Hemodynamic Consequences of Thoracic Artificial Lung Attachment Configuration: A Computational Model

Abstract

A thoracic artificial lung (TAL) is being developed to assist treatment of acute and chronic pulmonary dysfunction. The TAL is attached directly to the pulmonary circulation. Depending on pathophysiology, the TAL may be attached in series with the natural lungs (NLs), in parallel with the NLs, or in an intermediate, hybrid configuration. We developed a computational model to study hemodynamic consequences of TAL attachment configuration under pathologic conditions. The pulmonary and systemic circulations, heart, and TAL are modeled as interconnected compliances and conductances, some valved. Time-varying cardiac compliance drives the system and generates pressures and flow rates. The model includes blood pressure feedback from the sympathetic nervous system, renin-angiotensin system, and renal volume control mechanism. We used previously published results from porcine experiments to verify model accuracy. We modeled normal physiology and four disease states. A hybrid configuration with 100% cardiac output through the TAL and 40% through the NLs would deliver maximal blood flow, 3.6 to 4.6 l/min, to the TAL and be tolerated by the right ventricle. Additionally, the model suggests that reducing the large “minor loss” resistances at the graft anastomoses to the pulmonary artery would improve the hemodynamics of all TAL attachment configurations.

Lung disease may result from gas exchange or hemodynamic pathology and may be acute or chronic in nature. The standard treatment for acute lung disease is mechanical ventilation and that for chronic lung disease is oxygen therapy. The only treatment for end-stage pulmonary failure is lung transplantation. Delivery of sufficient oxygen, however, may require high tidal volume ventilation, which can overdistend the lungs and exacerbate lung injury,¹ or a highly elevated oxygen concentration, which can be toxic.^2,3 A thoracic artificial lung (TAL) attached to the pulmonary circulation thus is being investigated as an alternative or complementary treatment for lung disease.^4–9 The present work is a computational model developed to study the potential hemodynamic effects of attaching an artificial lung to a pathologic pulmonary system. The model is intended to aid in selecting an appropriate TAL attachment configuration and TAL design. Optimal attachment configuration depends on the specific pathology.

The TAL is an oxygenator with a low resistance to blood flow and a compliant housing.⁴ It is attached directly to the pulmonary circulation in series with the natural lungs (NLs), in parallel with the NLs, or in an intermediate configuration that is a hybrid of series and parallel⁵ (Figure 1). The TAL has access to the cardiac output (CO) from the right ventricle (RV) and requires no pump. Artificial lung attachment can increase or decrease pulmonary system resistance (PSR), which is the resistance of the combined TAL/NL system and equal to pulmonary vascular resistance (PVR) in the absence of a TAL. Artificial lung attachment can thus increase or decrease RV afterload. Series TAL attachment elevates PSR but may be appropriate with normal PVR and impaired NL gas exchange. Parallel TAL attachment decreases PSR by acting as an oxygenated shunt and may be appropriate with elevated PVR. Hybrid TAL attachment may be appropriate with combined NL gas exchange impairment and elevated PVR. The TAL could be used as a temporary assist in acute lung disease or as a bridge to transplant in chronic lung disease.

Figure 1.: Thoracic artificial lung (TAL) attachment to the pulmonary circulation. TAL is attached to the pulmonary circulation with an inlet graft from the proximal main pulmonary artery (PA) and outlet grafts to the distal main PA and left atrium (LA). TAL can be attached in full or partial series with the natural lungs (NLs) by clamping or partially occluding, respectively, the PA between the two grafts and by clamping the LA outlet graft. TAL can be attached in parallel with the NLs by clamping the PA outlet graft and, if desired, partially occluding the PA between the two grafts. TAL can be attached in an intermediate hybrid configuration by clamping or partially occluding the PA between the two grafts and by partially occluding the LA outlet graft.

The primary considerations in choosing an appropriate TAL attachment configuration are blood flow rates through the TAL, the NLs, and the TAL outlet graft to the left atrium (LA), which shunts blood around the NLs. With reduced NL gas exchange, TAL blood flow rate will largely determine pulmonary system gas exchange. Some degree of NL blood flow is important for maintenance of nonrespiratory NL functions. The NLs metabolize vasoactive molecules, act as an embolic sieve, and provide compliance to the pulmonary circulation.^10–12 Shunt flow around the NLs allows for embolic passage to the systemic circulation and should be minimized in any configuration. Series TAL attachment reduces CO due to increased PSR but delivers total CO to both TAL and NLs. Parallel TAL attachment increases CO due to reduced PSR but splits CO between TAL and NLs. How TAL attachment configuration affects pulmonary system flow rates is unclear.

To predict the effect of TAL attachment configuration on pulmonary hemodynamics, we developed the computational model described here. The model includes a normal or diseased pulmonary circulation, TAL, systemic circulation, and cardiac chambers, including a normal or diseased RV. Time-varying compliance of the cardiac chambers drives the system such that CO is an output of, not an input to, the model. Particular emphasis is placed on RV function. For a TAL to be effective, the RV must be capable of pumping through the combined TAL/NL system created by TAL attachment. Our group previously modeled TAL incorporation into the pulmonary system with the RV modeled as a flow source.¹³ Right ventricular failure, however, is a major cause of death in lung disease.^14,15 We therefore incorporate into the present model a reactive right ventricle.

Materials and Methods

The basic model of the natural/artificial circulatory system includes the natural vasculature, the heart, and the TAL. It is a network of compliant chambers connected by conductive links, some with valves, as shown in Figure 2. It is based on a model developed by Peskin and Tu¹⁶ to study congenital heart disease. As in the Peskin and Tu model, time-varying compliance of the cardiac chambers drives the system. The governing differential equations for the system are solved at each time point according to the method of Peskin and Tu.¹⁶ Each cardiac cycle is broken into 100 time steps, and the program is run through 200 cycles, a sufficient number to reach steady state. Significant decrease of the time step under a variety of model conditions does not significantly affect the model results. In addition to the components of the Peskin and Tu model, the present model includes separate atrial chambers; the TAL; an oxygen balance on the RV myocardium; short-, mid-, and long-term blood pressure control mechanisms; and models of pulmonary vascular and RV pathology. The present model was verified against results from a series of TAL attachment experiments in a porcine preparation. The model was then run to predict the consequences of TAL attachment under potential clinical conditions.

Figure 2.: Model of thoracic artificial lung inserted into circulatory system. Symbols: Symbol Compliance; Symbol Conductance; Symbol Valve. RA indicates right atrium; RV, right ventricle; MPA1, 2, and 3, main pulmonary artery sections 1, 2, and 3; COMP, compliance chamber proximal to thoracic artificial lung (TAL) inlet; TAL I and II, portions of TAL housing compliance proximal and distal to fiber bundle; PA, pulmonary arteries distal to main pulmonary artery; PV, pulmonary veins; LA, left atrium; LV, left ventricle; SA, systemic arteries; and SV, systemic veins.

Symbol

Symbol

Symbol

Model of the Vasculature

The model of the natural circulatory system is represented by the outside loop of the schematic in Figure 2. For convenience, the main pulmonary artery is modeled separately from the rest of the pulmonary arteries and is broken into three segments. The conductance, G, of each segment, is calculated according to the Poiseuille relation,

in which Q is flow rate, ΔP is pressure drop, r is radius, μ is blood viscosity, and l is length. The assumed values for r, μ, and l are 1.375 cm, 3.15 × 10^–3 Pa·s, and 1.69 cm, respectively. The values for r and l are based on data from Caro et al.¹⁷

The compliance, C, of the main pulmonary artery (PA) segments is calculated according to the relation for a thin walled elastic cylinder,

in which h is wall thickness and E is Young’s modulus of the tissue. An r/h ratio of 22 is used¹⁸ and Young’s modulus is calculated¹⁸ as

in which E_p is the pressure strain elastic modulus that is itself calculated¹⁸ as

An age of 50 years is assumed.

Pulmonary vascular resistance varies, as a result of capillary recruitment, with blood flow rate through the pulmonary vasculature. A relation between conductance of the pulmonary capillaries, G_PC, in (l/min)/mm Hg, and blood flow rate through the natural lungs, Q_NL, in l/min, is based on a model by Gorback¹⁹ and shifted upward to yield a normal PVR²⁰ of 1.24 mm Hg/(l/min) at a flow rate of 5.3 l/min:

The remainder of the vasculature is represented more generally by lumped compliances for the pulmonary arteries distal to the main pulmonary artery, pulmonary veins, systemic arteries, and systemic veins and by lumped conductances for the left and right atrial inlets, cardiac valves, main pulmonary artery sections, and systemic capillaries. Values for these parameters are based on values used in previously published numerical models of the circulatory system.^16,21 The values for all natural circulatory system parameters are listed in Table 1.

Table 1: Natural Circulatory System Parameters

Model of the Heart

The four cardiac chambers are represented by four time-varying compliance chambers. The compliance functions are based on those used by Peskin and Tu.¹⁶ The systolic and diastolic compliances, C_S and C_D, are calculated as

and

in which C_min and C_max are the minimum and maximum values of compliance for a given cardiac chamber, k is the reciprocal of the time constant for a given cardiac chamber in systole or diastole, T_S and T_D are the duration of systole and diastole, and t is the time from the start of systole or diastole. These functions are applied to atrial systole and diastole distinctly from ventricular systole and diastole. The values of the constant parameters for each cardiac chamber^16,21–24 are given in Table 2.

Table 2: Cardiac Chamber Parameters

The duration of systole, T_S, in seconds, is approximated as²³

in which HR is heart rate in beats per minute. The duration of atrial systole is approximated as one-third that of ventricular systole.²⁵ The conductances at the outlet of each cardiac chamber switch between a value representing an open valve when proximal pressure exceeds distal pressure and a value of zero, representing a closed valve, when the reverse is true.

Model of Thoracic Artificial Lung

The model of the TAL comprises a “minor loss” conductance at the inlet graft anastomosis to the PA, an inlet graft conductance, an inlet compliance chamber that is proximal to the TAL proper, a TAL inlet conductance, a proximal TAL housing compliance, a fiber bundle conductance, a distal TAL housing compliance, a TAL outlet conductance, two outlet graft conductances, and a “minor loss” conductance at the distal PA anastomosis. The inlet graft originates from the second main PA section; outlet grafts return to the third main PA section and to the LA. Varying the conductances of the main PA section between the two graft anastomoses and of the two outlet grafts permits simulation of any attachment configuration.

“Minor losses” across the PA anastomoses were measured in an in vitro mock circuit, using 1.6-cm inner diameter T-junctions to replicate the right angled geometry of the proximal and distal PA anastomoses. The circuit contained a water/glycerin mixture with a viscosity of 3.1 × 10^–3 Pa·s. The mechanical energy loss across the proximal anastomosis, ΔE_Ap, varied with percent diversion of CO to the TAL and was measured to be:

in which Q_Gin is flow rate through the TAL inlet graft in l/min (see Figure 2). The mechanical energy loss across the distal PA anastomosis, ΔE_Ad, varied with the relative fractions of flow coming directly from the RV, via the pulmonary artery, and from the TAL, via the outlet graft returning to the distal main PA. It was measured to be:

in which Q_GoutPA is flow rate through the TAL outlet graft to the distal main PA in l/min (see Figure 2). The conductance, G_A, of the proximal or distal anastomosis is calculated as Q_Gin/ΔE_Ap or Q_GoutPA/ΔE_Ad, respectively. At each time point in the simulation, the flow-dependent anastomosis conductance, G_A, is combined in series with the flow-independent Poiseuille conductance, G_Pois, to yield the total conductance, G_graft, of the proximal or distal PA graft:

The value of G_Pois is determined according to equation 1, with r equal to 0.85 cm and l equal to 20 cm. The conductance of the TAL outlet graft to the LA is equal to G_Pois.

The inlet to and outlet from the TAL are modeled as flow-dependent conductances caused by “minor losses” associated with sudden expansion and contraction at these two locations, respectively. These “minor losses” are described by the relation

in which ΔP is the pressure drop across the inlet or outlet, k_ml, is a minor loss coefficient and Q is the flow rate through the inlet or outlet. Conductance is calculated for the TAL inlet and outlet according the relation

Compliances and conductances of the TAL were measured in vitro in prototype TALs. Details of the measurement techniques are reported elsewhere.⁴ TAL parameter values are given in Table 3.

Table 3: Thoracic Artificial Lung (TAL) Parameters

Oxygen Balance on the Right Ventricular Myocardium

Oxygen supply to and consumption by the RV myocardia are approximated and compared to anticipate situations that may cause RV ischemia. Oxygen supply to the RV free wall is approximated²⁶ as

in which So₂ is the supply of oxygen delivered by the right coronary artery to the RV free wall in ml/min, 1.34 ml of oxygen are carried per gram of hemoglobin, 0.14 grams of hemoglobin is contained per ml of blood with a normal hematocrit of approximately 0.40, approximately 4/7 of total right coronary artery flow is directed toward the RV free wall,²⁷ Q_R.cor is total flow rate through the right coronary artery in ml/min, and 0.99 is the assumed oxygen saturation of the blood.

Although the right and left coronary arteries are not modeled as distinct elements, either from one another or from the systemic circulation as a whole, flow rates through the right and left coronary arteries, Q_R.cor and Q_L.cor, are calculated at each time point, based on the total coronary resistance, R_cor, the driving pressures in the right and left coronaries, ΔP_R.cor and ΔP_L.cor, the period of the cardiac cycle, T, and HR as follows:

and

The driving pressure in either coronary artery is the difference between systemic arterial pressure and downstream pressure, as dictated by the waterfall model of a rigid tube followed by a short collapsible distal segment.²⁸ Downstream pressure is equal to the greater of venous or tissue pressure. Tissue pressure in the thick-walled left ventricle (LV) is taken to be 80% of left intraventricular pressure;^21,28 tissue pressure in the thin-walled RV is taken to be equal to right intraventricular pressure.²⁸

The resistance of the total coronary circulation is R_cor and the conductance is the reciprocal, 1/R_cor. The factors preceding the integrals in the numerators of equations 14a and 14b are the fractions of total coronary conductance attributed to the right and left coronary arteries. These values are set such that the right and left coronary vessels receive 25% and 75% of the coronary flow, respectively, under normal physiologic conditions.²³ Total coronary resistance is set to be 296 mm Hg/(l/min), such that total coronary flow is 236 ml/min, or 4.5% of a 5.3 l/min cardiac output, under normal physiologic conditions.²³

Demand for oxygen by the RV, Vo₂, has been shown to be linearly dependent on the right ventricular pressure volume area (PVA),^29,30 which is shown in Figure 3. The intercept of the Vo₂-PVA relation, call it I-1, is itself dependent on maximum RV elastance, E_max, which is the reciprocal of minimum RV compliance, C_min, and a measure of RV contractility.³⁰ The dependence of I-1 on E_max is assumed to be linear,^26,30 with its own intercept, call it I-2. Right ventricular oxygen consumption is calculated as

Figure 3.: Right ventricular pressure volume area (PVA). Right ventricular PVA is the area bounded by the end-systolic pressure-volume line (ESPVL, solid gray line), the end-diastolic pressure-volume curve (EDPVC, dashed gray and black line), and the systolic portion of the right ventricular pressure (P)-volume (V) loop (solid black line).

or

in which the units for PVA are mm Hg·ml and the units for E_max are ml/mm Hg. The sum of the first two terms on the right side of equation 15a is I-1 and the first term alone is I-2. The denominator of the third term normalizes the PVA per 100 g of myocardia for an assumed 120 g weight of the RV free wall and interventricular septum combined.^24,30 The constants of the second and third terms on the right side of equation 15a, which relate normalized data, are based on results from canine experiments.³⁰ The first term constant, I-2, is set such that under normal conditions oxygen consumption is 70% of oxygen supply.²³ In equation 15b, normalized Vo₂ from equation 15a is multiplied by heart rate and by 0.502, the assumed 50.2 g weight of the RV free wall²⁴ normalized per 100 g of myocardium, to determine Vo₂ for the RV free wall in ml/min.

Blood Pressure Control

The model is designed to generate physiologically accurate pressures and flow rates under normal physiologic conditions. Modeling pulmonary pathophysiology or incorporating the TAL into the model alters the impedance to flow through the pulmonary system. Pulmonary impedance, in turn, affects left ventricular filling and, via the Frank-Starling mechanism, an intrinsic part of the model, mean sytemic arterial pressure (MAP). The blood pressure regulatory mechanisms of the body that maintain a constant MAP are thus simulated. The actions of the sympathetic nervous system (SNS), renin-angiotensin system (RAS), and kidney blood volume control mechanism (VOL) are mimicked to model the short-, mid-, and long-term responses, respectively, to a change in MAP.

The SNS reacts within seconds to a sudden change in blood pressure, sensed by baroreceptors in the systemic arteries, and corrects the initial change by 61.5%.^23,31 The SNS effects this correction through changes in cardiac contractility, systemic vascular resistance (SVR), and heart rate. The maximum elastance of each cardiac chamber, E_max, is a measure of contractility. To simulate sympathetic stimulation, the values of E_max, SVR and HR are each multiplied by an appropriate factor. Because data are not available on the relative degrees to which sympathetic stimulation increases contractility, SVR and HR, however, we used a commercially available simulation of the circulatory system (SimBioSys, Chicago, IL) to derive the relations

and

The Contractility Multiplier is set equal to a SNS variable in the simulation program. The normal value of the SNS variable is one. With the SNS variable equal to 1, all three multipliers equal 1, and there is no sympathetic stimulation. If the SNS variable/contractility multiplier is altered from 1, so too are the SVR and HR multipliers, according to equations 16a and 16b. The simulation is initially run to steady state, with the SNS variable equal to 1. The amount by which MAP deviates from the normal 100 mm Hg is used to calculate a new target MAP, according to a 61.5% correction of the offset. The SNS variable is then altered such that running the simulation to steady state results in the targeted MAP value.

The RAS, acting over the course of minutes to hours, maintains the correction in MAP initiated by the SNS.³¹ The RAS maintains blood pressure with roughly proportionate increases in PVR and SVR³²; an RAS variable in the simulation is used as a multiplier for both PVR and SVR. By the time the RAS is activated, the SNS has nearly but not fully returned to normal. Residual SNS activation is modeled as the level necessary to make the percent elevation in HR 21% of the percent elevation in PVR.^32,33 In the mid-term simulations, the SNS variable is manually reset to 1 and the residual level of SNS activity is calculated automatically based on the value of the RAS variable. Target MAP for the mid-term RAS simulations is the same as for the short-term SNS simulations.³¹

Only the VOL mechanism, acting over the course of days to weeks, returns blood pressure to normal through the retention of salt and water by the kidneys.³¹ Blood volume is defined implicitly in the model, determined by pressure and compliance. A VOL variable is used as a multiplier for all initial pressures input to the simulation. The RAS, unlike the SNS, does not fatigue over time,³¹ so its setting from the mid-term simulation is maintained in the long-term simulation. The VOL variable is used to return MAP to the normal 100 mm Hg.

The model is run from initial conditions to steady state four times to approximate the time course of blood pressure control. Blood pressure control is thus modeled in an open-loop fashion. The simulation is first run without any blood pressure control mechanism and then three additional times, using blood pressure control mechanisms to simulate the short-, mid-, and long-term responses to the change in MAP of the original run.

Disease States

In addition to the normal physiologic state, four disease states are simulated (Table 4). To simulate the hemodynamic disturbances associated with acute pulmonary disease, the G_PC-Q_NL curve is shifted downward,

Table 4: Disease States Simulated

such that PVR is about five times normal^34,35 and pulmonary arterial compliance is reduced to 70% of normal.³⁶ To simulate acute pulmonary disease with the administration of inotropic agents, contractility is increased by one third^30,37 in all cardiac chambers in addition to the changes of the acute disease state. To simulate right ventricular hypertrophy in chronic pulmonary disease, the contractility of the RV alone is increased by a third in addition to the changes of the acute disease state. The last disease state is a model of chronic pulmonary disease with severely elevated PVR, such as can occur in primary pulmonary hypertension.³² The G_PC-Q_NL curve is shifted further downward,

such that PVR is about 11 times normal, in addition to the changes of the chronic pulmonary disease state.

In acute disease and acute disease with inotropic treatment, the long-term blood pressure response to the pathophysiology is a model of disease progression without treatment, to which results from TAL attachment simulations can be compared. The simulated pathology is acute, however, and we simulate the case in which the TAL is attached soon after disease onset. The blood pressure control mechanisms are set back to a baseline value of one before the TAL attachment configuration simulations are begun.

Implicit in the model of RV hypertrophy, in the simulated chronic disease states is an advanced pathophysiologic state to which the long-term blood pressure control mechanism has responded. In the chronic disease states, only the SNS mechanism, which fatigues, is reset to its baseline value of one before simulation of TAL attachment. The RAS and VOL mechanisms are left at their levels from the long-term pathophysiologic simulation when the TAL attachment simulations are begun.

Experimental Verification and Prediction Simulation Protocol

The computational model was first verified against results from a series of porcine experiments. The prototype TAL and experimental details are described elsewhere.^4,5 Briefly, the animal was anesthetized, the chest was opened by left thoracotomy, and 1.8-cm inner diameter Gore-Tex grafts were sewn, end-to-side, to the proximal main PA, distal main PA and LA. The TAL inlet was attached to the proximal PA graft. A Y-connector at the TAL outlet was attached to the distal PA and LA grafts. Selective banding of the main PA section between the two graft anastomoses and of the TAL outlet grafts enabled alteration of TAL attachment configuration. Hemodynamic data were recorded at baseline and with the TAL attached for 20 min in each of four configurations: full series–1–1, hybrid–1–2/3, hybrid–2/3–2/3, and parallel–2/3–1/3, where the fractions after each configuration indicate the fraction of CO directed through the TAL and NLs, respectively. The prototype TALs were not sterilized. The unsterilized TALs caused an inflammatory response in the pigs, resulting in a moderately elevated PVR of 3.3 ± 1.3 (SD) mm Hg/(l/min).

The parameters of the prototype TAL⁴ were used for model verification (Table 3), and the same attachment configurations as were tested experimentally were simulated (Table 5). The reported pressure drops across the TAL inlet and outlet were corrected for pressure changes caused by Bernoulli effects to yield energy losses across the inlet and outlet. The compliance of the prototype TAL housing was linearized for use in the verification simulations. To model the elevated PVR/reduced G_PC of the experiments, the following relation was used in the verification simulations:

Table 5: Simulated Thoracic Artificial Lung Attachment Configurations

This reduction in G_PC decreased MAP from 100 to 90 mm Hg at baseline in the verification simulations. Because baseline MAP was depressed as the result of anesthesia in the experiments,⁵ however, no blood pressure control mechanisms were applied to correct this drop in MAP. Rather, an MAP of 90 mm Hg was used as the set point with respect to which further changes in MAP, caused by simulation of TAL attachment, were corrected by application of the blood pressure control mechanisms.

The prototype TAL used experimentally had strengths and weaknesses. Bundle resistance was low and housing compliance high; however, minor losses at the inlet and outlet were high, and there was no separate inlet compliance chamber proximal to the TAL inlet. After verifying the model, parameters for a feasible, improved TAL (Table 3) were used in the prediction simulations that are the purpose of this study. The porcine experiments also revealed some advantages and disadvantages of TAL attachment configurations.⁵ Full series–1–1 and hybrid–1–2/3 delivered total CO to the TAL for gas exchange but were tolerated by the RV only with difficulty. Hybrid–1–2/3 would be improved by allowing greater shunt around the NLs to unload the RV. Hybrid–2/3–2/3 was experimentally challenging and provided less benefit than anticipated. A partial series configuration with partial CO to the TAL and total CO to the NLs might be suitable in some cases. For the prediction simulations, we tested the configurations: full series–1–1, partial series–1/2–1, hybrid–1–2/5, and parallel–2/3–1/3 (Table 5). We simulated TAL attachment in all four configurations in each of the four disease states. For each combination of TAL attachment configuration and disease state, we applied the complete series of blood pressure control mechanisms.

Results

Model Validation

Verification simulations: model validation against experimental data.

Model-predicted hemodynamic responses to TAL attachment are compared with results from porcine experiments⁵ in Figure 4. Experimental results for MAP, PSR, CO, and mean PA pressure are presented from three animals as mean ± standard deviation. Pulmonary system resistance is the resistance of the combined TAL/NL system and is equal to (mean PA pressure – mean LA pressure)/CO. The attachment configurations investigated experimentally were full series–1–1 (FS), hybrid–1–2/3 (HII), hybrid–2/3–2/3 (HI), and parallel–2/3–1/3 (Pa), where the fractions after each configuration indicate the fraction of CO passing through the TAL and NLs, respectively. The data used were obtained at the end of a 20-minute period, during which each configuration was assessed. The data are compared with results of simulations run with the short- and mid-term blood pressure control mechanisms applied.

Figure 4.: Model verification against experimental data. a, Mean systemic arterial pressure, MAP; b, pulmonary system resistance, PSR; c, cardiac output, CO; d, mean pulmonary artery pressure, MPAP. All values normalized by baseline (BL) values. Data (mean ± SD) from 20-minute time point into each artificial lung attachment configuration plotted with model predictions for short- and mid-term blood pressure control effected by the sympathetic nervous system (SNS) and renin-angiotensin system (RAS), respectively. Pulmonary vascular resistance mildly elevated in model to match experimental condition. Attachment configurations: BL, full series (FS), hybrid II (HII), hybrid I (HI), and parallel (Pa).

Mean systemic arterial pressure was used as the set point for the simulations. A 61.5% correction in MAP offset was applied in the short- and mid-term simulations. Agreement between the experimental and simulated results validates our use of this blood pressure control strategy. The model simulates change in PSR well, indicating accurate simulation of the pulmonary hemodynamic environment with TAL attachment. The model mildly underpredicts CO, except in the full series configuration, and provides a reasonable estimate of mean PA pressure.

Further, we used the model to approximate the difference between oxygen supply to and consumption by the RV for each experimental condition. With simulation of the experimentally tested prototype TAL, the So₂-Vo₂ difference was 0.81 ml O₂/min at baseline and –0.84, –0.70, –0.11, and 0.88 mL O₂/min in FS, HII, HI, and Pa, respectively, with application of renin-angiotensin blood pressure control. Experimentally, FS and HII were challenging to the RV.

Prediction Simulations: Predicted Results for a Redesigned TAL

Response to TAL attachment for acute pulmonary disease.

Figure 5 shows MAP, PSR, CO, and mean PA pressure results for a model of acute pulmonary disease. Within each graph are shown, from left to right, results for the physiologic state (Ph), pathophysiologic state (PPh), full series–1–1 (FS) TAL attachment, partial series–1/2–1 (PS) TAL attachment, hybrid–1–2/5 (H) TAL attachment, and parallel–2/3–1/3 (Pa) TAL attachment. No blood pressure control was needed for the physiologic state. For the pathophysiologic state and all TAL attachment configurations, four bars are shown. The first bar (white) shows the response to disease or TAL attachment in the absence of blood pressure control. The shaded bars show the effects of applying the blood pressure control mechanisms: short-term sympathetic nervous system activation (light gray), mid-term renin-angiotensin activity (dark gray) and long-term kidney volume control (black).

Figure 5.: Simulation results for acute pulmonary disease state. a, Mean systemic arterial pressure, MAP; b, pulmonary system resistance, PSR; c, cardiac output, CO; d, mean pulmonary artery pressure, MPAP. Attachment configurations: physiologic (Ph), pathophysiologic (PPh), full series (FS), partial series (PS), hybrid (H), and parallel (Pa). Blood pressure control: none, short-term sympathetic nervous system (SNS), mid-term renin-angiotensin system (RAS), and long-term kidney blood volume control mechanism (VOL).

The four bars for the pathophysiologic condition represent the response to the diseased pulmonary system. If the disease state did not develop instantaneously, the SNS would not be expected to play a large role; the second bar of the group, representing SNS activity, would be unimportant. The third and fourth bars, representing RAS and VOL activity, respectively, depict mid- and long-term adaptations to untreated disease.

Mean systemic arterial pressure was used as a set point for the simulations. Artificial lung attachment typically adds resistance to the pulmonary system, causing RV output and LV preload to decrease. Decreased LV preload leads in turn to a decrease in MAP. Sympathetic nervous system activity corrects the change in MAP by 61.5%, and RAS activity maintains that correction. The VOL system, with its infinite gain, returns MAP to 100 mm Hg.

Series attachment adds the greatest resistance to the pulmonary circulation, causing an increase in PSR above the pathophysiologic level. In partial series, PSR exceeds the pathophysiologic level. In hybrid, PSR equals the pathophysiologic level. Parallel attachment reduces PSR below the pathophysiologic level, nearly to the normal physiologic level. The increase in PSR in full series reduces CO and elevates mean PA pressure. The decrease in PSR in parallel increases CO and reduces mean PA pressure toward normal physiologic levels. The increase in total vascular volume in the long term tends to increase both CO and mean PA pressure above mid-term levels.

Figure 6 shows RV pressure-volume loops for the acute disease state with mid-term blood pressure control. The more severe TAL attachment configurations of full series and hybrid markedly increase maximum RV pressure. These configurations also increase average RV volume, indicating RV congestion, while decreasing RV stroke volume. Parallel TAL attachment returns the RV pressure-volume loop from the pathophysiologic state nearly to the normal physiologic state. These changes in the RV pressure-volume loop cause changes in the RV pressure volume area (Figure 3), to which RV oxygen consumption is correlated (equation 15). The PVA is 1.6 mm Hg·l in the physiologic state. The acute disease state with mid-term blood pressure control increases the PVA to 2.3 mm Hg·l. Partial series, hybrid, or full series TAL attachment with mid-term blood pressure control further raise the PVA to 2.2, 2.4, or 2.5 mm Hg·l, respectively. Parallel TAL attachment with mid-term blood pressure control reduces the PVA to 1.6 mm Hg·l.

Figure 6.: Right ventricular pressure-volume loops. Results are shown for mid-term blood pressure control in acute disease state. Attachment configurations: physiologic (Ph), parallel (Pa), pathophysiologic (PPh), partial series (PS), hybrid (H), and full series (FS).

Right ventricular congestion in full series and hybrid is accompanied by LV underfilling (data not shown in figure), indicating an imbalance in fluid distribution throughout the circulatory system. In the normal physiologic state, mean RV and LV volumes are both 64 ml, thus the ratio of RV to LV volume is 1.0. With long-term blood pressure control in the acute, acute plus inotropes, chronic and chronic plus severe PVR disease states, mean RV volume is 86, 74, 74, and 97 ml, respectively, and mean LV volume is 53, 50, 58, and 48 ml, respectively. The acute disease state thus increases the RV to LV volume ratio to 1.62. Inotropic therapy decreases the ratio to 1.49. Selective RV hypertrophy in the chronic disease state is more effective in combating the imbalance and reduces the ratio to 1.29. The chronic disease state with severely elevated PVR increases the ratio to 2.03. The TAL attachment configurations have as marked an effect as the disease states. Compared with the long-term pathophysiologic ratio of 1.62 in the acute disease state, full series TAL attachment increases the ratio to 2.39 in the long-term, whereas parallel TAL attachment reduces it toward the physiologic level to 1.15.

Figure 7 shows flow rates in the pulmonary system. In TAL attachment configurations such as parallel, in which banding only partially occludes the PA, the majority of the RV output passes through the PA into the compliance of the pulmonary arterial system during systole. In diastole, a portion of that flow reverses, passing retrogradely through the PA and entering the TAL inlet graft. The compliance of the NLs enables large RV output when the PA is patent. In contrast, in TAL attachment configurations such as hybrid, in which the PA is fully occluded, all RV output must enter the TAL inlet graft directly. In hybrid, maximum RV output is markedly reduced but significant RV output is sustained throughout the second half of systole. Overall, CO is less in hybrid than in parallel.

Figure 7.: Pulmonary system flow rates. a, Mid-term response to parallel artificial lung attachment in acute disease state; b, mid-term response to hybrid artificial lung attachment in acute disease state. Flow rates: Pulmonary valve (PVlv), pulmonary artery section between two graft anastomoses (PA), inlet graft to artificial lung (IG), pulmonary capillaries (PC), and pulmonary valve in physiologic state without artificial lung (PVlv, phys).

The minor loss at the TAL inlet graft anastomosis to the PA increases with increasing flow rate. Additionally, this minor loss is located proximal to the compliance of the inlet compliance chamber. With full PA occlusion, the inlet anastomosis minor loss markedly reduces RV output. With parallel TAL attachment in the acute disease state, the average resistances of the inlet PA anastomosis and TAL are 0.64 and 0.74 mm Hg/(l/min), respectively, in the mid-term (data not shown in figure). With hybrid TAL attachment, in contrast, the resistances of the inlet PA anastomosis and TAL are 1.2 and 0.95 mm Hg/(l/min), respectively. Passage of total, albeit reduced, CO through the TAL in hybrid increases both the anastomosis and TAL resistances. The anastomosis resistance is more sensitive to flow rate, however. With total CO entering the inlet graft and TAL in hybrid, the anastomosis resistance exceeds that of the entire TAL.

Response to TAL attachment for chronic disease state with severe PVR.

Figure 8 shows MAP, PSR, CO, and mean PA pressure for a chronic pathology with severely elevated PVR. Both hybrid and parallel TAL attachment unload the RV compared with the pathophysiologic state, but parallel more so than hybrid. In this extreme disease state, parallel TAL attachment is particularly beneficial in returning pulmonary hemodynamics toward the physiologic state.

Figure 8.: Simulation results for chronic pulmonary disease with severely elevated pulmonary vascular resistance. a, Mean systemic arterial pressure, MAP; b, pulmonary system resistance, PSR; c, cardiac output, CO; d, mean pulmonary artery pressure, MPAP. Attachment configurations: Physiologic (Ph), pathophysiologic (PPh), full series (FS), partial series (PS), hybrid (H), and parallel (Pa). Blood pressure control: none, short-term sympathetic nervous system (SNS), mid-term renin-angiotensin system (RAS), and long-term kidney blood volume control mechanism (VOL).

Effects of alternative TAL design features.

Artificial lung design details, in addition to attachment configuration, affect pulmonary system hemodynamics. A single design improvement marginally decreases PSR. With hybrid TAL attachment for acute respiratory disease, for instance, halving the TAL inlet and outlet resistances reduces PSR 6% in the long term (data not shown). Halving the resistances of the proximal and distal PA anastomoses, on the other hand, reduces PSR by 13%. The two types of improvements together have a larger effect. Halving all four resistances at once, TAL inlet and outlet and proximal and distal anastomoses, reduces PSR by 20%. The corresponding increases in CO for these design changes are 2%, 6%, and 8%, respectively. The corresponding decreases in mean PA pressure are 3%, 5%, and 9%, respectively.

Effect of TAL attachment on RV oxygen supply and demand.

Figure 9a shows, for acute respiratory disease, traces for the driving pressure in the right coronary arteries. The hemodynamically severe attachment configurations of full series and hybrid each simultaneously decrease MAP and increase right ventricular pressure (RVP) during ventricular systole. These changes both tend to decrease the driving pressure in the right coronary arteries. Parallel TAL attachment, in contrast, by elevating MAP and decreasing RVP, returns driving pressure in the right coronary arteries toward the normal physiologic state.

Figure 9.: Oxygen supply to and oxygen balance for right ventricle. a, Driving pressure in right coronary arteries (ΔP_{R, corr}). Data are shown for mid-term response to artificial lung attachment in acute disease state. b, Difference between oxygen supply to (So₂) and consumption by (Vo₂) right ventricular free wall. Data are shown for acute disease state. Attachment configurations: Physiologic (Ph), parallel (Pa), pathophysiologic (PPh), partial series (PS), hybrid (H), and full series (FS).

Figure 9b indicates the difference between oxygen supply to (Figure 9a) and consumption by (Figure 6) the RV free wall. When this difference is positive, the oxygen demands of the RV free wall are being met, but when it is negative, there is potential for ischemia. In the physiologic state, the RV has a reserve of 1.9 ml/min of oxygen. In the pathophysiologic state, the balance is negative with SNS activity in the short term but becomes positive in the mid and long terms. With TAL attachment, the balance is likewise generally negative with SNS activity in the short term, due to an elevated heart rate of, for example, 166 beats/min with full series TAL attachment. Overstimulation of the SNS in the short term can be minimized, however, with a gradual transition to flow through the TAL. Over time, as the SNS fatigues, the potential for ischemia decreases markedly. In the mid and long terms, the balance is negative in full series, marginally positive in partial series and hybrid, and at a nearly physiologic positive level in parallel.

In the chronic disease state (data not shown), increased RV contractility models RV hypertrophy and causes an increase in RV output. In the short term, increased RV contractility moderates the SNS response and increases the So₂-Vo₂ difference parameter above the levels in the acute disease state. In the mid and long terms, however, the So₂-Vo₂ difference parameter is less in the chronic than in the acute disease state. Hypertrophy thus increases RV power and output but leaves the RV with reduced energy reserves.

Effect of TAL attachment on blood flow rates.

The full series and hybrid attachment configurations are hemodynamically challenging to the RV. Because of the passage of total CO through the TAL, however, these configurations may provide the greatest oxygenation potential. Figure 10 shows flow rates through the TAL and NLs. Results are shown for TAL attachment in all four disease states. Hybrid provides the greatest TAL flow rate: 3.6 and 4.1 l/min in the mid and long term, respectively, in the acute disease state and between 3.9 and 4.6 l/min in the mid and long terms of all other disease states. Full series provides nearly as much TAL flow as hybrid, with the added benefit of total CO passing through the NLs for embolic clearance. Parallel, by increasing total CO, likewise provides significant TAL flow despite passing only two thirds of CO through the TAL. Partial series, with only one half of CO to the TAL, provides markedly less TAL flow than any other configuration. However, partial series, like full series, forces total CO through the NLs for embolic clearance. Increased cardiac contractility, whether caused by inotrope administration or RV hypertrophy in chronic disease, marginally increases flow rates above those in acute disease. Severely elevated PVR decreases TAL flow rates slightly. Flow through the NLs is at least 1.4 l/min in all configurations. In parallel and hybrid with mid- or long-term blood pressure control, flow rate through the NLs ranges from 1.4 to 1.9 l/min. In full or partial series, the NLs receive total CO.

Figure 10.: Blood flow rates through (a) artificial lung, Q_TAL, and (b) natural lungs, Q_NL. Data are shown for four disease states that model acute pulmonary disease, acute pulmonary disease plus inotrope administration, chronic pulmonary disease, and chronic pulmonary disease with severely elevated pulmonary vascular resistance (PVR). Dashed line indicates normal physiologic flow rate through the natural lungs. Attachment configurations: Pathophysiologic (PPh), full series (FS), partial series (PS), hybrid (H), and parallel (Pa). Blood pressure control: None, short-term sympathetic nervous system (SNS), mid-term renin-angiotensin system (RAS), and long-term kidney blood volume control mechanism (VOL).

Discussion

This modeling study was designed to predict the hemodynamic response to TAL use in the treatment of pulmonary disease. It is intended to guide selection of optimum TAL attachment configuration for specific pathologic conditions. Having learned from our series of porcine experiments, we used the model to predict the efficacy of a feasible, redesigned TAL attached to the pulmonary circulation in four promising configurations.

Choice of Attachment Configuration

We chose to simulate the TAL attachment configurations of full series–1–1, partial series–1/2–1, hybrid–1–2/5, and parallel–2/3–1/3 for the following reasons. (i) Full series and hybrid advantageously deliver total CO to the TAL for gas exchange. (ii) Full and partial series pass total CO through the NLs, where emboli that might be generated in the artificial portion of the circuit would likely be cleared. (iii) Parallel, and possibly hybrid, can reduce PSR, unload the RV, and increase CO. In the parallel and hybrid configurations that were simulated, the blood flow rate through the NLs was at least 25% of baseline CO. This reduction in NL flow may be tolerated.¹¹ Further, with activation of the kidney volume control mechanism, all flow rates including that through the NLs would increase over time. If at any time more NL flow were required, however, a greater degree of TAL outflow could be routed back to the NLs. The drawback to forcing more flow through the NLs is a reduced ability to unload the RV. Unloading the RV is particularly important with pathologically elevated PVR.

Although both hybrid and parallel TAL attachment allow significant NL bypass, only parallel reduced PSR to 2.0 mm Hg/(l/min) from 5.3 mm Hg/(l/min) in the model of acute pulmonary disease. Hybrid did not alter PSR from the pathophysiologic state (Figure 5b). Parallel had the advantage of maintaining RV access to the NL compliance, which tended to reduce PSR and increase CO (Figures 5 and 7). Hybrid might benefit from maintaining a small flow through the PA to allow say 10% of CO to pass directly from the RV to the NLs. We attempted a similar hybrid configuration in our porcine experiments.⁵ That configuration was complicated, however, by the tendency for retrograde shunt flow through the distal PA graft to the TAL outlet and from the TAL outlet directly to the LA, bypassing both the TAL and the NLs. The other significant difference between parallel and hybrid was that greater flow to the TAL in hybrid increased the resistance of the minor loss at the inlet graft anastomosis. The resistance of the proximal PA anastomosis was problematic in our experimental studies, as well.⁵ Reduction of this inlet minor loss would benefit the hemodynamics of all attachment configurations, and particularly those such as full series and hybrid in which total CO is diverted to the TAL. With reduced resistance of the inlet minor loss and/or TAL, hybrid could partially unload the RV.

As indicated above, the design of the proximal anastomosis is especially important if TALs are to be widely applicable. In full series and hybrid, the proximal PA anastomosis resistance exceeded the TAL resistance. Even in parallel, the proximal anastomosis resistance magnitude was 86% that of the TAL. Angled graft attachment to the PA might reduce the “minor loss” at the proximal anastomosis. There have been numerous studies of angled anastomoses of coronary bypass grafts.^38–40 Most of the studies are of the distal anastomosis, where both a smaller attachment angle and the presence of some flow through the bypassed arterial section reduce wall shear stress in the distal attachment region. Because of the Reynolds numbers involved, however, the results of coronary bypass studies are not easily translated into predictions for resistance measurements across PA anastomoses. Thoracic artificial lung use thus would greatly benefit from a detailed study of PA anastomosis geometry.

Predicting RV performance with TAL attachment was a major motivation behind this study. An increase in PSR simultaneously increases RV afterload and, by reducing, in turn, return to the LV, LV preload and systemic arterial pressure, reduces RV perfusion (Figures 5, 6 and 9, and data on RV to LV volume ratio). The RV thus has a decreased energy supply with which to work against an increased afterload. Right ventricular ischemia is not typically a problem in moderate pulmonary disease. In fact, the RV is reported to be generally well protected against ischemia.⁴¹ In our porcine experiments, nevertheless, full series–1–1 and hybrid–1-/23 were not well tolerated by the RV and were difficult to maintain.⁵ Whether the experimental RV difficulty in those configurations was limited by RV strength or by oxygen supply was not determined. However, the present model predicted markedly negative So₂-Vo₂ differences for those two configurations. For the hybrid–2/3–2/3 configuration, which despite experimental complications was tolerated by the RV, the model in contrast predicted a marginally negative So₂-Vo₂ difference. These results validate the So₂-Vo₂ difference as an index of RV function. Thus, the RV should tolerate the hybrid–1–2/5 configuration modeled with a redesigned TAL in our predictive simulations, which had a marginally positive So₂-Vo₂ difference (Figure 9).

Among attachment configurations tolerated by the RV, the optimal configuration would be that which delivers the greatest blood flow to the TAL. With severely impaired NL oxygenation, flow rate through the TAL will determine the gas exchange capability of the combined TAL/NL system. Hybrid TAL attachment provided 3.6 l/min of blood flow to the TAL in the mid-term response to acute pulmonary disease and at least 4 l/min in all other cases. Full series and parallel each provided at least 3 l/min to the TAL. With severe disruption of NL oxygenation, a TAL flow rate of 4 l/min would be desirable for support of about 80% of the basal gas exchange requirement.⁴² The model underpredicts CO compared with the values measured in porcine experiments (Figure 4). The predicted TAL flow rates thus are conservative, and hybrid may provide 4 l/min to the TAL even in the mid-term with acute pulmonary disease. Partial series was included in these simulations for its embolic clearance potential. Without significant residual NL oxygenation, however, partial series probably would not support significant gas exchange. Hybrid, which supplied the greatest flow to the TAL and had a PSR between that of full series and parallel, should feasibly support significant gas exchange without overloading the RV. Further, any decrease in inlet anastomosis resistance or TAL resistance may enable hybrid to partially unload the RV in a manner similar to parallel. Thus, hybrid is a compromise that provides good gas exchange and acceptable hemodynamic alterations.

Modeling Considerations

The present model simulated the blood pressure control activities of the pressor receptors, the renin-angiotensin enzyme pathway, and the kidneys, which would come into play with altered vascular blood pressures and volumes. Other physiologic blood pressure control mechanisms were not incorporated into the model.

Because the model did not incorporate blood gas levels, neither of the two blood pressure control mechanisms sensitive to blood gas levels were included. These mechanisms are chemosensor activity and local blood flow regulation. Chemosensors tend to act in unison with baroreceptors, which were included in the model, and the two systems are somewhat redundant.²³ Omission of chemosensors probably is not significant.

Omission of local blood flow regulation may be a more crucial shortcoming. Decreased CO causes tissue hypoxia and, in turn, decreased systemic vascular resistance. The decrease in SVR increases CO. Local tissue flow control thus minimizes change in CO via negative feedback. The model predicts that an increase in RV afterload causes a decrease in CO. The lack of local blood flow regulation, however, may contribute to an under prediction of CO by the model (Figure 4c).

Additionally, proper function of the renin-angiotensin system is dependant on sufficient flow through the natural lungs. The potent vasoconstrictor angiotensin II is activated principally in the lungs.²³ In attachment configurations that minimize natural lung flow, particularly parallel and hybrid, the model may overestimate the ability of the renin-angiotensin system to increase blood pressure. Despite these shortcomings, however, the Guyton et al.³¹ model, on which we based the blood pressure control mechanisms in the present model, describes well the MAP data from our porcine experiments (Figure 4a).

We used the So₂-Vo₂ difference as an index of RV function. This index had limitations. Autoregulation, or the tendency of the coronary circulation, like most organ circulations, to minimize fluctuation of flow in response to fluctuation in pressure, was not modeled. The supply of oxygen thus might not drop as much as predicted by the model in response to the decrease in MAP in the short and mid terms. On the other hand, oxygen saturation, used in calculating supply of oxygen to the RV, was assumed to be 99% (equation 13). Oxygen saturation, however, would vary with attachment configuration and would decrease in a configuration such as partial series that had a lower TAL flow rate. The oxygen balance might be more negative than predicted in partial series. The oxygen balance thus was intended to suggest trends, not to predict definitively the onset of ischemia. Nevertheless, our use of this index was supported by correlation between experimental RV dysfunction and markedly negative So₂-Vo₂ difference values in our validation simulations.

Finally, the model includes resistances and compliances but does not include fluid inertia. Fluid inertia is not thought to play a large role in the compliant, low-resistance, natural pulmonary circulation. In our porcine studies, however, the TAL/NL system included a noncompliant TAL inlet graft with significant inertial impedance. With full PA occlusion in hybrid or series, this inertia, along with the inlet graft minor loss, was located proximal to all compliance in the TAL/NL system. Incorporating compliance into the inlet graft, as was done in an earlier study,⁸ should mitigate the effect of graft inertia. In the present model, peak flow rates through the pulmonary (Figure 7) and aortic (data not shown) valves were mildly elevated above physiologic levels. These high peak flow rates were not necessary for simulation of physiologic pressures in the natural circulatory system. They were necessary, however, to avoid underprediction of CO and mean PA pressure with TAL attachment in the verification simulations. Proximal inertia may increase peak pressure at the start of systole, when it opposes cardiac ejection, and may increase flow rate later in systole, when it contributes to flow and resists a decrease in flow. Had inertia been included in the model, sufficiently high pressures and flows might have been generated with lower peak ventricular ejection rates. Nevertheless, the model produced generally accurate pressures and flows and trends in pressures and flows when compared with both the physiologic circulatory system and the TAL attachment configurations in our experimental study.

Conclusion

This model is intended to guide selection of TAL attachment configuration in response to specific pulmonary hemodynamic and gas exchange pathology. The model suggests that parallel implantation is hemodynamically best in that it unloads the RV and increases CO. Hybrid TAL attachment with total CO to the TAL and 40% of CO to the NLs, however, provides the greatest TAL blood flow rate. Hybrid would thus maximize oxygen delivery while maintaining acceptable hemodynamics that should be tolerated by the RV. Across all TAL attachment configurations, a large “minor loss” resistance at the inlet graft anastomosis to the PA is problematic. Angled graft attachments to the PA and further reductions in minor losses at the TAL inlet and outlet would enable hybrid also to unload partially the RV. Additionally, right ventricular ischemia is a concern in full series and with sympathetic stimulation in any TAL attachment configuration; change in configuration should be made gradually to avoid excessive sympathetic stimulation. A hypertrophied RV in chronic lung disease bolsters cardiac output and partially re-equilibrates fluid volume between right and left ventricles but leaves the RV with less oxygen reserve. The model suggests that a TAL attached to the pulmonary system in a hybrid configuration would be tolerated by the right ventricle and capable of supplying significant gas exchange.

Acknowledgments

This study was supported by National Institutes of Health grant R01 HL 59537. The authors are grateful to Dr. Anthony Makarawicz for his consultation on the early development of this model and Dr. Keith Cook for his assistance in determining the hemodynamic parameters of the artificial lung.

Appendix

Abbreviations and Symbols

BL, Baseline state, verification simulations; C, compliance; C_D, diastolic cardiac chamber compliance; C_max, maximum cardiac chamber compliance; C_min, minimum cardiac chamber compliance; C_S, systolic cardiac chamber compliance; CO, cardiac output; COMP, compliance chamber proximal to artificial lung; E, Young’s modulus; E_max, maximum elastance of cardiac chamber; E_p, pressure-strain elastic modulus; ΔE_Ad, mechanical energy loss across distal pulmonary artery anastomosis; ΔE_Ap, mechanical energy loss across proximal pulmonary artery anastomosis; EDPVC, end-diastolic pressure-volume curve; ESPVL, end-systolic pressure-volume line; FS, full series artificial lung attachment configuration, verification and prediction simulations; G, conductance; G_A, conductance of proximal or distal pulmonary artery anastomosis; G_graft, total conductance of artificial lung inlet or outlet graft; G_PC, conductance of pulmonary capillaries; G_Pois, Poiseuille conductance of artificial lung inlet or outlet graft; h, wall thickness; H, hybrid artificial lung attachment configuration, prediction simulations; HI, hybrid artificial lung attachment configuration, verification simulations; HII, hybrid artificial lung attachment configuration, verification simulations; HR, heart rate; I-1 and I-2, intercepts in Equation 15a; IG, artificial lung inlet graft; k, reciprocal of time constant for cardiac chamber contractility; k_diastole, reciprocal of diastolic time constant for cardiac chamber; k_ml, “minor loss” coefficient for artificial lung inlet or outlet; k_systole, reciprocal of systolic time constant for cardiac chamber; l, length; LA, left atrium or left atrial; LV, left ventricle or left ventricular; μ, viscosity; MAP, mean systemic arterial pressure; MPA 1, 2 or 3 main pulmonary artery section 1, 2 or 3; MPAP, mean pulmonary artery pressure; NL, natural lung; P, pressure; ΔP, pressure drop; ΔP_L.cor, pressure drop in left coronary arteries; ΔP_R.cor, pressure drop in right coronary arteries; Pa, parallel artificial lung attachment configuration, verification and prediction simulations; PA, pulmonary artery; PC, pulmonary capillaries; Ph, physiologic model, prediction simulations; PPh, pathophysiologic model for prediction simulations; PS, partial series artificial lung attachment configuration, prediction simulations; PSR, pulmonary system resistance; PVlv, pulmonary valve; PV, pulmonary veins; PVA, right ventricular pressure volume area; PVR, pulmonary vascular resistance; Q, blood flow rate; Q_Gin, blood flow rate through artificial lung inlet graft; Q_GoutPA, blood flow rate through artificial lung outlet graft to distal pulmonary artery; Q_L.cor, blood flow rate through left coronary artery; Q_NL, blood flow rate through natural lungs; Q_R.cor, blood flow rate through right coronary artery; Q_TAL, blood flow rate through artificial lung; Q_TALin, blood flow rate through artificial lung inlet; Q_TALout, blood flow rate through artificial lung outlet; r, radius; R_cor, resistance of total coronary circulation; RAS, renin-angiotensin system; RV, right ventricle or right ventricular; RVP, right ventricular pressure; So₂, oxygen supply to right ventricular free wall; SA, systemic arteries; SNS, sympathetic nervous system; SV, systemic veins; SVR, systemic vascular resistance; t, time; T, cardiac period; T_D, duration of diastole, atrial or ventricular; T_S, duration of systole, atrial or ventricular; TAL, thoracic artificial lung; TAL I or II, portion of artificial lung housing compliance proximal or distal to gas exchange bundle; V, volume; Vo₂, oxygen consumption by right ventricular free wall; VOL, kidney blood volume control mechanism.

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