Electrolyte and pH Dependence of Heart Rate During Hemodialysis: A Computer Model Analysis

Electrolyte and pH Dependence of Heart Rate DuringHemodialysis: A Computer Model Analysis

Stefano Severi and Silvio Cavalcanti

Biomedical Engineering Laboratory, DEIS, University of Bologna, Italy

Abstract: The influence of hemodialysis-induced modifi-cations in extracellular fluid characteristics on heart ratewas investigated by using a detailed computer model ofsinus-node electrical activity. Changes similar to those oc-curring in the course of hemodialysis in extracellular con-centrations of sodium (from 138 to 140 mM), potassium(from 6 to 3.3 mM), and calcium (from 1.2 to 1.5 mM) ionsas well as in pH (from 7.31 to 7.4) and intracellular volumewere simulated. The model predicted that such changesmay largely influence the rhythm of the sinoatrial nodepacemaker, causing the heart rate to range from 69 to 86bpm. Heart rate increases after removing potassium (up to7 bpm) and also after calcium perfusion (up to 11 bpm)whereas restoring pH slows heart beat (up to 6 bpm). Ex-

tracellular sodium has no significant influence, but theheart rate strictly depends on intracellular sodium concen-tration (5 bpm/mM). A complex dependence of heart rateon electrolytes and pH was also recognized. Providing ex-tracellular potassium concentration is maintained above 5mM, heart rate exhibits low sensitivity to changes in cal-cium and potassium. When potassium concentration is re-duced below 4.5 mM, heart rate sensitivity to calcium andpotassium increases significantly to 10 and 30 bpm/mM,respectively. A sustained increase in heart rate always cor-responds to an increase in intracellular sodium concentra-tion. Key Words: Sinoatrial node—Artificial kidney—Cardiac period—Heart rate sensitivity—Ion concentration.

The artificial kidney is largely used to control thetotal volume and composition of body fluids in pa-tients suffering from severe renal failure. In particu-lar, the artificial kidney removes excess body wateraccumulated during the interdialytic period andtakes electrolyte concentrations and pH as close aspossible to their physiological values. Plasma waterdepletion as well as electrolyte and pH changes havea large impact on the cardiovascular system in asmuch as mechanisms devoted to controlling cardio-vascular functions can fail to maintain hemodynamicstability during hemodialysis. Cardiovascular insta-bility with symptomatic hypotension is still one ofthe most important and frequent complications ofhemodialysis, occurring in 30% of dialyses overall(1–4), and cardiovascular disturbances are one of themain causes of morbidity and mortality of patientsundergoing chronic hemodialysis.

The chief stress for the cardiovascular system, dur-ing hemodialysis, is the reduction in circulating

blood volume (about 15%) caused by ultrafiltrationof plasma water through the dialyzer. Such volumereduction also depends on osmotic refilling deter-mined by electrolyte concentrations, and circulatingblood volume can reduce dramatically due to theosmotic shift of extracellular fluid into the intracel-lular compartment, as occurs in the disequilibriumsyndrome. In order to prevent excessive blood vol-ume reductions and osmotic disequilibrium, noveltechniques that profile the ultrafiltration rate anddialysate electrolytic concentrations in the course ofhemodialysis are being attempted (5–8).

Besides their influence on the osmotic vascularrefilling, electrolyte concentrations and pH areknown to strongly affect the electrical activity of ex-citable cells such as nervous and muscular fibers,which play a pivotal role in maintaining cardiovas-cular stability. As far as hemodialysis is concerned,this aspect has been poorly investigated, and its pos-sible importance is not entirely clear. In particular,there are no quantitative studies about the directinfluence of hemodialysis-induced modifications inelectrolytes and pH on heart and circulation func-tions. This lack of knowledge is mainly due to the

Received April 1999; revised August 1999.Address correspondence and reprint requests to Dr. Silvio Cav-

alcanti, D.E.I.S., Viale Risorgimento, 2, I-40136, Bologna, Italy.E-mail: [email protected]

Artificial Organs24(4):245–260, Blackwell Science, Inc.© 2000 International Society for Artificial Organs

245

complex relationships between several factors affect-ing cardiovascular response. In particular, in thecourse of hemodialysis, cardiovascular functions aresimultaneously influenced by the autonomic re-sponse elicited by blood volume depletion. More-over, the effectiveness of the autonomic response isitself dependent on electrolyte concentrations andpH because they also modulate the electrical activityof nervous fibers.

From this point of view, it does not seem easy toelucidate the role of electrolytes and pH in the car-diovascular response to hemodialysis merely on theindication of clinical data because the complex inter-play with neural controls makes in vivo studies dif-ficult. For a preliminary analysis, therefore, it isworth approaching the problem by using computermodeling enabling simulation of conditions that aredifficult to investigate experimentally. As a peculiaradvantage, numerical simulation provides easy sepa-ration of each considered effect. In particular, it isinteresting to analyze the influence on heart activityof hemodialysis-induced changes in electrolytic con-centrations, pH, and intracellular volume apart fromthe influence of the autonomic nervous system.

This study focuses on the electrolytes and pH de-pendence of the sinoatrial (SA) node electrical ac-tivity, which is the first determinant of heart rate. Adetailed computer model of the cardiac pacemakerbased on the Di Francesco-Noble model (9,10) wasused to analyze the influence of extracellular con-centrations of sodium, potassium and calcium ions,pH, and intracellular volume on pacemaker activity.Similar to the Hodgkin-Huxley model, the SA nodeaction potential is reconstructed on the basis of time-and voltage-dependent membrane currents derivedfrom in vitro studies. The model incorporates activetransport mechanisms such as Na–K and Na–Ca ex-change pumps, and accounts for the effects of intra-and extracellular sodium, potassium, and calciumconcentrations on the different membrane currents.By applying computer simulation, it was possible tospan the range from a detailed description of thesingle membrane current to the study of the pace-maker’s overall functionality. In particular, it waspossible to investigate the hemodialysis-inducedchanges on heart rate by decomposing the effects ofsingle ionic species, pH, and intracellular volume.

METHODS

Sinoatrial node cell modelThe computer model used in the present study is

based on the modified formulation of the Di Fran-cesco and Noble (D-N) model (9,10) published by

Dokos et al. (11). According to the schematic dia-gram in Fig. 1, the total membrane current I resultsfrom the sum of 10 different ionic currents:

I = If + IK + IK1 + INa + INa,b + INaK + Isi + ICa,b

+ INaCa + IACh (1)

where If is the hyperpolarizing-activated current, IK

and IK1 are the time-dependent and the time-inde-pendent potassium currents, respectively, INa andINa,b the rapid and the background sodium currents,INaK and INaCa the Na–K and Na–Ca exchange pumpcurrents, ICa,b the background calcium current, Isi

the slow inward current, and IACh the acetylcholine-activated potassium current.

Each of the currents is a function of transmem-brane potential, Um, and of intra- and extracellularionic concentrations. Many of these currents also in-clude time-dependent gating variables whose open-ing and closing kinetic rates are functions of trans-membrane potential (see the Appendix for acomplete list of model equations).

IACh is the outward potassium current activated bythe release of acetylcholine (ACh) from the vagusnerve terminations. At rest, vagal tone prevails onsympathetic tone resulting in a negative chrono-tropic effect on the SA node activity. By includingIACh in the model, it is possible to take into accountthis effect by fixing the resting value of the AChconcentration. Because we are interested in simulat-

FIG. 1. The schematic diagram of the SA node cell model isshown. Ionic currents are fully defined in the Appendix. The sar-coplasmic reticulum (SR) is divided into two subcompartments:network SR (NSR) and junctional SR (JSR). Electrolyte concen-trations in the intracellular compartment are also indicated; dy-namic changes in intracellular concentrations of Ca2+, Na+, andK+ were simulated with the model considering extracellular Ca2+,Na+, K+, and pH as model input.

S. SEVERI AND S. CAVALCANTI246

Artif Organs, Vol. 24, No. 4, 2000

ing the effects on heart rate of changes in ionic con-centrations separately from neural control, the AChconcentration is held constant with respect to theresting value (1 mm) in all the simulations.

Transmembrane potential, Um, depends on mem-brane current, I, according to the equation:

dUm

dt= −

I

C(2)

where C is the capacitance of the membrane.Intracellular ionized calcium is modeled as three

compartments including sarcoplasmic reticulum up-take (NSR) and release (JSR) stores (Fig. 1) becausenot all calcium that enters the cell exists in free formin the cytosol. Calcium concentration in each com-partment changes according to the mass preserva-tion equations:

d@Ca2+#i

dt=

−Isi,Ca − ICa,b+ 2INaCa + Irel − Iup

2 ViF(3)

d@Ca2+#NSR

dt=

Iup − Itr

2 VNSRF(4)

d@Ca2+#JSR

dt=

Itr − Irel

2 VJSRF(5)

where Vi, VNSR and VJSR are compartment volumesand F is the Faraday constant.

Time variation of intracellular sodium and potas-sium concentrations is described by two additionaldifferential equations:

d @Na+#i

dt=

−INa + INa,b + 3 INaK + 3 INaCa + If,Na + Isi,Na

ViF(6)

d @K+#i

dt= −

IK + IK1 − 2 INaK + If,K + Isi,K + IACh

ViF(7)

As noted by Dokos et al (11), simulations with theoriginal D-N model produce a continual cycle-by-cycle accumulation or depletion of ionic species. Inother words, pre- and postcycle values of ionic con-centrations are not equal, as one might expect for asteady state condition. Dokos et al. succeeded in sta-bilizing the model by slightly modifying the value oftwo parameters, the maximum Na-K pump current(INaK,max) and the intracellular calcium uptake timeconstant (tup); these modified values are also as-sumed in the present study. The complete list ofmodel parameters is reported in the Appendix.

We also introduced in the model the effect of ex-tracellular pH on the Na-K pump activity. In accor-dance with Scharfetter et al. (12), the linear relation-ship:

INaK,max~pH! = 4.8543 10−8~0.517 pH − 2.927!(8)

is used to describe the dependence of maximum INaK

pump current on pH. This is consistent with the factthat the activity of Na+-K+-ATPase increases lin-early with pH in the physiological range (13). Theequation for INaK thus becomes

INaK

= INaK,max~pH! S @Na+#i

@Na+#i + KmNaD S @K+#o

@K+#o + KmKD(9)

where KmNa and KmK bring about pump sensitivityto intracellular sodium and extracellular potassiumconcentrations. An increase in intracellular Na wasfirst described in erythrocytes from some patientswith chronic renal failure some decades ago (14).Subsequently, alterations in Na–K transport havebeen seen in erythrocytes (15), leukocytes (16),muscle cells (17), the intestines (18), and the brain(19) in uremic subjects as well. Therefore, the valuesof KmNa and KmK were assigned to higher valuesthan in the D-N model (Appendix, Eq. 22) in orderto bring intracellular sodium concentration withinthe observed range for uremic patients’ erythrocytes(20) and to decrease intracellular potassium with re-spect to normal value (21).

Extracellular [Ca2+], [Na+], [K+], pH, and the per-cent intracellular volume, V, referring to the end-hemodialysis condition were considered as inputs tothe model and were assigned according to the con-ditions to be simulated. Model differential equationswere implemented in SIMULINK (Mathworks Inc.,Natick, MA, U.S.A.) and numerically integrated bythe Adams-Gear method (22). The model time-basewas scaled in order to reproduce heart rate withinthe human physiological range.

Simulation conditionsIn order to characterize the effects of hemodialysis

on the SA node electrical activity, typical values forelectrolyte concentrations, pH, and intracellular vol-ume at the beginning and at the end of hemodialysissessions were considered (Table 1). Electrolyte con-centrations and pH were determined (1600 BloodGas Electrolytes System, Instrumentation Labora-tory Corp., Barcelona, Spain, 550A pH-meter, OrionResearch Inc., Beverly, MA, U.S.A.) from bloodsamples taken immediately before and after hemo-

EXTRACELLULAR FLUID COMPOSITION AND HEART RATE 247


dialysis in the course of a previous clinical trial (datanot published) and agree with typical data for uremicpatients (23). Because the exchange of ions betweencapillary blood and interstitial fluid is very rapid(24), we considered plasmatic concentrations to beclose approximations of interstitial concentrations.

Intracellular volume at the beginning of hemodi-alysis was considered 5% greater than at the endbecause of hyperhydration, which characterizes thebeginning of the hemodialysis session. Initial intra-cellular volumes equal to ±10% of end hemodialysiswere also simulated.

Extracellular concentrations and pH were keptconstant during the simulations, considering the ex-tracellular compartment as an infinite volume, untilthe transient was extinct. Time varying inputs wereconsidered in sensitivity analysis by simulating theresponses to a very slow (quasi-steady state) and toa step change in extracellular electrolyte concentra-tions. The latter, which is a nonphysiological situa-tion, was simulated in order to elucidate the dynami-cal behavior of the SA node and to estimate theleading time constant of the response to fast varia-tions in electrolyte concentrations and pH.

Heart rate was computed from simulated beat-to-beat action potentials by using a threshold (−0.2 mV)crossing detection.

RESULTS

Overall effects of simultaneous variation ofelectrolytes, pH, and intracellular volume

The self-sustained action potential predicted bythe SA node computer model depends significantlyon the simulated changes in electrolytes, pH, andpercent intracellular volume (Fig. 2). Action poten-tial computed by assigning to electrolytes, pH, andintracellular volume values relative to the beginningof hemodialysis (Fig. 2, thin line) shows a period of0.852 s with a maximal diastolic potential of −60 mVand a slow depolarization phase of about 0.37 s. Ac-tion potential simulated by assigning the values cor-responding to the end of hemodialysis (Fig. 2, thick

line) exhibits a deeper hyperpolarization reaching amaximal diastolic potential of −69 mV (+15%). Be-cause of the lowering of the threshold for the actionpotential onset, the slow depolarization phase is re-duced to 0.25 s (−32%), and the simulated cardiacperiod at the end of hemodialysis is 0.746 s only.Therefore, as far as the differences between the be-ginning and the end of hemodialysis correspond tothose simulated (Table 1), the model predicts an in-crease in the heart rate from 70 to 80 bmp (+14%)merely as an effect of the changes in electrolyte con-centrations, pH, and intracellular volume.

The differences between beginning and end hemo-dialysis conditions also significantly influence so-dium and potassium concentrations in the cytoplasm(Fig. 3). Intracellular sodium increases at the end ofdialysis. Conversely, intracellular potassium is higherat the beginning than at the end of dialysis. Changesin intracellular sodium and potassium due to varia-tions in extracellular electrolyte concentrations, pH,and volume are much larger than the beat-to-beatoscillations due to the single action potential. (Notethe changes in the scale when comparing beginningand end curves in Fig. 3.) On the contrary, the in-tracellular calcium time course during the action po-tential exhibits large variations that prevail over thechanges induced by considering the beginning andend of hemodialysis conditions (Fig. 4). An appre-ciable difference concerns the calcium peak, which islower at the end than at the beginning. During thediastolic phase when the cell is maximally hyperpo-larized, the minimum intracellular calcium is slightlyhigher at the end than at the beginning (0.066 versus0.039 mm). It is worth noting that such a small dif-ference can strongly influence heart rate through the

TABLE 1. Simulated extracellular electrolyteconcentrations, pH, and intracellular volume at the

beginning and at the end of hemodialysis

Initial End

[Na+]o 138 mM 140 mM[K+]o 6.0 mM 3.3 mM[Ca2+]o 1.2 mM 1.5 mMpH 7.31 7.40V 105% 100%

V: intracellular volume.

FIG. 2. The graph compares the simulated SA node action po-tentials at the beginning (thin line) and at the end (thick line) ofhemodialysis. The cardiac period at the beginning of hemodialy-sis is significantly longer than at the end.



mechanism of calcium-induced calcium release, aspresented in the Discussion.

Separate effects of single variations in electrolytes,pH, or intracellular volume

The effects on heart rate of variation in extracel-lular sodium, potassium, calcium, pH, and intracel-

lular volume were decomposed by simulation. Foreach of these quantities, we first simulated a hypo-thetical hemodialysis that changes only it from itsinitial to end value (Table 1) whereas the otherswere fixed to their initial value. The resulting heartrate calculated by means of these simulations arelisted in Table 2 (initial column). Second, we simu-lated a hypothetical hemodialysis that changes allthese quantities from the initial to the end value,except the one that was kept to its initial value. Theresults of these simulations are listed in the end col-umn of Table 2. These latter simulations also corre-spond to simulations in which all the quantities areset to their end value whereas only one is convertedback to the initial value.

As a general result, it is evident that the simulatedvariations in ionized calcium, potassium, and pHhave the greatest influence on heart rate because byonly changing these quantities, heart rate signifi-cantly differs from initial (70 bpm) or end hemodi-alysis (80 bpm) heart rate considered as referenceconditions. On the contrary, the changes in sodiumand intracellular volume seem to have only minoreffects (Table 2). Moreover, the simulated increasein ionized calcium as well as the decrease in potas-sium accelerate heart beats whereas restoring physi-ological pH slows them down.

Extracellular ionized calcium concentration wassupposed to rise from 1.2 to 1.5 mM in the course ofhemodialysis, that is, with a 25% increase. Whensimulating a condition in which only this variationtakes place while the other quantities remain equalto those at the beginning of dialysis, an increase inheart rate from 70 to 76 bpm is predicted (Table 2,initial column). This increase corresponds to 60% ofthe total increase occurring when all the inputs were

FIG. 4. Shown is the time course of simulated calcium intracel-lular concentration at the beginning (thin line) and at the end(thick line) of hemodialysis. Intracellular calcium exhibits largevariations during the action potential that prevail over thechanges induced by considering the beginning and end of hemo-dialysis conditions. An appreciable difference concerns the cal-cium peak, which is lower at the end than at the beginning.

FIG. 3. Shown is the time course of simulated sodium (bottompanel) and potassium (upper panel) intracellular concentrationsat the beginning (thin lines) and at the end (thick lines) of hemo-dialysis. Sodium concentration increases at the end compared tothe beginning of dialysis; potassium concentration is higher at thebeginning than at the end. Changes in intracellular sodium andpotassium due to variations in extracellular electrolyte concen-trations, pH, and volume are much larger (about 500 times) thanthe beat-to-beat oscillations due to the single action potential.

TABLE 2. Simulated heart rate when only a quantity ischanged with respect to initial or end-hemodialysis

conditions. Differences from the reference initial andend-hemodialysis heart rate are also indicated

Initiala

(bpm)Endb

(bpm)

[Na+]o 71x (+1) 81L (+1)[K+]o 74+ (+4) 73, (−7)[Ca2+]o 76* (+6) 69n (−11)pH 69h (−1) 86 (+6)V 71 (+1) 79 (−1)Reference heart rate 70s 80❂

a All the inputs were at their initial values (Table 1) except forthe quantity indicated in the first column, which was at the end-hemodialysis value (Table 1).

b All inputs were at their end values (Table 1) except for thequantity indicated in the first column, which was at the initialvalue (Table 1).

x, L, +, ,, *, n, h, s, ❂: Symbols used to indicate the correspondingvalues in Figs. 5–8.



modified to their end hemodialysis values (from 70to 80 bpm). By simulating the end-of-hemodialysiscondition for all the quantities, with only calciumequal to its initial value, heart rate decreases to 69bpm (Table 2, end column), even less than heart ratewhen all the quantities correspond to the initial con-dition (70 bpm). These results show that the sameincrease in extracellular ionized calcium producesdifferent increases in heart rate depending on thevalues of the other quantities. Heart rate increasesby 6 bpm when calcium variation occurs, the otherquantities being in their initial condition. Instead,heart rate increases by 11 bpm (from 69 to 80 bpm)when end-of-hemodialysis conditions are assigned tothe other quantities.

Similar effects can be noted by changing extracel-lular potassium from 6 to 3.3 mM (Table 2). Heartrate increases by 4 bpm (70 versus 74 bpm) when theother quantities correspond to the beginning of di-alysis whereas the increase is larger (from 73 to 80bpm) when the end-of-hemodialysis condition issimulated.

The pH influence also strongly depends on thevalue of the other quantities. At the beginning ofhemodialysis, pH restoration only (from 7.31 to 7.4)slightly decreases heart rate (70 versus 69 bpm). Thesame variation in pH causes a large reduction inheart rate (86 versus 80 bpm) when applied at theend of hemodialysis.

In all the considered cases, variations in heart ratedue to the change in ionized calcium, potassium, orpH are greater when considering for the other fixedquantities at the end of hemodialysis rather than theinitial conditions. (Compare initial and end heartrate changes within brackets, Table 2.)

Heart rate sensitivity to electrolyte concentrationsThe sensitivity of heart rate to extracellular ion-

ized calcium, potassium, and sodium was assessed bysimulating a continuous, slow change in the corre-sponding input. The change rate of input was smallenough to maintain the model in a quasi-steadystate. Different fixed values were assigned to theother quantities in order to simulate several condi-tions: beginning of dialysis, end of dialysis, and con-ditions in which there is one parameter at the end-of-hemodialysis value and the others are all at initialvalues.

Extracellular ionized calcium increasing in therange 0.8–1.6 mM causes a significant rise in heartrate (Fig. 5). The change in heart rate with respect tocalcium changes; that is, the heart rate sensitivity tocalcium depends on the calcium concentration itselfwith a typical nonlinear behavior. It is also evident

that the same variation in the ionized calcium con-centration gives rise to a significantly differentchange in heart rate depending on the potassiumconcentration and pH. Heart rate sensitivity to cal-cium increases when extracellular potassium or pH isdecreased. Over the considered calcium range, heartrate exhibits a very large increase (27 bpm) at a lowpotassium concentration ([K+] 4 3.3 mM, Fig. 5,dash-dot line). The heart rate increase is lower (11bpm) when the potassium is at the higher initialvalue ([K+] 4 6 mM, Fig. 5, thin line). After pH isrestored (pH 4 7.4), the calcium-induced heart rateincrease is 6 bpm only (Fig. 5, dotted line). For highextracellular potassium and for low pH, as occurs atthe beginning of dialysis, heart rate sensitivity, thatis, the heart rate curve slope, is reduced with respectto the end of dialysis (compare thin and thick curves,Fig. 5).

It is worth noting that by lowering potassium con-centration, heart rate increases or decreases depend-ing on the calcium concentration (compare thin anddash-dot curves, Fig. 5). For low calcium concentra-tions (e.g., 0.8 mM), potassium depletion causes a 7bpm reduction in heart rate while for high calciumconcentrations (e.g., 1.6 mM), the same potassiumdepletion raises the heart rate by 9 bpm.

Variation in pH from 7.31 to 7.4 causes a decreasein heart rate as well as in sensitivity to extracellularcalcium. This is evident considering the heart ratecurve obtained for the beginning dialysis condition

FIG. 5. The graph shows the sensitivity of simulated heart rate toextracellular calcium. Four different conditions were simulated(Table 1): beginning of dialysis (thin line), end of dialysis (thickline), a condition in which only pH is at the end of hemodialysis(dotted line), and a condition in which only potassium concentra-tion corresponds to the end value (dash-dot line). Symbols indi-cate the simulation results shown in Table 2. Heart rate sensitivityto calcium depends on the calcium concentration itself, with atypical nonlinear behavior. The same variation in the calciumconcentration gives rise to a significantly different change in heartrate depending on potassium concentration and pH.



(Fig. 5, thin line) and the heart rate curve obtainedfor the end-of-dialysis pH (dotted line). The sameeffect can also be noted by comparing the hemodi-alysis end curve (Fig. 5, thick curve) with the curverelated to the only potassium set to the end value(Fig. 5, dash-dot line). In this case, the appreciablereduction in heart rate also is mainly caused by thedifference in pH. The effect of pH is larger for highercalcium concentration.

Slow variation of extracellular potassium in therange of 3–8 mM originates a biphasic heart rateresponse (Fig. 6). For a low potassium concentration(<4.5 mM), heart rate significantly decreases whenpotassium increases. Conversely, at a higher potas-sium concentration, heart rate rises slightly againwhen potassium increases further. This nonlinearcharacteristic is evident in both the curves obtainedsimulating the beginning and the end-of-hemodial-ysis conditions (Fig. 6, thin and thick lines, respec-tively). In the case of pH restoration (Fig. 6, dottedline), the heart rate curve moves down to lower heartrate values especially for low potassium concentra-tions. When the increased calcium condition is simu-lated (Fig. 6, dashed line), the heart rate curve riseswith an increase in heart rate of 4–11 bpm dependingon potassium concentration. Moreover, for high po-tassium concentrations, the heart rate curve exhibitsa plateau, which denotes a null sensitivity of heartrate to potassium changes.

The computer model predicts very low heart rate

sensitivity to extracellular sodium (Fig. 7). Simulat-ing such a large variation of sodium concentrationfrom 132 to 146 mM, heart rate slightly increases (3bpm) if initial hemodialysis conditions are consid-ered while heart rate slightly decreases (2 bpm)when the end-of-hemodialysis conditions are exam-ined. The slope of the heart rate curve turns frompositive to negative when reduced potassium con-centration is considered (compare thin and dash-dotline in Fig. 7). In the case of increased, end-of-he-modialysis calcium, no evident changes in heart rateoccur when sodium changes (Fig. 7, dashed line).Setting only pH to the end-of-hemodialysis value,the heart rate does not change significantly from thebeginning-dialysis curve.

Although heart rate is not influenced by extracel-lular sodium, it has a high sensitivity to intracellularsodium concentration (Fig. 8). Open-loop simula-tions were performed with intracellular sodiumslowly changing as an input quantity instead of as anintegration of the sodium mass balance equation(Eq. 6). In all simulated conditions, the heart rateshows significant sensitivity to intracellular sodium,increasing considerably when sodium concentrationrises. Note that sensitivity curves are very similarwhen initial and end-of-hemodialysis conditions aresimulated. The sensitivity curve shifts toward higherheart rate values, maintaining the same slope, whenextracellular calcium only is set to its end-of-hemo-dialysis value (Fig. 8, dashed line). The contrary oc-curs when extracellular potassium concentration isconsidered as in end-of-hemodialysis conditions(Fig. 8, dash-dot line). Setting only pH to the end

FIG. 6. The sensitivity of simulated heart rate to extracellularpotassium is shown. Four different conditions were simulated(Table 1): beginning of dialysis (thin line), end of dialysis (thickline), a condition in which only pH is at the end of hemodialysis(dotted line), and a condition in which only calcium concentrationcorresponds to the end value (dashed line). Symbols indicate thesimulation results shown in Table 2. Heart rate sensitivity to po-tassium depends on the potassium concentration itself, with abiphasic behavior. The same potassium variation gives rise to asignificantly different change in heart rate depending on calciumconcentration and pH.

FIG. 7. The graph shows the sensitivity of simulated heart rate toextracellular sodium. Four different conditions were simulated(Table 1): beginning of dialysis (thin line), end of dialysis (thickline), a condition in which only potassium concentration is at theend of hemodialysis (dash-dot line), and a condition in which onlycalcium concentration corresponds to the end value (dashedline). Symbols indicate the simulation results shown in Table 2.Heart rate sensitivity to extracellular sodium is very low.



value heart rate slight decreases with respect to thebeginning curve.

Heart rate response to electrolyte and pHstep changes

A step change in extracellular ionized calciumconcentration from initial to the end-of-hemodialysisvalue (from 1.2 to 1.5 mM) causes a simultaneousrapid increase in heart rate (Fig. 9). After this abruptchange in the correspondence of the calcium step, aslow increase towards the steady condition occurs.

The initial rapid increase in heart rate depends onthe conditions assigned to the other fixed inputs.This increase is about 6% (from 70 to 74 bpm) forthe initial hemodialysis condition (Fig. 9, thin curve)and 9% (from 69 to 75 bpm) for the end-of-hemodi-alysis (thick curve). The successive transient phase ischaracterized by a very long time constant, andabout 50 min are necessary to reach steady state. Therapid change in heart rate reveals its direct depen-dence on the extracellular calcium concentration.The slow dynamic is instead due to the accumulationor depletion of intracellular ions, which requires along time because the net cycle-by-cycle fluxes arevery small. Note that the same perturbation in cal-cium concentration gives rise to significantly differ-ent responses in heart rate depending on the con-centrations of the other electrolytes.

The heart rate response shows a biphasic responsewhen a rapid reduction in potassium concentration(from 6 to 3.3 mM) is simulated (Fig. 10). Simulta-neously with potassium depletion, an abrupt fall inheart rate occurs. After this rapid change in heartrate, a slow increase with a very long time constanttakes place. The step in heart rate was about 6%(from 70 to 66 bpm) under the initial conditions and5% (from 73 to 69 bpm) under the end-of-hemodi-alysis conditions.

A step in pH from the initial to the end-of-hemo-dialysis value (from 7.31 to 7.4) causes a decrease inheart rate. The amplitude and the transient timecourse of heart rate decrease depend on the condi-tions assigned to the other inputs (Fig. 11). Startingfrom the initial conditions (thin line), the change inpH causes a simultaneous decrease in heart rate of

FIG. 8. Shown is the sensitivity of simulated heart rate to intra-cellular sodium in the open-loop conditions. Four different condi-tions were simulated (Table 1): beginning of dialysis (thin line),end of dialysis (thick line), a condition in which only potassiumconcentration is at the end of hemodialysis (dash-dot line), and acondition in which only calcium concentration corresponds to theend value (dashed line). Symbols indicate the simulation resultsshown in Table 2. In all simulated conditions, heart rate shows ahigh sensitivity to intracellular sodium, increasing considerablywhen sodium concentration rises.

FIG. 9. The time course of heart rate after a rapid change incalcium extracellular concentration is depicted. The vertical seg-ment indicates the time in which the sudden input change oc-curred. The same calcium step (from 1.2 to 1.5 mM) was simu-lated by considering the beginning- and end-of-hemodialysisvalues for the other quantities (thin and thick lines, respectively).

FIG. 10. The time course of heart rate after a rapid change inpotassium extracellular concentration is shown. The vertical seg-ment indicates the time in which the sudden input change oc-curred. The same potassium step (from 6 to 3.3 mM) was simu-lated by considering the beginning- and end-of-hemodialysisvalues for the other quantities (thin and thick lines, respectively).



only 1 bpm. Starting from the conditions at the endof hemodialysis except for pH (thick line), thechange in pH causes a negligible rapid change inheart rate. Then, a significant (5 bpm) but very slowdecrease in heart rate begins, reaching steady condi-tion only after 1 h.

Heart rate sensitivity to Na–K pump parametersBecause Na–K pump current has a great influence

on intracellular electrolyte concentrations and thenon membrane electrical activity, we also analyzedheart rate sensitivity to pump parameters. In particu-lar, we simulated changes in pump sensitivity to so-dium and potassium concentrations. Both initial andend-of-hemodialysis conditions were considered af-ter imposing a ±10% change in KmNa or KmK pumpparameters (Eq. 9). An increase in each of theseparameters causes a decrease of pump current. Theresults of these simulations are listed in Table 3.

As a comprehensive result, a decrease in pumpcurrent induces an increase in heart rate (Table 3,rows corresponding to +10%). The contrary occurs

after increasing pump current, that is, decreasingKmNa or KmK parameters.

Heart rate increase from initial to end-of-hemodi-alysis conditions (DHR) is very sensitive to pumpparameters. At basal values of pump parameters,DHR is equal to 10 bpm. By increasing KmNa by 10%with respect to its basal value, DHR rises to 15 bpmwhile it drops to 2 bpm when this parameter is re-duced by 10%. Similar effects can be observed aftervarying KmK, but with less marked changes with re-spect to basal conditions: DHR increases to 13 bpmwhen KmK is increased, and it drops to 7 bpm whenKmK is decreased.

DISCUSSION

Electrolytes and pH play a pivotal role in the elec-trical activity of excitable membranes, and changesin these quantities may considerably affect nervousand muscular cell functions. Hemodialysis has alarge impact on such quantities and, during the ex-tracorporeal blood purification, significant changesof electrolyte concentrations and pH may occur. Theeffects of these changes on cardiovascular functionsof patients undergoing conventional hemodialysisare complex and poorly understood. As a matter offact, hemodialysis instability is a frequent side effectof hemodialysis and particularly affects patients withcardiovascular diseases or autonomic deficiency.

Up to now, hemodialysis-induced changes in bodyfluid composition have been particularly investi-gated concerning osmotic fluid shift between extra-and intracellular spaces (7,25–27). A physiological,adequate, circulating blood volume is of the highestimportance in preventing acute hypotension duringhemodialysis. In order to prevent excessive reduc-tions in circulating blood volume, new techniques forperforming hemodialysis were recently proposed, forexample, profiled hemodialysis, in which dialysatesodium concentration is modulated during dialysis(6,7), or hemodialysis with biofeedback, in which theultrafiltration rate is controlled in a closed loop bymeasuring blood volume changes (8). These tech-niques aim to minimize or prevent hemodynamic in-stability otherwise frequently observed in conven-tional hemodialysis. However, no full explanationexists on why they give rise to a better cardiovascularstability with a reduced percentage of collapses andother symptoms related to arterial hypotension. Adeeper understanding of such techniques, as well astheir effective use, probably calls for detailed knowl-edge of cellular response to modifications of bodyfluid composition.

In this study, we have focused on the influence of

FIG. 11. Shown is the time course of heart rate after a rapidchange in pH. The vertical segment indicates the time in whichthe sudden input change occurred. The same pH step (from 7.31to 7.4 mM) was simulated by considering the beginning- andend-of-hemodialysis values for the other quantities (thin and thicklines, respectively).

TABLE 3. Influence of changes in Na–K pump currentparameters on simulated heart rate at the beginning and

at the end of hemodialysis

Initial HR(bpm)

End HR(bpm)

DHR(bpm)

KmNa +10% 74 89 15−10% 69 71 2

KmK +10% 71 84 13−10% 70 77 7

Reference conditions 70 80 10

HR: heart rate.



extracellular fluid composition on sinus-node electri-cal activity, which is the main determinant of heartrate. As an advantage, heart rate is easy to measureand can be considered a representative marker ofcardiac pump activity. Although sinus-node activityis under autonomic control, we suggest that it shouldshow signs of hemodialysis-induced changes in bodyfluid composition.

It is known that patients generally terminate thehemodialysis session with slight tachycardia (28). Ina previous study (29), we found an increase of about10% in heart rate when comparing the beginningand the end of hemodialysis. Such an increase inheart rate can be ascribed to two distinct causes:first, the hemodialysis-induced blood volume de-crease that could cause a baroreflex increase in heartrate and second, the hemodialysis-induced changesin body fluid composition. In the previous study (29),heart rate and heart rate variability were comparedin 2 groups of patients with a different past history ofacute hypotension. In hypotension-resistant patients,heart rate increased by 7 ± 11 bpm; in hypotension-prone patients, the increase was 9 ± 13 bpm with nosignificant difference. Conversely, heart rate vari-ability was clearly different when comparing the 2groups, thus revealing different autonomic responsesin accordance with the opposite hemodynamic pa-tient classification. Because this difference was notfound when comparing heart rate, one could specu-late that the hemodialysis-induced increase in heartrate was primarily imposed by the change in fluidcomposition that was, indeed, similar in the 2 groups.We found it difficult to separate the autonomic-mediate contribution from contributions due to thechanges in fluid composition with an in vivo test.Therefore, as a preliminary approach, we addressedthis problem by using numerical simulation.

The computer model used in this study to simulatethe SA action potential and pacemaker activity wasoriginally developed by Di Francesco and Noble. Itwas initially proposed as a model of rabbit Purkinjefiber electrical activity (9); later it was applied tovoltage-clamp data from samples of SA node tissue(10). Dokos et al. modified this model to simulate invivo conditions (11) and applied it to the study ofvagal stimulation on the SA node. Because this modi-fied model represents one of the most up-to-date mod-els of SA node electrical activity, it was chosen as thebasis for the present study. As a unique, notable,model difference, we included the dependence ofNa–K pump activity on extracellular pH.

Extrapolation of data meant to simulate rabbit SApacemaker activity, with a higher resting heart ratethan in humans, is a limitation of our study. Never-

theless, model-based analysis produces a theoreticalframework providing a comprehensive prediction ofwhat could be experimentally observed in terms oftrends and behavior in mammalian sinoatrial cells.Of course, simulation results are not to be regardedas a precise numerical prediction; however, they de-mand a specific in vivo validation. Moreover, somelimitations to the reliability of presented resultscould come from phenomena, such as the influenceof removal of pump inhibitors by dialysis or themodifications in nonionic solute, not included in themodel formulation.

The computer model predicts an increase in heartrate from the beginning to the end of hemodialysis ofabout 10 bpm, which is very close to the values thatwe observed in a previous clinical trial (29). Actu-ally, this increase is considerably dependent on thesimulated condition, and in particular, extracellularionized calcium and potassium concentrations aswell as pH have a strong influence on heart rate. Byvarying these quantities within typical ranges foruremic patients, the simulated heart rate ranges from60 to 90 bpm.

As a comprehensive result, heart rate rises afterionized calcium concentration is increased or potas-sium concentration is reduced. Increased calciumand decreased potassium are typical conditionsreached at the end of extracorporeal purification,and this could justify tachycardia observed in manypatients at the end of hemodialysis. Restoration ofpH by hemodialysis slows the heart rate in contrastwith the increase due to electrolyte effect. However,model-based prediction indicates that the accelerat-ing effects of potassium and calcium overcome theslowing effect of pH providing the end-of-hemodial-ysis conditions are those simulated. In all simula-tions, extracellular sodium as well as intracellularvolume seems to have only a slight influence onheart rate. The negligible influence of intracellularvolume on heart rate was confirmed by further simu-lations with different initial intracellular volume(±10% of the final value), which led to minorchanges in heart rate (±1 bpm). This makes the as-signment of this parameter noncritical.

An interesting result concerns the complexity ofthe dependence of heart rate on electrolytes and pH.In particular, note the interrelationship betweenelectrolytes themselves as well as between electro-lytes and pH. Such complex dependence reveals thenonlinear nature of the underlying relationships,which makes heart rate sensitivity strongly depen-dent on the simulated condition. In particular, pro-viding potassium concentration is maintained above5 mM, the heart rate exhibits low sensitivity to



changes in potassium (Fig. 6) and calcium (Comparethin and dash-dot lines, Fig. 5.) When potassium isreduced to below 4.5 mM, heart rate sensitivity tocalcium and potassium increases significantly. In par-ticular, when potassium is in the 3 to 4 mM range,heart rate sensitivity is about 10 bpm/mM to potas-sium and about 30 bpm/mM to calcium. Analogouselectrolyte dependence of heart rate sensitivity tovagal stimulation was found earlier by Toda andWest (30,31).

According to physiological knowledge, computeranalysis predicts that the main determinant ofchanges in the heart period is the duration of slowdiastolic depolarization. In fact, at the end of hemo-dialysis, this depolarization phase is shorter than atthe beginning by about 0.12 s, which is almost equalto the variation in the cardiac period. During thisphase, membrane depolarization is mainly caused byNa–Ca exchange pump current, background sodiumcurrent, and background calcium currents. Amongthese, calcium currents directly influence heart ratebecause intracellular ionized calcium concentrationis the first determinant of rapid calcium release fromthe sarcoplasmic reticulum that, in turn, triggers theaction potential upstroke by rapidly increasing INaCa

(32). In fact, it is known that under a variety of cir-cumstances the diastolic intracellular calcium con-centration is closely associated with the pacemakerfiring rate (33). Note that although INaCa acts to ex-trude calcium from the cell, it is a net inward currentdue to the larger inward flux of sodium as part of itsexchanger function.

Simulation of a step increase of extracellular ion-ized calcium causes an initially rapid and subse-quently slow heart rate increase (Fig. 9). This modelprediction is in accordance with the decrease in cyclelength observed in multicellular preparations afterelevation of calcium concentration in the bathingfluid (31,34). The first phase of the model step re-sponse is due to the direct influence of extracellularcalcium concentration on the calcium currents. Infact, after the increase in extracellular calcium, theintracellular calcium diastolic uptake rises, thus in-creasing the minimum intracellular calcium (0.066versus 0.039 mm). As a consequence, the thresholdcalcium level, at which the calcium-induced calciumrelease is triggered, is reached early causing the in-crease in heart rate. Intracellular calcium uptake isalso indirectly influenced by the intracellular sodiumconcentration through the Na–Ca pump. After anincrease in extracellular calcium, the sodium systolicinflow through the pump rises, causing a slow cycle-by-cycle intracellular sodium accumulation. By in-creasing intracellular sodium, the Na–Ca pump in-

creases calcium diastolic inflow, consequentlyincreasing heart rate. This mechanism underlies theslow increase of heart rate after the rapid changes inextracellular calcium (Fig. 9).

Heart rate decrease is a well-known effect of ex-tracellular potassium excess (23,35). When extracel-lular potassium concentration diminishes, heart ratedrops immediately (Fig. 10). In this condition, themembrane undergoes increased hyperpolarizationbecause of the increased outward potassium currentduring the repolarization phase. This extra hyperpo-larization causes a reduction in Na–Ca pump calciumdiastolic inflow because it depends on membrane po-tential. After extracellular potassium reduction,therefore, intracellular calcium uptake is reducedand thus significantly slows heart rate. At the sametime, after depletion of extracellular potassium,Na–K pump current drops causing a cycle-by-cycleslow increase in intracellular sodium. Consequently,Na–Ca pump slowly raises calcium inflow and heartrate increases with a long transient (Fig. 10). Whenintracellular sodium is clamped, as in the case of Fig.8, Na–K pump feedback is blocked, and thus theslow heart rate increase does not occur. In this case,the only effect of extracellular potassium depletion isa rapid reduction in heart rate (compare the de-creased potassium dash-dot curve and the initial di-alysis thin curve in Fig. 8). On the contrary, when thepump is active the prevailing effect, after the tran-sient is extinguished, is an increase in heart rate(compare the 3 mM potassium curve and the initialdialysis curve in Figs. 5 and 7).

Kodama and Boyett (36) found a decrease in thecycle length of SA node pacemaking when decreas-ing extracellular potassium from 8 to 4 mM. In theseconditions, the model predicts a heart rate variationthat depends on the other electrolyte concentrationsand pH (Fig. 6). In particular, heart rate increases forhigh calcium concentration, which is in accordancewith the cited study in which the SA fibers wereplaced in a solution with 1.8 mM calcium.

The sensitivity analysis with respect to Na–Kpump parameters (Table 3) confirms the pump cur-rent’s pivotal role on determining heart rate. Asnatural behavior, the Na–K pump responds to he-modialysis-induced extracellular potassium deple-tion by reducing potassium inflow and sodium out-flow, which causes a reduction in intracellularpotassium and an increase in intracellular sodium(Fig. 3). In the model, Na–K pump current is de-scribed by a nonlinear relationship with saturationcharacteristics with respect to both extracellular po-tassium and intracellular sodium (Eq. 9). Saturationmakes the sensitivity of the pump particularly de-



pendent on extracellular potassium. When the po-tassium concentration is much greater than the valueof parameter KmK, as at the beginning of dialysis,([K+]o 4 6, KmK 4 1.75 mM), the pump is saturatedand its sensitivity is very low. In this condition, thepump is unable to control efficiently, and a change inextracellular potassium does not cause significant al-terations in intracellular electrolyte concentrations,in particular in sodium. Thus, heart rate has low sen-sitivity to extracellular potassium (Fig. 6). By de-creasing extracellular potassium toward KmK, thepump desaturates and is once again efficient. Underthese conditions, which correspond to the end of he-modialysis ([K+]o 4 3.3 versus KmK 4 1.75 mM),changes in extracellular potassium cause significantchanges in pump current and then in intracellularelectrolyte concentrations. In this condition, heartrate exhibits high sensitivity to extracellular potas-sium (Fig. 6).

Changes in extracellular sodium concentrationsimulated in this study do not affect intracellularelectrolyte concentrations. In fact, the Na–K pump ishighly sensitive to intracellular sodium, and it effi-ciently limits intracellular changes with respect toextracellular sodium perturbations. Therefore, heartrate does not exhibit evident changes when extracel-lular sodium is changed, according to van Kuijk et al.(37). The contrary occurs when intracellular sodiumis changed.

Hemodialysis-induced pH restoration increasesNa–K pump current (Eq. 8). Consequently, sodiumoutflow from the cell increases, and intracellular so-dium concentration decreases with a reduction inheart rate. This effect is enhanced when a low extra-cellular potassium concentration is considered (Fig.6, dotted line).

The simulation of rapid changes in electrolyte con-centrations reveals that from 40 min to 1 h are re-quired to reach a steady heart rate (Fig. 10), whichdepends on the time needed for the underlyingchanges in intracellular concentrations, and denotesthat stable equilibrium conditions for intracellularspace is reached only after a sufficiently long time.This is in agreement with the typical rebound effectsafter the end of hemodialysis (28), which denote anonequilibrium condition between intra- and extra-cellular spaces.

The results of this study indicate that hemodialy-sis-induced changes in body fluid composition maylargely influence the rhythm of the SA node pace-maker. Based on this result, similar influencesshould exist also for all excitable muscular fibers,thus contributing to clarifying the large impact ofhemodialysis on the cardiovascular system. Concern-

ing the SA node, the effect of hemodialysis seems tobe complex because of the interdependence of theinfluences of different ionic species. However, onecan briefly summarize that potassium removal aswell as a rise in ionized calcium accelerates heartbeat and, on the contrary, restoring pH slows it. It isinteresting to note that extracellular sodium has nosignificant influence, but heart rate strictly dependson intracellular sodium concentration. As a possibleimplication, a sustained increase in heart rate duringhemodialysis should denote an increase in intracel-lular sodium concentration.

The present simulation study is not specificallyoriented to model the kinetics of electrolytes and pHduring dialysis, being focused on the impact of suchkinetics on the heart rate. In particular, we tried toelucidate what contribution each ionic componentcan give to heart rate changes, setting aside the mu-tual interaction between plasma electrolytes, pH,and osmolarity. From this point of view, simulationsdoes not reproduce how dialysis works. The rel-evance of simulation results to the real situation is aquantitative evaluation of the comparative weight ofeach considered factor. Such information could beimportant, for instance, in the management of theprofiled hemodialysis in which the plasma ionic con-centrations are affected in the course of the treat-ment by acting on dialysate composition.

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APPENDIX 1

Membrane potential

dUm

dt= −

Itot

C(10)

Total membrane current

Itot = If + IK + IK1 + INa + INa,b + INaK + Isi + ICa,b

+ INaCa + IACh (11)

with C 4 0.018 mF

Hyperpolarizing-activated current

If = y S @K+#o

@K+#o + KmfD ~gf,Na ~Um − ENa!

+ gf,K ~Um − EK!! (12)

with gf,Na = gf,K 4 6 mS, Kmf 4 45 mM and y definedaccording to

dy

dt= ay ~1 − y! − by y (13)

ay = 0.0167 e−0.067 ~Um+52! (14)

by =0.3333 ~Um+52!

1−e−0.2 ~Um+52!(15)

Time-dependent K+ current

IK = x IK,max S@K+#i − @K+#o e−Um F

RT

140D (16)

with IK,max 4 20 nA and x defined according to

dx

dt= ax ~1 − x! − bx x (17)

ax =0.1667 ~Um + 22!

1 − e−Um + 22

5

(18)



bx =0.0593 ~Um + 22!

eUm + 22

15 − 1(19)

Time-independent K+ current

IK1 = gK1 S @K+#o

@K+#o + Km1D S Um − EK

1 + e2 F ~Um − EK + 10!

R TD(20)

with gK1 4 10 mS and Km1 4 210 mM

Background sodium current

INa,b = S@Na+#o

140 D gNa,b ~Um − ENa! (21)

with gNa,b 4 0.07 mS

Na-K exchange pump current

INaK

= INaK,max~pH! S @Na+#i

@Na+#i + KmNaD S @K+#o

@K+#o + KmKD

(22)

INaK,max~pH! = 4.8543 10−8 ~0.517 pH − 2.927! (23)

with KmNa 4 56 mM and KmK 4 1.75 mK

Na–Ca exchange current

INaCa = KNaCa1@Na+#i

3 @Ca2+#o egUm F

2 RT

− @Na+#o3 @Ca2+#i e

~g−1! Um F

2 RT

1 + dNaCa ~@Na+#o3 @Ca2+#i

+ @Na+#i3 @Ca2+#o!

2 (24)

with KNaCa 4 0.02, dNaCa 4 0.001 and g 4 0.5

Fast sodium current

INa = m3 h gNa ~Um − Emh! (25)

with gNa 4 1.25 mS and m, h defined according to

dm

dt= am ~1 − m! − bm m (26)

am =66.6667 ~Um + 41!

1 − e−0.1 ~Um+41!(27)

bm = 2666.7 e−0.056 ~Um+66! (28)

dh

dt= ah ~1 − h! − bh h (29)

ah = 6.6667 e−0.125 ~Um+75! (30)

bh =666.6667

1 + 320 e−0.1 ~Um+75!(31)

Background calcium current

ICa,b = gCa,b ~Um − ECa! (32)

with gCa,b 4 0.01 mS.

Acetylcholine-activated K+ current

IACh = u gK, ACh ~Um − EK! (33)

with gK,ACh = 0.27 µS and u defined according to

du

dt= au ~1 − u! − bu u (34)

au =4.1067

1 +0.0042

@ACh#

(35)

bu = 3.3333 e0.0133 ~Um+40! (36)

Slow-inward current

Isi = Isi,Ca + Isi,K + Isi,Na (37)

with Isi,Ca , Isi,K and Isi,Na defined as

Isi,Ca = 4 d f fCa Psi

SUm − 50D F

RT

S@Ca2+#i eS100 F

RT D− @Ca2+#o e−~Um − 50! 2 F

RT D1 − e−

~Um − 50! 2 F

RT

(38)

Isi,Na = 0.01 d f fCa Psi

~Um − 0.05! F

RT

S@Na+#i e0.05 F

RT − @Na+#o eS−~Um − 0.05! F

RT DD1 − e−

~Um − 0.05! F

RT

(39)

Isi,K = 0.01 d f fCa Psi

~Um − 0.05! F

RT

S@K+#i e0.05 F

RT − @K+#o eS−~Um − 0.05! F

RT DD1 − e−

~Um − 0.05! F

RT

(40)

where Psi 4 7.5 and d, f, and fCa defined according to

d~d!

dt= ad ~1 − d! − bd d (41)

ad =10 ~Um + 24!

1 − e−0.25 ~Um+24!(42)



bd =4 ~Um + 24!

e0.1 ~Um+24!−1 (43)

df

dt= af ~1 − f! − bf f (44)

af =2.0833 ~Um + 34!

e0.25 ~Um+34!−1 (45)

bf =16.6667

1 + e−0.25 ~Um+34!(46)

dfCa

dt= afCa

~1 − fCa! − bfCa@Ca2+#i fCa (47)

where afCa4 3.3333 and bfCa

4 3333.3.

Equilibrium sodium potential

ENa =RT

Fln S@Na+#o

@Na+#iD (48)

Equilibrium potassium potential

EK =RT

Fln S@K+#o

@K+#iD (49)

Equilibrium calcium potential

ECa =RT

2 Fln S@Ca2+#o

@Ca2+#iD (50)

Reversal potential for fast sodium current

Emh =RT

Fln S0.12 @K+#o + @Na+#o

0.12 @K+#i + @Na+#iD (51)

Intracellular sodium concentration

d@Na+#i

dt=

−INa + INa,b + 3 INaK + 3 INaCa + If,Na + Isi,Na

ViF(52)

Intracellular potassium concentration

d@K+#i

dt= −

IK + IK1 − 2 INaK + If,K + Isi,K + IACh

ViF(53)

Intracellular ionized calcium concentrations

d@Ca2+#i

dt=

−Im,Ca + Irel − Iup

2 Vi F(54)

d@Ca2+#NSR

dt=

Iup − Itr

2 VNSR F(55)

d@Ca2+#JSR

dt=

Itr − Irel

2 VJSR F(56)

where

Im,Ca = Isi,Ca + ICa,b − 2 INaCa (57)

Iup = aup@Ca2+#i ~@Ca2+#NSR,max − @Ca2+#NSR! (58)

Itr = atr p ~@Ca2+#NSR − @Ca2+#JSR! (59)

Irel = arel @Ca2+#JSR S @Ca2+#i2

@Ca2+#i2 + Km,Ca

D (60)

with [Ca2+]NSRmax 4 5 mM, Km,Ca 4 0.000001 mMand aup, p, atr, arel defined according to

aup =2 F Vi

tup @Ca2+#NSR,max

(61)

dp

dt= ap ~1 − p! − bp p (62)

ap =0.2083 ~Um + 34!

e0.25 ~Um+34! − 1(63)

bp =1.6667

e−10.25 ~Um+34!+1 (64)

atr =2 F VJSR

ttr(65)

arel =2 F VJSR

trel(66)

where tup 4 0.0118 s, ttr 4 0.6 s, trel 4 0.03 s, Vi 42.8274 10−3 mm3 (in normal conditions) and

VNSR = 0.05 Vi (67)

VJSR = 0.02 Vi (68)

Faraday constant F was set to 96,500°C/mol, the gasconstant R was 8.32 J/mol°K, and temperature T wasset to 310°K.

APPENDIX 2

[A]o and [A]i 4 extracellular and intracellular con-centrations of ion A, respectively

ACh 4 acetylcholineC 4 membrane capacitanceCa2+ 4 calcium iondNaCa 4 denominator constant for INaCa

EA 4 Nerst potential of ionic species AEmh 4 reversal potential for fast sodium

currentF 4 Faraday’s constant



gi 4 maximum membrane conductance tothe current Ii

IACh 4 acetylcholine-activated potassium cur-rent

ICa,b 4 background calcium currentIf 4 hyperpolarizing-activated currentIf,K 4 K+ component of If

If,Na 4 Na+ component of If

IK 4 time-dependent potassium currentIK,max 4 maximum value of time-dependent po-

tassium currentKK1 4 time-independent potassium currentIm,Ca 4 total membrane calcium currentINa 4 fast sodium currentINa,b 4 background sodium currentINaCa 4 Na–Ca exchange currentINaK 4 Na–K exchange pump currentINaK,max 4 maximum value of Na–K exchange

pump currentIrel 4 calcium release from JSR to myoplasmIsi 4 slow inward currentIsi,Ca 4 calcium component of slow-inward

currentIsi,K 4 potassium component of slow-inward

currentIsi,Na 4 sodium component of slow-inward

currentItot 4 total membrane currentItr 4 calcium transfer from NSR to JSRIup 4 calcium uptake from myoplasm to

NSRJSR 4 junctional sarcoplasmic reticulum (re-

lease store)K+ 4 potassium ionKm,Ca 4 Km for Ca binding to release siteKmf 4 Km for extracellular K activation of If

KmK 4 Km for K activation of Na–K pumpKmNa 4 Km for Na activation of Na–K pumpKm1 4 Km for K activation of IK1

KNaCa 4 scaling factor for INaCa

Na+ 4 sodium ionNSR 4 network sarcoplasmic reticulum (up-

take store)Psi 4 permeability of slow-inward current

channelR 4 gas constantSR 4 sarcoplasmic reticulumT 4 absolute temperatureUm 4 membrane potentialVi 4 myoplasm volumeVJSR 4 JSR volumeVNSR 4 NSR volumeau 4 opening rate coefficient of u gating

variablebu 4 closing rate coefficient of u gating vari-

abletrel 4 time constant of intracellular calcium

release from JSR to myoplasmttr 4 time constant of intracellular calcium

transfer from NSR to JSRtup 4 time constant of intracellular calcium

uptake into sarcoplasmic reticulum

Initial Conditions

[ACh] 4 0.001 mM[Ca2+]i 4 0.000029 mM[Ca2+]JSR 4 0.6424 mM[Ca2+]NSR 4 1.9456 mMd 4 0.0001f 4 0.9995fCa 4 0.7486h 4 0.094[K+]i 4 137.1 mMm 4 0.0649[Na+]i 4 9.5 mMp 4 0.8156u 4 0Um 4 −64.4 mVx 4 0.4776y 4 0.0041



Electrolyte and pH Dependence of Heart Rate During Hemodialysis: A Computer Model Analysis

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Transcript of Electrolyte and pH Dependence of Heart Rate During Hemodialysis: A Computer Model Analysis