A mathematical model of the four cardinal acid-base disorders

Alhaji Cherif; Vaibhav Maheshwari; Doris Fuertinger; Gudrun Schappacher-Tilp; Priscila Preciado; David Bushinsky; Stephan Thijssen; Peter Kotanko; Alhaji Cherif; Vaibhav Maheshwari; Doris Fuertinger; Gudrun Schappacher-Tilp; Priscila Preciado; David Bushinsky; Stephan Thijssen; Peter Kotanko

doi:10.3934/mbe.2020246

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 4457-4476. doi: 10.3934/mbe.2020246

Previous Article Next Article

Research article Special Issues

A mathematical model of the four cardinal acid-base disorders

1.
Renal Research Institute, New York, NY, USA
2.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA
3.
Fresenius Medical Care Germany, Bad Homburg, Germany
4.
Department of Mathematics, University of Graz, Graz, Austria
5.
University of Rochester, Rochester, NY, USA
6.
Icahn School of Medicine at Mount Sinai, New York, NY, USA

Received: 14 February 2020 Accepted: 15 June 2020 Published: 23 June 2020

Precise maintenance of acid-base homeostasis is fundamental for optimal functioning of physiological and cellular processes. The presence of an acid-base disturbance can affect clinical outcomes and is usually caused by an underlying disease. It is, therefore, important to assess the acid-base status of patients, and the extent to which various therapeutic treatments are effective in controlling these acid-base alterations. In this paper, we develop a dynamic model of the physiological regulation of an HCO₃^-/CO₂ buffering system, an abundant and powerful buffering system, using Henderson-Hasselbalch kinetics. We simulate the normal physiological state and four cardinal acidbase disorders: Metabolic acidosis and alkalosis and respiratory acidosis and alkalosis. We show that the model accurately predicts serum pH over a range of clinical conditions. In addition to qualitative validation, we compare the in silico results with clinical data on acid-base homeostasis and alterations, finding clear relationships between primary acid-base disturbances and the secondary adaptive compensatory responses. We also show that the predicted primary disturbances accurately resemble clinically observed compensatory responses. Furthermore, via sensitivity analysis, key parameters were identified which could be the most effective in regulating systemic pH in healthy individuals, and those with chronic kidney disease and distal and proximal renal tubular acidosis. The model presented here may provide pathophysiologic insights and can serve as a tool to assess the safety and efficacy of different therapeutic interventions to control or correct acid-base disorders.
- mathematical model,
- acid-base homeostasis,
- bicarbonate,
- acidosis,
- alkalosis
Citation: Alhaji Cherif, Vaibhav Maheshwari, Doris Fuertinger, Gudrun Schappacher-Tilp, Priscila Preciado, David Bushinsky, Stephan Thijssen, Peter Kotanko. A mathematical model of the four cardinal acid-base disorders[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4457-4476. doi: 10.3934/mbe.2020246

Related Papers:

Abstract

Precise maintenance of acid-base homeostasis is fundamental for optimal functioning of physiological and cellular processes. The presence of an acid-base disturbance can affect clinical outcomes and is usually caused by an underlying disease. It is, therefore, important to assess the acid-base status of patients, and the extent to which various therapeutic treatments are effective in controlling these acid-base alterations. In this paper, we develop a dynamic model of the physiological regulation of an HCO₃^-/CO₂ buffering system, an abundant and powerful buffering system, using Henderson-Hasselbalch kinetics. We simulate the normal physiological state and four cardinal acidbase disorders: Metabolic acidosis and alkalosis and respiratory acidosis and alkalosis. We show that the model accurately predicts serum pH over a range of clinical conditions. In addition to qualitative validation, we compare the in silico results with clinical data on acid-base homeostasis and alterations, finding clear relationships between primary acid-base disturbances and the secondary adaptive compensatory responses. We also show that the predicted primary disturbances accurately resemble clinically observed compensatory responses. Furthermore, via sensitivity analysis, key parameters were identified which could be the most effective in regulating systemic pH in healthy individuals, and those with chronic kidney disease and distal and proximal renal tubular acidosis. The model presented here may provide pathophysiologic insights and can serve as a tool to assess the safety and efficacy of different therapeutic interventions to control or correct acid-base disorders.

References

[1]	W. B. Busa, R. Nuccitelli, Metabolic regulation via intracellular pH, Am. J. Physiol., 246 (1984), R409-R438.
[2]	T. E. DeCoursey, The intimate and controversial relationship between voltage-gated proton channels and the phagocyte NADPH oxidase, Immunol. Rev., 273 (2016), 194-218.
[3]	E. K. Hoffmann, L. O. Simonsen, Membrane mechanisms in volume and pH regulation in vertebrate cells, Physiol. Rev., 69 (1989), 315-382.
[4]	A. Roos, W. F. Boron, Intracellular pH, Physiol. Rev., 61 (1981), 296-434.
[5]	A. Schonichen, B. A. Webb, M. P. Jacobson, D. L. Barber, Considering protonation as a posttranslational modification regulating protein structure and function, Annu. Rev. Biophys., 42 (2013), 289-314.
[6]	K. F. Atkinson, S. M. Nauli, pH sensors and ion Transporters: Potential therapeutic targets for acid-base disorders, Int. J. Pharma Res. Rev., 5 (2016), 51.
[7]	W. F. Boron, Acid-base transport by the renal proximal tubule, J. Am. Soc. Nephrol., 17 (2006), 2368-2382.
[8]	V. Fencl, J. Vale, J. Broch, Respiration and cerebral blood flow in metabolic acidosis and alkalosis in humans, J. Appl. Phys., 27 (1969), 67-76
[9]	L. L. Hamm, N. Nakhou, K. S. Hering-Smith, Acid-base homeostasis, Clin. J. Am. Soc. Nephrol., 10 (2015), 2232-2242.
[10]	D. Hornick, An approach to the analysis of arterial blood gases and acid-base disorders, Virtual Hospital, University of Iowa Health Care [On-line], 2003.
[11]	M. Levitzky, Pulmonary Physiology, McGraw-Hill Book Company, 2003.
[12]	C. Lote, Principles of Renal Physiology, Kluwer Academic Publishers, 1999.
[13]	N. A. Masco, Acid-base homeostasis, J. Infusion Nurs., 39 (2016), 288-295.
[14]	R. Mitchell, M. Singer, Respiration and cerebrospinal fluid pH in metabolic acidosis and alkalosis, J. Appl. Phys., 20 (1965), 905-911.
[15]	G. T. Nagami, L. L. Hamm, Regulation of acid-base balance in chronic kidney disease, Adv. Chronic kidney Dis., 24 (2017), 274-279.
[16]	R. Pitts, Physiology of the Kidney and Body Fluids, Year Book Medical Publishers Inc, 1970.
[17]	J. Poppell, P. Vanamee, K. Roberts, H. Randall, The effect of ventilatory insufficiency on respiratory compensations in metabolic acidosis and alkalosis, J. Lab. Clin. Med., 47 (1956), 885-890.
[18]	R. Quigley, Acid-base homeostasis. Clin. Pediatr. Nephrol., 2016, 235.
[19]	W. B. Schwartz, J. J. Cohen, The nature of the renal response to chronic disorders of acid-base equilibrium, Am. J. Med., 64 (1978), 417-428.
[20]	L. A. Skelton, W. F. Boron, Y. Zhou, Acid-base transport by the renal proximal tubule, J. Nephrol., 23 (2010), S4.
[21]	J. Lemann Jr., D. A. Bushinsky, L. L. Hamm, Bone buffering of acid and base in humans, Am. J. Physiol. Renal Physiol., 285 (2003), F811-F832.
[22]	H. Davenport, The ABC of acid-base chemistry, The University of Chicago, Chicago, IL, 1974.
[23]	R. Hainsworth, Acid-Base Balance, Manchester University, UK, 1986.
[24]	J. B. West, Respiratory Physiology, 9th edition, Lippincott Williams and Wilkins, Philadelphia, PA, 2012.
[25]	J. Widdicombe, A. Davies, Respiratory Physiology, Edward Arnold Publishers, 1983.
[26]	W. Lang, R. Zander, Prediction of dilutional acidosis based on the revised classical dilution concept for bicarbonate, J. Appl. Phys., 98 (2005), 62-71.
[27]	M. B. Wolf, E. C. DeLand, A mathematical model of blood-interstitial acid-base balance: Application to dilution acidosis and acid-base status, J. Appl. Phys., 110 ( 2011), 988-1002.
[28]	K. Annan, Mathematical modeling of the dynamic exchange of solutes during bicarbonate dialysis, Math. Comput. Modell., 55 (2012), 1691-1704.
[29]	L. Coli, M. Ursino, A. De Pascalis, C. Brighenti, V. Dalmastri, G. La Manna, et al., Evaluation of intradialytic solute and fluid kinetics, Blood Purif., 18 (2000), 37-49.
[30]	R. K. Dash, J. B. Bassingthwaighte, Simultaneous bloodtissue exchange of oxygen, carbon dioxide, bicarbonate, and hydrogen ion, Ann. Biomed. Eng., 34 (2006), 1129-1148.
[31]	S. Marano, M. Marano, Frontiers in hemodialysis: Solutions and implications of mathematical models for bicarbonate restoring, Biomed. Signal Process. Control, 52 (2019), 321-329.
[32]	N. K. Martin, E. A. Gaffney, R. A. Gatenby, R. J. Gillies, I. F. Robey, P. K. Maini, A mathematical model of tumour and blood pHe regulation: The HCO3-/CO2 buffering system, Math. Biosci., 230 (2011), 1-11.
[33]	J. A. Sargent, M. Marano, S. Marano, F. J. Gennari, Acidbase homeostasis during hemodialysis: New insights into the mystery of bicarbonate disappearance during treatment, Semin. Dial., (2018), 1-11.
[34]	O. Thews, H. Hutten, A comprehensive model of the dynamic exchange processes during hemodialysis, Med. Prog. Technol., 16 (1990), 145-161.
[35]	M. Ursino, L. Coli, C. Brighenti, L. Chiari, A. De Pascalis, G. Avanzolini, Prediction of solute kinetics, acid-base status, and blood volume changes during profiled hemodialysis, Ann. Biomed. Eng., 28 (2000), 204-216.
[36]	K. A. Hasselbalch, Die berechnung der wasserstoffzahl des blutes aus der freien und gebundenen kohlensure desselben, und die sauerstoffbindung des blutes als funktion der wasserstoffzahl, Biochem. Z., 78 (1917), 112-144.
[37]	L. J. Henderson, Concerning the relationship between the strength of acids and their capacity to preserve neutrality, Am. J. Physiol., 21 (1908), 173-179.
[38]	C. Chegwidden, E. Edwards, Carbonic Anhydrases: New Horizons, Birkhauser Press, 2000.
[39]	J. J. Batzel, F. Kappel, D. Schneditz, H. T. Tran, Cardiovascular and Respiratory Systems: Modeling, Analysis, and Control, SIAM, Philadelphia, 2007.
[40]	Memorang, Compensation Reactions to Acid/Base Imbalance [online]. Available from: https://www.memorangapp.com/flashcards/94371/Compensation+Reactions+to+Acid/2FBase+Imbalance (last accessed 24 April 2020).
[41]	M. S. Albert, R. B. Dell, R. W. Winters, Quantitative displacement of acid-base equilibrium in metabolic acidosis, Ann. Int. Med., 66 (1967), 312-322.
[42]	D. A. Bushinsky, F. L. Coe, C. Katzenberg, J. P. Szidon, J. H. Parks, Arterial PCO2 in chronic metabolic acidosis, Kidney Int., 22 (1982), 311-314.
[43]	K. Engel, R. B. Dell, W. J. Rahill, C. R. Denning, R. W. Winters, Quantitative displacement of acid-base equilibrium in chronic respiratory acidosis, J. Appl. Phys., 24 (1968), 288-295.
[44]	S. B. Gonzlez, G. Menga, G. A. Raimondi, H. Tighiouart, H. J. Adrogu, N. E. Madias, Secondary response to chronic respiratory acidosis in humans: A prospective study, Kidney Int. Rep., 3 (2018), 1163-1170.
[45]	A. Hasan, The Analysis of Blood Gases: Handbook of Blood Gas/Acid-Base Interpretation, Springer London, (2013), 253-266.
[46]	S. Javaheri, N. S. Shore, B. Rose, H. Kazemi, Compensatory hypoventilation in metabolic alkalosis, Chest, 81 (1982), 296-301.
[47]	K. Roberts, J. Poppell, P. Vanamee, R. Beals, H. Randall, Evaluation of respiratory compensation in metabolic alkalosis, J. Clin. Invest., 35 (1956), 261-266.
[48]	S. M. Blower, H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, Int. Stat. Rev., 2 (1994), 229-243.
[49]	S. Marino, I. B. Hogue, C. J. Ray, D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196
[50]	D. A. Bushinsky, T. Hostetter, G. Klaerner, Y. Stasiv, C. Lockey, S. McNulty, et al., Randomized, controlled trial of tRC101 to increase serum bicarbonate in patients with CKD, Clin. J. Am. Soc. Nephrol., 13 (2018), 26-35.
[51]	R. C. De Sousa, J. T. Harrington, E. S. Ricanati, J. W. Shelkrot, W. B. Schwartz, Renal regulation of acid-base equilibrium during chronic administration of mineral acid, J. Clin. Invest., 53 (1974), 465-476.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)