Multivariable optimal control for hemodialysis: A physiologically-grounded simulation study

Redemtus Heru Tjahjana; Ratna Herdiana; Zani Anjani Rafsanjani HSM; Yogi Ahmad Erlangga; Redemtus Heru Tjahjana; Ratna Herdiana; Zani Anjani Rafsanjani HSM; Yogi Ahmad Erlangga

doi:10.3934/mbe.2025088

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 9: 2409-2433. doi: 10.3934/mbe.2025088

Previous Article Next Article

Research article Special Issues

Multivariable optimal control for hemodialysis: A physiologically-grounded simulation study

1.
Department of Mathematics, Faculty of Science and Mathematics, Diponegoro University, Indonesia
2.
Department of Mathematics, College of Health and Natural Sciences, Zayed University, Abu Dhabi United Arab Emirates

Received: 07 April 2025 Revised: 28 May 2025 Accepted: 20 June 2025 Published: 22 July 2025

This study introduces a novel multivariable optimal control framework for hemodialysis, which uniquely integrates five physiological states (blood urea concentration, fluid volume, blood pressure, electrolytes, and hemoglobin) with three clinically adjustable inputs (ultrafiltration rate, blood flow, and dialysate composition). By employing the limited-memory Broyden-Fletcher-Goldfarb-Shanno-B (L-BFGS-B) algorithm with patient-specific box constraints, the model enforces patient-specific physiological safety limits while dynamically balancing clinical targets. Numerical simulations demonstrate the stabilization of key parameters within ±5% of clinical benchmarks (e.g., KDIGO guidelines), though deviations in the hemodynamic responses underscore the need for adaptive control in real-world scenarios. Urea clearance trajectories align with efficacy patterns observed in practice, while blood pressure fluctuations reveal systematic offsets that require protocol refinement. This work bridges control theory with hemodialysis dynamics, thus offering a simulation-driven foundation for future clinical validation and personalized treatment optimization.
- optimal control,
- hemodialysis modeling,
- physiological variables,
- L-BFGS-B algorithm,
- personalized treatment
Citation: Redemtus Heru Tjahjana, Ratna Herdiana, Zani Anjani Rafsanjani HSM, Yogi Ahmad Erlangga. Multivariable optimal control for hemodialysis: A physiologically-grounded simulation study[J]. Mathematical Biosciences and Engineering, 2025, 22(9): 2409-2433. doi: 10.3934/mbe.2025088

Related Papers:

Abstract

This study introduces a novel multivariable optimal control framework for hemodialysis, which uniquely integrates five physiological states (blood urea concentration, fluid volume, blood pressure, electrolytes, and hemoglobin) with three clinically adjustable inputs (ultrafiltration rate, blood flow, and dialysate composition). By employing the limited-memory Broyden-Fletcher-Goldfarb-Shanno-B (L-BFGS-B) algorithm with patient-specific box constraints, the model enforces patient-specific physiological safety limits while dynamically balancing clinical targets. Numerical simulations demonstrate the stabilization of key parameters within ±5% of clinical benchmarks (e.g., KDIGO guidelines), though deviations in the hemodynamic responses underscore the need for adaptive control in real-world scenarios. Urea clearance trajectories align with efficacy patterns observed in practice, while blood pressure fluctuations reveal systematic offsets that require protocol refinement. This work bridges control theory with hemodialysis dynamics, thus offering a simulation-driven foundation for future clinical validation and personalized treatment optimization.

References

[1]	A. Abdelrasoul, H. Westphalen, S. Saadati, A. Shoker, Hemodialysis biocompatibility mathematical models to predict the inflammatory biomarkers released in dialysis patients based on hemodialysis membrane characteristics and clinical practices, Sci. Rep., 11 (2021), 1–16. https://doi.org/10.1038/s41598-021-01660-1 doi: 10.1038/s41598-021-01660-1
[2]	H. Akbarialiabad, S. Kavousi, A. Ghahramani, B. Bastani, N. Ghahramani, COVID-19 and maintenance hemodialysis: A systematic scoping review of practice guidelines, BMC Nephrol., 21 (2020), 470. https://doi.org/10.1186/s12882-020-02143-7 doi: 10.1186/s12882-020-02143-7
[3]	P. Bachhiesl, H. Scharfetter, H. Hutten, F. Kappel, M. Hintermuller, Efficient computation of optimal controls for the exchange processes during the dialysis therapy, Comput. Optimiz. Appl., 18 (2001), 161–174. https://doi.org/10.1023/A:1008726605165 doi: 10.1023/A:1008726605165
[4]	S. Chander, S. Luhana, F. Sadarat, O. Parkash, Z. Rahaman, H. Y. Wang, et al., Mortality and mode of dialysis: Meta-analysis and systematic review, BMC Nephrol., 25 (2024), 1. https://doi.org/10.1186/s12882-023-03435-4 doi: 10.1186/s12882-023-03435-4
[5]	L. Coli, M. Ursino, V. Dalmastri, F. Volpe, G. La Manna, G. Avanzolini, et al., A simple mathematical model applied to selection of the sodium profile during profiled hemodialysis, Nephrol. Dial. Transplant., 13 (1998), 404–416. https://doi.org/10.1093/oxfordjournals.ndt.a027838 doi: 10.1093/oxfordjournals.ndt.a027838
[6]	M. G. M. Guimaraes, F. P. M. Tapioca, N. R. dos Santos, F. P. d. C. T. Ferreira, L. C. S. Passos, P. N. Rocha, Hemodiafiltration versus Hemodialysis in end-stage kidney disease: A systematic review and meta-analysis, Kidney Med., 6 (2024), 100829. https://doi.org/10.1016/j.xkme.2024.100829 doi: 10.1016/j.xkme.2024.100829
[7]	J. J. Joseph, T. J. Hunter, C. Sun, D. Goldman, S. R. Kharche, C. W. McIntyre, Using a human circulation mathematical model to simulate the effects of hemodialysis and therapeutic hypothermia, Appl. Sci., 12 (2022), 1–16. https://doi.org/10.3390/app12010307 doi: 10.3390/app12010307
[8]	V, Kovacic, L. Roguljic, V. Kovacic, Metabolic acidosis of chronically hemodialyzed patients, Am. J. Nephrol., 23 (2003), 158–164. https://doi.org/10.1159/000070205 doi: 10.1159/000070205
[9]	C. J. Lee, Y. L. Chang, A new therapeutic parameter and computer control to approach optimal hemodialysis, Comput. Methods Programs Biomed., 30 (1989), 33–42. https://doi.org/10.1016/0169-2607(89)90120-X doi: 10.1016/0169-2607(89)90120-X
[10]	O. Aydogdu, M. L. Levent, Kalman state estimation and LQR assisted adaptive control of a variable loaded servo system, Eng. Technol. Appl. Sci. Res., 9 (2019), 4125–4130, https://doi.org/10.48084/etasr.2708 doi: 10.48084/etasr.2708
[11]	K. G. Aktas, I. Esen, State-Space Modeling and active vibration control of smart flexible cantilever beam with the use of finite element method, Eng. Technol. Appl. Sci. Res., 10 (2020), 6549–6556. https://doi.org/10.48084/etasr.3949 doi: 10.48084/etasr.3949
[12]	C. V. Nguyen, M. T. Nguyen, H. T. Tran, M. L. Trinh, H. M. La, H. T. T. Nguyen, Trajectory tracking control for a quadcopter under external disturbances, Eng. Technol. Appl. Sci. Res., 14 (2024), 17620–17628. https://doi.org/10.48084/etasr.8449 doi: 10.48084/etasr.8449
[13]	V. Maheshwari, M. Ferris, G. Filler, P. Kotanko, Novel extracorporeal treatment for severe neonatal jaundice: A mathematical modeling study with Allo-hemodialysis, Sci. Rep., 14 (2024), 21910. https://doi.org/10.1038/s41598-024-72256-8 doi: 10.1038/s41598-024-72256-8
[14]	M. Pietribiasi, J. K. Leypoldt, Modeling acid-base transport in hemodialyzers, Biocybern. Biomed. Eng., 41 (2021), 1150–1161. https://doi.org/10.1016/j.bbe.2021.07.006 doi: 10.1016/j.bbe.2021.07.006
[15]	L. Pstras, J. Waniewski, Mathematical Modelling of Haemodialysis: Cardiovascular Response, Body Fluid Shifts, and Solute Kinetics, Springer International Publishing, 2019.
[16]	S. Raoofi, F. Pashazadeh Kan, S. Rafiei, Z. Hoseinipalangi, S. Rezaei, S. Ahmadi, et al., Hemodialysis and peritoneal dialysis—health-related quality of life: Systematic review plus meta-analysis, BMJ Support. Palliat. Care, 13 (2023), 365–373.
[17]	S. Rogg, D. H. Fuertinger, S. Volkwein, F. Kappel, P. Kotanko, Optimal EPO dosing in hemodialysis patients using a non-linear model predictive control approach, J. Math. Biol., 79 (2019), 2281–2313. https://doi.org/10.1007/s00285-019-01429-1 doi: 10.1007/s00285-019-01429-1
[18]	W. Stecz, R. Pytlak, A. Rymarz, S. Niemczyk, Application of dynamic optimisation for planning a haemodialysis process, BMC Nephrol., 20 (2019), 236. https://doi.org/10.1186/s12882-019-1409-8 doi: 10.1186/s12882-019-1409-8
[19]	F. Sanmarchi, C. Fanconi, D. Golinelli, D. Gori, T. H. Boussard, A. Capodici, Predict, diagnose, and treat chronic kidney disease with machine learning: A systematic literature review, J. Nephrol., 36 (2023), 1101–1117. https://doi.org/10.1007/s40620-023-01573-4 doi: 10.1007/s40620-023-01573-4
[20]	J. Waniewski, Mathematical modeling of fluid and solute transport in hemodialysis and peritoneal dialysis, J. Membr. Sci., 274 (2006), 24–37. https://doi.org/10.1016/j.memsci.2005.11.038 doi: 10.1016/j.memsci.2005.11.038
[21]	T. S. Wong, Q. Chen, T. Liu, J. Yu, Y. Gao, Y. He, et al., Patients, Healthcare providers, and general population preferences for hemodialysis vascular access: A discrete choice experiment, Front. Public Health, 12 (2024), 1–10. https://doi.org/10.3389/fpubh.2024.1047769 doi: 10.3389/fpubh.2024.1047769
[22]	M. Ziółko, J. A. Pietrzyk, J. Grabska-Chrzastowska, Accuracy of hemodialysis modeling, Kidney Int., 57 (2000), 1152–1163. https://doi.org/10.1046/j.1523-1755.2000.00942.x doi: 10.1046/j.1523-1755.2000.00942.x
[23]	J. H. Salazar, Overview of urea and creatinine, Lab. Med., 45 (2014), 19–20. https://doi.org/10.1309/LM920SBNZPJRJGUT doi: 10.1309/LM920SBNZPJRJGUT
[24]	S. Ookawara, K. Ito, T. Uchida, K. Tokuyama, S. Kiryu, T. Suganuma, et al., Hemodialysis crossover study using a relative blood volume change-guided ultrafiltration control compared with standard hemodialysis: The BV-UFC study, Ren. Replace Ther., 6 (2020), 45. https://doi.org/10.1186/s41100-020-00295-8 doi: 10.1186/s41100-020-00295-8
[25]	E. Niemann, S. Blatt, Windkessel Model Analysis: Blood Pressure in Response to Blood Flow, School of Biomedical Engineering, Drexel University, (2018).
[26]	N. De Paola, M. P. Burns, Fluid Biomechanics and Circulation in Biomechanics, (eds. M. Doblare and J. Merodio), Eolss Publisher Co. Ltd/UNESCO, (2015), 78–85.

Reader Comments

Your name:*

Email:*
© 2025 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)