Optimal switching time control of the hyperbaric oxygen therapy for a chronic wound

Dan Zhu; Qinfang Qian; Dan Zhu; Qinfang Qian

doi:10.3934/mbe.2019419

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 8290-8308. doi: 10.3934/mbe.2019419

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Optimal switching time control of the hyperbaric oxygen therapy for a chronic wound

Dan Zhu ¹,
Qinfang Qian ^{2
,
,}

1.
Department of Cardiology, Hangzhou Hospital of Traditional Chinese Medicine, Hangzhou, Zhejiang 310007, China
2.
Central Sterile Supply Department, Zhejiang Hospital, Hangzhou, Zhejiang 310013, China

Received: 26 May 2019 Accepted: 06 September 2019 Published: 17 September 2019

Chronic wounds, defined as those wounds which fail to heal through the normally orderly process of stages and remain in a chronic inflammatory state, are a significant socioeconomic problem. This paper considers an optimal switching time control problem of the hyperbaric oxygen therapy for a chronic wound. First, we model the spatiotemporal evolution of a chronic wound by introducing oxygen, neutrophils, invasive bacteria, and chemoattractant. Then, we apply the method of lines to reduce the partial differential equations (PDEs) into ordinary differential equations (ODEs), which lead to an ODE optimization problem with the changed time switching points. The time-scaling transformation approach is applied to further transform the control problem with changed switching time into another new problem with fixed switching time. The gradient formulas of the cost functional corresponding to the time intervals are derived based on the sensitivity analysis. Finally, computational numerical analysis demonstrates the effectiveness of the proposed control strategy to inhibit the growth of bacterial concentration.
- hyperbaric oxygen therapy,
- chronic wounds,
- optimal control,
- time-scaling transformation approach,
- partial differential equations
Citation: Dan Zhu, Qinfang Qian. Optimal switching time control of the hyperbaric oxygen therapy for a chronic wound[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 8290-8308. doi: 10.3934/mbe.2019419

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Abstract

Chronic wounds, defined as those wounds which fail to heal through the normally orderly process of stages and remain in a chronic inflammatory state, are a significant socioeconomic problem. This paper considers an optimal switching time control problem of the hyperbaric oxygen therapy for a chronic wound. First, we model the spatiotemporal evolution of a chronic wound by introducing oxygen, neutrophils, invasive bacteria, and chemoattractant. Then, we apply the method of lines to reduce the partial differential equations (PDEs) into ordinary differential equations (ODEs), which lead to an ODE optimization problem with the changed time switching points. The time-scaling transformation approach is applied to further transform the control problem with changed switching time into another new problem with fixed switching time. The gradient formulas of the cost functional corresponding to the time intervals are derived based on the sensitivity analysis. Finally, computational numerical analysis demonstrates the effectiveness of the proposed control strategy to inhibit the growth of bacterial concentration.

References

[1]	R. G. Frykberg and J. Banks, Challenges in the treatment of chronic wounds, Adv. Wound Care, 4 (2015), 560-582.
[2]	R. Nunan, K. G. Harding and P. Martin, Clinical challenges of chronic wounds: searching for an optimal animal model to recapitulate their complexity, Dis. Model. Mech., 7 (2014), 1205-1213.
[3]	N. A. Richmond, A. D. Maderal and A. C. Vivas, Evidence-based management of common chronic lower extremity ulcers, Dermatol. Ther., 26 (2013), 187-196.
[4]	A. Rondas, J. Schols, E. Stobberingh, et al., Prevalence of chronic wounds and structural quality indicators of chronic wound care in dutch nursing homes, Int. Wound J., 12 (2015), 630-635.
[5]	S. McDougall, J. Dallon, J. Sherratt, et al., Fibroblast migration and collagen deposition during dermal wound healing: mathematical modelling and clinical implications, Philos. Trans. R. Soc. A, 364 (2006), 1385-1405.
[6]	B. D. Cumming, D. McElwain and Z. Upton, A mathematical model of wound healing and subsequent scarring, J. R. Soc., Interface, 7 (2009), 19-34.
[7]	E. Gaffney, K. Pugh, P. Maini, et al., Investigating a simple model of cutaneous wound healing angiogenesis, J. Math. Biol., 45 (2002), 337-374.
[8]	J. A. Thackham, D. McElwain and I. Turner, Computational approaches to solving equations arising from wound healing, Bull. Math. Biol., 71 (2009), 211-246.
[9]	F. Vermolen, Simplified finite-element model for tissue regeneration with angiogenesis, J. Eng. Mech., 135 (2009), 450-460.
[10]	J. A. Flegg, Mathematical Modelling of Chronic Wound Healing, Ph.D thesis, Queensland University of Technology, 2009.
[11]	S. Enoch, J. E. Grey and K. G. Harding, Recent advances and emerging treatments, Br. Med. J., 332 (2006), 962-965.
[12]	E. A. Ayello and J. E. Cuddigan, Conquer chronic wounds with wound bed preparation, The Nurse Practitioner, 29 (2004), 8-25.
[13]	A. A. Tandara and T. A. Mustoe, Oxygen in wound healing more than a nutrient, World J. Surg., 28 (2004), 294-300.
[14]	J. A. Thackham, D. L. S. McElwain and R. J. Long, The use of hyperbaric oxygen therapy to treat chronic wounds: a review, Wound Repair Regen., 16 (2008), 321-330.
[15]	L. K. Kalliainen, G. M. Gordillo, R. Schlanger, et al., Topical oxygen as an adjunct to wound healing: a clinical case series, Pathophysiology, 9 (2003), 81-87.
[16]	P. K. Gajendrareddy, C. K. Sen, M. P. Horan, et al., Hyperbaric oxygen therapy ameliorates stressimpaired dermal wound healing, Brain, Behav., Immun., 19 (2005), 217-222.
[17]	D. E. Eisenbud, Oxygen in wound healing: nutrient, antibiotic, signaling molecule, and therapeutic agent, Clin. Plast. Surg., 39 (2012), 293-310.
[18]	A. Gill and C. Bell, Hyperbaric oxygen: its uses, mechanisms of action and outcomes, QJM, 97 (2004), 385-395.
[19]	P. Kranke, M. H. Bennett, M. Martyn-St James, et al., Hyperbaric oxygen therapy for chronic wounds, Cochrane Database Syst Rev., (2015), 1-69.
[20]	G. Han and R. Ceilley, Chronic wound healing: a review of current management and treatments, Adv. Ther., 34 (2017), 599-610.
[21]	S. Guffey, Application of a Numerical Method and Optimal Control Theory to a Partial Differential Equation Model for a Bacterial Infection in a Chronic wound, Masters Theses & Specialist Projects. Western Kentucky University, 2015.
[22]	R. Schugart, A. Friedman, R. Zhao, et al., Wound angiogenesis as a function of tissue oxygen tension: a mathematical model, Proc. Natl. Acad. Sci., 105 (2008), 2628-2633.
[23]	C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, Proc. Natl. Acad. Sci., 106 (2009), 16782-16787.
[24]	J. A. Flegg, D. L. S. McElwain, H. M. Byrne, et al., A three species model to simulate application of hyperbaric oxygen therapy to chronic wounds, PLoS Comput. Biol., 5 (2009), e1000451.
[25]	S. Song, J. Hogg, Z. Peng, et al., Ensemble models of neutrophil trafficking in severe sepsis, PLoS Comput. Biol., 8 (2012), e1002422.
[26]	B. C. Russell, Using Partial Differential Equations to Model and Analyze the Treatment of a Chronic Wound With Oxygen Therapy Techniques, Honors College Capstone Experience/TesisProjects, 2013.
[27]	D. V. Zhelev, A. M. Alteraifi and D. Chodniewicz, Controlled pseudopod extension of human neutrophils stimulated with different chemoattractants, Biophys. J., 87 (2004), 688-695.
[28]	W. E. Schiesser, The Numerical Method of Lines: Integration of Partial Differential Equations, Academic Press, London, 2012.
[29]	R. M. Leach, P. J. Rees and P. Wilmshurst, Hyperbaric oxygen therapy, Br. Med. J., 317 (1998), 1140-1143.
[30]	Z. Ren, Z. Zhou, C. Xu, et al., Computational bilinear optimal control for a class of onedimensional MHD flow systems, ISA Trans., 85 (2019), 129-140.
[31]	R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929.
[32]	T. Chen and C. Xu, Computational optimal control of the Saint-Venant PDE model using the time-scaling technique, Asia-Pac. J. Chem. Eng., 11 (2016), 70-80.
[33]	P. T. Boggs and J. W. Tolle, Sequential quadratic programming, Acta Numerica, 4 (1995), 1-51.
[34]	D. Huang and J. Xu, Steady-state iterative learning control for a class of nonlinear PDE processes, J. Proc. Control., 21 (2011), 1155-1163.
[35]	T. Chen, C. Xu and Z. Ren, Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels, J. Ind. Manag. Optim., 14 (2018), 1251-1269.
[36]	T. Chen, S. Zhou, Z. Ren, et al., Optimal open-loop control for 2-D colloid transport in the deadend microchannel, IEEE Trans. Control Syst. Technol., (2018), 1-9.
[37]	Z. Zhao, X. He, Z. Ren, et al., Boundary adaptive robust control of a flexible riser system with input nonlinearities, IEEE Trans. Syst. Man Cybern. A, 49 (2018), 1971-1980.

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