Modeling cardiovascular autoregulation of the preterm infant

Transcription

1 Modeling cardiovascular autoregulation of the preterm infant Marco Dat BMTE 1.18 November 8th 21 ID nr: Faculty of medical engineering Department of cardiovascular biomechanics Supervisors: ir. W. Jennekens (TU/e, MMC) dr.ir. P.H.M. Bovendeerd (TU/e) dr.ir. N.A.W. van Riel (TU/e) dr. P. Andriessen (MMC) Prof.dr.ir. P.F.F. Wijn (TU/e, MMC) Prof.dr.ir. F.N. v.d. Vosse (TU/e)

2 2 September 8, 21

3 i Abstract In the Netherlands, approximately 1% of all live births are preterm (gestational age (GA)<37 weeks). Of these preterm infants, 2.2 are born very prematurely (GA<32 weeks). Approximately 2% of infants of very low birth weight (<15 g) or infants of less than 35 weeks GA have a high grade type of brain hemorrhages. These hemorrhages can be caused by poor cerebral autoregulation, which is mainly dependent on blood pressure control. Blood pressure is regulated by the autonomic nervous system which is capable of changing blood pressure by adapting effectors, e.g. heart period. The autonomic nervous system may be underdeveloped in preterm infants. A measure of the development of the autonomic nervous system can be obtained from frequency analysis of blood pressure and heart period fluctuations. The goal of this research is to build a model of the cardiovascular circulation with autonomic nervous system regulation of a preterm infant of 28 weeks GA and a body weight of 1 gram at the end of the first week and with a closed patent ductus arteriosus. This model should be based on perinatal cardiovascular physiology and be able to reproduce blood pressure and heart period fluctuations caused by autonomic nervous system regulatory mechanisms. This model is used to investigate if and how, it is possible to obtain information about the condition of the autonomic nervous system using the heart period and blood pressure signals. Models described in literature are combined and adjusted to preterm infant dimensions. The resulting model is verified with independent literature. The developed model is capable of simulating the hemodynamics and baroregulation of a preterm infant in homeostasis. The model produces the same frequencies in blood pressure and heart period, with comparable amplitudes as measured in clinical data, which also leads to comparable gains and delays between these frequencies. A simulated atropine administration suggest model refinement.

4

5 i List of abbreviations ANS BP BR BV C CNS CO D E ECG G GA HF HP HR L LF LSR P san p PDA PNS PVR q R RR SBP SV SVR V n V V VLF Autonomic nervous system Blood pressure Baroreflex Blood volume Compliance Central nervous system Cardiac Output Delay Elastance Electrocardiogram Gain Gestational age High frequency Heart period Heart rate Inertance Low frequency Lung stretch receptor Reference of systemic arterial pressure Pressure Patent ductus arteriosus Peripheral nervous system Pulmonary vascular resistance Flow Resistance Interval between R peaks in ECG Systolic blood pressure Stroke volume Systemic vascular resistance Reference value of lung stretch Unstressed volume Volume Very low frequency

6 ii subscripts of used abbreviations abd ao av duc ep ev la lv max min mv pa pp pv ra rv sa sa,e sa,i sp ep sv sv,e sv,i th tv Abdominal Aorta Aortic valve (patent) ductus (arteriosus) Extrasplanchnic periphery Extrasplanchnic vein Left atrium Left ventricle Maximum Minimum Mitral valve Pulmonary artery Pulmonary periphery Pulmonary vein Right atrium Right ventricle Systemic artery Extrathoracic artery Intrathoracic artery Splanchnic periphery Extrasplanchnic periphery Systemic veins Extrathoracic systemic veins Intrathoracic systemic veins Thoracic Tricuspid valve September 8, 21

7 CONTENTS i Contents 1 Introduction 1 2 Physiology The cardiovascular system The autonomic nervous system Physiological basis of cardiovascular rhythms Obtaining information about the ANS from clinical data Rebuilding the Sa Couto model of neonatal hemodynamics Introduction Methods Results & Discussion Rebuilding the Ursino model of adult baroregulation Introduction Methods The cardiovascular model The baroreflex Simulations Results & Discussion Simulated hemorrhage Understanding the origin of model outcome Building the premature neonatal model with baroregulation Introduction Methods The preterm infant cardiovascular model Coupling the baroregulation model to the cardiovascular model New baroreflex parameter values Simulations Results & Discussion Homeostatic outcomes Sensitivity of the model The effect of atropine General discussion 49 7 Conclusion 51

8 ii CONTENTS 8 Recommendations 53 Bibliography 6 A Review of models 61 A.1 Model criteria A.2 Morrison & Bekey model A.3 Seydnejad & Kitney Model A.4 Ursino model A.5 Sa Couto Model A.6 Van Roon model A.7 Consideration of other models A.8 Review summary A.9 Model choice B Parameters of the models 72 C Original results of the atropine test 75 September 8, 21

9 1 Chapter 1 Introduction In the Netherlands, 8% of all live births were preterm (<37 weeks gestational age (GA)) in 21. Of these preterm infants, 2.2 are born very prematurely (<32 weeks GA) and this number is increasing [1]. Preterm birth is a serious health problem. Preterm infants have increased risk of mortality and morbidity related to breathing problems, infection and feeding intolerance. Mortality is strongly related to gestational age. Very preterm infants of weeks GA have a mortality of 72%. This strongly decreases with GA, 2% for weeks GA, 3% for weeks GA and less than 1% for weeks GA [1]. These preterm infants also face an increased risk of mental or motor disabilities, which are related to cerebral hemorrhages or ischemia. Approximately 2% of very low birth weight infants (<15 g) or infants of less than 35 weeks GA have a high grade type of brain hemorrhage [2]. These hemorrhages can be caused by altered cerebral blood flow secondary to poor cerebral autoregulation, systemic hypo- or hypertension [3]. Cerebral perfusion is mainly dependent on blood pressure control, which is regulated by the autonomic nervous system through adjusting effectors, e.g. heart period. It is assumed that abnormal fluctuations are caused by an underdeveloped autonomic nervous system. To gain more insight in the development of the autonomic nervous system of a preterm infant, these fluctuations are studied with frequency analysis. The power density spectra of blood pressure and heart period show two major frequency bands. It is generally assumed that these originate from the regulation by the autonomic nervous system [4]. The goal of this research is to build a model of the cardiovascular circulation with autonomic nervous system regulation of a preterm infant of 28 weeks GA and a body weight of 1 gram at the end of the first week and with a closed patent ductus arteriosus. This model should be based on perinatal cardiovascular physiology and be able to reproduce blood pressure and heart period fluctuations caused by autonomic nervous system regulatory mechanisms. This model is used to investigate if and how, it is possible to obtain information about the condition of the autonomic nervous system using the heart period and blood pressure signals. Outline Before describing the development of this model, the physiology and the baroreflex induced heart period and blood pressure fluctuations are described in more detail in Chapter 2. Different types of models which are available in literature for human blood pressure regulation are investigated, of which an overview is given in appendix A. A good model has to meet certain criteria as a good physiological basis, simple and clear, validation in human or animal studies, the model has to be well described and it must be possible to adjust to neonatal dimensions because no models for preterm infants baroregulation are found in literature. A difference is expected between preterm infants and adults, because of the different preterm infant dimensions and a possible immature autonomic nervous system.

10 2 Introduction Based on the literature review, a model of term hemodynamics is used as starting point for the cardiovascular model. This model is rebuilt to verify correct implementation, which is described in Chapter 3. This model will be combined with a model for adult baroregulation. A description of rebuilding this model is described in Chapter 4. To simulate the preterm cardiovascular system under baroregulation the parameters are adapted to represent the cardiovascular system of a 28 week gestational age infant of 1 week old. This model is combined with the adult baroregulation model, which also needs to be adapted to preterm infant dimensions. The combining and adjustments of these models are described in Chapter 5. This chapter also describes the validation with published clinical data obtained at the neonatal intensive care unit of the Maxima Medical Center (MMC) in Veldhoven. This study concerns the effect of atropine on baroregulation. In Chapter 6 a general discussion is presented, which leads to a conclusion, presented in Chapter 7. In Chapter 8 a number of possibilities for future research are given. September 8, 21

11 3 Chapter 2 Physiology In this section the physiology will be explained, on which the model developed in this study is based. The focus will be on the central nervous system and the baroreceptor reflex, as this is the regulator for blood pressure and heart period on short term. First the hemodynamical interactions in the cardiovascular system are described. 2.1 The cardiovascular system Figure 2.1: Left figure shows an already simplified representation of the human circulation including blood flow to and from a number of important organs. This can be simplified to the right figure, which distinguishes an intrathoracic and extrathoracic compartment. These figures show the systemic and pulmonary circulation, with the arteries in red, veins in blue and the capillaries. Figures adapted from [5]. The cardiovascular circulation in Figure 2.1 consists of a systemic and pulmonary circulation, with the left and right part of the heart, respectively, as driving force for pressure build-up. In both loops, flow from the ventricle enters the large arteries. From here it goes into the periphery of all organs and finally into the venous compartments, which end up in the heart again. Valves prevent flow reversal during contraction. The left figure in Figure 2.1 shows an already simplified representation of the circulation. For

12 4 Physiology the eventual model, the circulation is further simplified into the right figure. All organs are represented by one periphery. The pulmonary circulation and a part of the systemic circulation are intrathoracic. The periphery is extrathoracic, this is where most internal organs are located. The intrathoracic compartments are affected by varying extravascular pressure caused by respiration. The blood pressure (BP) in the extrathoracic compartment is influenced by abdominal pressure fluctuations. The main artery in the body is the aorta. It originates from the left ventricle. From the aorta blood is divided into smaller arteries which end up in the periphery. Maintaining an adequate BP in the aorta is very important. If BP is too low, blood flow to the organs will be insufficient, if BP is too high, the vascular system can be damaged. The BP results by an interaction between heart and vascular system. The heart can vary stroke volume and ventricle pressure. The resistance to blood flow in the peripheral vessels, caused by their small diameter, is the main cause of pressure build up in the large arteries. To keep the BP in range, it is regulated by the so called baroreflex. BP is measured in the aorta by baroreceptors and interpreted by the brain, which in turn can affect parts of the body which are able to control the BP. 2.2 The autonomic nervous system The nervous system consists of the central nervous system (CNS) and the peripheral nervous system (PNS). The CNS consists of the brain and the spinal cord. The PNS consists of afferent nerves, which transmit information from receptors throughout the body to the CNS and efferent nerves, which transmit information from the CNS throughout the body. The nervous system is also divided in a somatic (voluntary) and autonomic (involuntary) part. The autonomic nervous system (ANS) innervates all internal organs. The ANS can be divided into two counteracting parts: the parasympathetic and sympathetic system. Influence of both nerve systems on heart, arterioles and bronchi is given in Table 2.1. Table 2.1: Effects of sympathetic and parasympathetic activity on heart, arterioles and bronchi. organ sympathetic effect parasympathetic effect heart -rate acceleration decelerations -contractility increase decrease -rate of relaxation increase decrease -conduction velocity increase decrease arterioles constriction generally no effect bronchi dilatation constriction One of the most important efferent parasympathetic nerves is the vagal nerve which innervates, among others, the heart and the lungs. The aortic nerve innervates baroreceptors in the aortic arch to the CNS. Baroreceptors are specialized pressure sensors. The most important ones are located in the aortic wall and the carotid sinuses. These receptors are connected via specific afferent nerves to the part of the brain that translate afferent information to efferent activity. With efferent nerves the signal goes to the effector. With this information the effector will adapt to control BP. Measuring the BP, processing this information and adapting organs to control BP is a feedback loop called the baroreflex (BR), graphically represented in Figure 2.2. This figure also shows that the BR is affected by respiration through lung stretch receptors. The function of the BR is to stabilize short-term disturbances on perfusion pressure. A diagram describing the relations between BP and the effectors is presented in Figure 2.3. September 8, 21

13 The autonomic nervous system 5 Figure 2.2: Simplified representation of the baroreflex, with fast parasympathetic nerves and slower sympathetic nerves, which send information to the cardiovascular system. Input for the baroreflex is obtained from lung stretch receptors and baroreceptors. Adapted from McSharry et al. [6]. K C I J H A J? D H A? A F J H I I O F = J D A J E? ) H J A H E = I J D K I? A I? I J H E? J E = J = J E = H J A H E = I O I J A I A J F E J I A J F E J + 5 I O F = J D A J E? I O F = J D A J E? 8 A K I I J D K I? A I + = E =? K I? A I? I J H E? J E = J = J E? J H =? J E E J O L A K I I O I J A 8 4 D A = H J : : = H J A H E = F H A I I K H A I O F = J D A J E? F = H = I O F = J D A J E? 5 A D A = H J H = J A = B B A H A J =? J E L E J O > = H H A? A F J H I Figure 2.3: The closed loop control of systemic arterial blood pressure. Consisting of the central nervous system (CNS) as controller, a negative feedback by baroreceptors and pressure regulation through the four effectors (arterial and venous smooth muscles, cardiac muscle and the SA node). The arterial smooth muscles and the SA node are also influenced by respiration. Abbreviations: cardiac output (CO), stroke volume (SV), venous return (VR), total peripheral resistance (TPR). Based on Ten Voorde [7].

14 6 Physiology As can be seen in Figure 2.3, the BR is a negative feedback control system. Input of the baroreceptor is BP, and its derivative to time. Output is a change of the effectors, which adapt the BP. There are four important effectors: Sino-atrial node which controls the HP Cardiac muscle contractility which affects stroke volume and pressure Peripheral smooth muscle cell tone, which affects the total peripheral resistance Venous smooth muscle cell tone, which controls the total venous compliance The input-output curve of each effector pathway is sigmoidal, with a threshold where output is minimal and a saturation level where output is maximal. As an example, the sigmoidal function of heart period (HP) is shown in Figure 2.4. The slope of the curve is the gain, which is dependent on the input. This gain is a estimate of the sensitivity of BR system to adapt e.g. HP. The operating point is the equilibrium point at which pressure is optimal for the system. Figure 2.4: Heart period [s] vs. mean pressure [mmhg]. Dashed line indicates the equilibrium point. Figure is adapted from Cavalcanti et al. [8]. The brain compares the afferent receptor signal to a pressure reference value. The difference between these values is the driving signal for different types of effectors to regulate BP. Another input for baroregulation is the lung-stretch receptor, which responds to changes in lung volume. This receptor indirectly influences systemic peripheral resistance and HP. The effect effect of the lung stretch receptors on heart period is called Sinus Arrhythmia (RSA) and causes oscillations with respiratory respiratory frequency in heart period and BP. It is assumed that fast fluctuations (respiratory associated) in HP are parasympathetically mediated. Low frequency fluctuations are jointly mediated by the parasympathetic and sympathetic nervous system [9]. The parasympathetic nervous system can react more rapidly (within 2 ms) than the sympathetic nervous system (latency of 2-5 seconds) [1], due to differences in pathways and structure. Because of the wide spread availability of ECG data, from which the heart period can be determined, BR mediated HP control has been studied more than any other effector. This is accomplished through various methods as applying a pressure pulse to the neck which will change the transmural pressure to September 8, 21

15 2.2 The autonomic nervous system 7 the carotid sinus region [11], or administering vaso-active drugs [12, 13]. With these methods the BP will deviate from its equilibrium point and the input-output curve can be studied. Phenylephrine is used to decrease BP below equilibrium point and sodium nitroprusside is used to increase BP above equilibrium point, this way baroreceptor sensitivity to change HP is estimated to be 1 ms/mmhg and 3 ms/mmhg for BP fall and rise respectively, for human adults [14]. The higher baroreceptor sensitivity during BP rise above equilibrium point, can be ascribed to a relative lower vagal cardioinhibitory tone. This means, if the system is in rest, a relative low vagal cardioinhibitory tone (so heart rate increases easier) causes the BP to restore faster from a sudden pressure rise, than from a pressure fall. Although these methods are useful to estimate BR sensitivity, they cannot be used for preterm infants because of ethical considerations. A possible alternative is spectral analysis of arterial BP and HP [15]. This method is based on spontaneous fluctuations in BP and HP. The method can be used to determine BR sensitivity as well as give more information about sympathetic and parasympathetic nervous system activity. This will be explained in detail in Section Limited and conflicting data is available about maturation of baroreceptor reflex function in the preterm infant, likely due to differences in population e.g. gestational age, methodology and sleep state. In more recent studies BR sensitivity is estimated to be 4 ms/mmhg for preterm infants (mean gestational age of 3 weeks) [16, 17]. For full-term neonates this is estimated at 1 ms/mmhg [16, 18, 19]. A postnatal increase in BR sensitivity with increase in postnatal age is observed in very preterm infants, however these infants reach lower BR sensitivity values reaching term compared to full-term neonates [16] Physiological basis of cardiovascular rhythms Spectral analysis of BP and R-R peak interval as shown in Figure 2.5 can provide information about physiological processes. Figure 2.5: Example of the power spectral density of the heart period of a human adult, with a peak at.1 Hz (LF) and a peak around.2 Hz (HF). Adapted from Ursino et al. [2]. The relatively low frequencies (LF) of.1 Hz in adults [21],.7 Hz for neonates [22], are commonly ascribed to BR activity and are assumed to represent both sympathetic and parasympathetic activity [4, 9, 23, 24]. This oscillation of.1 Hz is often referred to as Mayer wave. Cross spectral analysis of BP and R-R interval at this LF band (.4-.15Hz [21]) gives a reliable estimate of BR sensitivity [25]. Although there are different opinions about the origin of these Mayer waves, e.g. very low frequencies (caused by

16 8 Physiology e.g. the renin-angiotensin system, circadian rhythm, thermoregulation or peristaltic activity [2, 4, 26, 27]), or a central oscillator in the brain [28], the baroreceptor is a commonly accepted cause. High frequencies (HF) activity is also found (adults.25 Hz, neonates Hz). which is associated with respiratory activity and assumed to be under parasympathetic control only [24], as the sympathetic system is too slow to react to these high frequencies. Cross spectral analysis of this HF band ( Hz for adults [21]), gives an estimate for parasympathetic activity. Comparing the LF and HF band should give information about the autonomic balance [21, 29], however controversy exists about the value of this comparison, because the LF band is dependent on both parasympathetic and sympathetic nervous system [3], without knowing its relative contribution. This ratio has never been validated for preterm infants. Another controversy for preterm infants is the exact HF band to use. It is mostly assumed to be around Hz [31]. Another possibility is to use a bandwidth which is dependent on the subject and takes percentiles of the respiration frequency, e.g. Andriessen et al. uses the 1 th and 9 th percentile [17]. Others claim that a respiratory peak is not detectable in neonates [22, 32]. Frequencies below the LF band also exist. These are called very low frequencies (VLF). It is supposed that these VLF are related to sympathetic nerve activity only [33]. 2.3 Obtaining information about the ANS from clinical data To evaluate the LF and HF bands, heart period is acquired from R peak detection of the an electrocardiogram (ECG). Typical HP and BP signals for a preterm neonate are shown in Figure 2.6. The LF with higher amplitude and a wavelength of approximately 1 seconds and the HF with less amplitude and a wavelength of approximately second is visible, both in HP and BP. From these signals power density spectra are calculated to evaluate the frequency bands shown in Figure 2.7. These figures show the LF and HF band as explained in Section 2.2.1, of which the LF band has more power compared to the HF band, especially for the HP. The HF band is around the respiration frequency, in this case approximately.8 Hz, which is 75 breaths a minute. Figure 2.6: Time series of RR-interval and systolic blood pressure of a preterm infant during quiet sleep state at a postconceptional age of 3 weeks. Adapted from Andriessen et al. [17]. September 8, 21

17 2.3 Obtaining information about the ANS from clinical data 9 Figure 2.7: Spectral power curves of the RR-interval (A) and systolic blood pressure (B) of a preterm infant during quiet sleep state at a postconceptional age of 31 weeks. Adapted from Andriessen et al. [18]. From these power spectra the coherence is calculated. Coherence shows the degree of resemblance between two sets of time series and ranges from (no resemblance at all) to 1 (perfectly resembled). Coherence level higher than.5 indicates a relevant relationship between the two signals, based on 5% shared variance. The advantage of coherence function over correlation coefficient is that coherence is a function of frequency. In other words, it can show that at which frequencies two sets of time series data are coherent and at which frequencies they are not. The gain and phase of the LF and HF bands of the two signals are calculated where the coherence in the appropriate band is highest. Figure 2.8 shows the coherence, gain and delay of each frequency of the signals shown in Figure 2.6.

18 1 Physiology Figure 2.8: Coherence (A), gain (B) and phase delay (C) between the RR-interval power spectrum and the power spectrum of systolic blood pressure of the same preterm infant as shown in Figure 2.7. Adapted from Andriessen et al. [17]. The LF and HF gains are estimated to be 2.2 and 1. ms/mmhg, respectively. The LF and HF delays are estimated to be -1.5 and -.5 seconds, respectively, indicating that the BP lead HP fluctuations. These powers, gains and delays are used to validate the model which is made in this thesis, as will be explained in Chapter 5. More information about calculating coherence, gain and delay can be found in [34]. September 8, 21

19 11 Chapter 3 Rebuilding the Sa Couto model of neonatal hemodynamics 3.1 Introduction First models found in literature are evaluated as starting point for the eventual model. A review of models is given in appendix A. A summary of this review is shown in Table 3.1 compared to important criteria for a good model: Reliable estimation of the required variables As simple as possible Validated with human or animal studies Well described and explained Adjustable to preterm infant dimensions Table 3.1: Summary of the literature review criteria Morrison [35] Seydnejad [28] van Roon [36] Sa Couto [37] Ursino [2] reliable simple validation description adjustable The model of Morrison et al. has a poor description of the parameter estimation and is therefore unreliable. The model of Seydnejad et al. does not have a physiology based cardiovascular model and is thus unsuitable for the purpose of this study. The model of Van Roon et al. is a good model. However the purpose of his study deviates from this study, which results in an unnecessary complicated model as a starting point for the purpose of this model. From this review the model of Sa Couto et al. [37] is chosen as starting point for the cardiovascular model, which will be the focus of this chapter. Sa Couto is chosen because of their use of neonatal dimensions and the possibility to create a ductal flow, which is often seen in preterm infants. The Sa Couto model does not give an accurate outcome for the eventual needed gains and delays because the baroreflex is oversimplified for the purpose of this study. So the cardiovascular model of Sa Couto will be extended with the baroregulation model of Ursino, which will

20 12 Rebuilding the Sa Couto model of neonatal hemodynamics be the focus of Chapter 4. In Chapter 5 these two models will be combined and adapted to simulate preterm infant hemodynamics under baroreflex control. The found literature uses lumped parameters to model a circulation with baroregulation. Another possibility would be a finite element model. However this would unnecessarily cause extra complexity, compared to a lumped parameter model. 3.2 Methods To build a lumped parameter model of the circulation, a hydrodynamic equivalent has to be made. The model separates the function of blood storage and blood flow into compartments and segments, respectively. The storage compartments represent specific parts of the circulation. A compartment consists of a compliance and an unstressed volume. The different compartments are connected through segments that describe the flow of blood in between. A segment consists of a resistance and optionally an inertance, which are described later on. Pressure in the compartments and flow between compartments are given by constitutive equations and the volume of a compartment is based on conservation of mass. All variables are time-dependent. These equations are derived from the Navier-Stokes equations, which describe the motion of fluids. They arise from applying Newton s second law to fluid motion, together with the assumption that the fluid stress is the sum of a viscous term (proportional to the gradient of velocity) and a pressure term. The pressure p(t) in a compartment is based on the constitutive equation p(t) = V (t) V, (3.1) C with volume V, unstressed volume (volume at zero pressure) V and compliance C (see Figure 3.2 for a graphical representation). The compliance is a measure of the flexibility of the vessel wall which makes it possible to store more blood at higher pressures. Pressure is developed by an extent of blood volume over the compliance of the vessel at zero pressure. If the vessel wall is very flexible (high C) the pressure rise will be low with increasing blood volume, A low value for compliance makes a vessel rigid, so if blood volume increases, pressure will rise. The pressure in the heart is modeled as a time-varying elastance (see Figure 3.2 for a graphical representation), proposed by Suga et al. [38]. This model is used for both ventricles and atria. The relation between heart chamber pressure and heart chamber volume is given by p(t) = (E pas + a(t)(e max E pas )) (V (t) V )), (3.2) with E max and E pas the maximum (full contraction) and passive (no contraction) elastance of the cardiac wall, respectively. Elastance is the inverse of compliance. a(t) is an activation function, ranging between and 1 and is given by a(t) = { (sin(π t delay t act )) 2 t < t act + delay & t > delay else, (3.3) with delay the time between atrial contraction and ventricle contraction (delay is zero for atrial contraction) and t act the duration of contraction of the atria or ventricle. The activation curve is shown in Figure 3.1. of a compartment equals the difference between inflow q in (t) and out- The change of volume flow q out (t): dv (t) dt dv (t) dt = q in (t) q out (t) (3.4) September 8, 21

21 3.2 Methods 13 normalized activation 1.5 activation curve atria ventricles time (t) Figure 3.1: Normalized activation curve of the simulated atria and ventricles in the Sa Couto model. Flow between compartments, q(t) is given by: q(t) = p in(t) p out (t) (3.5) R Flow is dependent on the pressure difference p in (t) p out (t) over the segment. The amount of flow that goes through the segment depends on the resistance R (see Figure 3.2 for a graphical representation) of the vessel. The most important factor for resistance is the diameter of the vessel. Small diameter vessels (the peripherals) have a high resistance. In larger arteries the resistance of the vessel wall on blood flow will have less effect. However in the case of large arteries, inertance L (see Figure 3.2 for a graphical representation) will play a more important role. Inertance is less dependent on vessel diameter and more dependent on blood density. In this case the flow described in equation 3.5 is extended with inertance, a resistance to change in flow, according to q(t) = p in(t) p out (t) + L dq(t) (3.6) R dt If the heart contracts the blood flows only in one direction using valves (see Figure 3.2 for a graphical representation). These valves prevent reversal flow. This is simulated by a flow restraint q mv = { pin (t) p out(t) R p in (t) p out (t) p in (t) < p out (t), For example, in the case of the mitral valve, p in (t) is the pressure in the left atrium, and p out (t) the pressure in the left ventricle. R is the resistance of the mitral valve. The flow through the systemic extrathoracic vein differs for backward flow and forward flow. This is because veins have small valves to decrease backward flow. Resistance to backward flow is simulated 1 times larger than to forward flow. This is simulated by using an additional restraint q(t) = { pin p out R p in p out (3.7) p in p out (3.8) 1 R p in < p out The model has a distinction between extrathoracic and intrathoracic arteries and veins. Difference between these compartments is the effect of thoracic pressure (P th ) of -3 mmhg, caused by pressure in the lungs on the intrathoracic compartments. A constant abdominal pressure (P abd ) n the extrathoracic compartment is set to mmhg. With these building blocks the model of Sa Couto has been rebuilt in Matlab 7.5. (The Mathworks Inc.). An electrical equivalent of the model is shown in Figure 3.3. Simulations are performed with parameter settings for a term infant of 35 gram without and with an open patent ductus arteriosus (PDA). The parameters can be found in appendix B. Hemodynamic results are obtained after the model is in steady state, which is defined as a maximum difference of.2 ml between the stroke volume of two consecutive beats. Time steps are.5 ms. The hemodynamic results are compared with the results published by Sa Couto et al. [37].

22 L 14 Rebuilding the Sa Couto model of neonatal hemodynamics 4 A I E I J =? A + F E =? A 8 = H O E C - = I J =? A 1 A H J E = L = L A Figure 3.2: Graphical representations of a resistance, compliance, time varying elastance, inertance and a valve. 2 J D + F = + F L 4 F F 2 F = 2 F L G F F 2 J D 4 F = G F L = G = 4 F L - = J - H L J F F L = F K = H O L = L A K? K? E J H = L = L A 2 = 2 J D 2 J D 2 J D - H = J 4 J L 2 H L G J L J H E? K I F L = L A 2 H K? G = H J E? L = L A G I = E 4 L - L J 2 L F = 4 I = E + I = E 2 J D 2 J D + I L E 4 I L E G H = 2 I L E 2 I = E I = A 2 J D 4 I L A G I L A G I = A 4 I = A 2 = + I L A 2 I L A 4 I F G I F 2 I = A + I = A 2 = Figure 3.3: An electrical equivalent of the Sa Couto model. First letter of abbreviation is p; pressure, q; flow, R; resistance, C; compliance E; time varying elastance. L; inertance, Following letters: lv; left ventricle, rv; right ventricle, la; left atrium, ra; right atrium, pva; pulmonary valve, mv; mitral valve, tv; tricuspid valve, ao; aorta, sa,i; intrathoracic systemic artery, sa,e; extrathoracic systemic artery, sp; systemic periphery, sa,e; extrathoracic systemic vein, sv,i; intrathoracic systemic vein, pa; pulmonary artery, pp; pulmonary, pv; pulmonary vein, th; thoracic, abd; abdominal, duc; ductus. The dashed line represents the possibility to simulate a patent ductus arteriosus. September 8, 21

23 3.3 Results & Discussion Results & Discussion Figure 3.4 shows the results of a simulation without a PDA. A pressure is built up in the ventricles by contraction of the heart. If the pressure in the ventricle becomes higher than in the artery (aorta or pulmonary artery), the valve in between will open and blood will flow (q ao or q pa, respectively) into the artery, resulting in a pressure build up in the artery. After contraction the heart relaxes, leading to pressure decrease in the ventricles. The valve closes when the pressure in the ventricle becomes lower than in the artery and there is no more flow through the valve. As soon as the pressure in the ventricle is lower than in the atrium, blood will flow (q mv or q tv, respectively) into the ventricle. The filled ventricle will contract and the new cycle begins. pressure [mmhg] plv pao pra prv ppa pla pressure [mmhg] LV RV time [ms] volume [ml] flow [l/min] qmv qao qsp qla flow [l/min] qtv qpa qpp qra time [ms] time [ms] Figure 3.4: Two consecutive beats of the rebuilt neonatal model of Sa Couto. First letter of abbreviation is p; pressure, q; flow. Following letters: lv; left ventricle, ao; aorta, ra; right atrium, rv; right ventricle, pa; pulmonary artery, la; left atrium, mv; mitral valve, sp; systemic periphery, tv; tricuspid valve, pp; pulmonary periphery This same cycle can be seen in the pressure volume loop which clearly shows the isovolumetric contraction, which creates a pressure build-up. If pressure is higher than in the artery, the valve opens, blood will flow and the ventricle volume decreases because of continuing contraction. After the end of contraction (maximum in the pressure volume loop) and pressure in the ventricle is lower than in the aorta, the valve closes and isovolumetric relaxation of the heart is visible. As soon as pressure in the ventricle drops below atrial pressure, the valve in between will open and blood will flow into the ventricle and the volume increases again. At the minimum of the pressure volume loop, the heart is totally relaxed and filled with blood from the atrium, both active and passive.

24 16 Rebuilding the Sa Couto model of neonatal hemodynamics The pressure becomes twice as high in the left ventricle and aorta, compared to right ventricle and pulmonary artery. This is caused by the higher systemic resistance. Because of this, pressure needs to be higher to develop the same flow as in the pulmonary circulation. The aortic flow q ao reaches its maximum in a shorter amount of time compared to the pulmonary artery flow (13 ms vs. 16 ms), which also has a lower maximum. The aortic flow increases more rapidly because the pressure in the left ventricle also increases faster than in the right. Compared to the aortic flow, the pulmonary flow lasts longer because the blood flows earlier in the cardiac cycle in the pulmonary artery. This is because there is less pressure difference over the valve. The surface under the aortic and pulmonary artery flow is similar, which means stroke volume of the left and right ventricle is the same. The nett flow into the atria q la becomes negative during contraction because blood will flow back into the vein caused by the higher pressure in the atrium, compared to the vein. Normally, contraction of the atrium leads to a flow through the mitral and tricuspid valve into the ventricle (- 1 ms), which is only seen in the simulation in the right part of the heart. However, this flow lasts too long, because the tricuspid valve is still open when the pulmonary artery valve is also open. So the right atrium contracts too strong. This also causes a large backflow into the systemic vein during the start of contraction of the heart ( to 1 ms). The pressure volume shows similar stroke volume for the left and right ventricle as expected. Stroke volume is defined as the difference between end diastolic and end systolic volume. pressure [mmhg] plv pao pra prv ppa pla pressure [mmhg] LV RV time [ms] volume [ml] flow [l/min] qmv qao qsp qla qduc flow [l/min] qtv qpa qpp qra time [ms] time [ms] Figure 3.5: Two consecutive beats of the rebuilt neonatal model of Sa Couto including a patent ductus arteriosus. Abbreviations are the same as in Figure 3.4 and the abbreviation duc is ductus. September 8, 21

25 3.3 Results & Discussion 17 Figure 3.5 shows the results of a simulation of the neonatal cardiovascular system with inclusion of a PDA. The maximum pressure in the aorta remains unchanged compared to the situation without PDA. The maximum pressure in the pulmonary artery increases strongly, it almost doubles compared to the situation without the PDA. This is because of the extra volume load caused by the PDA. To keep the aortic pressure at the same level, the left ventricle is modeled with a higher maximum cardiac elastance to pump out more blood. This blood goes through the PDA and the pressure in the aorta drops more rapidly because flow goes to the pulmonary artery where pressure is lower. The pulmonary valve opens later (at 7 ms vs 3 ms after start of atrial contraction) as pulmonary flow starts later in the cardiac cycle. This is because the afterload is higher causing the pressure difference to last longer. The valve will not open until this difference is canceled out. In the case of a PDA, stroke volume of the left and right ventricle are not the same. The blood pumped out of left ventricle can bypass the right ventricle through the PDA, which is a shortcut back to the left ventricle after passing the pulmonary periphery. Because of this, the left ventricle needs to produce more output, to maintain an sufficient flow into the aorta. Table 3.2 shows the most important cardiovascular variables produced by the rebuilt model for simulations without and with a PDA compared to the variables given by Zijlmans et al. [39], who corrected minor errors in the implementation of the Sa Couto model. Table 3.2: Vital sign values of an a-term neonate with and without a PDA, compared to the results obtained by Zijlmans et al.[39] Rebuilt model Zijlmans simulation cardiovascular variable without PDA with PDA without PDA with PDA heart rate [beats/min] left systolic ventricle pressure [mmhg] left diastolic ventricle pressure [mmhg] systemic systolic pressure [mmhg] systemic diastolic pressure [mmhg] mean systemic pressure [mmhg] right systolic ventricle pressure [mmhg] 29 52? 52 right diastolic ventricle pressure [mmhg] 3 3? 3 pulmonary systolic pressure [mmhg] pulmonary diastolic pressure [mmhg] mean pulmonary pressure [mmhg] 17 3? 3 systemic cardiac output [ml/min] pulmonary cardiac output [ml/min] In conclusion, the rebuilt model corresponds well with the data published data by Sa Couto et al. and Zijlmans et al. The model dynamics are as expected and the hemodynamical variables of blood pressure and flows are verified by Sa Couto. This model is assumed to be a good model of a term infant and therefore used as a basis for the preterm infant, which will be explained in Chapter 5.

26 18 Rebuilding the Sa Couto model of neonatal hemodynamics September 8, 21

27 19 Chapter 4 Rebuilding the Ursino model of adult baroregulation 4.1 Introduction The pressure regulating part of the model is based on the baroreceptor model used by Ursino [2] because of the extensive validation and possibilities to extend the baroreflex model of Ursino for different purposes. For example, the baroreflex model proposed in 22 [4] introduces carotid chemoreceptors responding to O 2 and CO 2 pressure changes. This is introduced to evaluate the effect of hypoxia and hypercapnic stimuli. This option is interesting for future research as hypoxia is often seen in preterm infants. The total model has been duplicated to verify a correct implementation, after which the autonomic control part can be extracted. The rebuilt model is validated by simulating a hemorrhage in Section The effects of respiration, abdominal pressure, very low frequencies and the baroreflex itself and the influence of only the baroreceptors or only the lung stretch receptors on the frequency response are investigated in Section Methods The model of the baroreflex described in the publication of 23 [2] is used because it includes vagal activity affecting the HP and respiration which affects the blood pressure directly, and through the nerve system. It also has less parameters than the baroreceptor reflex model used in 1998 [41], but still is sufficient for the purpose of this research. Fewer parameters means fewer estimations when adapting the model to preterm infant dimensions. Unfortunately, the cardiovascular model used in the 23 article introduces an unquantified extra compartment for the intrathoracic vein, compared to the 1998 model on which it is based. Next to this, the cardiovascular outcome of 1998 model is poorly described in the articles of Ursino. The rebuilt model does give a similar aortic pressure, however the atrial volumes are not given in the article and cannot be verified. The rebuilt model gives very high values for these volumes. A number of parameters are adapted to give normal atrial volumes with a similar aortic pressure. First the cardiovascular model used by Ursino will be explained shortly The cardiovascular model The cardiovascular model of Ursino has been rebuilt with the same constitutive laws and conservation of mass as the Sa Couto model described in Chapter 3. The difference with the Sa Couto model is a distinction between a splanchnic and extrasplanchnic part for the systemic peripherals and systemic

28 2 Rebuilding the Ursino model of adult baroregulation veins. The difference between these compartments is the effect of varying abdominal pressure (P abd ) between and -2.5 mmhg on the splanchnic part, and the effect of autoregulation for both parts. All other compliances are affected by the thoracic pressure (P th ), which varies between -4 and -9 mmhg. P abd and P th are explained in appendix B. This model also uses a time varying elastance for the left and right ventricle like Sa Couto. The atria are modeled as passive elastances. No distinction is made between an intra- and extrathoracic part. Ursino adds noise in the range of to.12 Hz to the extrasplanchnic resistance to mimic the VLF fluctuations caused by VLF and LF sources as humoral and thermal control or vasomotion. The extrasplanchnic resistance R ep (t) is given by R ep (t) = R ep,con (t) + A R rand (t), (4.1) with R ep,con the controlled parameter resulting from the baroreflex equations and R rand a normalized uniformly distributed noise band, which is amplified by a factor A. This factor is not given by Ursino and is set to approximately 1 % of total systemic resistance. The noise is applied with a low passed filter for the sake of simplicity, instead of the proposed linear decreasing filtering band of Ursino. Also no quantification of slope of this band is given. The difference between these filters is expected to be in the very low frequencies, which are not the aim of this study. Figure 4.1 shows the electrical equivalent of the Ursino cardiovascular model. Values of the parameters of the model can be found in appendix B. A number of values are changed to give physiologically correct volumes, especially for the atria. Compliances of the left and right atria are reduced according to the values given by Beneken et al. [42], as used in the adult version of the Sa Couto model. The unstressed volumes are estimated by adjusting these values to give pressures [43] and volumes [44] in a physiological range. The new values for the atria can be found in Table 4.1. Note that the unstressed volumes are only values of the intersection with the x-axis of the assumed linear relation between pressure and volume around its working point. Because of this reason these values can be negative. Also, the unstressed volume for the systemic arteries is increased by 15 ml and the pulmonary artery by 1 ml. The total blood volume is reduced by 2 ml. This is done because with the new values for the atria, BP becomes too high. Table 4.1: Adjusted values for the hemodynamic model of Ursino. The total blood volume is reduced from 53 ml to 51 ml. compliance [ml/mmhg] unstressed volume [ml] orginal new orginal new left atrium right atrium systemic artery pulmonary artery The baroreflex With the input of lung stretch receptors and baroreceptors, the BR adjusts the effectors (heart rate, cardiac elastance, peripheral resistance and venous unstressed volume). The effectors are adjusted using gains, pure time delays and low-pass-filters as shown in Figure 4.2 and 4.3. The gains represent the relative contribution of the baroreceptors and lung stretch receptors to a change of 1 mmhg BP and 1 liter of lung volume, respectively. A sigmoidal shaped function is used because the organs have a maximum range to what extent they can adapt. The sigmoidal function of each effector is shown in Figure 4.4. The time delay is used to simulate the time the signal travels from the receptors to the effector. The low pass filter September 8, 21

29 L 4.2 Methods 21 2 J D 2 F F 4 F F 2 F L 2 J D + F F G F F + F L 4 F = G F = G = 4 F L 2 J D + F = F = 2 F = E J H = L = L A 2 = + = 2 J D - H L J 4 H L F K = H O L = L A G F L G 4 L - L J 2 L 2 J D 2 J D 2 H L G = L 4 L 4 J F G J F = H J E? L = L A + I = + H = J H E? K I F L = L A 2 I = 2 J D 2 H = I = 2 = + A L 4 A L G A L 4 I L G I L 2 I L + I L 4 I F G I F 4 A F G I = 4 I = 2 I F + I F 2 = 2 A L G A F + A F Figure 4.1: An electrical equivalent of the hemodynamic model of Ursino using the graphical representations as explained in Figure 3.2.

30 22 Rebuilding the Ursino model of adult baroregulation is used because an organ cannot react instantaneously to a signal. The parameters of the components can be found in appendix B. Figure 4.2: Sympathetic regulation mechanism for a generic effector θ. P sa and P san are instantaneous systemic arterial pressure and its reference value, respectively. V L and V Ln indicate lung volume and its end-expiration reference volume, respectively. G is the maximum gain, which is different for the two receptors. After this information is summed, it is filtered by a static sigmoidal function to determine the effective gain for the effector. Than, a pure time delay and low pass filter are applied. Figure adapted from [2]. Figure 4.3: Sympathetic and vagal regulation mechanism for the heart period. P sa and P san are instantaneous systemic arterial pressure and its reference value, respectively. V L and V Ln indicate lung volume and its end-expiration reference volume, respectively. G is the maximum gain, which is different for the two receptors and two nerves. After this information is summed, it is passed through a pure time delay and low pass filter. The vagal and sympathetic activity are summed and filtered by a static sigmoidal function to determine the effective gain for the effector. Figure adapted from [2]. For sympathetic regulated mechanisms for effector θ, the deviation of systemic arterial pressure P sa from its reference value, P san and the lung volumev L from its reference value V Ln is translated into, x θ = G aθ (P sa P san ) + G pθ (V L V Ln ), (4.2) with G aθ and G pθ the gain of the input for the baroreceptors and lung stretch receptors, respectively and x θ is the input for the sigmoidal function σ, σ θ = θ min + θ max e ±x θ/k θ 1 + e ±x θ/k θ, (4.3) September 8, 21