Sargent et al.1 have recently proposed a relatively simple, homogeneous, extracellular fluid (EC) compartment model that closely predicts clinical measurements of bicarbonate kinetics in chronic renal failure patients during hemodialysis (HD) treatment. The proposed mechanisms are diffusive and convective bicarbonate and acetate transport, acetate metabolism and intracellular H+ generation. Quantitative conclusions from their model were: 1) 35% of the bicarbonate added to the patient during HD was because of patient influx of acetate from the HD fluid and subsequent metabolism and 2) about 90% of the bicarbonate delivered to the patient was neutralized by H+ from cellular sources, the latter a central feature of their newly hypothesized H+-mobilization model. Leypoldt et al.2 have used this model to describe HD-induced plasma bicarbonate and total carbon dioxide kinetics with bicarbonate and lactate HD fluid buffers.
The major problem with the Sargent et al. approach1 is that bicarbonate is an ion; hence, it cannot be treated as an individual species like urea, for example. Bicarbonate transport between HD fluid and plasma must be accompanied by an electrically-neutralizing cation(s) as dictated by the equilibrium principle of fluid electroneutrality. Gamble et al.3 showed in 1923 that in plasma, the sum of the electrical concentrations of cations and anions must be equal. Later studies by van Slyke et al.4 and Siggaard-Andersen,5,6 whose aims were to devise analytical methods and steady-state models to diagnose and treat acid-base disorders, also recognized the importance of the electroneutrality principle, even though their focus was principally on the role of bicarbonate in acid-base balance.7
More recently, Stewart8 and Adrogué and Madias9 used the electroneutrality principle along with the equilibrium-based Henderson–Hasselbalch (HH) equation to devise a mathematical model that predicted equilibrium pH changes in plasma resulting from changes in PCO2, total electrical concentration difference of small cations and anions, called the “Strong-ion difference” or SID and the electrical concentrations of albumin and inorganic phosphates, denoted as Atot. These three quantities are independent variables in isolated plasma, but the latter two change dependently when plasma communicates chemically with other body fluids. An important conclusion of Stewart's work is that changes in plasma bicarbonate do not cause acid-base alterations, but they are only the result of changes in these three quantities.
Stewart,8 along with others10–12 devising acid-base models, have not considered body-fluid, acid-base kinetics that would occur during HD. Their models implicitly assume that acid-base changes can be described as a series of steady-states, each of which obey electrochemical equilibrium principles at every time point. In contrast, the Sargent et al. model1 simulates ionic transport processes, which move small ions between HD fluid and a generalized EC. Their model assumes an equilibrium state between EC values of PCO2, pH and bicarbonate concentration, the latter derived using the HH equation; however, their model does not include the electroneutrality constraint on their EC compartment.
Consequently, the first aim of the current study is to devise a mathematical model of patient, body-fluid kinetics during HD treatment similar to that of Sargent et al.,1 but one that incorporates the electroneutrality principle. The second aim is to propose reasonable hypotheses to explain the different findings of the new model from those of Sargent et al. relative to the mechanisms leading to acid-base homeostasis during HD.
Materials and Methods
Patient Electroneutrality Model
- The patient model consists of a uniform EC compartment containing water, bicarbonate (HCO3−) and acetate (Ac−) ions, similar to the Sargent et al. model,1 except the small ions, Na+, K+, Cl−, and net unidentified small ions (NUIs), are included. Also present is a volume-dependent, fixed negative charge attributable to albumin.
- The resulting EC-water (ECW) compartment is electrically neutral at all times.
- Ultrafiltration occurs from the ECW at a constant rate throughout HD.
- Transport of small ions, but not albumin, can occur between HD fluid and ECW.
- Mass balance obtains in the ECW compartment for all small ions, other than bicarbonate, because of its interconversion with CO2 (see section “Discussion”).
- The effective concentration of small ions in the HD fluid is modified by a Gibbs–Donnan (GD) equilibrium factor as in the Sargent et al. model.1 This factor increases HD fluid anion concentrations by 1.05 and decreases cation concentrations by the same factor.
- Ac− is metabolized in the ECW compartment exponentially during the HD treatment, as in the Sargent et al. model,1 except the formulation is somewhat different.
- For transport purposes, all small-ion concentrations in the EC compartment of the new model are those in water, requiring a decrease by a factor of 0.93 to convert to plasma. In contrast, the Sargent et al.1 model implicitly assumes that their measured mM plasma concentrations exist throughout their EC compartment.
- pH is determined from the HH equation with CO2 solubility of 0.0302 mM/mm Hg, consistent with the value used by Sargent et al.1 to calculate plasma [HCO3−] from their blood PCO2 and pH measurements.
- All parameter values are those of Sargent et al.,1 except where noted otherwise.
Equations (all concentrations are mEq/lwater, except where otherwise denoted)
Electroneutrality in ECW compartment
where all concentrations are in mEq/lECW.
Small-ion transport between HD fluid (d) and ECW
where Jion is the ionic flux (mEq/min), D is the dialysance (l/min), Q is the flow (l/min), UF and B refer to the ultrafiltered water and to blood, respectively, and d refers to the HD fluid.
ECW volume change
where V is the volume (l), superscript 0 refers to the predialysis volume, and t is the time (min) from beginning of HD treatment.
Small-ion mass balance in ECW (except for HCO3−)
where M is the mass (mmol).
Acetate metabolism in ECW
Henderson–Hasselbalch equation in EC compartment
where α is the solubility, P is the partial pressure (mm Hg), and EC refers to extracellular (not corrected to water concentration).
Experimental Data for Model Comparison
The experimental data used were from Sargent et al.1 They measured blood pH and PCO2 and calculated plasma [HCO3−] using a rearrangement of the HH equation (equation 7) at various time points during a 4 hour HD procedure in 14 patients undergoing HD using a HCO3−- and Ac− -containing HD fluid. Sargent et al. did not give CO2 solubility values used in their plasma [HCO3−] calculations, but it was determined that their data given were consistent with a CO2 solubility of 0.0302 mmol/lEC/mm Hg, the value used in equation 7 in the present patient model.
Parameter Selection and Estimation
Sargent et al.1 did not give measurement values for important quantities, other than the ones from arterial blood gas analysis; however, those required to apply the electroneutrality principle of equation 1 are arterial-plasma ion concentrations. Assumed predialysis values for the latter quantities are shown in the first column of Table 1. The [NUI] factor is included because patients on dialysis often have metabolic acidosis because of accumulation of various organic acids.13 The bottom rows of column 1 show the arterial blood gas measurements of Sargent et al. The second column shows the ionic concentrations given by Sargent et al. for the HD fluid and the last column shows the Sargent et al. dialysance (D) values. Note that the D value for Cl− was assumed to be that for Na+ and the value for NUI was assumed to be the same as for Ac−. Other Sargent et al. data used in the present simulation were the 12.3 l of predialysis ECW, a 209 min treatment time and 1.8 l of ultrafiltration, at a constant rate of 8.6 ml/min, over the treatment duration.
The set of assumed ion concentrations in Table 1 was sufficient for the model to predict the predialysis [HCO3−]EC value measured by Sargent et al.1 (see section “Results”), while preserving EC electroneutrality. It is likely that other data sets would yield the same result, but coupled with the D values used, this set was also able to closely predict the measured postdialysis [HCO3−]EC value, as well as the measured kinetic values during the HD treatment (see sections “Results” and “Discussion”).
Table 1. -
New Model Predialysis Conditions and Other Values
||Patient Predialysis Arterial Plasma (mM)
||Hemodialysis Fluid* (mmol/lwater)
||Dialysance (D) Values (l/min)
|PCO2 (mm Hg)
*From Sargent et al.
†Assumed predialysis value.
‡Assumed equal to Na+ value.
§Adjusted to achieve predialysis electroneutrality and experimentally measured [HCO3−].
¶Assumed equal to Ac− value.
∥Postdialysis, from Sargent et al.
NUI, unidentified small ion.
Sargent et al.1 suggested that Ac− in the HD fluid plays an important role in determining acid-base changes in blood. The model was used to examine this role by manipulating its concentration and those of Na+, Cl−, and HCO3− in the HD fluid.
The computer simulation program used (Altair Embed, Altair Engineering, 1820 E. Beaver Road, Troy, MI) has the ability to solve the dynamic model equations required for the unknown value of , as constrained by electroneutrality. The solution procedure took <10 sec, using a step size of 0.005 min, which was sufficient to maintain constraint values sufficiently small so as to minimally affect the computed results.
The open circles with SD bars in Figure 1, A, are the plasma [HCO3−] measurements by Sargent et al.,1 and the dashed line shows the Sargent et al. model predictions. The new model predictions, as converted to plasma concentrations, are shown by the solid line. As seen, the new-model predictions are well within the SDs of the measured values and are almost as close as the model predictions by Sargent et al. Similar close predictions of their pH measurements (open circles) are shown in Figure 1, B. These results confirm that the present model has the capability of closely predicting the experimental data of Sargent et al., a primary aim of the current study (see section “Discussion”).
Sargent et al.1 suggested that Ac− plays an important role in adding HCO3− to the body fluids. This assertion was tested using the present model. Table 2 shows the results. Column A shows the end-dialysis effects on acid-base variables for the HD fluid, ion concentrations shown, the same as used for the model predictions seen in Figure 1. Substituting HCO3− for Ac− (column B) has little impact, other than further increasing the ECW HCO3− content by 6.4 mmol, whereas substituting Cl− for Ac− (column C) decreases pH by 0.03 units by decreasing HCO3− content by about 23 mmol. Decreasing HD fluid [Na+] to maintain [HCO3−] (column D) causes similar effects as the latter condition, but actually decreases HCO3− content even more. Consequently, including Ac− in the HD fluid can be useful for patients with metabolic acidosis, but the simpler solution is to substitute some HCO3− for Ac−.
Using the principle of electroneutrality in models of body-fluid chemistry shifts the focus from the importance of HCO3− to that of small ions that do not interact with H+, the “Strong” ions, as defined by Stewart.9 He first pointed out that it is the difference in electrical concentrations of these small ions, the “Strong-ion difference” or SID, and its changes from normal that are a major factor in causing acid-base disorders, but changes in PCO2 and Atot are also important. Consequently, complete models of patient acid-base chemistry kinetics during dialysis must incorporate these factors.
The recent Sargent et al.1 computer model focuses on predicting the kinetic [HCO3−] measurements in plasma during an HD treatment and relating these kinetics and those of Ac− to H+ generation from intracellular sources. Their central hypothesis is that HCO3− is the primary buffer neutralizing the H+ produced by intracellular metabolic processes. A secondary buffer is the Ac− contained in the HD fluid transferring to the EC, but this effect is temporary due to subsequent Ac− metabolism. Their model assumes that the total increase in body-fluid HCO3− content is because of its transport from HD fluid to the EC compartment, less the amount due to H+ generation. The latter mechanism was simulated as an empirically determined, constant parameter value (mH+) times the increase in plasma [HCO3−] from predialysis values. They found that optimal mH+ values determined for each patient varied greatly, from 0.08 to 0.44 L/min. The resulting H+ mobilization model has been subsequently used by Leypoldt et al.2 to predict kinetic changes in EC total-CO2 content during and after HD for various HD treatment prescriptions.
The model developed in the current study is more complete than that of Sargent et al.1 in that it includes other important ions and requires an electroneutrality constraint on all ions in the EC. Its predictions, shown in Figure 1, demonstrate that it is a feasible alternative to the Sargent et al. model in that it also closely predicts [HCO3−] kinetics during the HD treatment, but by some quite different processes. To obtain these predictions in the new model, the predialysis [HCO3−]EC was matched to the plasma value of Sargent et al. by assuming a set of predialysis values for concentrations of Na+, K+, and Cl− in the EC compartment, a fixed electrical charge due to albumin, which changed only as compartment volume changed and an unidentified anionic species (NUI), likely organic acids, which are transported between HD fluid and the ECW compartment (see Table 1). No other parameter values were changed to achieve the results of Figure 1, which gives confidence in the model’s ability to accurately predict other data, such as the effects of manipulating Ac− and other small ions in the HD fluid, as described in Table 2.
Table 2. -
Effect of Acetate on Patient Acid-Base Balance (End-Dialysis Values)
||HD Fluid Ion Concentrations (mEq/lwater)
The present model does not have a specific H+ generation factor; hence, the [HCO3−] curve shape of Figure 1 must be due to other factors. The present hypothesis is that it is actually due to the electroneutrality principle (see equation 1), which constrains the kinetics of [HCO3−]EC; hence, its compartmental content, to the net electrical kinetics of the other small ions in the EC compartment (see equations 1–5). Figure 2 shows the predicted EC HCO3−-content kinetics (solid line) during the HD treatment assuming the compartmental electroneutrality constraint. Quantitatively, the magnitude of the curve depends upon the D values used (see Table 1), which are mainly from Sargent et al.1 The hypothesis of Sargent et al. is that HCO3−--kinetics are primarily due to convective and diffusive transport of HCO3− between the HD fluid and EC compartment. To test their hypothesis using the present model, the long-dashed line in Figure 2 shows the predicted content curve using their assumed HCO3−D value; it reached a near constant level after about 120 min, whereas the electroneutrality constraint curve (solid line) reaches a peak and then rapidly declines.
The experimental results of Morel et al.14 suggest that calculated values of HCO3− dialysance decrease linearly over the HD treatment time (THD), but their experimental data had considerable scatter. The short-dashed curve in Figure 2 shows that assuming D equals (0.198 – 0.08 × THD/209) l/min, produces a result somewhat nearer to the electroneutrality constraint curve, but still far from a good match.
Another possibility to explain the difference between the constant dialysance and electroneutrality curves is that HCO3− is not actually transported, because there is sufficiently rapid interconversion of CO2 and HCO3− in the body fluids to always satisfy the electroneutrality constraint, without depleting the large reservoir of CO2 in the lungs and body fluids. This postulate would explain the ability of complex body-fluid models,15,16 which incorporate the electroneutrality constraint, to closely describe acid-base changes due to fluid and electrolyte infusions used to correct acid-base disturbances.
The quantitative conclusions derived from this study critically depend upon the assumptions of the new model, many of which are similar to those in the Sargent et al. model.1 Most important is the assumption of a homogeneous single compartment to represent the body fluids, which simplifies the model greatly, in contrast to representing the body fluids by more complicated models. Many of these models, however, do not include an electroneutrality constraint; hence, they would not be able to accurately portray the changes in body-fluid composition stemming from acid-base disorders. One exception is the model of Ursino et al.,17 who devised a 3-compartment model (no red-cell compartment) designed to study the effects of profiled HD on changes in acid-base status and blood volume; however, electroneutrality was only applied to the ionic transport between interstitial and intracellular compartments. Casagrande et al.18 did not apply a body-fluid electroneutrality constraint in their multicompartment HD model. In contrast, Wolf15,16 has developed detailed multicompartment, fluid and electrolyte models, with electroneutrality constraints on each compartment, which could be applicable to further study of the effects of HD-induced kinetics on body-fluid, acid-base issues.
The ideas proposed in the present HD model and in my previous model of peritoneal-dialysis-induced, acid-base kinetics19 can have profound implications for the understanding of the body’s ability to maintain acid-base homeostasis. Rapid intracompartmental CO2 to bicarbonate conversion, rather than transmembrane bicarbonate transport, becomes the primary mechanism, which could lead to a reexamination of long-held beliefs concerning homeostatic acid-base mechanisms.
1. Sargent JA, Marano M, Marano S, Gennari FJ: Acid-base homeostasis during hemodialysis: New insights into the mystery of bicarbonate disappearance during treatment. Semin Dial. 31: 468–478, 2018.
2. Leypoldt JK, Pietribiasi M, Ebinger A, Kraus MA, Collins A, Waniewski J: Acid-base kinetics during hemodialysis using bicarbonate and lactate as dialysate buffer bases based on the H(+) mobilization model. Int J Artif Organs. 43: 645–652, 2020.
3. Gamble JL, Ross GS, Tisdall FF: The metabolism of fixed base during fasting. J Biol Chem. 58: 633–95, 1923.
4. Van Slyke DD, Wu H, McLean FC: Studies of gas and electrolyte equilibria in the blood. V. Factors controlling the electrolyte and water distribution in the blood. J Biol Chem. 56: 765–849, 1923.
5. Siggaard-Andersen O: The Acid-Base Status of the Blood, Copenhagen, Munksgaard, 1963.
6. Siggaard-Andersen O: The van Slyke equation. Scand J Clin Lab Invest Suppl. 146: 15–20, 1977.
7. Siggaard-Andersen O: Acid-base balance. In: Encyclopedia of Respiratory Medicine, Boston G, ed, Amsterdam, Academic Press, 2006, pp. 5–10.
8. Stewart PA: How to Understand Acid-Base, New York, Elsevier North Holland, 1981.
9. Adrogué HJ, Madias NE: Assessing acid-base status: Physiologic versus physicochemical approach. Am J Kidney Dis. 68: 793–802, 2016.
10. Andreassen S, Rees SE: Mathematical models of oxygen and carbon dioxide storage and transport: Interstitial fluid and tissue stores and whole-body transport. Crit Rev Biomed Eng. 33: 265–298, 2005.
11. Lang W, Zander R: Prediction of dilutional acidosis based on the revised classical dilution concept for bicarbonate. J Appl Physiol (1985). 98: 62–71, 2005.
12. Russell CD, Roeher HD, DeLand EC, Maloney JV Jr: Acute response to acid-base stress. Ann Surg. 187: 417–422, 1978.
13. Morgan TJ: Unmeasured ions and the strong ion gap. Kellum JA, Elbers PWG, eds. In: Stewart’s Textbook on Acid-Base, 2nd ed. Amsterdam, The Netherlands, AcidBASE.org / Paul WG Elbers, 2009, pp. 323−37.
14. Morel H, Jaffrin MY, Lux C, et al.: A comparison of bicarbonate kinetics and acid-base status in high flux hemodialysis and on-line post-dilution hemodiafiltration. Int J Artif Organs. 35: 288–300, 2012.
15. Wolf MB: Whole body acid-base and fluid-electrolyte balance: A mathematical model
. Am J Physiol Renal Physiol. 305: F1118–F1131, 2013.
16. Wolf MB: Physicochemical models of acid-base. Semin Nephrol. 39: 328–339, 2019.
17. Ursino M, Colí L, Brighenti C, Chiari L, de Pascalis A, Avanzolini G: Prediction of solute kinetics, acid-base status, and blood volume changes during profiled hemodialysis. Ann Biomed Eng. 28: 204–216, 2000.
18. Casagrande G, Bianchi C, Vito D, et al.: Patient-specific modeling of multicompartmental fluid and mass exchange during dialysis. Int J Artif Organs. 39: 220–227, 2016.
19. Wolf MB: Mechanisms of acid-base kinetics during peritoneal dialysis: A mathematical-model study. ASAIO J, 67: 809–816, 2021.