\text{2+}}$]${}_{\text{i}}$ available for cardiac myocyte contraction is not supported by our simulations.

In our simulations, there is an increase in peak [Ca${}^{\text{2+}}$]${}_{\text{i}}$ at all levels of hypernatremia, yet it is most prominent at 50% hypernatremia. For example, in the BPS2020 model, with a beating rate of 75 bpm, the peak [Ca${}^{\text{2+}}$]${}_{\text{i}}$ is 0.704 µmol/L at 50% hypernatremia compared to 0.412 µmol/L at baseline (+71%; Fig. 2D). Yet, the associated changes in I${}_{\text{CaL}}$ and I${}_{\text{NaCa}}$ do not seem to be large enough to explain this increase in [Ca${}^{\text{2+}}$]${}_{\text{i}}$ fully. Early in the AP, there is an increase in reverse mode I${}_{\text{NaCa}}$, making more Ca${}^{\text{2+}}$ ions enter the cell, but on the other hand, there is a decrease in peak I${}_{\text{CaL}}$ (Fig. 2F,K). In this regard, it should be noted that the increase in [Ca${}^{\text{2+}}$]${}_{\text{i}}$ with increasing levels of hypernatremia is largely due to the associated decrease in cell volume. The free Ca${}^{\text{2+}}$ ions reside in a 32% smaller myoplasmic volume compared to baseline, which per se results in a 47% increase in [Ca${}^{\text{2+}}$]${}_{\text{i}}$. Thus, even a reduced amount of Ca${}^{\text{2+}}$ ions entering the cell or released from the sarcoplasmic reticulum can still increase [Ca${}^{\text{2+}}$]${}_{\text{i}}$.

It should be noted that in all of the above, the focus is on acute hypernatremia, with the associated abrupt cell shrinkage; however, the more common hypernatremia in the clinic is a more gradually developing and chronic hypernatremia. In this context, the changes in intracellular volume and osmolarity and those in individual membrane currents will be much smaller. It is likely that, as a consequence, the effects of this type of hypernatremia will be much less pronounced than in our simulations. Unfortunately, there are no cellular electrophysiological data on this type of hypernatremia to support this hypothesis. Furthermore, in the clinical management of the more gradually developing hypernatremia, it is important to consider not only the absolute level of the serum Na${}^{+}$ concentration but also the time of development of the hypernatremia because too slow or too rapid corrections of the hypernatremia are both associated with a poor patient prognosis [47, 48].

Another point of attention is the extreme 50% hypernatremia we used in our simulations. Such a high level of hypernatremia is clinically limited to a few cases reported worldwide [10, 11, 12, 13, 14, 15] and some of the cases in the commonly referenced study by Finberg et al. [49], in which they describe the dramatic mix-up of salt and sugar in the preparation of feedings that were received by 14 hospitalized infants, six of whom died. The highest level of Na${}^{+}$ observed in the latter study was as high as 274 mmol/L (with a non-fatal outcome).

Our computer simulations were performed using the default versions of the ToR–ORd and BPS2020 models. It should be noted that both models represent endocardial cardiomyocytes with their default settings. However, epicardial and mid-myocardial versions of each model are also available. In both models, changing the cell type affects the kinetics and the amplitude of I${}_{\text{to}}$, the amplitude of I${}_{\text{CaL}}$, I${}_{\text{Kb}}$, I${}_{\text{Kr}}$, I${}_{\text{Ks}}$, I${}_{\text{K1}}$, I${}_{\text{NaCa}}$, I${}_{\text{NaK}}$, and I${}_{\text{NaL}}$, and parameters related to the uptake and release of Ca${}^{\text{2+}}$ ions by the sarcoplasmic reticulum. Given the highly similar simulation data obtained using the two models, despite differences in the amplitude and kinetics of individual ion currents, it is highly unlikely that substantially different simulation results will be obtained when repeating our simulations with the epicardial or mid-myocardial versions of each model.

It should be recognized that experimental data on the effects of acute exposure to a hyperosmotic solution on individual cardiac ion currents are scarce and obtained at different levels of hyperosmolarity (Table 1), so we had to estimate several of the values for use in our simulations (Table 2). We did this using linear interpolation and extrapolation, which we considered the best possible option. However, there is no evidence that the observed changes in individual ion currents are actually linearly dependent on the level of hyperosmolarity over the range studied.

One might anticipate cell shrinkage to affect the membrane capacitance. However, Ogura et al. [22] carried out specific experiments on this issue in which the membrane capacitance of seven guinea pig ventricular cardiomyocytes was monitored during sequential superfusion with control, 50% hyperosmotic, and 50% hypoosmotic solutions (varying sucrose levels). The superfusion with the anisosmotic solutions caused cell shrinkage and swelling but did not affect the membrane capacitance. Based on the experimental data of Ogura et al. [22], we did not change the membrane capacitance in our simulations of hypernatremia. However, it may well be that human ventricular cardiomyocytes show a change in membrane capacitance upon acute hypernatremia that we have not accounted for in our simulations.

It should be noted that our study is limited to the electrophysiological effects of acute hypernatremia on a single ventricular cardiomyocyte (for which experimental data on the effects of the hyperosmolarity are available, as summarized in Table 1). As such, it is difficult to predict the effects of acute hypernatremia on the tissue or whole-heart level from our simulations. For that, data are required on the effects of acute hypernatremia on other cell types (sinoatrial, atrial, atrioventricular, Purkinje) and intercellular coupling.

It can be argued that hypernatremia-induced cell shrinkage affects stretch-activated ion channels in the cell membrane, thereby affecting the AP and ion flow across the cell membrane. However, although very comprehensive, the two models we used do not include stretch-activated channels. To overcome this limitation, we should have implemented the various stretch-activated ion channels in the two models and calibrated the resulting extended models to experimental data, as was recently performed for the ToR–ORd model by Buonocunto et al. [50].

Both hypoosmotic and hyperosmotic stress can induce the cardiac T-tubules to seal, which could dramatically alter the Ca${}^{\text{2+}}$ handling and AP propagation [51], although this was not considered in our study. However, this sealing is a threshold-dependent process, as observed by Uchida et al. [52] in a study on isolated mouse left ventricular cardiomyocytes. In the case of hyperosmolarity induced by the addition of extra NaCl to the extracellular Tyrode’s solution, the threshold for the sealing effect is at a hyperosmolarity of $\approx$65 mOsm/L [52], i.e., at $\approx$22% hypernatremia. This suggests that our simulation results obtained at hypernatremia levels $>$20% should be considered with some caution regarding the potential role of T-tubular sealing.