Cycle-Averaged Models of Cardiovascular Dynamics

Cycle-Averaged Models of Cardiovascular Dynamics by Jolie L. Chang Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Engineering in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2002 @ Jolie L. Chang, MMII. All rights reserved. MASSACHUSEMS ITTEUTE OF TECHNO.OGY JUL 3 1 2002 I Ud LIBRARIES The author hereby grants to MIT permission to reproduce distribute publicly paper and electronic copies of this thesis document in whole or in part. BARKER Author . Department of Electrical Engineering and Computer Science May 24, 2002 Certified by .............. George C. Verghese Professor .Thnis Supervisor :. . ..... Arthur C. Smith Chairman, Department Committee on Graduate Students Accepted by ...... 2 Cycle-Averaged Models of Cardiovascular Dynamics by Jolie L. Chang Submitted to the Department of Electrical Engineering and Computer Science on May 24, 2002, in partial fulfillment of the requirements for the degree of Master of Engineering in Electrical Engineering and Computer Science Abstract A computational, circuit-based, hemodynamic model has been developed by Heldt et al. in order to study the cardiovascular system's response to different situations and conditions. When studying the model's transient response over many heartbeat cycles, to perturbations such as lower body negative pressure or standing up, a cycleaveraged version of the model would seem to be sufficient. Such a model would suppress details of the periodic pulsing waveform of the heart beat, instead capturing just the cycle-to-cycle dynamics. Cycle-averaged circuit models are also expected to be more computationally efficient, and can be used to define and study control structures within the system that respond to averaged rather than instantaneous quantities. This thesis is focused on developing and studying such cycle-averaged models. In addition, we demonstrate that both pulsing and cycle-averaged models are very straightforward to implement in HSPICE, a commonly used circuit simulation package. A simple circuit-based, one-ventricle, pulsing heart model is used throughout the thesis. The pulsing action is captured in a time-varying capacitor, and the valve action through two diodes. This model was set up and studied in HSPICE. We derived from this pulsing model two cycle-averaged models, one involving three dependent sources, and the other with four dependent sources. Both averaged models turned out to be linear and time invariant in their responses to initial conditions, and were therefore analytically very tractable when set up in state-space form. When simulated in HSPICE, both cycle-averaged models produced responses that followed, with reasonable accuracy, the actual cycle-averaged waveforms derived from the corresponding responses of the pulsing model. Due to the suppressed intra-cycle detail, the cycle-averaged models typically needed less than a tenth of the number of computed samples required by the pulsing model, and ran almost twenty times as fast. The four-source averaged model was also used to represent the arterial baroreceptor reflex, with feedback control of heartbeat period, zero-pressure filling volume, and peripheral resistance. Each of these controls, taken one at a time, was successful in counteracting changes in the average arterial pressure from its desired steady-state value. Future work can include: improving the accuracy of the averaged models still 3 further, by refining the assumptions made in their derivation; extending these models to the full hemodynamic model; and using these models for further study of dynamics, control, regulation, and identification. Thesis Supervisor: George C. Verghese Title: Professor 4 Acknowledgments First, I would like to thank my thesis advisor, Professor George Verghese, for all of his continued support and guidance throughout the last year. He was always available to help me with any problems I encountered along the way. His insight, experience, and enthusiasm made this project possible. Thanks to Professor Roger Mark, my academic advisor and thesis co-supervisor, for his advice on my thesis, classes, and life at MIT and beyond. Special thanks to Thomas Heldt for encouraging me and helping me to see how my project related to the bigger picture. He was always there to help provide insightful and useful suggestions. Thanks to Siebel Systems for selecting me for the Siebel Scholars Program, which funded me for part of my M.Eng year. I would like to thank all of the wonderful people I have met at MIT, which have made my experiences interesting and lively. Specifically, thanks to Nori Yoshida for believing in me and for being there to support me for the last four years. He has helped me stay sane along every step of the way. Thanks to Alex Park for answering my endless questions about LaTEX, 6.003, thesis writing, and life in general. Thanks to Fong Keng for her constant and contagious smile during the past year. Thanks to the 6.003 Spring 2002 teaching staff for helping me through my last semester. I had a lot of fun teaching for the first time. Thanks to all of the lifelong friends I have made at MIT: Janice, Kathie, Wan-jen, Rich, Alan, Jack, Xuemin, Yi-fung, Eugene, Sanjay, Jessica, Cat, and Stephanie. And thanks to the countless people that have made the last five years memorable. Finally, and most importantly, much thanks and love to Mom, Dad, and Anita for supporting me throughout all of my endeavors. 5 6 Contents 1 Computational Models and Biological Systems 15 1.1 16 1.2 2 Background and Previous Work ..................... . . . . . . . . . . . . . . . . . . . 16 . . . . . . . . . . . . . . 17 . . . . . . . . . . . 18 . . . . . . . . . . . . . . . . . . . . . 22 Goals and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.1.1 The Cardiovascular System 1.1.2 Modeling the Cardiovascular System 1.1.3 The Computational Hemodynamic Model 1.1.4 Time-Averaged Circuits A Simplified Model 25 2.1 Using HSPICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 A Simple RC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Simple RC Circuit with a Current Source . . . . . . . . . . . . . . . . 28 2.4 The Pulsing Heart Model . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.1 How it works . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.2 Simulation in HSPICE . . . . . . . . . . . . . . . . . . . . . . 31 2.4.3 Changing R 2 2.5 . . . . 34 Using the Pulsing Model . . . . . . . . . . . . . . . . . . . . . . . . . 35 . . . .. . . . . . . . .. . . . . .. 3 Developing the Cycle-Averaged Model 37 3.1 Introduction of a Transient . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Calculating the Time-Average Waveform . . . . . . . . . . . . . . . . 38 3.3 Finding Ceff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Calculations with Elastance . . . . . . . . . . . . . . . . . . . 40 3.3.1 7 Ceff in a Simple RC circuit . . . . . . . . . . . . 42 Replacing the Diodes . . . . . . . . . . . . . . . . . . . . 46 3.4.1 Voltage Analysis with Switching Functions . . . . 46 3.4.2 Current through the Diodes . . . . . . . . . . . . 48 The Preliminary Cycle-Averaged Model . . . . . . . . . . 48 3.5.1 Time-averaged Current . . . . . . . . . . . . . . . 48 3.5.2 Time-averaged Voltage V . . . . . . . . . . . . .. 51 3.5.3 The Initial Three-Source Cycle-Averaged Model . 51 3.5.4 Simulation of the Three-Source Model . . . . . . 53 . . . . . . . . . 55 3.6.1 Defining the Dependent Sources . . . . . . . . . . 56 3.6.2 Approximating < qV > 57 3.6.3 Simulation of the Four-Source Model in HSPICE 3.3.2 3.4 3.5 3.6 4 The Four-Source Cycle-Averaged Model . . . . . . . . . . . . . . Studying the Cycle-Averaged Model 4.1 61 State-Space Model and its Eigenvalues and Eigenvectors 61 4.1.1 Deriving the State-Space Model . . . . . . . . . . 61 4.1.2 Eigenvalues and Eigenvectors 63 4.1.3 Using the Steady-State Eigenvector to Match the Calculated . . . . . . . . . . . < Vh > . . . . . . . . . . . . . . . . . . . . . . . . 4.2 57 Computational Efficiency. .. .. .. 64 . . . .. . . . . . . 69 . . .. ... 4.2.1 Timesteps in HSPICE .. . . . . . . 69 4.2.2 Default Dynamic Internal Timesteps . . . . . . . 70 4.2.3 Larger Timesteps . . . . . . . . . . . . . . . . . . 70 5 Modeling the Arterial Baroreceptor Reflex 75 5.1 Feedback Control with Venous Zero-Pressure Filling Volume 75 5.2 Control with the Period T . . . . . . . . . . 78 5.3 Control with R 3 . . . . . . . . . . . . . . . . 82 5.4 The Feedback Models . . . . . . . . . . . . . 84 8 6 Conclusion 85 A HSPICE Code 89 A.1 The Cycle-Averaged Three-Source Model ....... . ..... 89 . . . . . . . . . . . . 90 A.3 Feedback Models (Based on the Four-Source Model) . . . . . . 91 A.3.1 Control with V . . . . . . . . . . . . . . . . . . . . . . 91 A.3.2 Control with T . . . . . . . . . . . . . . . . . . . . . . 92 A.3.3 Control with R3 . . . . . . . . . . . . . . . . . . . . . . 93 A.2 The Cycle-Averaged Four-Source Model B New Approximations for < qV 2 > and Vhsys B.1 Approximation for Vhay, 95 95 . . B.2 Approximation for < qV 2 > 97 B.3 A New Model .......... 98 C Eigenvalues of the Four-Source Model 101 D Further Simplification based on Slow/Fast Decomposition 105 9 10 List of Figures 1-1 The circuit-based hemodynamic model. . . . . . . . . . . . . . . . . . 19 1-2 Time-varying capacitance waveform. . . . . . . . . . . . . . . . . . . 20 1-3 Response of the hemodynamic model to a stand test. . . . . . . . . . 21 2-1 Simple RC circuit with time-varying capacitor. . . . . . . . . . . . . . 26 2-2 Simple RC circuit response to initial voltage of 1OV, simulated in H SP IC E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 . . . . . . . . . . . . . . . . 28 2-3 Simple RC circuit with a current source. 2-4 Response of a simple RC circuit with a current source and initial voltage of 1OV. ....... 2-5 29 ................................ The simple heart model with capacitance waveform (parameter values m odeled on those in [3]). . . . . . . . . . . . . . . . . . . . . . . . . . 30 2-6 Simulation of the pulsing model in HSPICE. . . . . . . . . . . . . . . 33 2-7 Simulation of the pulsing model with a larger value of R 2 = 1Q. . . 35 3-1 Response of the pulsing model to a transient in R 3 . . . . . . . . . . . 38 3-2 The calculated time-averaged waveforms of Vh and V2. ... . . . . . . 39 3-3 Time-varying elastance waveform. . . . . . . . . . . . . . . . . . . . . 40 3-4 RC circuit with a step in resistance. . . . . . . . . . . . . . . . . . . . 42 3-5 Response of RC circuit with a step in resistance. . . . . . . . . . . . . 43 3-6 RC circuit with an exponential current source. . . . . . . . . . . . . . 44 3-7 Response of RC circuit with an exponential current source. . . . . . . 45 3-8 The pulsing model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3-9 Switching functions q and q'.. . . . . . . . . . . . . . . . . . . . . . . . 47 11 . 3-10 Closeup of key waveforms. . . . . . . . . . . . . . . . . . . . . . . . . 49 3-11 Preliminary three-source cycle-averaged model . . . . . . . . . . . . . 52 3-12 The response of voltage < Vh > in the Three-Source Time-Averaged M odel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3-13 The response of voltage < V2 > in the three-source cycle-averaged model. 54 3-14 The four-source cycle-averaged model. . . . . . . . . . . . . . . . . . 55 3-15 The response of voltage < Vh > in the four-source time-averaged model. 58 3-16 The response of voltage < V2 > in the four-source time-averaged model. 59 3-17 The response the four-source time-averaged model to two steps in R 3 . 4-1 Simulations of the Three-Source Model using original and new initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 60 66 HSPICE simulations showing Vh from the pulsing model and < Vh > from teh averaged model, using both the default internal timestep settings and larger internal timesteps. . . . . . . . . . . . . . . . . . . . 72 5-1 Four-Source Cycle-Averaged Model with V1. . . . . . . . . . . . . . . 76 5-2 The response of the four-source model with controlled V/. . . . . . . . 77 5-3 The response of the four-source model with controlled period T. . . . 81 5-4 The Four-Source Model response with controlled period R 3 due to a step in V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 B-1 One period of V2 from the pulsing model . . . . . . . . . . . . . . . . 96 B-2 Responses of the three-source model using the new approximations for < qV 2 > and Vh y.... . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 D-1 Cycle-averaged model with source Vheart. . . . . . . . . . . . . . . . . 108 D-2 Response of the cycle-averaged model with source Vheart. . . . . . . . 109 12 List of Tables 3.1 Constant capacitance values. . . . . . . . . . . . . . . . . . . . . . . . 4.1 Samples and CPU time for simulations in HSPICE using default internal timesteps (30s transient analysis). . . . . . . . . . . . . . . . . . . 4.2 70 Samples and CPU time for simulations in HSPICE using large internal timesteps (30s transient analysis). . . . . . . . . . . . . . . . . . . . . 5.1 42 71 Voltage values at steady-state of the four-source model with and without V control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Steady-state voltage values of the four-source model with controlled T. 80 5.3 Steady-state voltage values of the four-source model with controlled R 3 . 83 13 14 Chapter 1 Computational Models and Biological Systems A computational model is a quantitative representation that captures the parameters and dynamics of a physical system. Some of the most exciting physical systems being studied include complex biological systems within the human body. Scientists use computational tools, and mathematical descriptions in the form of computational models, in order to gain an understanding of how complex physiological systems function. These approaches are currently being used at MIT to model systems such as the diffusion of ions through a neuron cell membrane, the acoustic properties of the inner ear, and cardiovascular system responses to orthostatic stress. The goal of developing computational models is to provide a good match to the actual physical system. A good computational model should respond in ways similar to the expected behavior of the system being modeled. Since a biological system consists of a complex array of many different interactions, models are used to help organize how we think of the components within the system. Therefore, the models can be used to gain knowledge about the functionality of a system by providing a framework for both testing hypotheses and developing new questions to study. Computational models often involve electrical and mechanical descriptions. Specifically, circuit-based representations provide a quantitative explanation and allow the model to be analyzed using advanced circuit design and simulation tools. The ability 15 to capture the features of a biological system within an electrical model also allows more probing and closer examination of system parameters than possible from experimental human studies. Specifically, we will see how using circuit models will help us describe the dynamics of the cardiovascular system. 1.1 1.1.1 Background and Previous Work The Cardiovascular System The cardiovascular system is a very complex and dynamic system. Physiologists have studied portions of the system, making assumptions about certain sections. In general, blood flows through vessels of the cardiovascular system from regions of high pressure to regions of lower pressure. The difference in pressure between two points determines the rate of blood flow. The rate of flow is measured as volume of blood per unit of time. Blood flow through a vessel is impeded by blood viscosity and by frictional effects that depend on the length of the blood vessel and its radius. These factors determine the friction between adjacent layers of flowing fluid, and the friction between blood and the vessel walls. In the human cardiovascular system, variations in the radii of the blood vessels contribute most to the changes in flow resistance. Equation 1.1 defines the relation between blood flow F, pressure difference AP, blood viscosity r, vessel length 1, and vessel radius r in laminar flow: AP = F (1.1) This expression can be further simplified and rewritten as R = F , (1.2) where R is the resistance to flow between two points in a blood vessel, AP is the pressure difference between these two points, and F is the blood flow rate between them. 16 The heart is a pump that recirculates blood throughout the body. The heart itself cycles through two phases. When the left ventricle of the heart contracts in systole, blood is ejected into the aorta. During diastole the ventricle relaxes and the heart fills with blood. This pumping action produces the pressure which drives blood flow through the system. Blood pressure in cavities of the cardiovascular system is also effected by the elasticity of the cavity walls. 1.1.2 Modeling the Cardiovascular System Using the description of how blood flows through vessels, shown in Equation 1.2, we can see how a circuit analogy for a computational model of the cardiovascular system falls into place. Blood flow rate could be analogous to an electrical current within a circuit. Current flows from regions of high potential to regions of lower potential and is impeded by circuit resistances. Thus, in a circuit-based model, blood flow rate would map to electrical current, pressure difference to voltage, and flow resistance to electrical resistance. The elasticity of compartment walls within the cardiovascular system can also be modeled, with electrical capacitors. One way to describe this analogy is to relate the distending blood volume BV in a cavity to the elastance E of the cavity walls and the pressure difference AP between the inside and outside of the cavity: 1 BV = - AP . E (1.3) In general, a distending volume within a compartment will generate a pressure in that cavity, and the pressure generated is dependent on the elasticity of the compartment's walls. The relation in Equation 1.3 is similar to a description of charge q on a capacitor with capacitance C in electrical circuits: q=CV. (1.4) Capacitance evidently functions as the inverse of elastance. The next section will 17 discuss how the pumping action of the heart can be modeled via a compartment with periodically varying elastance. 1.1.3 The Computational Hemodynamic Model Heldt et al. at MIT have developed a circuit-based computational hemodynamic model to help analyze cardiovascular function during conditions such as head-up tilt and lower body negative pressure [3]. The model, as shown in Figure 1-1, divides the human cardiovascular system into twelve compartments. Each compartment is represented with resistors, capacitors, diodes, voltage sources, and/or time-varying capacitors. This lumped parameter circuit model can be simulated on a computer and used to explain how the cardiovascular system responds to different situations and conditions. The key components of the model are the time-varying capacitors, which capture the pumping action of the heart. The two time-varying capacitors C,(t) and C1 (t) represent the right and left ventricles of the heart. The contraction of the ventricles generates the pressure that drives the blood flow through the system. The capacitance waveform with respect to time for these capacitors is shown in Figure 1-2. The highcapacitance state is analogous to the relaxed diastolic state in the heart, while the lowcapacitance state is analogous to the contracted systolic state of the heart's pumping cycle. The sudden drop in capacitance value, which corresponds to the contraction of the heart, causes the voltage across the capacitor to increase, thereby causing current to flow out of the time-varying capacitor and into the rest of the system. The rise in capacitance, which matches the relaxation of the ventricle, causes the voltage across the capacitor to drop, and marks the point where current begins to flow into the capacitor. Two diodes separate each time-varying capacitor from the rest of the circuit. These diodes represent the valves between the atrium and the ventricle, and between the ventricle and either the pulmonary trunk (right ventricle) or the aorta (left ventricle). The diodes prevent the backflow of current in the circuit, just as valves in the heart prevent the backflow of blood in the system. 18 RAv pa P Rr Tpa th - SM(C plv th C,(t) C (t h th P p P, I C." Csu-I-Vh Cup Rkidl "sup kd2 TfCkid Rab Ca 0 p ab Inf If A Ra Csp R2 I Pbasli,4 7 N~biasl Figure 1-1: The circuit-based hemodynamic model. 19 p'4 - 10 9 8 7 E 44 3 2 0 0.5 1 Time (sec) 1.5 2 Figure 1-2: Time-varying capacitance waveform. The voltage sources labeled Pth simulate the intrathoracic pressure, which is a reference for all the compartments within the chest; Pth varies in value with respiration. The rest of the circulatory system is modeled with linear resistors and capacitors. In general, each capacitor represents a different compartment in the body. As we can see, the overall hemodynamic model captures the fact that the cardiovascular system is an interconnected circulatory system. Our figure omits various feedback control loops that the body uses the regulate the cardiovascular system. Specifics of parameter values and how the parameters were determined are described in Heldt's paper [3]. The transient response of heart rate to a stand test in the hemodynamic model is shown in Figure 1-3. The small ripples in the figure represent heart beats. The large, slowly varying transient curve is the response of the model to a perturbation of the system. When studying the cardiovascular system and its response to perturbations, such as lower body negative pressure or standing up, a cycle-averaged version of the hemodynamic model would appear to be sufficient to describe the average response over time. Circuit averaging is done because the local average behavior of a nearly 20 100 90 E ~80 cc C 70 - C) 60 - 50' j 0 50 Time (sec) 100 Figure 1-3: Response of the hemodynamic model to a stand test. 21 periodic waveform is often the curve of interest when studying the general response over long periods of time. These time-averaged models become especially useful when the periodic ripples in heart-rate are not as important as the overall transient response. Most importantly, an averaged circuit can also be simulated and computed more efficiently and simply than the instantaneous case. 1.1.4 Time-Averaged Circuits Circuit averaging is used in power electronics in order to analyze the dynamic behavior of many types of circuits containing switches that are operated in nearly periodic fashion. The idea involves using circuit techniques to develop a general averagedcircuit model that produces the local average response of variables within the circuit to any initial conditions or perturbations introduced to the system. Circuit averaging is often done in power circuits that contain switches and LTI (linear time invariant) components. A time-averaged version of a circuit should produce a response that captures the smooth variations in the local average behavior, with the switching ripple of the original response removed or attenuated. The cycle-averaged model should especially react similarly to the original model in its ability to follow the circuit's response to transients and perturbations. Often, it is these average values of voltages and currents in the circuit that are of interest during analysis. For example, when studying the transient response of the hemodynamic model, as in Figure 1-3, a time-averaged model would just capture the overall transient without the small periodic ripples. Circuit averaging can be done to increase computational efficiency. The averaged quantities are also the values needed for regulation and feedback control purposes. Overall, the approach taken to developing a time-averaged circuit involves replacing all voltages and currents in the circuit by their cycle-averaged values, keeping all the LTI components unchanged, and replacing the time-varying components in the system with appropriately chosen, and often approximate, equivalents [4]. 22 1.2 Goals and Outline This thesis will explore the characteristics of a simple version of the hemodynamic model, designed to represent a prototype cardiovascular circuit. Analysis and simulation in HSPICE [1], a circuit analysis software tool, will be done to show the effectiveness of using HSPICE as a tool for analyzing computational models that involve potential-driven flows. Next, a cycle-averaged circuit based on the instantaneous hemodynamic model will be designed to describe the average responses of our simple cardiovascular system. The motivation behind designing a cycle-averaged circuit includes computational efficiency. We will use HSPICE to characterize the time-averaged model and to show that it requires less computations than the instantaneous case. Lastly, we will use the developed cycle-averaged model to analyze and simulate some feedback control loops present in cardiovascular dynamics within the body. 23 24 Chapter 2 A Simplified Model The characteristics of the hemodynamic model are first analyzed piece by piece. The most unique and important feature of the model is the time-varying capacitor, a component that is not commonly seen in electrical circuits. Therefore, studies were first done with simulations in HSPICE of simple RC circuits with time-varying capacitors. Nodal analysis was performed for each of these preliminary models in order to understand the characteristics of the time-varying capacitor. Finally, the pulsing simple heart model, which consists of a one-chamber heart, provides the primary circuit used to model a simple cardiovascular system. This pulsing model will be the basis for the time-averaged design. 2.1 Using HSPICE The primary step in studying our computational model of the cardiovascular system involves implementing pieces of the system in HSPICE and studying the behavior of these components. HPSICE is a numerical circuit simulation tool. The current version of the hemodynamic model is not implemented in HSPICE. The motivation behind simulating such models in HSPICE includes the ability to take advantage of a developed and popular circuit design tool used by many engineers. As a highly developed program that is flexible and easy to use, HSPICE has the capability to implement complex circuits using very few lines of code. HSPICE is also designed to 25 solve all the equations necessary to plot and analyze the transient characteristics of any circuit model. All of the models described will be simulated and analyzed using this software tool [1]. 2.2 A Simple RC Circuit We begin by simulating in HSPICE a simple RC circuit with a time-varying capacitor. The circuit parameters are defined in Figure 2-1. The capacitance waveform has a C 1OF R = 1 ohm C time (s) 0.66 1 Figure 2-1: Simple RC circuit with time-varying capacitor. period of 1 second, and is in the high capacitance (10F) state for 0.66 seconds and in the low capacitance (0.4F) state for 0.33 seconds. This waveform is an approximation to the capacitance waveform shown in Figure 1-2. HSPICE has no direct implementation of a time-varying capacitor. However, it does allow the user to define a capacitance based on the voltage value at any node. The program written to implement the RC circuit with a time-varying capacitor defines a periodic voltage source separate from the rest of the circuit. This voltage source is controlled by a pulse function of defined amplitude, period, and duration to produce the desired time-varying waveform. The capacitance was then defined as a function of this voltage source. The HSPICE code for implementing the simple RC circuit is shown below: 26 * Title RC Circuit .options post * Required .op ** Initial voltage 10V across Capacitor .ic V(ni)= 10 ** Pulsing voltage source used to define the capacitance waveform * Starts at 1bV, at 0.66s drops to 0.4V for 0.33s, repeats every is Vcap n2 0 pulse(10 0.4 0.6666 in in 0.3333 1) Rgrd n2 0 1meg Cv n1 0 C = 'V(n2)' CTYPE=1 ** Define C interms of Vcap R2 n1 0 1 **Transient analysis for 10s with timestep=0.Ois .tran 0.01s 10s UIC .plot tran V(ni) .end The circuit in Figure 2-1 was simulated with an initial voltage of 10V across the capacitor and produced the response shown in Figure 2-2. The response shows that the voltage peaks at the points in time when the time-varying capacitor drops in value from 1OF to O.4F. The drop in capacitance corresponds to a sharp increase in the voltage across the capacitor because the charge within the capacitor cannot change instantaneously (without impulsive currents). The voltage at the node decays after each peak since the capacitor discharges through the resistor when the capacitance value is constant. The heights of the voltage peaks also decrease with time, as expected, since no additional charge is being placed on the capacitor by any sources or recirculated to it from other capacitors. The circuit heads towards a steady state of OV at the node. 27 - 2 40 - - -... -.. -.... - ....- - .--. - 220 200 10 IS0 140 I, --. 120 - -.-. . 100 ........ -. so s0 40 20 0 0 a 2 11mnm (1in) (T-nIE) 10 Figure 2-2: Simple RC circuit response to initial voltage of 10V, simulated in HSPICE. 2.3 Simple RC Circuit with a Current Source Next, a current source of 100A was added to the simple RC circuit studied in Section 2.2. The circuit and capacitance waveform are shown in Figure 2-3. The current C 1 OF I = 100A C Chigh R = 1 ohm O.4F ClowI . 0.66 1 1 I time (s) Figure 2-3: Simple RC circuit with a current source. source supplies the circuit with additional charge. Therefore, the response of this system to an initial voltage of 10V does not decay to a steady state of OV as shown in Figure 2-4. Within two periods of the pulsing capacitance, the voltage across the 28 -q 350 Soo mo0 10 50 0 Figure 2-4: Response of a simple RC circuit with a current source and initial voltage of 1oV. capacitor settles into a steady state with the voltage consistently peaking at around 360V at every transition to the Clow capacitance state. The nature of the time-varying capacitance controls the periodicity of the voltage response behavior. During the Clo time interval, the voltage across the capacitor is high and it decays in value with a time constant of T = Clw* R = 0.4s as the capacitor discharges through the resistor. In the Chigh time interval, the voltage is in a low state and it increases with a longer time constant of Chigh * R = 10s as the capacitor stores charge from the current source. Again, we can see how the time-varying capacitor coupled with a source of charge begins to capture the periodic, pulsing nature seen in the pumping behavior of the heart. 2.4 The Pulsing Heart Model Next, a simplified version of the hemodynamic model was implemented in HSPICE. In this simple, single-ventricle, three-compartment model, shown in Figure 2-5, the central time-varying capacitor is analogous to the heart in the cardiovascular system. 29 R3 Ri = 0.03 Q Ri Di D2 R2 R2 = 0.01 Q Vh V2 Vi +± C CV C Ca CV= 100 F Ca = 2 F Cd = 10 F Cs = 0.4F Td = 0.66 s T= 1 s Cd Cs I I Td I I I I T time Figure 2-5: The simple heart model with capacitance waveform (parameter values modeled on those in [3]). 30 The diodes D1 and D 2 are in place to prevent the backflow of current in the system. These diodes represent the valves in the heart. The voltage V2 denotes arterial pressure; the voltage V represents the venous pressure; Vh corresponds to the pressure within the heart; R 3 is analogous to the resistance to blood flow in the body. This simple model captures the behavior of a closed-loop circulatory system driven by a time-varying capacitor. 2.4.1 How it works When C is in its high-capacitance state Cd, the voltage Vh is low. This corresponds to diastole when the chamber is in its filling state with a corresponding low heart pressure. During the diastolic portion of the cycle, the voltage V across the time-varying capacitor is low, current flows through the diode D 1 , diode D 2 is non-conducting, and Vh increases only slightly as charge is stored in C. During the transition from diastole to systole, the heart contracts, causing the pressure in the heart to jump and blood to flow out of the heart. During systole in the circuit, C is in its low capacitance state and Vh is high. Consequently, current flows from the capacitor C through diode D2 into the rest of the system, and D, becomes non-conducting. Therefore, Vh decreases in value as charge leaves the timevarying capacitor, corresponding to the behavior seen during the systolic cycle of the heart. 2.4.2 Simulation in HSPICE The model in Figure 2-5 was implemented in HSPICE. The entire code for the circuit simulation is shown below: Pulsing Heart Model * Title .options post .op ***Initial Conditions: Vi = 9V, Vh = 7V, V2 = 56V .ic V(n2)=9 V(n4)=7 V(n6)=56 31 ***Define the voltage Source Vcap which implements the pulsing waveform * for the time-varying capacitor: Vcap nc 0 pulse(10 0.4 0.6666 lu lu 0.3333 1) Rgrd nc 0 1meg ***Define the Circuit Connections Cv n2 0 C = 100 R3 n2 n6 R = 1 R1 n2 n3 R =0.03 Ch n4 0 C = 'V(nc)' CTYPE=1 *Use the voltage source defined above R2 n5 n6 R =0.01 Ca n6 0 C = 2 D1 n3 n4 diodel D2 n4 n5 diodel .model diodel D level=1 IS=le-14 CJO=le-12 VJ=.783 ***Transient Analysis over 10s with steps of .tran 0.Ols 0.Ols 10s .end These few lines of code define the connectivity between the parts of the model, the initial conditions, the type of analysis, and the pulsing capacitance waveform. Due to the pulsing capacitance, we will henceforth refer to this model as the pulsing model. Figure 2-6 shows the simulation of the pulsing model using the initial values V1 = 9V, Vh = 7V, V2 = 56V. Since the capacitance C starts off in the diastolic state, the initial value of Vh was chosen to have a low value compared to V and V2 . The plots of the voltages at each node follow the expected characteristics. All of the waveforms are periodic with a period of T = 1s. The first 0.66 seconds of each period correspond to a state of diastole where Vh, corresponding to the pressure in the heart, is low compared to V1 and V2. The voltage Vh grows with an approximate time constant of Cd * R, = 0.3s as current flows through the diode D1 and into the capcitor C. During diastole, no current flows through R 2 since diode D 2 is non- conducting. However, charge does leave the capacitor Ca through R 3 , causing V2 on the arterial side to decrease with a time constant of about C * R3 = 2s. Lastly, since 32 200 -- 18O 16O 140 120 ao 100 h 8o 60 - . ... .. . 40 20 0 0 6 4 2 (iiri) TIme 8 (TIMVU) Figure 2-6: Simulation of the pulsing model in HSPICE. 33 10 C, is so large, the voltage V on the venous side remains relatively constant. By the end of diastole, the voltage Vh is approximately equal to V1 . The last 0.33 seconds of each period correspond to the state of systole in the heart. During systole, Vh is very large, causing the diode D 2 to start conducting and diode D1 to become non-conducting. Current flows from capacitor C through R 2 with a fast time constant of about R 2 * C, = 0.004s. This fast time constant produces the sharp peaks seen in Figure 2-6. The fast time constant also causes Vh to drop rapidly until it equals V2 . When Vh equals V2, they both start decreasing with a time constant of about R 3 * Ca = 2s as Ca discharges through R 3 . At the end of one period, the circuit switches back to the diastolic state, causing the cycle to repeat. Note that the very high peaks of the Vh curve are not seen in an actual physiological response of the cardiovascular system. The sharpness of the peaks are due mainly to the small values of R 2 and C, and the abrupt step changes in the capacitance waveform. The charge q stored in a capacitor C with voltage Vh is given by: q = CV (2.1) At the step change in C from Cd to C8 , the pulsing model switches from the diastolic to the systolic state. The sudden change in C causes a sharp change in the value of Vh since the charge stored in the capacitor remains constant during the transition. At the transition from Cd to C8 , Vh sharply increases from Vh(-) to Vh(+) = Vh(-)Cd C., i.e.,to 40 times its diasolic level. At the transition from systole back to diastole, C steps from C, to Cd and Vh drops to 0.04 times the value it was at the end of systole. After one to two periods, the simulation essentially falls into a steady state where Vh peaks at the same value of 193V every cycle. 2.4.3 Changing R2 When the resistor R 2 in the pulsing model is increased from O.O1Q to 1Q, the circuit produces the simulated behavior shown in Figure 2-7. This increase in R 2 could be analogous to high resistance in the outlet valve of the heart, which could result from 34 aortic stenosis. The differences between Figure 2-6 and Figure 2-7 are pronounced. The increased vascular resistance causes a slower decay of the voltage Vh across the time-varying capacitor. As a result, the peaks in Vh are not as sharp and the rise time of V2 on the arterial side is much slower. 200 180 160 140 120 Ca 100 Vh 80 60 40 20 0 0 4 2 Time 6 (lin) 8 Figure 2-7: Simulation of the pulsing model with a larger value of R 2 2.5 10 (TIVI) = 1Q. Using the Pulsing Model The pulsing model will be the basis for the time-averaging of circuit variables and the development of the cycle-averaged model. The model is simple enough that there are only a few circuit variables and parameters to study, but it can still model the basic characteristics of a cardiovascular system. A cycle-averaged version of the pulsing model should be computationally more efficient and should capture the transient responses of the system to disturbances or perturbations. 35 36 Chapter 3 Developing the Cycle-Averaged Model The goal of a cycle-averaged model is to capture the overall cycle-to-cycle transient behavior of a given heart model while supressing the intra-cycle details. The key to developing such a model is to replace the time-varying capacitor and the diodes in the heart model with elements that only depend on cycle-averaged quantities within the circuit. The design should produce time-averaged responses that are smoothed, averaged curves without the sharply pulsatile nature of the detailed pulsing heart model responses. The cycle-averaged model will be more computationally efficient when intra-cycle details are not of interest, and can be used to define control structures within the system that only involve locally averaged rather than instantaneous variables. 3.1 Introduction of a Transient In order to study the time-averaged response to changes within the system, a transient was introduced to the pulsing heart circuit of Figure 2-5. This transient was produced by a step at 10 seconds in the resistance R 3 , from 1Q to 5Q. The response of the model to the step in resistance is shown in Figure 3-1. The step increase in resistance produces an increase in the average values of the voltages Vh and V2 . This transient 37 will be used to analyze the behavior of the time-averaged model. 80 1 1 i 70 Vt 60 C,, 0 50 CZ 40 - V 20 10 0' ) 10 5 20 15 25 30 Time (seconds) Figure 3-1: Response of the pulsing model to a transient in R 3. 3.2 Calculating the Time-Average Waveform The (symmetric) cycle-averaged value of a waveform V is a time-average over each period T of the cycle, and is defined by: =T < V(t) > V(T)dT . =+- Tt- (3.1) T A property of this definition is that: dV\ dt d_<V> dt (3.2) The cycle-averaged waveforms of Vh and V2 from the transient responses in Fig38 ure 3-1 were calculated in MATLAB using the trapezoidal rule to approximate the integral in Equation 3.1, with a timestep of 0.01 seconds. The calculated time-averaged waveforms are shown as the dotted lines < V1 > and < V2 > in Figure 3-2. These 80 1 1 1 1 I _11x I 70 601-1 0 5040 I V 30 20 I 10 01 /I- I/ VKKK VK C 5 /I~ 10 'i _ I-, [I--, 15 20 LJ 25 30 Time (seconds) Figure 3-2: The calculated time-averaged waveforms of Vh and V2. curves are generally smooth and lack the pulsatile nature of the original voltage waveforms. The slight ripples in the curve near the transient are due to the non-periodicity of the waveforms during the transient. (The zero values at the beginning and end are from the limitations to the time-frame used in HSPICE.) During the change in R 3 , the calculated cycle-averaged voltages < Vh > and < V2 > increase in value, consistently with the behavior seen in the original model. The cycle-averaged circuit models that we are aiming to develop should produce responses that are good approximations to the cycle-averaged waveforms in Figure 3-2 for this same experiment (namely, a step change in R 3 ), as well as in other such experiments. Therefore, during the development of the cycle-averaged model, the 39 results we obtain will be compared to this calculated average. 3.3 Finding Ceff In order to create a cycle-averaged model, we first study how to replace the timevarying capacitor in the pulsing model with a constant, effective capacitance value. 3.3.1 Calculations with Elastance It will be more convenient for our purposes to work with the reciprocal of capacitance, namely elastance, 1 (3.3) C For the time-varying capacitance waveform seen in Figure 2-5 in Chapter 2, we can draw the elastance waveform as in Figure 3-3. E We will make use shortly of the 1/Cs 1 /Cd Td Ts I T time Time in Systole: Ts = T - Td Time in Diastole: Td Figure 3-3: Time-varying elastance waveform. time-average of this elastance waveform, or the effective elastance, namely 1 T 40 Td Cd + -T. CS (3.4) Since V = Eq from the combination of Equations 2.1 and 3.3, where q is the charge on the capacitor, taking the time-averaged values produces: < V > = < Eq > (3.5) The charge q has relatively low ripple, and is approximately constant compared to E over any interval of length T. We can therefore make the approximation < V > r < E >< q > . (3.6) The current through a constant capacitance C is given by dq(t) dt )_ 1 dV dV . dt E dt (3.7) The time-average of the current can then be written as: < > = d(t) dt d (< V >) - <E > dt 1 d d < V >. < E >dt - (3.8) 39 (3.9) (3.10) Equation 3.8 follows from 3.2, Equation 3.9 follows from 3.6, and 3.10 follows because < E > is constant since E is periodic. Since the result in Equation 3.10 is similar to the expression for current in Equation 3.7, and is in the form expected for a timeaveraged circuit, one possibility for replacing the time-varying elastance would be to use < E >, the time-averaged value of the elastance E. Therefore, in a cycleaveraged circuit model, the equivalent elastance of the time-varying capacitor would be the value of < E > in Equation 3.4. The corresponding effective capacitance is Ceff .T3.1 = S(3.11) 41 3.3.2 Ceff in a Simple RC circuit A simple RC circuit similar to the one described in Chapter 2, Section 2.2 was used to analyze the behavior of a system which incorporates the value of Ceff. First, the simple RC circuit with a step change in resistance at 6 seconds, shown in Figure 3-4, was simulated in HSPICE. The response with a time-varying capacitor C is displayed as the pulsing waveform in Figure 3-5 . The step change in resistance caused the voltage to increase with an exponential transient. The time-average response was calculated and is depicted as the dotted line in Figure 3-5. C 1 OF I = 100A C Chigh R = Step at 6s from 1ohm to 5 ohm Clow O.4F I I 0.66 1 time (s) Figure 3-4: RC circuit with a step in resistance. Next, the time-varying capacitor in Figure 3-4 was replaced with a constant capacitance. Four simulations were done, each with a different constant capacitance value. The values chosen include the high capacitance value Chigh, the low capacitance value Clow, the average value of the high and low capacitances Cag, and the effective capacitance Ceff derived from the time-averaged elastance and specified in Equation 3.11. The values are displayed in Table 3.1. The simulation with each of Capacitance Chigh Value lOF Cow O.4F Cavg Ce!! 5.2F 1.11F Table 3.1: Constant capacitance values. 42 70C - 60C 1 Calculated <V: Using Ceff - Using Cavg 1 50C- 40Ca) CD) 0 o 30C- .- 9oN 20Cloc- 20~ 0 - - - -- 10C - 2 6 4 8 10 Time (Seconds) Figure 3-5: Response of RC circuit with a step in resistance. 43 - 12 these values is shown in Figure 3-5. We can see that the curve which results from using the value of Ceff most closely matches the calculated time-average of the original voltage < V >. To exhibit the behavior of Ce!f for other types of transients, the RC circuit in Figure 3-6 with an exponentially varying current source was simulated. The exponential current source provided the transient in the circuit. This exponential function increased with a time constant of is towards a value of lOGA and at t = 2s fell with a time constant of 2s back towards GA. The voltage waveforms derived from simulation of this circuit are displayed by the pulsing curve in Figure 3-7. I = exp(O 100A 0 ls 2s 2s) C 10F S C / The plot shows os- Chigh R=1ohm I 0.66 I 1 I time (s) Figure 3-6: RC circuit with an exponential current source. cillations in the voltage across the time-varying capacitor. The general response also captures the exponential nature of the current source. The calculated time-average of the voltage is plotted as the dotted line. The response of the same circuit, except with a constant capacitance of Ce!f equal to 1.11F in place of the time-varying capacitor, is shown as the solid curve. Again, replacing the time-varying capacitor by a constant capacitance Ce!! produces a circuit that follows the time-average of the voltage curve. This result further solidifies the reasoning behind using Ceff as the effective capacitance to replace the time-varying capacitor. 44 300 I II II - Original Calculated <VUsing Ceff 25C- 20CU) a0) 1 50- V from Circ jit with Ceff 0 0 2 6 4 8 10 Time (Seconds) Figure 3-7: Response of RC circuit with an exponential current source. 45 12 3.4 Replacing the Diodes The next step in developing a cycle-averaged circuit model involves replacing the diodes in the original pulsing model, shown again in Figure 3-8, with elements that depend only on averaged variables within the circuit. R3 Ri = 0.03 Q R1 D1 D2 R2 1R2 V2 V1iV = 0.01 Q R3 = 1 92 Cv = 100 F Ca = 2 F CV / Vi C V Ca Cs = 0.4F - C Td = 0.66 s T=1 s Cd Cs i Td Itime T Figure 3-8: The pulsing model. 3.4.1 Voltage Analysis with Switching Functions In order to characterize the voltages across each diode, the voltages Vi and V in Figure 3-8 of the pulsing model were studied. Each diode is in one of two states: conducting or non-conducting. During diastole, diode D, is conducting current while D 2 is non-conducting. During systole, diode D1 is non-conducting and D2 is conducting. We also define two switching waveforms q and q' = 1 - q as shown in Figure 3-9. The waveform q is 1 during the diastolic portion of the cycle, while q' is 1 during the 46 q q' 1 0 1 0- time T Td T Td time Ts Ts Td = 0.667 seconds Ts = 0.333 seconds T = 1 second Figure 3-9: Switching functions q and q'. systolic part of the cycle. Also we denote the (constant) time-average value < q > by a and < q' > by a'; note that a is the fraction of each cycle that the heart is in diastole. For the parameter values in Figure 3-9, a equals 0.667 and a' equals 0.333. The voltage Vi equals the voltage Vh across the time-varying capacitor during diastole when the diode Di conducts current, and equals V during systole when D1 is non-conducting. This relation is described using Equation 3.12: V, qVh + qV1 (3.12) For a heart rate of 1 cycle/second with the values of Td = 0.66s and T, = 0.33s, Equation 3.12 says that the voltage Vi is equal to Vh during the first 0.66 seconds of the cycle, and is equal to V during the last 0.33 seconds of the cycle. Similarly, the voltage V equals Vh during systole when D 2 is conducting, and equals V2 during diastole when D 2 is turned off. This produces Equation 3.13: V = qV2 + q'V . (3.13) Taking the cycle-average of Equations 3.12 and 3.13 gives us: < V > = < qVh > + < q'V > 47 (3.14) <VO > = < qV 2 > + < q'Vh > .( (3.15) These equations will become useful in building the cycle-averaged model. What we seek are approximations that allow the averages of products, as on the right sides of Equations 3.14 and 3.15, to be written purely in terms of single averaged variables. 3.4.2 Current through the Diodes The current through diode D1 equals the current il through resistor R 1 at all times, and the current through diode D 2 equals the current 12 through R 2 . The current i1 is zero during the systolic phase of the cycle, and the current i 2 is zero during the diastolic portion of the cycle. We shall find an approximate expression for < i1 > below. 3.5 The Preliminary Cycle-Averaged Model We begin by using the waveforms for Vh, V1 , and V2 seen in Figure 3-10 to make approximations to the equations describing the time-averaged voltages, < V > and < Vi >, and the currents, and . Note first that because C, is so large, V is essentially constant over any interval of length T, so we shall feel free to replace Vi by < V > in all that follows. 3.5.1 Time-averaged Current Plotting a couple of periods of the Vh cycle in steady state, shown in Figure 3-10, helps describe the current il based on other variables and parameters in the pulsing model. If we assume that during systole Vh decays with a fast time constant (T = 0.004s) towards V2, then early in systole V becomes equal to the value of V2 . If we denote the value of Vh at the end of systole by Vhsy, and its value at the beginning of diastole by Vhdia, then 1Vhda = - Vhy, since the time-varying capacitor changes in value from C, to Cd. We can then describe the initial current, 48 iinitiaI, through D1 at the beginning 200 18016C diastole 14C C,) 0) (C, diastole systole 12C 10C Vhsys ------------- - 0 80 60 V2 40 Vhdia Vh 20 Vi 0 3 3.5 4 4.5 5 Time (seconds) Figure 3-10: Closeup of key waveforms. 49 5.5 of diastole to be: V1 . - V - 1 -d (3.16) Vhdia CsVsys (3.17) R, (3.18) VhsysCs RjCd V1Cd - This current decays during diastole with a time constant of T RlCd= 0.3s. So the current i1 during the diastolic state can be written as: i1(t) = (VlCd-VhsysCs) e RCd (3.19) Note that i 1 only flows during diastole. Also, the value of Vsys is approximated as being equal to < V2 >, the average value of V2 . This approximation is based on the assumption that V2 has relatively low ripple and is perhaps the poorest of the approximations we make, given that V2 has a 25%-30% peak-to-peak ripple. Better approximations can be made, at the cost of making the model somewhat more complex, but these refinements are left to future work (see Appendix B for a preliminary study). If we assume that by the end of diastole the current il has essentially decayed to zero, with Vh essentially equal to V1, we can compute the time-averaged current as: Cd- ~ V Cd - VhsysCs 50 Rj~d dT (3.20) R1Cd )RTd(3.21) < V2 > Cs < V1 > Cd - < V2 > Cs T )e 3.2 (3.22) 3.5.2 Time-averaged Voltage V Next, using Equation 3.15 and considering the waveform V2 in Figure 3-10 to be relatively low ripple, we approximate: < V > ~ a < V2 > + < q'Vh> . (3.23) . (3.24) Furthermore, the time-average < q'V > is the same as: < q'Vh > < Vh> - < qV> During diastole, the voltage V/ rises exponentially, with a time constant of 0.3s, to meet the value of V seen in Figure 3-10. Estimating that V is a relatively small, constant value, we then make the preliminary approximation: < qVh > ~~a < V> - (3.25) Combining these approximations gives an expression for < V, >: <V> 3.5.3 ~ <V 2 >+<Vh> -a<Vl > . (3.26) The Initial Three-Source Cycle-Averaged Model The initial cycle-averaged model proposed is shown in Figure 3-11. This model captures the desired average current flow through the circuit by placing the dependent current source I, between the capacitors C, and Ceff. The value of Ceff was derived in Equation 3.11 and calculated to be 1.11F, and I is given by the average value < i1 > determined in Equation 3.22. The dependent source V, captures the value of < V > given in Equation 3.26, and the current source I, simply follows the time-averaged current through R 2 . Combining these with the circuit values introduced in the pulsing model shown in Figure 3-8, the definitions of the dependent 51 <i3> <i1> <Vh> * R2 <V1 > CV <2> <V2> Ceff + la Ca Va 0.01 Q 1 - 5 Q (Step at 1 Oseconds) Cv = 100 F Ca =2 F Cen = 1.11 F Cd = 1OF Cs = 0.4F a= 0.66 T= 1 s Is = (Cd<V1> - Cs<V2>)/ T R2 = la = <i2> Va = CC<V2> + <Vh> - c<V1> R3 = Figure 3-11: Preliminary three-source cycle-averaged model. 52 sources in Figure 3-11 are numerically: a < V2 > + < V > -a < V1 > (3.27) = 0.66 < V2 > + < Vh> -0-66 < V > (3.28) = < i2 > (3.29) Va Is = 3.5.4 Cd < V > -Cs i >= =< < V2 > (3.30) T (3.31) 10 < V 1 > -0.4 < V2 > . Simulation of the Three-Source Model The three-source model depicted in Figure 3-11 was simulated in HSPICE using initial conditions closer to the steady state values in diastole, with < V > at 5.56V, < V1 > at 1.45V, and < V2 > at 35.4V (Appendix A). The response of the voltage < V1 > in the model is shown in Figure 3-12 (solid line) along with the calculated time-average of the voltage waveform Vh from the pulsing model (dotted line). The response MI - Calculated <Vt> 70 1) 60 K 50 IL 40 II $11 *D u 3C ii tdkh h>frI I 3 soy!ecr oe 1ce 00 I*i~ 2C It tC l=t~LH 1iC (7 ~0 VI _________ 5 V 10 15 20 25 30 Time (seconds) Figure 3-12: The response of voltage < Vh > in the Three-Source Time-Averaged Model. follows the general shape of the calculated time-average values. During the transient 53 0 -CIcae _ <V>I -Calculated Simulated <V2> from 3 source model eV2,1 70 -- U 60 - > 5040- -- 30- 20-1 10-- 1'5 S51'0 20 25 30 Time (seconds) Figure 3-13: The response of voltage < V2 > in the three-source cycle-averaged model. caused by the step increase in R 3 , the three-source model is also capable of tracking the change in voltage. Before the transient, the steady-state value of < Vh > from the model is about 2.5V (16.46%) higher than the calculated average of 15.19V. After the transient, the value of < Vh > is 4.1V (16.91%) higher than the calculated value of 24.24V. The response of < V2 > in the model is shown in Figure 3-13. As with < Vh >, the voltage < V2 > follows the general shape of the calculated average, but offset to a greater value. The steady-state value of < V2 > from the simulation of the model before the transient is 8V (24.28%) higher than the calculated value of 33.08V, and after the transient it is 12V (18.92%) higher than the calculated average of 63.44V. Overall, the smooth responses of < Vh > and < V2 > exhibit the characteristics expected from a cycle-averaged response. The next goal involves bringing the simulated voltage waveform from the model closer to the actual calculated average curves. 54 3.6 The Four-Source Cycle-Averaged Model The three-source model was capable of capturing the desired time-averaged response in a general way. However, in building the three-source model, we lost some of the characteristics of the pulsing model, such as the presence of the resistor R 1 . In order to keep all of the LTI components present in the original pulsing model, a four-source cycle-averaged model was built. This circuit model is shown in Figure 3-14. R <i3> i <Vi> t- R1 R2 <Vh> <12> <V2> Cv Vi Ceff Va+ Vs = 0.732510 <V1> + 0.01 06996<V2> Is = <ii> la = <i2> Va = -0.399177<V1>+ <Vh> + 0.65597<V2> Ri = 0.03 Q R2 = 0.01 Q R3 = 1 Q Cv = 100 F Ca = 2 F Cef = 1.11 F Td= 0.66 s T= 1 s Figure 3-14: The four-source cycle-averaged model. 55 Ca 3.6.1 Defining the Dependent Sources The current sources and < Ia > are simply equal to the average currents through < R1 > and < R 2 > in the model, so we can write: (3.32) is I =< 2> (3.33) We use approximations to the equations for < V > and < V > in order to determine the values for the dependent voltage sources. The value of the source V, should be equal to the time-average value < V > as shown below, repeating Equation 3.14: Vs <V>=<qVh>+<q'V1 > (3.34) Since the waveform for V is essentially constant over any averaging interval, we can write: V~ < qVh > +' <V1 > -(3-35) where a' is equal to the time-average value of < q' > in Figure 3-9. The approximation used for < qVh > from the previous section in Equation 3.25 does not work here because the value of V would evaluate to < V >, as shown below: V = a < V > +a' < V > = < V > (3.36) In the four-source model, if the dependent source V equals < V1 >, then no current will flow through R 1 , making < i1 > equal to zero. Without steady current flow, the model breaks down. Consequently, we need a more accurate approximation for the value of < qVh >. 56 3.6.2 Approximating < qVh > Figure 3-10 shows that the diastolic portion of the curve Vh, captured by the expression qV, grows exponentially with a time constant of about T = RlCd = 0.3s. At the beginning of diastole, Vh starts at a value of By estimating that the value jqVhy. of Vh8 y, is equal to the average value of < V2 > and using Equation 3.1, we can write the expression for < qV > as: 1 = <qVh > - T aT C (-Va JOCd 8 + (V1 C Vs) - - e)1 )) eRcd Cd I = IaT < V > + ( < V >C CS dt (3.37) -aT < V2 >) R1Cd(e RCd - 1)] (3.38) Notice that unlike in the three-source model, we do not approximate that Vh reaches the value of V by the end of diastole. Using the values in the pulsing model, Equation 3.38 evaluates to: < qV > = 0.399177 < V1 > +0.0106996 < V2 > . (3.39) Incorporating the new approximation for < qV >, we can define the values of the sources V, and V using Equations 3.23, 3.24, and 3.35: V = - Va = = 3.6.3 < qVh > +a' < Vi > (3.40) 0.732510 < V1 > +0.0106996 < V2 > (3.41) a <V2>+<Vh>-<qV> (3.42) -0.399177 < V > + < Vh > +0.65597 < V2 > . (3.43) Simulation of the Four-Source Model in HSPICE The four-source model was simulated in HSPICE by using dependent voltage sources that were defined in terms of the voltages < V >, < Vh >, and < V2 > of the circuit. The dependent current sources were written in terms of the currents < i1 > and of the circuit. The HSPICE code for the model is shown in Appendix A. 57 The voltage responses produced are shown in Figures 3-15 and 3-16. Compared 3011 - from 3-Source Model Calculated <Vh> |<Vh> <Vh> from 4-Source Model 25 -- Calculated <Vh> 20-0 Z' 1 5 - - - - - - - - - - - 10- - 5-1 55 1'0 1'5 Time (Seconds) 20 25 30 Figure 3-15: The response of voltage < Vh > in the four-source time-averaged model. to the responses of the three-source model, the four-source model produces curves that are closer in value to the calculated time-averaged waveforms. The response of < Vh > in the four-source model is about 1.8V closer in value to the calculated average before the transient, or 0.67V (4.38%) higher than the calculated average. After the transient, < Vh > is 3.05V (12.59%) above the calculated average. There is also a difference of about 2.15V between the curves for < V2 > from the two models before the transient, corresponding to a relative error of 17.70% (5.85V). After the transient, the response for < V2 > is 10.27V (16.19%) higher than the calculated average. The response of the four-source model is still offset from the desired calculations of < Vh > and < V2 >. More accurate approximations are required to bring this value down to match the calculations. The four-source model is capable of tracking the general local average response of the pulsing model to a variety of transients. A different transient step in R 3 , up from 1Q to 5Q at 10s then a step back down to 1Q at 15s, was created. We can see the response of the cycle-averaged model follows the behavior of the original model 58 - <V2> Calculated <V2> from 3-Source Model <V2> from 4-Source Model_ 70 - 6005040-- 30- 20-1- 10-- O 5 10 15 20 25 0 Time (Seconds) Figure 3-16: The response of voltage < V2 > in the four-source time-averaged model. in Figure 3-17. The four-source and the three-source models will both be used for characterizing time-averaged responses and studying feedback responses in the rest of the thesis. Both models are linear in terms of their responses to initial conditions. Improvements to these models will be discussed in Chapter 6. The three-source model uses sources with more simply defined values than the four-source model. Therefore, the threesource model will be used to characterize and derive the approximate behavior of a cycle-averaged system with state-space equations. The more accurate four-source model will be used in simulations to discuss computational efficiency and feedback models. The next chapter will look more closely at the properties and parameters in the cycle-averaged model. 59 Simple-Heart Model Four-Source Model - 100 80 0 -/ / - , 60 0) v?> 40 20- 0 5 10 15 20 25 30 Time (Seconds) Figure 3-17: The response the four-source time-averaged model to two steps in R 3 - 60 Chapter 4 Studying the Cycle-Averaged Model Both the three-source and the four-source models developed in the last chapter, shown in Figure 3-11 and Figure 3-14, responded to initial conditions and perturbations of the system in a manner similar to the behavior of the pulsing model. We have essentially developed a linear, time-invariant system from the nonlinear time-varying pulsing model. This chapter characterizes the cycle-averaged circuit and its computational efficiency. 4.1 State-Space Model and its Eigenvalues and Eigenvectors 4.1.1 Deriving the State-Space Model We first characterize the system using the setup of the three-source model because of its simplicity. A similar method can be used for the four-source model, as shown in Appendix C. Using the three-source model, the algebraic expressions for the sources 1s, Ia, and V are repeated below: Va = a< V2 >+ < Vh> -a 61 < Vl > (4.1) Ia Is (4.2) < i2 > =< ii >= Cd < V1 > -Cs < V2 > (4.3) T Now the currents <ZiI >, , and < i3 > in the model are defined to be: Cd < V1 > -C, <> < V2 > (4.4) T < V2 > Va- (4.5) < V2 > - < V > < i3 > (4.6) Finally, the state-space equations of the circuit can be obtained by expressing the currents through the capacitors in terms of the candidate state variables < V >, < V2 >, and < Vh>. For this, we first note that C,< V1 > < i3 > = Ceff< Vh > Is - Ca<V 2 > = < Z1 > - >- (4.8) (4.9) > and then substitute for Equations 4.1, 4.4, 4.5, and 4.6. Placing the result in matrix form, we obtain <V 1 > < > (4.10) < Vh > <Y2h> <V 2 > where -1 RsCv A- _ a + Cd TCe ff R2Cef f 1 R 3 Ca + 1 d TC, R 3 Cv -1 R 2 Cef5 a R 2 Ca 62 -C TCeff _ TC, a-1 R2Ceff 1 a-1 1 R 2 Ca R 2 Ca R3 C. (4.11) 4.1.2 Eigenvalues and Eigenvectors Using the values for the components described in the pulsing model in Figure 2-5, (note in particular that T = 1), we determined the numerical values for the A matrix: A -0.11 0 0.014 69.069 -90.090 29.6697 -32.833 50 -17.167 (4.12) The eigenvalues and eigenvectors of this A matrix were calculated in MATLAB to be: -106.6606 (4.13) 0 A -0.7064 ev 0.0001 0.1163 -0.0224 0.8730 0.3900 0.2993 -0.4877 0.9134 0.9539 . (4.14) The negative reciprocals of the three eigenvalues are the time constants of the exponentials that comprise of the response of the model to initial conditions. The eigenvectors describe the relative effects of the values of < V >, < Vh >, and < V2 > on the response from each eigenvalue. The eigenvalue at zero with its corresponding eigenvector show that the voltages at the nodes can have non-zero values when the system is in steady state. The eigenvalue of -106.66 is irrelevent to our averaged model because it corresponds to a time constant much smaller than our averaging interval T; moreover, its effects are too fast to observe in our simulations. We then expect to see the voltage values change with an exponential related to the last eigenvalue of A3 = -0.7064, which corresponds to a slower time constant of about 1.42s. If we examine the responses of the three-source model, such as in Figure 3-12, we can see that the curves for < Vh > and < V2 > both respond to transients exponentially. This exponential behavior occurs at the beginning of the simulation, in response to 63 the initial conditions. The exponential settles to a steady state value after about 3 time constants, a time of 4s. This suggests a time constant of around 1.3s, which is consistent with the value of 1.42s computed above. Using ideas from singular perturbation (slow/fast decomposition) theory [2], it can be shown (Appendix D) that good approximations to the slow and fast eigenvalues of matrix A are: Afast ~ = Aslow 4.1.3 -90.09 -1 I (R 3Cv = (4.15) R 2 Ceff -0.81 (4.16) + RCa Cd d TCv . CS CaT) (4.17) (4.18) Using the Steady-State Eigenvector to Match the Calculated < Vh > As discussed in Section 3.5, the three-source model produced responses that do not quite match the calculated cycle-averaged voltage values. The calculated cycle- averaged voltages of the pulsing model at steady state, before the transient in R 3 , are: <V >cal 5.4792V (4.19) <V 15.1905V (4.20) >calc = < V2 >clc = 33.076V . (4.21) The three-source model produced voltages for < Vh > and < V 2 > that were higher than these average values. The eigenvector, ev 2 , that matches the zero eigenvalue, describes the relative nodal 64 voltage values at steady state: 0.1163 ev 2 0.3900 = (4.22) 0.9134 We used this relation to define new inital conditions for the three-source model. Specifically, to match the value of < Vh > in the model to the calculated average, we set the initial condition < V1 >initial to be 15.1905V, the value of the steady state calculated average. Then using the proportions of the eigenvector ev 2 , we found new initial conditions for < V > and < V2 >: 15.1905 * 0.1163 (4.23) 0.3900 2V>initia >initia = 4.5299 (4.24) = 15.1905 * 0.9134 0.3900 (4.25) 35.3769 . (4.26) - Note that these new initial conditions are different from the ones used for simulating the pulsing model and the cycle-averaged models in Chapter 3. Using different initial conditions will place a different amount of total charge in the circuit. Previously, the cycle-averaged models and the pulsing model were both simulated with the same initial conditions in order to preserve the amount of charge in the system, allowing us to make the appropriate comparisons. Simulation in HSPICE with the new inital conditions produces the response shown in Figure 4-1. By using voltages set in proportion to the eigenvector ev 2 , we are starting the system off at steady state since: A * ev 2 = A2 * ev 2 = 0 . (4.27) (4.28) Therefore, the < Vh >new curve is flat before the transient in R 3 , lacking the initial 65 30 - Calculated Average <Vh Original 28- <Vh> 26<Vh>new using new initial conditions 24 -Calculated ' <Vh>cac 22- 0 ) 2018 - 16 - 14/ 12[-1 0 5 10 15 20 25 30 time (seconds) Figure 4-1: Simulations of the Three-Source Model using original and new initial conditions. 66 exponential behavior seen in the original simulation. Before the transient in R 3 , the difference between the simulated < Vii >ne, and calculated < Vh >calc curves is very small, at about 0.01V (0.07%). After the response to the step in R 3 , the steady-state values are still close to the calculated < V >calc values, the difference is now 0.15V (0.23%). Note that although the < V >ne, curve matches the calculated average very closely, the voltages < V >ne, and < V2 >ne, do not match the calculated versions, as seen by comparing the values in Equations 4.19 and 4.21 with Equations 4.24 and 4.26. To characterize the steady-state voltage values after the transient in Figure 41 caused by a step in R 3 , we can recompute the A matrix using the new value of R 3 = 5Q: At -0.102 0 0.006 69.069 -90.090 29.6697 -33.3223 50 -16.7667 . (4.29) The eigenvalues and eigenvectors of this new matrix At are depicted as matrices At and evt: -106.6011 At = 0 (4.30) -0.3576 0 evt = 0.0550 -0.0224 0.8738 0.3501 0.2983 -0.4863 0.9351 0.9542 . (4.31) The eigenvalue at -0.3576 corresponds to a time constant of about 2.8s. As seen in Figures 3-12 and 3-13, the transient voltage responses due to the step in R 3 take longer to settle into a steady state than the responses from the initial conditions at the beginning of the simulation. After about a period of 8 seconds, a length of time corresponding to three time constants, the voltages settle close to a steady-state 67 value. The relative voltage values in this steady state correspond to the values in the second column of evt. The actual voltage values will be a scale factor of this second eigenvector. To determine this scale factor we looked at the total charge in the system. The total charge qT is determined by the initial conditions in voltage set on the system, and is not affected by the step change in R 3 . The total charge can be calculated by adding the charge on each of the capacitors C, Ceff, and Ca, as follows: < V >initial = qT Cv Ceff Ca < Vh >initial (4.32) < V2 >initiai 4.5299 = 100 1.11 2 (4.33) 15.1905 35.3769 = 540.605C (4.34) . This total charge remains constant throughout a simulation due to the circuit's structure. Therefore, after the transient step in R 3 , the total charge in the system must still be the same. Using this idea, the scale factor K of the relative voltage values in the new steady state can be determined by solving the equation: qT [v Ceff Ca evt 2 K. (4.35) Numerically this gives us: 540.605 = 7.758811K (4.36) K = 69.6763 . (4.37) This constant factor multiplied by the eigenvector eVt2 defines what the steady-state voltage values should be after the transient in the simulation of the < V 68 >ne, curve shown in Figure 4-1. These voltages are: 3.8322 < V1 >t < Vh >t 24.3937 = . (4.38) 65.1543 < V2 >t Measuring these values in the HSPICE simulation produces: < V >t < V < >t 3.83 ~ 24.38 .(4.39) 65.12 V2 >t Comparing Equations 4.38 and 4.39 shows that the A matrix defined in Equation 4.12 successfully characterizes the three-source model at steady state for both parts of the transient simulation. The cycle-averaged model allows an easy characterization and derivation of the time-averaged behavior with eigenvalues and eigenvectors. 4.2 Computational Efficiency One of the motivations behind developing a cycle-averaged model is the computational efficiency expected in simulating the model, because of suppressed intra-cycle detail. Since the averaged model produces responses that are smoother curves in time, compared to the pulsing original model response, fewer timepoints need to be calculated in order to study its transient behavior, and larger timesteps can be taken for a specified numerical accuracy. The goal of this section is to quantitatively examine the difference in the number of computations required between the pulsing model and the four-source model. 4.2.1 Timesteps in HSPICE When HSPICE computes the voltage values at various time points during a transient analysis, it uses an internal timestep separate from the timestep specified directly 69 by the user. HSPICE calculates this internal timestep value using a dynamically changing timestep algorithm. Specifically the DVDT Dynamic Timestep Algorithm calculates the internal timestep based on the rate of change of a nodal voltage. For rapidly changing voltages, the algorithm uses a small timestep, while for slowly changing voltages it uses larger timesteps. In general, simulation speed has to be traded off against simulation accuracy. 4.2.2 Default Dynamic Internal Timesteps HSPICE was run for a 30s transient analysis simulation using the default timestep settings. The number of time points calculated with the dynamic internal timestep (using in the HSPICE file '.options itrprt' to generate a report) was 2051 samples for the pulsing simple heart model. The same simulation with the default settings was run on the four-source model, producing 638 samples, so the cycle-averaged model is more computationally efficient in this respect by a factor of 3.2. The number of samples and corresponding CPU time measured for each simulation in HSPICE is shown in Table 4.1. Note that the averaged model also shows a 7-fold advantage in simulation time here. Pulsing model 4-source model Samples 2051 638 CPU time (seconds) 1.46 0.18 Table 4.1: Samples and CPU time for simulations in HSPICE using default internal timesteps (30s transient analysis). 4.2.3 Larger Timesteps We next attempted to change the timestep options in order to force HSPICE to take larger internal timesteps. This was done by changing the variables slopetol, absvar, relvar, ft, fs, and rmax in the .options section of the HSPICE file. These control variables designate the amount of timestep reduction and when the timestep should 70 be dynamically altered. The descriptions for these variables can be found in Chapter 7 of the HSPICE manual [1]. The major control factor for setting larger timesteps was the rmax variable, which controls the maximum value of the internal timestep. Increasing the value of 'rmax' causes HSPICE to take larger timesteps, ultimately requiring fewer points for calculation. By setting the options in the HSPICE file to: .options post itrprt vntol=5OmV slopetol=2 absvar=20 relvar=2 ft=0.5 fs=0.7 rmax=20 we ran the transient simulation of the pulsing model with only 1060 samples in time. The same settings were also used on the four-source model, needing only 179 samples, as shown in Table 4.2. With larger timesteps we see an even greater advantage of Pulsing model 4-source model Samples 1060 179 CPU time (seconds) 0.56 0.08 Table 4.2: Samples and CPU time for simulations in HSPICE using large internal timesteps (30s transient analysis). the four-source model in the number of samples required to capture its behavior; the pulsing model required 6 times more samples than the cycle-averaged model. There is again a 7-fold advantage in simulation time as well. The larger timesteps cause a degradation in the simulation accuracy of the pulsing model. To show this, we plotted the voltage responses of both models from simulations with both the default timestep and the larger internal timestep, as shown in Figure 42. The plot of Vh from the pulsing model shows that the larger timestep reduces the resolution of the curves to the point that the simulation no longer captures the original waveforms. However, plotting the < Vh > curve from the four-source model using the larger timestep does not change the waveform shape at all. This shows that the cycle-averaged model can more afford to be simulated with larger timesteps and less calculations than the original pulsing model. 71 12CI - Default Internal Timestep Larger Internal Timestep 10C- 80- 0 600) clc 0 OO 5 10 15 20 time (seconds) 25 30 Figure 4-2: HSPICE simulations showing Vh from the pulsing model and < Vh > from teh averaged model, using both the default internal timestep settings and larger internal timesteps. 72 In conclusion, to preserve the accuracy of the waveforms, the pulsing model would need to be simulated with parameters roughly equal to the default values, whereas the four-source cycle-averaged model can be simulated with the larger timesteps without discernible loss of resolution. In view of this, and comparing the first row of Table 4.1 with the second row of Table 4.2, we see that the averaged model is a factor of 11 (reduction in the number of samples needed for a faithful simulation) to 18 (speedup in CPU time) times faster than the pulsing model. 73 74 Chapter 5 Modeling the Arterial Baroreceptor Reflex The full hemodynamic model, developed by Heldt et al., incorporates the arterial baroreceptor reflex of the body [3]. The baroreceptor reflex is a neural reflex that maintains arterial blood pressure homeostasis by dynamically adjusting the body's heart rate, peripheral resistance, systolic cardiac contractility, and venous zero-pressure filling volume. For example, if the arterial blood pressure dropped, the baroreceptor reflex would induce an increase in heart rate, an increase in cardiac contractility, arteriolar constriction to increase the peripheral resistance, and an increase in venous constriction in order to resist the drop in blood pressure [5]. In this chapter we will use the cycle-averaged four-source model to explore how to control arterial pressure represented by < V2 >. We will study the effects of controlling the peripheral resistance R 3 , the cycle period T, and the venous zero-pressure filling volume V. 5.1 Feedback Control with Venous Zero-Pressure Filling Volume In the body, blood vessels have the ability to hold a certain amount of blood before any pressure is exerted on the vessel walls. The circuit analogy would be a capacitor 75 that can hold charge, yet have zero voltage across it. We therefore modeled the venous zero-pressure filling volume as a (variable) voltage source V in series with the venous capacitance Co, as shown in Figure 5-1. Essentially, V1 allows us to directly adjust the venous voltage < V > in the system. We must be careful with implementing V in different situations, since a negative venous pressure can never physically occur. R3 <i3> R1 R2 <V><i1> <Vh> < 2> CCv VS + - Is Va Ceila C nVq - Vs = 0.732510 <V1> + 0.01 06996<V2> Is = <i1> Ia = <2> Va = -0.39858<V1> + <Vh> + 0.65597<V2> Ri = 0.03 Q R2 = 0.01 Q R3 = 1 Q Cv = 100 F 2F Ceft = 1.11 F Td = 0.66 s T= 1 s Ca = Figure 5-1: Four-Source Cycle-Averaged Model with V1. The four-source model responds to a step up in R 3 with an increase in the voltage < V2 >. If we decrease the value of < V > by the change in the value of the arterial pressure from the steady state condition, we can counteract the increase in voltage. Using this idea, an equation for V1 can be written: Vq = k(V2- < V2 >). (5.1) V2 represents the desired steady-state arterial voltage, and < V2 > is the average value of the arterial voltage at a particular time. Therefore, if the value of < V2 > deviates from the steady-state value, V1 will become non-zero. An increase in the arterial voltage causes V to become negative, which brings down the venous voltage 76 < V1 >. Decreasing < V > should also bring down the value of < V2 >, the arterial voltage we want to control. The new model with a voltage-dependent source V was simulated in HSPICE using Equation 5.1 with a gain of k = 1. Since the steady-state value of < V2 > is an important part of the definition of the V1 source, the steady-state values of all voltages were determined and used as the initial conditions: < V1 >initial < Vh >initial= < V2 >initial = = 5.33V (5.2) 15.86V (5.3) 38.93V (5.4) Therefore, the source V1 is dependent on the nodal voltage < V2 > and its steadystate value 38.93V. The response produced with this feedback model is shown in Figure 5-2. This model is capable of counteracting the transient increase in average 80 Original 4 Source Model -Model <V2> from Original Model with Vq Control 7060- 50<V2> with Vq Control 40 0! a) 30 24 <Vh> with Vq Control 10 --------------------------------I 0 5 10 <V1> with Va Control, 25 20 15 time (seconds) 30 Figure 5-2: The response of the four-source model with controlled Vlj. 77 arterial voltage < V2 >, after the step change in R 3 . Although the < V2 > curve was pulled close to the pre-transient voltage level, it was not pulled all the way back to the steady-state value of 38.93. To achieve complete regulation, one would have to use some form of integral control, rather than simply the proportional control in Equation 5.1. The values in Table 5.1 more quantitatively illustrate the difference in steady-state voltage values from before to after the transient for the model with feedback control in V4. The code for the HSPICE simulation is shown in Appendix A. The difference between the steady-state values of < V2 > from before to after the Voltage No V4 Control Before Transient 5.33V 15.86V 38.96V < V1 > < Vh > < V2 > No V1 Control After Transient 4.51V 27.30V 73.72V V With V Control Before Transient 5.33V 15.85V 38.93V -. 0035V With V4 Control After Transient 2.54V 15.43V 41.67V -2.74V Table 5.1: Voltage values at steady-state of the four-source model with and without V4 control. transient is about 2.74V, which represents a relative error of approximately 7.0%, compared with an error of 34.76V, or 89.2%, without any control. 5.2 Control with the Period T In the cardiovascular system, increasing the heart rate produces an increase in the arterial blood pressure. Heart rate is inversely proportional to the period T of the heart cycle. Therefore, an increase in the value of T should lead to a decrease in the arterial blood pressure. In order to study a system that uses T to control < V2 >, we first look at the algebraic expressions for the sources V, and V in the four-source model, which can be derived from Equations 3.14, 3.15, and 3.38: Vs= < qVh>+ <q'V1 > (5.5) 78 = Va <V > ) (e I + < - v 2 > (CRI- (5.6) < qV2 > + < q'Vh > (5.7) qV2 > + < Vh> - < qV> S< (5.8) <V > + < Vh > + - 1 - - < V2 > - 1 T+ (5.9) . (5.10) The equations show that V, and V do not depend linearly on the period T of the heart cycle. In order to get a linear equation, we use the second-order Taylor series approximation for e pe RlCd in order to get: aT ____ e RCd '1 - RlCd aT 2 \R1Cd +1 2 (5.11) We can now redefine the sources V and Va to be linear functions of T using Equation 5.11: Vs ~~ <V> Va <V 1 > 1-a+ ±a2T --2T < V2C> 2R1Cd) 2RCd)<V> < V2> (-aC Cd 8 TC + a 22C (5.12) 2R1Cd O-C,+ Cd a 2 C2 (5.13) 2R1Cd Using the component values from the pulsing model, the sources evaluate numerically to: < V1 > (0.33333 + 0.74074T)- < V2 > (-0.02667 + 0.02963T) (5.14) Vs Va S< V1 > (-0.74074T)+ < Vh > + < V2 > (0.64 + 0.02963T). (5.15) To build a system that controls the arterial voltage < V2 >, the period T should depend on the steady-state value V2, the value of the voltage < V2 >, and the steadystate value T = is of the period before the step change in R 3 . Since increasing the value of T should decrease the arterial voltage < V2 >, as expected in the arterial 79 baroreceptor reflex, we write the relation for T as: T = T+k(<V > -V 2 ) (5.16) = 1+k(<V2 > -V 2 ). (5.17) 2 For large values of the arterial voltage < V2 >, the period T will increase and attempt to counteract the voltage increase. Implementing the feedback model with T in HSPICE is more complicated than for the VK case. Now the sources V, and V depend upon the value of T and the values of the voltages < V >, < Vh >, and < V2 >. The HSPICE code is shown and described in Appendix A. The overall model looks like the original four-source model in Figure 3-14, but with the sources V, and V as defined in Equations 5.14 and 5.15. The response of the four-source model with feedback control using T, with a gain of unity and the initial conditions in Equations 5.2, 5.3, and 5.4, is shown in Figure 5-3. The model is capable of pulling down the < V2 > curve close to the desired steady-state pre-transient level. The quantitative steady-state values before and after the transient are shown in Table 5.2. Comparing the steady-state values to < V1 > < Vh > < V2 > T With T Control Before Transient 5.34V 15.73V 38.48V 0.548s With T Control After Transient 5.33V 16.34V 38.76V 0.829s Table 5.2: Steady-state voltage values of the four-source model with controlled T. the values from the V1 feedback model, we see that the model which uses T brings the value of < V2 > closer to the desired steady-state value of 38.93V. The change in < V2 > from before to after the transient is now only 0.28V or 0.7%. Therefore, the period T does a better job in pulling down the arterial voltage, for the particular choices we made for feedback gains; other choices may lead to a different picture. Heart rate in the cardiovascular system is also one of the more powerful factors in 80 80Original 4 Source Model - - <V2> from Original Model Model with T control 70- 60- 50- 40 - <V2> with T Control 4 co 0 30 20 <Vh> with T Control - 10 <V1> with T Control OC 5 15 10 20 25 30 time (seconds) Figure 5-3: The response of the four-source model with controlled period T. 81 controlling the arterial pressure. 5.3 Control with R 3 Peripheral resistance is another controlling factor in the arterial baroreceptor reflex. All previous simulations have derived transient responses from a step increase in the peripheral resistance R 3 . In order to study the effects of R 3 as a control for the voltage < V2 >, we must produce another type of transient response. The model shown in Figure 5-1 was used for simulation, except a step increase in V from OV to 2V at 10 seconds replaced the step in R 3 as the initiating disturbance. This step increase in V is analogous to an external compression of the veins in the cardiovascular system. The response of the four-source model to the increase in V is shown in Figure 5-4 as the dotted line. Lowering the peripheral resistance should lower the arterial blood pressure in the cardiovascular system. Therfore, the relation for R 3 can be written as: R3 caic = R 3 + k(V 2 - < V2 >) = 1+k( 2- < V2 >) (5.18) (5.19) where the steady state value of R 3 before the transient is 1Q. However, Equation 5.19 shows that for large values of < V2 >, the resistance will become negative, a situation that cannot physically occur. In order to prevent a negative resistance, the value of R 3 is set to be the maximum value of either zero or R3caic: R3 = max(0, Rcaic) (5.20) Using this value for R 3 , the feedback model was simulated in HSPICE with a step in V as described above. The code for the model is shown in Appendix A. The voltage responses of the controlled R 3 model are shown as the solid lines in Figure 5-4. The quantitative steady-state voltage values before and after the transient are depicted in Table 5.3. Again, the feedback model successfully counteracts the increase in 82 55. I I I Original 4 source model with R3 control -Model <V2> from I Original Model 5045<V2> with R3 Control 40 0 as 0) 3530- - 25- 20- ~ / <Vii> with R3 Control 154 10 <Vi> with R3 Control 0O 5 10 20 15 25 30 time (seconds) Figure 5-4: The Four-Source Model response with controlled period R 3 due to a step in V. Voltage < V1 > < Vh> < V2 > No R 3 Control Before Transient 5.33V 15.86V 38.96V No R 3 Control After Transient 7.03V 20.91V 51.35V R3 With R 3 Control Before Transient 5.33V 15.85V 38.93V 0.999Q With R 3 Control After Transient 7.31V 16.95V 39.31V 0.625Q Table 5.3: Steady-state voltage values of the four-source model with controlled R 3 . 83 arterial voltage < V2 >. The change in < V2 > due to the transient is now 0.38V, or 1.0%, compared to an error of 12.39V, or 31.8%, without any control. 5.4 The Feedback Models In general, the model using period T to control changes in voltage does the best job at pulling down the value of the arterial voltage < V2 > back to the desired value. The feedback models described in this chapter will also function to counteract a drop in arterial voltage. However, the model for controlling T must be altered to ensure that a negative period does not occur for small values of < V2 >. A falling arterial voltage < V2 > can be induced by a step decrease in either R 3 or Vq. The next step in studying the arterial baroreceptor reflex involves putting all of the controlled models together so they act cooperatively instead of independently. The models described also acted immediately to counteract any changes in the arterial voltage < V2 > from the steady-state values, whereas in the cardiovascular system each factor actually acts with a certain time delay. These delays can be modeled by changing the gain k in the control equations or by adding delay elements or, more simply, first order lags to the system. 84 Chapter 6 Conclusion We began by studying the characteristics of the time-varying capacitor in simple RC circuits, subsequently simulating the simple pulsing heart model in HSPICE. The response of our heart model includes a pulsing voltage Vh across the time-varying capacitor, with very high and sharp peaks. We approximated the time-varying capacitance waveform to a periodic square-pulse wave. Physiologically, the elastance cycle of the heart is more curved. Further studies should look at the responses of the simple heart model to a curved capacitance waveform. Theoretically, a less abrupt change in the capacitance should lead to less abrupt changes in the Vh curve. Next, we derived two cycle-averaged models from the pulsing model. The simulation of the cycle-averaged models produced voltage responses < Vh > and < V2 > that are consistently greater than the calculated cycle-averaged values. The causes of the offset of the cycle-averaged model responses are the approximations made during the model derivation. The three-source model approximates the current < ii > as having decayed to zero by the end of diastole, and also sets < qV >~ a < V >. Both of these approximations are corrected for in the derivation of the four-source model. The four-source model avoids both approximations by defining the value of < qVh > as the integral in Equation 3.38. The simulations showed that correcting these approximations improved the responses of the four-source model over the three-source model. Further improvements to other approximations should move the voltage responses of the cycle-averaged model closer to the calculated values. Specifically, both 85 the four-source model and the three-source model approximate V2 to be a relatively low-ripple signal, allowing the approximation < qV2 > z- a < V2 >. Looking at the pulsing model responses in Figure 2-6, we see that approximating V2 as a low-ripple signal may not be very accurate, and may be one of the main sources of error. Therefore, using a more accurate expression for < qV 2 > and Vh8 y, should improve the cycle-averaged model responses. The other more reasonable approximation involved estimating that V is relatively constant in time compared to Vh and V2 , allowing us to use the approximation < q'V >- a' < V >. One major characteristic of the cycle-averaged model responses is the exponential increase from the initial conditions at the beginning of every simulation. This exponential is characterized by the eigenvalues and eigenvectors of the state-space equations describing the model. The eigenvector corresponding to the eigenvalue at zero describes the relative average nodal voltages at steady state. However, the values in this eigenvector do not match the relative calculated cycle-averaged voltages due to the approximations made when deriving the cycle-averaged model. We did preliminary studies in choosing specific initial conditions to avoid the exponential increase. The cycle-averaged four-source model was capable of increasing the computational efficiency of the simulations by a factor of 11-18 times over the pulsing model. The HSPICE internal timestep parameters should be further analyzed to more completely characterize this increase in computational efficiency. We saw that increasing the internal timestep caused a significant (and probably unacceptable) decrease in the resolution of the voltage responses in the pulsing model, but not in the cycle-averaged model. The timestep limits should also be studied further. The cycle-averaged model was used to develop feedback models for the arterial baroreceptor reflex in the body. Three variables, the period T, the zero-pressure filling volume V, and the peripheral resistance R 3 , were each used to control the arterial voltage < V2 >. The next step in feedback analysis involves designing a system that incorporates all of these feedback loops with the appropriate delays. We mostly analyzed model responses to a step increase in the peripheral resistance 86 R 3 . Further analysis should be done to study transient responses from a variety of perturbations, including changes in R 2 and Cs, both of which are possible perturbations in the cardiovascular system. Other types of perturbations such as ramps or exponential increases or decreases of these system parameters can also be explored. HSPICE, a circuit simulation tool, was used to simulate all of the models. We demonstrated the ability of HSPICE to simulate various cardiovascular models, including the pulsing model, the two cycle-averaged models, and a group of feedback models. HSPICE is both powerful and easy to use as shown by the relatively short pieces of code required to simulate each model. It is also capable of simulating a variety of system components relevent to cardiovascular modeling and control. Overall, we designed two cycle-averaged circuit models that were capable of capturing the time-averaged responses of a simple pulsing heart model. The three-source time-averaged model was easily characterized by defining the state-space equations and studying the eigenvalues and eigenvectors. The four-source model was more computationally efficient than the pulsing model by a factor of 11-18, providing a strong motivation for using the cycle-averaged model in extensive simulations. The four-source model was also used to design and study three separate feedback loops using the period T of the heart cycle, the zero-pressure filling volume V, and the peripheral resistance R 3 to control the arterial voltage < V2 >. These feedback models successfully counteracted transient increases in the arterial voltage < V2 >. The next major step in these studies should involve expanding all models to match the full hemodynamic model. The hemodynamic model designed by Heldt et al. [3] can first be simulated in HSPICE to use the advantages of this circuit simulation tool. Then a cycle-averaged version of the entire hemodynamic model can be designed using the methods described in this thesis for deriving the three-source and four-source models of a simple, single ventricle heart. Then the cycle-averaged model can be used to develop a feedback model that incorporates all of the feedback loops in the arterial baroreceptor reflex. The responses to each of these expanded models can then be matched to the physiological cardiovascular system responses. Taking advantage of the computational efficiency of cycle-averaged models, a full version of the cycle87 averaged feedback model can be used to study the response of the cardiovascular system to various situations and conditions. 88 Appendix A HSPICE Code A.1 The Cycle-Averaged Three-Source Model Three-Source Model .options post .op *** Initial Conditions (Same as for Pulsing Model) on V1, Vh, and V2: .ic V(n2)=5.56 V(n4)=1.45 V(n6)=35.4 *** Voltage source that defines the step in R3 at 10s from 1 to 5 Ohms Vtran nr 0 pulse(1 5 10 lu lu 20 30) Rgrd2 nr 0 1meg *** Circuit Connections: Cv n2 0 C = 100 R3 n2 n6 R = 'V(nr)' *** Dependent current source Is = 10<V1> - 0.4<V2> Gv n2 n4 POLY(2) n2 0 n6 0 0 10 -0.4 C1 n4 0 C = 1.111 **Effective Capacitance R2 n5 n6a R = 0.01 Vbmeas n6a n6 DC 0 **Dummy voltage source to define i2 *** Dependent voltage source Va = <Vh> + 0.6667<V1> - 0.6667<V2> Ea n5 0 POLY(3) n4 0 n6 0 n2 0 0 1 0.6667 -0.6667 *** Dependent current source Ia depends on the current i2 through Vbmeas 89 Fa n4 0 Vbmeas 1 C = 2 Ca n6 0 *** Transient analysis .tran 0.01s 30s UIC .plot tran V(n4) .end A.2 The Cycle-Averaged Four-Source Model Four-Source Model .options post .op *** Initial Conditions (Same as for Pulsing Model) on V1, Vh, and V2: .ic V(n2)=5.56 V(n4)=1.45 V(n6)=35.4 *** Voltage source that defines the step in R3 at 10s from 1 to 5 Ohms Vtran nr 0 pulse(1 5 10 lu lu 20 30) Rgrd2 nr 0 1meg *** Circuit Connections: Cv n2 0 C 100 R3 n2 n6 R = = 'V(nr)' R1 n2a n2b 0.03 Vameas n2 n2a DC 0 ***Dummy source to define ii *** Dependent current source Is equal to <ii> through Vameas Fs 0 n4 Vameas 1 *** Dependent voltage source Vs = 0.732510<V1> + 0.01069958<V2> Es n2b 0 POLY(2) n2 0 n6 0 0 0.732510 0.01069958 C1 n4 0 C = 1.111 **Effective Capacitance R2 n5 n6a R = 0.01 Vbmeas n6a n6 DC 0 **Dummy voltage source to define i2 *** Dependent voltage source Va = -0.399177<V1> + <Vh> +0.65597<V2> Ea n5 0 POLY(3) n4 0 n6 0 n2 0 0 1 0.65597 -0.399177 90 *** Dependent current source Ia equal to <i2> through Vbmeas Fa n4 0 Vbmeas 1 Ca n6 0 *** C = 2 Transient analysis .tran 0.01s 30s UIC *Use initial conditions .plot tran V(n4) .end A.3 Feedback Models (Based on the Four-Source Model) A.3.1 Control with V Zero-pressure filling volume control .options post *** Initial Conditions .op (Start off in Steady-State) .ic V(n2)=5.33 V2ss = 38.93V V(n4)=15.86 V(n6)=38.93 Vtran nr 0 pulse(1 5 10 lu lu 20 30) Rgrd2 nr 0 1meg *** Circuit Connections: Cv n2 nv C = 100 **Connected in series with the voltage source Eq *** feedback control using Vq = V2ss - V2 Eq nv 0 POLY(1) n6 0 38.93 -1 R3 n2 n6 R = 'V(nr)' R1 n2a n2b 0.03 *** Dependent sources same as in four-source model Vameas n2 n2a DC 0 Fs 0 n4 Vameas 1 Es n2b 0 POLY(2) n2 0 n6 0 0 0.732510 0.01069958 C1 n4 0 C = 1.111 91 R2 n5 n6a R = 0.01 Vbmeas n6a n6 DC 0 Ea n5 0 POLY(3) n4 0 n6 0 n2 0 0 1 0.65597 -0.39858 Fa n4 0 Vbmeas 1 C = 2 Ca n6 0 .tran 0.Ols 30s UIC .plot tran V(nv) .end A.3.2 Control with T Feedback model with control in T .options post .op ***Initial Conditions based on Steady State Values V2ss=38.93V .ic V(n2)=5.33 V(n4)=15.86 V(n6)=38.93 ***Voltage Source for T = 1 + (V2 - V2ss) Et nt 0 POLY(1) n6 0 -37.93 1 ***Voltage Sources that depend on the value of T (used in defining Va and Vs) Eti nkl 0 POLY(1) nt 0 0 -.74074 Et2 nk2 0 POLY(1) nt 0 .64 .02963 Et3 nk3 0 POLY(1) nt 0 .333 .74074 Et4 nk4 0 POLY(1) nt 0 .026667 -0.02963 ***Step change in R3 Vtran nr 0 pulse(1 5 10 lu lu 20 30) Rgrd2 nr 0 1meg ***Circuit Connections Cv n2 0 C R3 n2 n6 R 100 = = 'V(nr)' R1 n2a n2b 0.03 Vameas n2 n2a DC 0 Fs 0 n4 Vameas 1 92 ***The source Vs depends of the voltages at nodes nk3 and nk4 defined above. Es n2b 0 POLY(2) n2 0 n6 0 0 'V(nk3)' 'V(nk4)' Ceff n4 0 C = 1.111 R2 n5 n6a R = 0.01 Vbmeas n6a n6 DC 0 *** The voltage source Va depends on the values at nodes nkl and nk2 Ea n5 0 POLY(3) n2 0 n4 0 n6 0 0 'V(nkl)' 1 'V(nk2)' Fa n4 0 Vbmeas 1 Ca n6 0 *** C = 2 Transient simulation .tran 0.01s 30s UIC .plot tran V(nt) .end A.3.3 Control with R3 Feedback model with control in R3 using a step perturbation in Vq .options post *** .op Initial Conditions in steady-state V2ss = 38.93V .ic V(n2)=5.33 V(n4)=15.86 V(n6)=38.93 *** Step up in Vq from OV to 2V at 10s Vq nv 0 pulse(0 2 10 lu lu 20 30) Rgrd2 nv 0 1meg *** Control Source R3 = 1 + V2ss - V2 Etran nr 0 POLY(1) n6 0 39.93 -1 *** Circuit Connections Cv n2 nv C = 100 ** Cv in series with Vq source *** Make sure R3 never goes negative (depends on the control source) R3 n2 n6 R = 'max(V(nr),0)' R1 n2a n2b 0.03 Vameas n2 n2a DC 0 93 Fs 0 n4 Vameas 1 Es n2b 0 POLY(2) n2 0 n6 0 0 0.732510 0.01069958 C1 n4 0 C = 1.111 R2 n5 n6a R = 0.01 Vbmeas n6a n6 DC 0 Ea n5 0 POLY(3) n4 0 n6 0 n2 0 0 1 0.65597 -0.39858 Fa n4 0 Vbmeas 1 Ca n6 0 *** C = 2 Transient Analysis .tran 0.Ols 30s UIC .plot tran V(nr) .end 94 Appendix B New Approximations for < qV 2 > and Vhsys The derivation of the cycle-averaged models depended on the assumption that V2 has low ripple, which is an approximation that can be improved. Correcting for the approximations of < qV 2 >~ a < V2 > and Vhsy, ~< V2 > should help reduce the error of the cycle-averaged model seen in Figures 3-12 and 3-13. In order to find a better approximation, we first examine the V2 curve from the pulsing model. One period T of V2 is drawn in Figure B-1 with T, equal to the systolic period and Td equal to the diastolic period. The values Viit, VhSys, < V 2 >, and V2 denote important points in the waveform. In Chapter 2 we explained that V2 decreased with a time constant of about r = Ca * R 3 = 2s during both systole and diastole as Ca discharges through R 3 . Since 2 decreases slowly compared to the period of T = Is, the curve in Figure B-1 can be approximated as a line with a slope of m = -Vn where Vinit is the value of V2 at the beginning of systole and < V2 > is the cycle-averaged (midpoint) value of the V2 line. B.1 Approximation for Vhy, The derivation in Section 3.5.1 denotes VSy, as the value of Vh at the end of systole. This point also equals the value of V2 at the end of systole as shown in Figure 3-10. 95 Vinit <V2> Vhsys V2d Ts systole -T/2 Td diastole 0 T/2 Figure B-1: One period of V2 from the pulsing model . 96 Figure B-1 shows that VSy, is actually greater than < V2 >. In order to derive an expression for Vsys,, we use the slope m and the systolic, diastolic, and total periods to write: Vinit(Td Vhsys (B.1) + < V2 > R 3 Ca The value of Vinit can be written in terms of < V2 >: Vinit < V2 > 1- = (.2 (B.2) T 2R 3 Ca Combining Equations B.1 and B.2 with the circuit values defined in the pulsing model of Figure 2-5 produces: ) R - < V2 > Vsys R 3 Ca - 2 1.111 < V2 > . (B.3) V = B.2 (B.4) Approximation for < qV 2 > Using the time-average integral from Equation 3.1, and the variables in Figure B-1, < qV2> can be approximated with < qV2 > = 1 T -J T T - V2d * 2 Td T' (B.6) , where V2d is the value of V2 at the midpoint of the diastolic period V2d (B.5) (T)dT d_- 2 Td. The value of can be determined using the slope of the line giving us: V2 = S< < V2 > V2 > _ (Ts + R 97 Td 2 - Z: )Vinit 2 R 3 Ca 3 -. R -Ca2 2a ) (B.7) (B.8) This leads to a new expression for < qV 2 > which evaluates numerically to: < qV2 > = < V2 > 2 R3Ca 2 T = 0.5926 < V2 > B.3 (B.9) (B.10) A New Model We can now place the new approximations for < qV 2 > and Vhsy, from Equations B.4 and B.10 into the three-source model, producing new expressions for the sources I, and Va, which can be written as: VI > -Cs1.111 < V2 > T 10 < V 1 > -0.444 < V2 > (B.11) < qV 2 > + < Vh> -a < V > (B.13) 0.5926 < V 2 > + < Vh> -0.66 < V > (B.14) Cd < Va (B.12) The solid curves in Figure B-2 are the responses < Vh > and < V2 > from the simulation of the three-source model with the dependent sources redefined as in Equations B.11 and B.14. These curves are compared with the calculated values of < Vh > and < V2 > from the pulsing model and the responses from the original three-source model in Chapter 3. We see that < V2 > from the new version of the three-source model moves down towards the calculated time-averaged waveform. However, < Vh > from the new model moves up away from the calculated timeaveraged waveform. After the transient, the steady-state value of < V2 > is now 7.79V (12.28%) greater than the calculated value of < V2 > at 63.44V, and the steady-sate value of < Vh > is now 7.92V (32.98%) greater than its calculated value of 24.236V. Using the same new approximations in the four-source model produces similar results. The correction for the approximations used in deriving the cycle-averaged models still needs to be studied further. 98 80 - Calculated Average - 3-Source Model - New Model 70- 7 2> from model with new approxi ation -Calculated <V2> 60- 50- ~ 40 - <Vh> from model with new approximatiod ---- 0 5 10 15 20 25 30 Time (seconds) Figure B-2: Responses of the three-source model using the new approximations for <CqV 2 and Vhsys t> 99 100 Appendix C Eigenvalues of the Four-Source Model Here, we use the same techniques shown in Chapter 4 to characterize the behavior of the four-source model. We begin by writing the algebraic expression for the voltage sources V, and V in the model by combining the approximation for < qV > found in Equation 3.38 with the equations for < Vi > and < V > in Equations 3.14 and 3.15. Vs= < qVh> + < q'V1 > v > (1 + S< Va = RT (e (C.1) < V 2 >(C RCd < > (- > / CsRi (a + C T (e RjC + < Vh > + RCd1 (C.5) aT R (C.6) 1Cd Now we can define the currents < i1 >, , and < i3 > in the model to be: < Zi > 2) (C.4) qV > + < Vh> - < qVh> S<V1 1)(C. (C.3) < qV 2 > + < q'Vh > -< (e RCd- = R> Ri 101 s(C.7) Va- < V2 > (C.8) R2 < V2 > - < VI > < i3 > (C.9) R3 Finally, the state-space equations of the currents through the capacitors can be defined in terms of the currents < i1 >, < i2 >, and < i3 >: Cv<v1 > = Ceff< Vh > -< -Is - Ia- Ca< V2 > = (C.10) il > < il > -< (C.11) < i2 > (C.12) i3 > By combining all of the above equations, we can now write the state-space equations such that they depend only on < Vi>, < Vh >, and < V 2 >. Placing these in matrix form looks like: < 1 > <y> < 11 > < Vh > < < 2 > 2 (C.13) > MyT Matrix A can be written by first defining ~3=(eRjCd-1) A=I Cd _ 1 TC, R 3 Cv aR1T+R Cd/3-CR1 R 2/8 TR1 R 2 Cef I -1 R 2 Ce1 1 R 2 T-aR3 T-R 1 RSCd3 TR 2 R3 Ca 1 R 2 Ca 1 0 -OC 8 R 3 C, TC, R1T-R1aT+CR1R 0-C R? 2 TR1 R 2 Ce5! aR 3 T-R 3 T-R1T+CsR 1 R 3 3 TR 2 R3 Ca (C. 14) Using the values for the components described in the simple-heart model in Figure 2-5, the eigenvalues and eigenvectors of A were calculated in MATLAB to be: -107.2188 A = (C.15) 0 -0.6721 102 V - 0.0001 0.1257 -0.0225 0.8731 0.3742 0.3145 -0.4876 0.9188 0.9490 (C.16) The three eigenvalues are each inversely proportional to the negatives of the time constants of an exponential function. These eigenvalues are very close to the ones determined for the three-source model. Therefore, both cycle-averaged models behave similarly. 103 104 Appendix D Further Simplification based on Slow/Fast Decomposition Using methods from singular perturbation theory [2], we will derive a simplifiction for the expressions of the eigenvalues from Chapter 4. Given a state-space description of the form ± = Ax, suppose the state variables in the vector x can be separated into a slowly varying set x, and the remaining rapidly varying set xf. [ A1 if A2 XS A3 A 4 Xf (D.1) Using Equation D.1, we can expand the equations for ai, and if: 3s = A 1 x, + A 2 xf (D.2) if = A 3 x,+ A 4 xf. (D.3) Since Xf is a signal with a very fast transient, if will be approximately zero after a short transient interval of time. Consequently, xf can be written simply in terms of x, using Equation D.3: Xf(t) ~ -A4 1A 3 xs(t) for t > short transient interval 105 (DA) Now Equation D.2 can be expanded to get an expression for 3s in terms of x,: A 2 A4 1A3x, (D.5) - A 2 A4'A 3 )xs (D.6) AIx, 1 -(A - Going back to matrix A in Equation 4.12, we first notice that the fast eigenvalue A1 , from Equation 4.13, is approximately equal to the value of -90.09 in the A matrix. This makes sense because Vh is the most rapidly varying signal in all of our simulations. The value of -90.09 corresponds to -1 f R2Ce in the matrix from Equation 4.11. Therefore, we get the first approximation for the fast eigenvalue A ast: (D.7) -1 Afast R 2 Ceff Next we can reorder the variables in Equation 4.10 so that < Vh > represents the fast represents the slow signal x,. signal xf and the vector < V2 > <y 1 > <Y12 > < V2 > <Yi> <Vh> A1, A2 A, A4 A (D.8) (D.9) The matrices A 1 , A 2 , A 3 , and A 4 are defined using Equation 4.11: -1 A1, A3 = C R 3 Cv TC, 1 __ R 3 Ca a R 2 C. + Cd TCeff a R 2 Ce f 1 R3 C, C" + TC, 1 a-1_ R 2 Ca - R 3 Ca C a-i TCejf Evaluating the matrix S = A1 - A 2 A I R2Cef5 'A 3 106 0 A2 A4 (D.10) = = -1 I R2Cef5 (D.11) from Equation D.6 by substitution from Equations D.10 and D.11, we get: S = A1 A 2 A 'A 3 - +Qs -_ _C 1 R 3 Cu TC, R 3 C, TC _C _ -1 R3Ca C, 1 _ CaT _R3Ca . (D.12) CaT] From Equation D.6 we also get an expression for the slow signals: V S = < V> (D.13) < V2 > 2 The eigenvalues of this S matrix include one eigenvalue at zero and another at Aslow. Since the sum of the eigenvalues of S equals its trace, we can easily solve for the value of Aslow: Aslow tr(S) = - R3Cv (D.14) + R3Ca + TCv + CaT . (D.15) Using the values from the simple heart model, this slow eigenvalue evaluates to -0.81, which approximately matches the third eigenvalue in Equation 4.13. Furthermore, by using the approximation in Equation D.4 and matching the fast signal to < Vh >, we can write an expression for < V > in terms of the slower signals < V1 > and < V2 >: < Vh > = -A4jA 3 [<w>] (D.16) L< V2 > R2Cv = R2C) + a) < V > -(R + (a - 1)) < V2 > . (D. 17) This suggests that the three-source model characterized with Equation 4.11 can be written using two state variables < V > and < V2 >, instead of three state variables. For this, we simply replace the capacitor Ceff in the three-source model with a dependent voltage source Vheart which is defined by Equation D.17. The resulting model is shown in Figure D-1, and the simulation of this model is shown in Figure D-2. The 107 response behaves similarly to the three-source model, but exhibits greater error from the calculated time-averaged curve. <i3> "M I <Vi> <i1> R2 IS<Vh> <12> 1 + Cv Vhear - <V2> - Ca Va ~~T I Is= T Va = <V2> + <Vh> - a<V1> Vh = (R2Cd/T + u )<Vi> - (CsR2/T + (cL - 1))<V2> (Cd<V1> - Cs<V2>)/ R2 = 0.01 Q R3 = 1 - 5 Q (Step at 1 Oseconds) Cv = 100 F Ca = 2 F Cd = 1OF Cs =0.4F a 0.66 T= 1 s Figure D-1: Cycle-averaged model with source 108 Vheart. - 80 - Calculated Time-Average 3-Source Model <V2> from Model with Vheart Model with Vheart 8;- - - --0 -- '<V2> from 3-Source Model 70alculated <V2> 60- :=0 5040 -- - 20 Calculated <Vh> 10 - 101 S5 10 15 20 25 30 Time (Seconds) Figure D-2: Response of the cycle-averaged model with source Vheart 109 110 Bibliography [1] Star-Hspice Manual. http://www.ee.washington.edu/class/cadta/hspice/, 2 edi- tion, July 1998. [2] Vahe A. Caliskan, George C. Verghese, and Aleksandar M. Stankovic. Multifrequency averaging of DC/DC converters. IEEE Transactionson Power Electronics, 14(1):124-133, January 1999. [3] Thomas Heldt, Eun B. Shim, Roger D. Kamm, and Roger G. Mark. Computational modeling of cardiovascular response to orthostatic stress. Journalof Applied Physiology, 92:1239 - 1254, March 2002. [4] John G. Kassakian, Martin F. Schlecht, and George C. Verghese. Principles of Power Electronics, chapter 11. Addison-Wesley, 1991. [5] Arthur J. Vander, James H. Sherman, and Dorothy S. Luciano. Human Physiology: The Mechanisms of Body Function. McGraw Hill, 1990. 111

Cycle-Averaged Models of Cardiovascular Dynamics

Related documents

Products

Support

Cycle-Averaged Models of Cardiovascular Dynamics

Related documents

Add this document to collection(s)

Add this document to saved

Suggest us how to improve StudyLib