zbMATH — the first resource for mathematics

Time delay in physiological systems: analyzing and modeling its impact. (English) Zbl 1244.92012
Summary: This article examines the functional and clinical impact of time delays that arise in human physiological systems, especially control systems. An overview of the mathematical and physiological contexts for considering time delays will be illustrated, from the system level to the cell level, by examining models that incorporate time delays. This examination will highlight how such delays in combination with other system structures and parameters influence system dynamics. Model analysis that reveals the influence of delays can also reveal related physiological effects which may have medical consequences and clinical applications.

92C30 Physiology (general)
92C50 Medical applications (general)
34K60 Qualitative investigation and simulation of models involving functional-differential equations
Full Text: DOI
[1] Abbiw-Jackson, R.M.; Langford, W.F., Gain-induced oscillations in blood pressure, J. math. biol., 37, 203, (1998) · Zbl 0903.92017
[2] Adimy, M.; Crauste, F.; Ruan, S., Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases, Bull. math. biol., 68, 2321, (2006) · Zbl 1296.92102
[3] Adimy, M.; Crauste, F.; Ruan, S., Periodic oscillations in leukopoiesis models with two delays, J. theor. biol., 242, 288, (2006)
[4] Agur, Z.; Arakelyan, L.; Daugulis, P.; Ginosar, Y., Hopf point analysis for angiogenesis models, Discrete contin. dyn. syst. - ser. B, 4, 29, (2004) · Zbl 1051.34059
[5] an der Heiden, U., Delays in physiological systems, J. math. biol., 8, 345, (1979) · Zbl 0429.92009
[6] Bachar, M.; Dorfmayr, A., HIV treatment models with time delay, CR biol., 327, 983, (2004)
[7] Banks, H.; Burns, J., Hereditary control problems: numerical methods based on averaging approximations, SIAM J. control optim., 16, 169, (1978) · Zbl 0379.49025
[8] Banks, H.T.; Bortz, D.M., A parameter sensitivity methodology in the context of HIV delay equation models, J. math. biol., 50, 607, (2005) · Zbl 1083.92025
[9] Batzel, J.J.; Goswami, N.; Lackner, H.K.; Roessler, A.; Bachar, M.; Kappel, F.; Hinghofer-Szalkay, H., Patterns of cardiovascular control during repeated tests of orthostatic loading, Cardiovasc. eng., 9, 134, (2009)
[10] Batzel, J.J.; Kappel, F.; Timischl-Teschl, S., A cardiovascular – respiratory control system model including state delay with application to congestive heart failure in humans, J. math. biol., 50, 293, (2005) · Zbl 1080.92035
[11] Batzel, J.J.; Tran, H.T., Stability of the human respiratory control system. part I: analysis of a two dimensional delay state-space model, J. math. biol., 41, 45, (2000) · Zbl 0999.92012
[12] Batzel, J.J.; Tran, H.T., Stability of the human respiratory control system. part II: analysis of a three dimensional delay state-space model, J. math. biol., 41, 80, (2000) · Zbl 0999.92013
[13] Bélair, J.; Mackey, M.C.; Mahaffy, J.M., Age-structured and two-delay models for erythropoiesis, Math. biosci., 128, 317, (1995) · Zbl 0832.92005
[14] Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press New York · Zbl 0118.08201
[15] Boese, F., Stability with respect to the delay: on a paper of K.L. cooke and P. Van den driessche, J. math. anal. appl., 228, 293, (1998) · Zbl 0918.34071
[16] Brandt, M.E.; Chen, G., Time-delay feedback control of complex pathological rhythms in an atrioventricular conduction model, Int. J. bifurcat. chaos, 10, 2781, (2000) · Zbl 0964.92020
[17] Burgess, D.E.; Hundley, J.C.; Li, S.G.; Randall, D.C.; Brown, D.R., First-order differential-delay equation for the baroreflex predicts the 0.4-hz blood pressure rhythm in rats, Am. J. physiol. regul. integr. comp. physiol., 273, 1878, (1997)
[18] Burić, N.; Mudrinic, M.; Vasović, N., Time delay in a basic model of the immune response, Chaos solitons fract., 12, 483, (2001) · Zbl 1026.92015
[19] Byrne, H.M., The effect of time delays on the dynamics of avascular tumor growth, Math. biosci., 144, 83, (1997) · Zbl 0904.92023
[20] Byrne, H.M.; Alarcon, T.; Owen, M.R.; Webb, S.D.; Maini, P.K., Modelling aspects of cancer dynamics: a review, Philos trans. A math. phys. eng. sci., 364, 1563, (2006)
[21] Campbell, S.A., Time delays in neural systems, (), 65
[22] Cavalcanti, S.; Belardinelli, E., Modeling of cardiovascular variability using a differential delay equation, IEEE trans. biomed. eng., 43, 982, (1996)
[23] Chaplain, M.A.; McDougall, S.R.; Anderson, A.R., Mathematical modeling of tumor-induced angiogenesis, Annu. rev. biomed. eng., 8, 233, (2006)
[24] Cherniack, N.S., Apnea and periodic breathing during sleep, New england J. med., 341, 985, (1999)
[25] Cherniack, N.S.; Longobardo, G.S., Mathematical models of periodic breathing and their usefulness in understanding cardiovascular and respiratory disorders, Exp. physiol., 91, 295, (2006)
[26] Cleave, J.P.; Levine, M.R.; Fleming, P.J.; Long, A.M., Hopf bifurcations and the stability of the respiratory control system, J. theor. biol., 119, 299, (1986)
[27] Collins, C.; Fister, K.R.; Williams, M., Optimal control of a cancer cell model with delay, Math. model. nat. phenom., 5, 63, (2010) · Zbl 1188.49038
[28] Cooke, K.L.; van den Driessche, P., On zeroes of some transcendental equations, Funkcial. ekvac., 29, 77, (1986) · Zbl 0603.34069
[29] Cooke, K.L.; Grossman, Z., Discrete delay, distributed delay and stability switches, J. math. anal. appl., 86, 592, (1982) · Zbl 0492.34064
[30] Cooke, K.L.; Krumme, D.W., Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations, J. math. anal. appl., 24, 372, (1968) · Zbl 0186.16902
[31] Cooke, K.L.; Turi, J., Stability, instability in delay equations modeling human respiration, J. math. biol., 32, 535, (1994) · Zbl 0807.92007
[32] De Gaetano, A.; Arino, O., Mathematical modelling of the intravenous glucose tolerance test, J. math. biol., 40, 136, (2000) · Zbl 0999.92016
[33] d’Onofrio, A.; Gandolfi, A., A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Math. med. biol., 26, 63, (2009) · Zbl 1157.92024
[34] d’Onofrio, A.; Gatti, F.; Cerrai, P.; Freschi, L., Delay-induced oscillatory dynamics of tumour – immune system interaction, Math. comput. mod., 51, 572, (2010) · Zbl 1190.34088
[35] Driver, R.D., Ordinary and delay differential equations, (1977), Springer-Verlag New York · Zbl 0374.34001
[36] Drobnjak, I.; Fowler, A.C.; Mackey, M.C., Oscillations in a maturation model of blood cell production, SIAM J. appl. math., 66, 2027, (2006) · Zbl 1197.76164
[37] Eckberg, D.L., Point:counterpoint: respiratory sinus arrhythmia is due to a central mechanism vs. respiratory sinus arrhythmia is due to the baroreflex mechanism, J. appl. physiol., 106, 1740, (2009)
[38] El’sgol’tz, L.E.; Norkin, S.B., Introduction to the theory and applications of differential equations with deviating arguments, (1973), Academic Press New York · Zbl 0287.34073
[39] Erneux, T., Applied delay differential equations, (2009), Springer NY · Zbl 1201.34002
[40] Fink, M.; Batzel, J.J.; Tran, H., A respiratory system model: parameter estimation and sensitivity analysis, Cardiovasc. eng. int. J., 8, 120, (2008)
[41] Finucane, C.; Boyle, G.; Fan, C.W.; Hade, D.; Byrne, L.; Kenny, R.A., Mayer wave activity in vasodepressor carotid sinus hypersensitivity, Europace, 12, 247, (2010)
[42] Foley, C.; Mackey, M.C., Dynamic hematological disease: a review, J. math. biol., 58, 285, (2009) · Zbl 1161.92338
[43] Fowler, A.C., Approximate solution of a model of biological immune responses incorporating delay, J. math. biol., 13, 23, (1981) · Zbl 0477.92009
[44] Fowler, A.C.; McGuinness, M.J., A delay recruitment model of the cardiovascular control system, J. math. biol., 51, 508, (2005) · Zbl 1091.92024
[45] Francis, D.P.; Willson, K.; Davies, L.C.; Coats, A.J.; Piepoli, M., Quantitative general theory for periodic breathing in chronic heart failure and its clinical implications, Circulation, 102, 2214, (2000)
[46] Garcia-Touchard, A.; Somers, V.K.; Olson, L.J.; Caples, S.M., Central sleep apnea: implications for congestive heart failure, Chest, 133, 1495, (2008)
[47] Ghazanshahi, S.D.; Khoo, M.C.K., Optimal ventilatory patterns in periodic breathing, Ann. biomed. eng., 21, 517, (1993)
[48] Glass, L.; Beuter, A.; Larocque, D., Time delays oscillations and chaos in physiological control systems, Math. biosci., 90, 111, (1988) · Zbl 0649.92008
[49] Gois, S.; Savi, M.A., An analysis of heart rhythm dynamics using a three-coupled oscillator model, Chaos solitons fract., 41, 2553, (2009) · Zbl 1198.37123
[50] Golbin, J.M.; Somers, V.K.; Caples, S.M., Obstructive sleep apnea, cardiovascular disease, and pulmonary hypertension, Proc. am. thorac. soc., 5, 200, (2008)
[51] Goldberger, A.L., Giles F. filley lecture. complex systems, Proc. am. thorac. soc., 3, 467, (2006)
[52] Grodins, F.S.; Buell, J.; Bart, A.J., Mathematical analysis and digital simulation of the respiratory control system, J. appl. physiol., 22, 260, (1967)
[53] Gyllenberg, M.; Heijmans, H.J.A.M., An abstract delay-differential equation modelling size dependent cell growth and division, SIAM J. math. anal., 18, 74, (1987) · Zbl 0634.34064
[54] Hadjiandreou, M.M.; Conejeros, R.; Wilson, D.I., Planning of patient-specific drug-specific optimal HIV treatment strategies, Chem. eng. sci., 64, 4024, (2009)
[55] Hairer, E.; Nørsett, S.; Wanner, G., Solving ordinary differential equations I. nonstiff problems, () · Zbl 0638.65058
[56] Halanay, A., Differential equations: stability, oscillations, time lags, (1966), Academic Press NY · Zbl 0144.08701
[57] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer Verlag New York · Zbl 0787.34002
[58] Hammer, P.E.; Saul, J.P., Resonance in a mathematical model of baroreflex control: arterial blood pressure waves accompanying postural stress., Am. J. physiol. regul. integr. comp. physiol., 288, R1637, (2005)
[59] Haurie, C.; Dale, D.C.; Mackey, M.C., Cyclical neutropenia and other periodic hematological disorders: a review of mechanisms and mathematical models, Blood, 92, 2629, (1998)
[60] Holstein-Rathlou, N.H.; Yip, K.P.; Sosnovtseva, O.V.; Mosekilde, E., Synchronization phenomena in nephron – nephron interaction, Chaos, 11, 417, (2001)
[61] Ito, K.; Kappel, F., Two families of approximation schemes for delay equations, Results math., 21, 93, (1992) · Zbl 0756.34078
[62] Ito, K.; Kappel, F., Evolution equations and approximations, (2002), World Scientific Singapore · Zbl 1014.34045
[63] Javaheri, S., A mechanism of central sleep apnea in patients with heart failure, N. engl. J. med., 341, 949, (1999)
[64] F. Kappel, Approximation of neutral functional differential equations in the state space \(\mathbb{R}^n \times L^2\), in: M. Farkas (Ed.), Qualitative Theory of Differential Equations, vol. I, vol. 30 of Colloquia Mathematica Societatis Janos Bolyai, Janos Bolyai Math. Soc. and North Holland Publ. Comp., 1982, p. 463.
[65] F. Kappel, Approximation of LQR-problems for delay systems: a survey, in: K. Bowers, J. Lund (Eds.), Computation and Control II, Progress in Systems and Control Theory, vol. 11, Birkhäuser, Boston, 1991, p. 187. · Zbl 0778.93050
[66] Karemaker, J.M., Counterpoint: respiratory sinus arrhythmia is due to the baroreflex mechanism, J. appl. physiol., 106, 1742, (2009)
[67] Khoo, M.C.K., A model-based evaluation of the single-breath CO_{2} ventilatory response test, J. appl. physiol., 68, 393, (1990)
[68] Khoo, M.C.K.; Anholm, J.D.; Ko, S.W.; Downey, R.; Powles, A.C.; Sutton, J.R.; Houston, C.S., Dynamics of periodic breathing and arousal during sleep at extreme altitude, Respir. physiol., 103, 33, (1996)
[69] Khoo, M.C.K.; Gottschalk, A.; Pack, A.I., Sleep-induced periodic breathing and apnea: a theoretical study, J. appl. physiol., 70, 2014, (1991)
[70] Khoo, M.C.K.; Kronauer, R.E.; Strohl, K.P.; Slutsky, A.S., Factors inducing periodic breathing in humans: a general model, J. appl. physiol., 53, 644, (1982)
[71] Krasovskij, N., Stability of motion: applications of lyapunov’s second method to differential systems and equations with delay, (1963), Stanford University Press Stanford, CA
[72] Krasovskij, N.N., Approximation of an optimal control problem for a system with delay, Sov. phys. dokl., 11, 219, (1966) · Zbl 0196.46202
[73] Lanfranchi, P.A.; Somers, V.K., Sleep-disordered breathing in heart failure: characteristics and implications, Respir. physiol. neurobiol., 136, 153, (2003)
[74] Layton, A.T.; Moore, L.C.; Layton, H.E., Multistable dynamics mediated by tubuloglomerular feedback in a model of coupled nephrons, Bull. math. biol., 71, 515, (2009) · Zbl 1165.92007
[75] Levine, M.; Cleave, J.P.; Dodds, C., Can periodic breathing have advantages for oxygenation, J. theor. biol., 172, 355, (1995)
[76] Levine, M.; Hathorn, M.K.; Cleave, J.P., Optimization of inspiratory work in periodic breathing in infants, Pediatr. res., 47, 256, (2000)
[77] Li, J.; Kuang, Y.; Mason, C.C., Modeling the glucose – insulin regulatory system and Ultradian insulin secretory oscillations with two explicit time delays, J. theor. biol., 242, 722, (2006)
[78] Liu, W.; Hillen, T.; Freedman, H.I., A mathematical model for m-phase specific chemotherapy including the G0-phase and immunoresponse, Math. biosci. eng., 4, 239, (2007) · Zbl 1123.92014
[79] Longobardo, G.S.; Cherniack, N.S.; Fishman, A.P., Cheyne – stokes breathing produced by a model of the human respiratory system, J. appl. physiol., 21, 1839, (1966)
[80] Longobardo, G.S.; Cherniack, N.S.; Gothe, B., Factors affecting respiratory system stability, Ann. biomed. eng., 17, 377, (1989)
[81] Longobardo, G.S.; Evangelisti, C.J.; Cherniack, N.S., Analysis of the interplay between neurochemical control of respiration and upper airway mechanics producing upper airway obstruction during sleep in humans, Exp. physiol., 93, 271, (2008)
[82] Longobardo, G.S.; Gothe, B.; Goldman, M.D.; Cherniack, N.S., Sleep apnea considered as a control system instability, Respir. physiol., 50, 311, (1982)
[83] Mackey, M.C., Unified hypothesis for the origin of aplastic anemia and periodic haematopoiesis, Blood, 51, 941, (1978)
[84] Mackey, M.C.; Ou, C.; Pujo-Menjouet, L.; Wu, J., Periodic oscillations of blood cell populations in chronic myelogenous leukemia, SIAM J. math. anal., 38, 166, (2006) · Zbl 1163.34055
[85] Magosso, E.; Biovatti, V.; Ursino, M., Role of the baroreflex in cardiovascular instability: a modeling study, Cardiovasc. eng., 1, 101, (2001)
[86] Mahaffy, J.M.; Bélair, J.; Mackey, M.C., Hematopoietic model with moving boundary condition and state dependent delay: applications in erythropoiesis, J. theor. biol., 190, 135, (1998)
[87] Milton, J.; Cabrera, J.L.; Ohira, T.; Tajima, S.; Tonosaki, Y.; Eurich, C.W.; Campbell, S.A., The time-delayed inverted pendulum: implications for human balance control, Chaos, 19, 026110:1, (2009) · Zbl 1309.92020
[88] Mishkis, A., Lineare differentialgleichungen mit nacheilendem argument, (1955), Deutscher Verlag d. Wissenschaften Berlin
[89] Müller, T.; Lauk, M.; Reinhard, M.; Hetzel, A.; Lücking, C.H.; Timmer, J., Estimation of delay times in biological systems, Annal. biomed. eng., 31, 1423, (2003)
[90] Olufsen, M.S.; Tran, H.T.; Ottesen, J.T.; Lipsitz, L.A.; Novak, V., Modeling baroreflex regulation of heart rate during orthostatic stress, Am. J. physiol. regul. integr. comp. physiol., 291, R1355, (2006)
[91] Ottesen, J.T., Modelling the baroreflex-feedback mechanism with time-delay, J. math. biol., 36, 41, (1997) · Zbl 0887.92015
[92] Panunzi, S.; De Gaetano, A.; Mingrone, G., Advantages of the single delay model for the assessment of insulin sensitivity from the intravenous glucose tolerance test, Theor. biol. med. model., 7, 9, 1, (2010)
[93] Panunzi, S.; Palumbo, P.; De Gaetano, A., A discrete single delay model for the intra-venous glucose tolerance test, Theor. biol. med. model., 4, 35, 1, (2007)
[94] Pinho, S.T.R.; Freedman, H.I.; Nani, F., A chemotherapy model for the treatment of cancer with metastasis, Math. comput. model., 36, 773, (2002) · Zbl 1021.92014
[95] Presnov, E.V.; Agur, Z., The role of time delays, slow processes, and chaos in modulating the cell-cycle clock, Math. biosci. eng., 2, 625, (2005) · Zbl 1079.92026
[96] Roessler, A.; Goswami, N.; Haditsch, B.; Loeppky, J.A.; Luft, F.C.; Hinghofer-Szalkay, H., Volume regulating hormone responses to repeated head-up tilt and lower body negative pressure, Eur. J. clin. invest., 41, 8, 863, (2011), doi:10.1111:j.1365-2362.2011.02476
[97] Sanga, S.; Sinek, J.P.; Frieboes, H.B.; Ferrari, M.; Fruehauf, J.P.; Cristini, V., Mathematical modeling of cancer progression and response to chemotherapy, Exp. rev. anticancer ther., 6, 1361, (2006)
[98] Schöll, E.; Hiller, G.; Hövel, P.; Dahlem, M.A., Time-delayed feedback in neurosystems, Philos. trans. A math. phys. eng. sci., 367, 1079, (2009) · Zbl 1185.34108
[99] Selgrade, J.F., Bifurcation analysis of a model for hormonal regulation of the menstrual cycle, Math. biosci., 225, 108, (2010) · Zbl 1193.92025
[100] Skeldon, A.C.; Purvey, I., The effect of different forms for the delay in a model of the nephron, Math. biosci. eng., 2, 97, (2005) · Zbl 1061.92026
[101] Slemrod, M., The flip-flop circuit as a neutral equation, (), 387
[102] Smith, H.L., Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study, Math. biosci., 113, 1, (1993) · Zbl 0797.92024
[103] Srividhya, J.; Gopinathan, M.S., A simple time delay model for eukaryotic cell cycle, J. theor. biol., 241, 617, (2006)
[104] Stepan, G., Delay effects in brain dynamics. introduction, Philos. trans. A math. phys. eng. sci., 367, 1059, (2009) · Zbl 1185.92025
[105] Stepan, G., Delay effects in the human sensory system during balancing, Philos. trans. A math. phys. eng. sci., 367, 1195, (2009) · Zbl 1185.92010
[106] Tolić, I.M.; Mosekilde, E.; Sturis, J., Modeling the insulin – glucose feedback system: the significance of pulsatile insulin secretion, J. theor. biol., 207, 361, (2000)
[107] Ursino, M.; Magosso, E., Role of short-term cardiovascular regulation in heart period variability: a modeling study, Am. J. physiol. heart circ. physiol., 284, H1479, (2003)
[108] Vielle, B., A new explicit stability criterion for human periodic breathing, J. math. biol., 41, 546, (2000) · Zbl 1002.92006
[109] Villasana, M.; Radunskaya, A., A delay differential equation model for tumor growth, J. math. biol., 47, 270, (2003) · Zbl 1023.92014
[110] Wang, H.; Li, J.; Kuang, Y., Mathematical modeling and qualitative analysis of insulin therapies, Math. biosci., 210, 17, (2007) · Zbl 1138.92021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.