@article{fink_batzel_tran_2008, title={A respiratory system model: Parameter estimation and sensitivity analysis}, volume={8}, ISSN={["1567-8822"]}, DOI={10.1007/s10558-007-9051-7}, abstractNote={In this paper we compare several approaches to identifying certain key respiratory control parameters relying on data normally available from non-invasive measurements. We consider a simple model of the respiratory control system and describe issues related to numerical estimates of key parameters involved in respiratory function such as central and peripheral control gains, transport delay, and lung compartment volumes. The combination of model-specific structure and limited data availability influences the parameter estimation process. Methods for studying how to improve the parameter estimation process are examined including classical and generalized sensitivity analysis, and eigenvalue grouping. These methods are applied and compared in the context of clinically available data. These methods are also compared in conjunction with specialized tests such as the minimally invasive single-breath CO2 test that can improve the estimation, and the enforced fixed breathing test, which opens the control loop in the system. The analysis shows that it is impossible to estimate central and peripheral gain simultaneously without usage of ventilation measurement and a controlled perturbation of the respiratory system, such as the CO2 test. The numerical results are certainly model dependent, but the illustrated methods, the nature of the comparisons, and protocols will carry over to other models and data configurations.}, number={2}, journal={CARDIOVASCULAR ENGINEERING}, author={Fink, Martin and Batzel, Jerry J. and Tran, Hien}, year={2008}, month={Jun}, pages={120–134} }
 @book{cardiovascular and respiratory systems modeling, analysis, and control_2007, publisher={Philadelphia: Society for Industrial and Applied Mathematics}, year={2007} }
 @misc{batzel_tran_2000, title={Modeling instability in the control system for human respiration: applications to infant non-REM sleep}, volume={110}, ISSN={["0096-3003"]}, DOI={10.1016/s0096-3003(99)00112-5}, abstractNote={Mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. The phenomena collectively referred to as periodic breathing (including Cheyne–Stokes respiration and apneustic breathing) have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of F.S. Grodins, J. Gray, A.I. Norins, R.W. Jones [J. Appl. Physiol. 1 (1954) 283–308] in the early 1950s. The purpose of this paper is to extend the model presented by M.C.K. Khoo, R.E. Kronauer, K.P. Strohl, A.S. Slutsky [J. Appl. Physiol. 53 (3) (1982) 644–659] in 1991 to include variable delay in the feedback control loop and to study the phenomena of periodic breathing and apnea as they occur during quiet sleep in infant sleep respiration at around 4 months of age. The nonlinear mathematical model consists of a feedback control system of five delay differential equations. Numerical simulations are performed to study instabilities in the control system and the occurrence of periodic breathing and apnea in the above case which is a time frame of high incidence of sudden infant death syndrome (SIDS).}, number={1}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Batzel, JJ and Tran, HT}, year={2000}, month={Apr}, pages={1–51} }
 @article{batzel_tran_2000, title={Stability of the human respiratory control system I. Analysis of a two-dimensional delay state-space model}, volume={41}, ISSN={["1432-1416"]}, DOI={10.1007/s002850000044}, abstractNote={A number of mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. Among them, periodic breathing (including Cheyne-Stokes respiration and apneustic breathing) is a collection of regular but involuntary breathing patterns that have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of Grodins et al. in the early 1950's [12]. The purpose of this paper is to study the stability characteristics of a feedback control system of five differential equations with delays in both the state and control variables presented by Khoo et al. [17] in 1991 for modeling human respiration. The paper is divided in two parts. Part I studies a simplified mathematical model of two nonlinear state equations modeling arterial partial pressures of O2 and CO2 and a peripheral controller. Analysis was done on this model to illuminate the effect of delay on the stability. It shows that delay dependent stability is affected by the controller gain, compartmental volumes and the manner in which changes in the ventilation rate is produced (i.e., by deeper breathing or faster breathing). In addition, numerical simulations were performed to validate analytical results. Part II extends the model in Part I to include both peripheral and central controllers. This, however, necessitates the introduction of a third state equation modeling CO2 levels in the brain. In addition to analytical studies on delay dependent stability, it shows that the decreased cardiac output (and hence increased delay) resulting from the congestive heart condition can induce instability at certain control gain levels. These analytical results were also confirmed by numerical simulations.}, number={1}, journal={JOURNAL OF MATHEMATICAL BIOLOGY}, author={Batzel, JJ and Tran, HT}, year={2000}, month={Jul}, pages={45–79} }
 @article{batzel_tran_2000, title={Stability of the human respiratory control system II. Analysis of a three-dimensional delay state-space model}, volume={41}, ISSN={["0303-6812"]}, DOI={10.1007/s002850000045}, abstractNote={A number of mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. Among them, periodic breathing (including Cheyne-Stokes respiration and apneustic breathing) is a collection of regular but involuntary breathing patterns that have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of Grodins et al. in the early 1950's [1]. The purpose of this paper is to study the stability characteristics of a feedback control system of five differential equations with delays in both the state and control variables presented by Khoo et al. [4] in 1991 for modeling human respiration. The paper is divided in two parts. Part I studies a simplified mathematical model of two nonlinear state equations modeling arterial partial pressures of O2 and CO2 and a peripheral controller. Analysis was done on this model to illuminate the effect of delay on the stability. It shows that delay dependent stability is affected by the controller gain, compartmental volumes and the manner in which changes in the ventilation rate is produced (i.e., by deeper breathing or faster breathing). In addition, numerical simulations were performed to validate analytical results. Part II extends the model in Part I to include both peripheral and central controllers. This, however, necessitates the introduction of a third state equation modeling CO2 levels in the brain. In addition to analytical studies on delay dependent stability, it shows that the decreased cardiac output (and hence increased delay) resulting from the congestive heart condition can induce instability at certain control gain levels. These analytical results were also confirmed by numerical simulations.}, number={1}, journal={JOURNAL OF MATHEMATICAL BIOLOGY}, author={Batzel, JJ and Tran, HT}, year={2000}, month={Jul}, pages={80–102} }