A framework for heart-lung interaction and its application to prone position in the acute respiratory distress syndrome

While both cardiac output (Qcirculatory) and right atrial pressure (PRA) are important measures in the intensive care unit (ICU), they are outputs of the system and not determinants. That is to say, in a model of the circulation wherein venous return and cardiac function find equilibrium at an ‘operating point’ (OP, defined by the PRA on the x-axis and Qcirculatory on the y-axis) both the PRA and Qcirculatory are, necessarily, dependent variables. A simplified geometrical approximation of Guyton’s model is put forth to illustrate that the independent variables of the system are: 1) the mean systemic filling pressure (PMSF), 2) the pressure within the pericardium (PPC), 3) cardiac function and 4) the resistance to venous return. Classifying independent and dependent variables is clinically-important for therapeutic control of the circulation. Recent investigations in patients with acute respiratory distress syndrome (ARDS) have illuminated how PMSF, cardiac function and the resistance to venous return change when placing a patient in prone. Moreover, the location of the OP at baseline and the intimate physiological link between the heart and the lungs also mediate how the PRA and Qcirculatory respond to prone position. Whereas turning a patient from supine to prone is the focus of this discussion, the principles described within the framework apply equally-well to other more common ICU interventions including, but not limited to, ventilator management, initiating vasoactive medications and providing intravenous fluids.


Introduction
Though evidence of benefit has existed for placing patients with moderate-tosevere acute respiratory distress syndrome (ARDS) in the prone position for some time, the coronavirus pandemic raised clinical awareness of this maneuver (Guérin et al., 2020). Guidelines currently recommend prone position for patients with ARDS and a partial pressure-to-fraction of inspired oxygen (P a O 2 /F i O 2 ) ratio of not more than 150 mmHg (Papazian et al., 2019). Furthermore, with this ARDS severity, patients should maintain the prone position for at least 12 h per day for optimal benefit (Guérin et al., 2013).
Turning a patient from the supine to prone position has salutary benefits on gas exchange as oxygenation and carbon dioxide elimination are both enhanced (Guérin et al., 2020). The mechanisms by which the prone position exerts its salubrious effects are OPEN ACCESS EDITED BY manifold. When the dorsal, de-gassed 'sponge lung' (Bone, 1993) is no longer gravity-dependent, it is recruited and the surface area for gas exchange increased. Critically, the newly-enlisted alveoli see no significant change in pulmonary blood flow (Henderson et al., 2013); as a consequence, the burden of low ventilation-to-perfusion (V/Q) lung units is reduced. In addition to alveolar recruitment, shifting to the prone position improves 'shape matching' between the pulmonary parenchyma and the chest wall (Gattinoni et al., 2013). In total, the result is that there is less pulmonary inhomogeneity (Cressoni et al., 2014;Cressoni et al., 2015) and, therefore, fewer 'stress-raisers' (Mead et al., 1970) that amplify radial traction forces upon the lungs and pulmonary vasculature (Broccard et al., 1998;Marini et al., 2003;Repessé et al., 2016). Furthermore, stiffening the chest wall with improved pulmonary compliance diminishes trans-pulmonary pressure (P TP ) as the pleural pressure is raised for any given airway pressure (Marini and Gattinoni, 2021). This reduces the mechanical power applied to the pulmonary parenchyma and mitigates West zone 1 and 2 conditions (Gattinoni and Quintel, 2016). All of the aforementioned changes in pulmonary physiology (i.e., improved oxygenation and carbon dioxide elimination, optimized perivascular pulmonary mechanics, diminished P TP ) minimize the afterload experienced by the right ventricle (RV), giving weight to the motto: 'what's good for the lung is good for the RV (Repessé et al., 2016). ' While the literature is replete with elegant investigations into the mechanical pulmonary pathophysiology of ARDS in both supine and prone positions, comparatively little is known about the hemodynamic effects. With a recent investigation exploring the determinants of venous return in the prone position (Lai et al., 2021) and an excellent related review (Lai et al., 2023), this overview will expand upon relevant concepts in clinical hemodynamics, propose a simplified geometrical model clarifying the determinants of cardiac output and right atrial pressure and then relate this to what is currently known about prone position in the ARDS patient (Table 1).

Venous return
The determinants of venous return from the peripheral circulation are, from Guyton's experiments: 1.) the mean circulatory filling pressure (P MCF ), 2.) right atrial pressure (P RA ) and 3.) the resistance to venous return (R VR ) (Guyton, 1955;Sylvester et al., 1983;Jacobsohn et al., 1997;Feihl and Broccard, 2009a). Together the P MCF and the P RA define the pressure gradient for venous return.

Introduction
A cursory overview of the mechanical effects of prone position (PP) on the injured lung. PP recruits both airspace and pulmonary vasculature. This improves pulmonary mechanics, gas exchange and reduces right ventricular outflow impedance Guyton primer The venous return (VR) subsection describes the: 1.) pressure gradient for VR (i.e., P MSF -P RA ), 2.) resistance to VR (R VR ) and 3.) how both (P MSF -P RA ) and R VR together describe VR.
The Guyton diagram subsection describes how VR and cardiac function form an equilibrium-the operating point (OP)-which is the dependent variable in the Guyton model. As the OP is a dependent variable, so too are its two coordinates (i.e., P RA and Q circulatory ). Thus, contrary to what is commonly taught, P RA is not an independent determinant of Q circulatory (i.e., total circulatory blood flow = venous return = cardiac output) Geometrical model To illustrate how P RA is not a determinant of Q circulatory , a simplified geometrical model is derived; the independent variables of the circulation are shown to be: 1.) P MSF, 2.) the pericardial pressure (P PC ), 3.) R VR and 4.) 'cardiac resistance' (R cardiac ) The circulation can be 'cardiac-' or 'venous-limited.' The former is synonymous with preload unresponsiveness. When the circulation is 'cardiac-limited', changing P MSF or R VR only alters P RA with no effect on Q circulatory . When 'venous-limited', changing cardiac function (R cardiac ) or P PC only alters P RA with no effect on Q circulatory from the sense that when the heart recommences circulatory flow, pressure in the arteries rise up from the P MCF while the pressure in the downstream veins fall below it; thus, the P MCF acts as a quasistatic 'pivot' around which pressures upstream and downstream rise and fall, respectively (Broccard, 2012). As discussed below, the P MCF is similar, but not equivalent to, the mean systemic filling pressure (P MSF ). The P MSF excludes the contributions of intrathoracic blood volume and compliance. The P MCF (or P MSF ) is determined by two related-and often confused-biophysical properties: capacitance and compliance (Rothe, 1986;Rothe, 1993;Tyberg, 2002). To understand capacitance, the reader must appreciate that the total circulatory volume is comprised of two distinct (though dynamic) 'types' of volume-the unstressed (V US ) and stressed (V S ) volumes (Gelman, 2008;Magder, 2012). The V US does not create a vascular elastic recoil pressure while the V S does. As an analogy, filling a waterbed requires water volume before the walls are stretched (i.e., the V US ); further volume generates a recoil pressure from the elastic walls (i.e., the V S ). As compared to a water balloon, a waterbed has a much larger capacitance because its V US is greater than the V US of the balloon. Compliance and its inverse, elastance, describe the relationship between changing vascular volume and changing recoil pressure (Rothe, 1993). It follows that compliance (or elastance) pertain to the V S ; the V US , by definition, generates no change in pressure (i.e., the V US has infinite compliance or zero elastance). Continuing with the analogy above, were the waterbed made from a poorly elastic (i.e., stiff) material, it would have a large capacitance, but low compliance (or high elastance). If the water balloon was made of a highly elastic material, it would have a low capacitance, but high compliance (or low elastance). Mathematically, the P MCF is determined by the volume of blood generating a recoil pressure (i.e., the V S , which is determined by total vascular volume and capacitance) divided by the vascular compliance (Tyberg, 2002;Magder, 2012) (Figure 1).
As noted above, the P MCF includes the cardiac and pulmonary vascular volumes and compliances (i.e., the total circulation) while the P MSF measures only the extra-thoracic, systemic, circulation (Rothe, 1993); they are very similar in value and often used interchangeably. In clinical practice, the methods for estimating this static, 'pivot pressure' reflect the systemic pressure (i.e., P MSF ) and this measure will be used throughout this review (Berger et al., 2016). For patients, there are three methods to estimate the P MSF : 1.) extrapolation to zero flow of the P RA -cardiac output relationship altered by ventilator-hold maneuvers (Pinsky, 1984;Maas et al., 2009), 2.) extremely rapid cuff insufflation on the arm with an ipsilateral arterial line (Maas et al., 2012) and 3.) mathematical modelling by the method of Parkin and Leaning (Parkin and Leaning, 2008). Though beyond the scope of this discussion, the ventilator-hold and arm-occlusion methods over-estimate P MSF (Maas et al., 2012;Berger et al., 2016) for a variety of reasons (Moller and Berger, 2023) whereas the mean systemic filling pressure analogue (P MSA ) (i.e., the method of Parkin and Illustration of capacitance and compliance. Vessel A has a relatively small capacitance because its unstressed volume (V US A ) is small. The compliance of the vessel (C A ) is the inverse of elastance on this graph; by rearrangement, the recoil pressure generated in this vessel (P A ) is equal to its stressed volume (V S A ) divided by its compliance. Vessel B shows a larger capacitance, but an increased elastance (i.e., reduced compliance, C B ) relative to vessel A. Vessel A and B are analogous to the 'water balloon' and 'waterbed,' respectively, described within the text. V S B is the stressed volume, V US B is the unstressed volume and P B is the recoil pressure of vessel B.

FIGURE 2
Pressure gradient for venous return. The effect of changing mean systemic filling pressure (in millimeters of mercury, mmHg) from a low (P MSF1 ) to a higher value (P MSF2 ) (e.g., volume infusion, decreased capacitance). The slope of the venous return curve is constant between the two curves meaning that the resistance to venous return is constant (see below). For a given right atrial pressure (P RA ), the lower P MSF1 (i.e., reduced pressure gradient for venous return, Δ P VR1 ) causes a diminished venous return on the y-axis (liters per minute, L/min). The same P RA in a system with P MSF2 (i.e., increased pressure gradient for venous return, Δ P VR2 ) results in a higher venous return on the y-axis. The P MSF is the pressure in the right atrium at zero flow (i.e., the x-intercept). V S and C are the stressed volume and average compliance, respectively, of the systemic vasculature. The flattening of the venous return curve is where the great veins collapse; this creates a maximal venous return in each state.
Leaning) accurately estimated absolute and changing values of P MSF in a porcine model (Werner-Moller et al., 2022). Because it is simply calculated from P RA , cardiac output and mean arterial pressure (Parkin and Leaning, 2008;Moller and Parkin, 2022), the P MSA is an attractive tool for guiding both prospective and retrospective research as well as clinical therapy (Moller and Parkin, 2022;Moller and Berger, 2023). Given the above, the importance of understanding and, arguably, measuring the P MSF is that it is a hemodynamic variable the clinician can target therapeutically. For example, a low P MSF intimates low V S which could be due to diminished total blood volume (e.g., hypovolemia, hemorrhage) and/or high venous capacitance (e.g., venodilation, sepsis). The clinician might rectify these pathological states by giving volume and/or administering alpha-agonists, respectively (Parkin and Leaning, 2008). Thus, the P MSF and its determinants are independent variables that can be adjusted for therapeutic control of the circulation; increasing P MSF raises venous return for any given right atrial pressure (P RA ) ( Figure 2).
Downstream from the P MSF is the P RA . In Guyton's original experimental set-up, P RA was studied as an independent variable, altered via the height of a collapsible tube (Guyton et al., 1957). Guyton observed that the P RA was inversely related to venous return; in other words, decreasing P RA increased venous return, linearly ( Figure 2). Consequently, the difference between P MSF and P RA is the pressure gradient for venous return (Δ P VR ); the value of this gradient is directly proportional to blood return to the right heart (Equation 1). More concretely, an increase in P MSF and/or decrease in P RA will augment venous return and vice versa (Magder, 2012).

The resistance to venous return
Guyton began with a mathematical approximation of the circulation, modeled after a system of distensible tubes (Jacobsohn et al., 1997). In this representation, the forces that resist total blood flow back to the heart are termed the 'resistance to venous return' (R VR ). While the R VR is often considered to be a purely Poiseuillean description of the venous circulation, this is not correct. The R VR , like the P MSF , is a weighted average of the system (i.e., including arterial components) (Jacobsohn et al., 1997). Each vascular bed faces a downstream resistance and has a unique compliance; the R VR is a summation of the downstream resistance encountered by each vascular bed, multiplied by its individual compliance relative to the total compliance of the system (Figure 3).
In this way the R VR can also be described by the time constant (i.e., the resistance multiplied by the compliance) of each vascular segment (Magder, 2016). This is clinically-important because diverting blood volume towards or away from a vascular bed with a long time constant (e.g., the splanchnic circulation) will increase or decrease the R VR , respectively (Caldini et al., 1974). The converse is true for vascular beds with a short time constant (e.g., kidneys, muscle) (Magder, 2016) (Figure 4). Accordingly, should an intervention in the ICU (e.g., prone position) alter the fraction of flow to vascular beds of differing time constants, R VR will be affected.
On the venous return curve, change in the R VR alters the slope for a given pressure gradient ( Figure 5). An increase in R VR reduces the slope, while a decrease in R VR steepens the slope.
In summary, venous return is directly proportional to the P MSF less the P RA and indirectly proportional to the R VR . If the P MSF increases and/or P RA falls, then venous return rises ( Figure 2). Similarly, decreased R VR facilitates blood return to the heart and vice versa ( Figure 5). The Ohmic representation of this relationship is as follows (Berger and Takala, 2018):

Venous return and cardiac function: the Guyton diagram
In addition to detailing the peripheral vascular determinants of blood returning to the heart, Guyton expanded our understanding of hemodynamics by adding to his analysis the cardiac determinants of blood flow from the heart. He did so by superimposing the venous return and Starling-Sarnoff curves (Guyton, 1955); this depiction is commonly referred to as the 'Guyton diagram.' These curves can be placed over each other because they both have P RA on the x-axis and blood flow on the y-axis ( Figure 6). Though it will be developed in more detail below, the Guyton diagram introduces an important distinction between intravascular and transmural pressures. The P RA and P MSF measured on the Guyton diagram are intravascular pressures. Thus, the pressure gradient for venous return is directly related to the difference between the intravascular P MSF and P RA (Equation 1). The Starling mechanism, however, is related to right atrial transmural pressure which is the pressure within the right atrium less its ambient pressure (i.e., the pericardial pressure). The transmural right atrial pressure is a static pressure that determines

FIGURE 3
The resistance to venous return (R VR ). R VR is composed of resistances (R x ) and compliances (C x ) for the entire circulation. This simplified model shows 3 vascular segments in series. The R VR is the sum of the resistance and compliance for each segment divided by the total compliance of the circulatory system (C TOT ). Resistance multiplied by compliance is the time constant (τ).
Frontiers in Physiology frontiersin.org cardiac myocyte stretch which servo-controls the ejected stroke volume to match the venous return inflow. Like any model, the value of the Guyton diagram is that it makes explicit the system's independent and dependent variables. Independent variables are those things the clinician can change or control (e.g., vascular volume and capacitance, airway pressure), whereas dependent variables are what the clinician wants to predict or study (e.g., cardiac output) by manipulating the independent variables. These distinctions are critical when considering the effects of any intervention in the ICU (e.g., prone position).
Nestled within the venous return curve are some of the independent variables of the circulatory system touched upon above: 1.) vascular capacitance, 2.) total vascular volume and 3.) the R VR . Thus, increasing total vascular volume via intravenous fluids and/or decreasing vascular capacitance via alpha-agonists both augment the V S and, therefore, P MSF . On the Guyton diagram, raising P MSF right-shifts the venous return curve such that there is increased blood flow to the heart for any given P RA . Similarly, beta-agonists (Green, 1977) and/or shunting blood from long to short time-constant vascular beds decreases the R VR (Caldini et al., 1974); this also enhances venous return for any given P RA . On the Guyton diagram, diminished R VR is manifested by an increased slope of the venous return curve ( Figure 6). The converse also holds, diminished blood volume, increased capacitance and/or increased R VR all reduce venous return for any given P RA . One clinically-important scenario wherein vascular capacitance rises (i.e., which decreases P MSF ) is reduced adrenergic tone (e.g., sedation, anesthesia, relief of hypoxemia) (Bressack and Raffin, 1987).
Found within the cardiac function curve are additional independent variables: heart rate, rhythm, valve function, afterload, inotropic and lusitropic states (Feihl and Broccard, 2009a;Feihl and Broccard, 2009b). Consequently, rate and rhythm control (e.g., cardioversion), afterload reduction (e.g., vasodilator therapy, pulmonary vascular recruitment), enhanced contractility and improved relaxation (e.g., epinephrine infusion) all increase the slope of the Starling-Sarnoff curve. With this, blood flow from the heart is enhanced for any given P RA . The converse also holds, for example, rapid atrial dysrhythmia coupled with torrential tricuspid regurgitation and severe pulmonary arterial hypertension decreases the slope of the cardiac function curve, that is to say, reduce cardiac output for any given P RA (Figure 6).
But what about the P RA itself? Is it an independent variable? In Guyton's experimental work on venous return, P RA was studied as an independent variable. However, on the Guyton diagram, which considers both venous return and cardiac function simultaneously, P RA is not independent. This was clearly stated by Guyton in his initial proposal: "right atrial pressure is not one of the primary determinants of cardiac output but, instead, is itself determined simultaneously with cardiac output" (Guyton, 1955). Later, Fiehl and Broccard expanded upon P RA as a dependent variable in their excellent review (Feihl and Broccard, 2009a). Accordingly, when analyzing venous return and cardiac function simultaneously, the dependent variable is the equilibrium formed at their The resistance to venous return with high and low time-constant segments in parallel. This is an expansion of Figure 3 with two representative segments in parallel-the non-splanchnic (NS) (i.e., low time constant, τ) and splanchnic (S) (i.e., high τ) segments. Here the fraction of total circulatory flow (Q circulatory ) to the splanchnic (i.e., Q S /Q circulatory ) versus non-splanchnic (i.e., Q NS /Q circulatory ) segments determines the R VR . If all blood diverted to the splanchnic segment (i.e., Q S /Q circulatory = 1.0; Q NS /Q circulatory = 0.0), its higher compliance (C S ) increases R VR . If all blood diverted to the nonsplanchnic segment (i.e., Q S /Q circulatory = 0.0; Q NS /Q circulatory = 1.0), its lower compliance (C NS ) decreases R VR (assuming all other resistances remain equal). C TOT is the total compliance of the system.

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frontiersin.org intersection-the operating point. Thus, both the x-(i.e., P RA ) and y-(i.e., cardiac output) Cartesian coordinates are equally dependent upon the system. This may be counterintuitive given the convention of placing the independent variable on the x-axis, however, with the Guyton diagram this is a vestige of his initial work on venous return. When it is understood that the operating point is the dependent variable, the circular and specious reasoning that the concept of venous return is incorrect because 'raising P RA reduces venous return per Guyton but augments cardiac output by Starling' becomes moot. Rather, at any given time (or in response to an intervention, such as the prone position) there are characteristics of the peripheral circulation and heart that, in tandem, produce a unique cardiac output and P RA (Guyton, 1955). To clarify this, a modified Guyton model is proposed below to disclose the clinicallyrelevant independent variables.

A geometrical model
This is a simplified geometric approximation of the principles discussed above. If we consider the intersection of cardiac function and venous return as two directly-opposed right triangles, then we can solve for the height of their shared apex at equilibrium (i.e., cardiac output or venous return presently identified as Q circulatory ) as a function of their bases and hypotenuse slopes (Figure 7). Q circulatory is numerically equivalent to cardiac output and/or venous return. It is used in the geometrical model to emphasize that total blood flow (i.e., Q circulatory ) is determined by the operating point-the intersection of both peripheral venous and cardiac function. This avoids the confusion that sometimes arises when 'cardiac output' is thought to be determined only by cardiac factors or when 'venous return' is thought entirely due to peripheral factors; 'Q circulatory ' circumvents this ambiguity.
The base of the left triangle rests on the x-axis and is defined by the pressure immediately surrounding the heart, within the pericardium (i.e., the P PC ) and the P RA ; this is the transmural pressure of the right atrium. The slope (i.e., hypotenuse) of this triangle is the change in cardiac output per mmHg of transmural right atrial pressure, or cardiac conductance (G cardiac ). This value is estimated to be 35 mL/min/kg per 1 mmHg (Rothe, 1993). Multiplying the base of this triangle (i.e., P RA -P PC ) by the slope of the hypotenuse (G cardiac ) gives the height of this triangle (i.e., total circulatory flow, Q circulatory ).
Similarly, the base of the rightmost triangle is defined by the P MSF and the P RA ; this is the pressure gradient for venous return (the difference between two intravascular pressures along a hypothetical length of vessel), as above. The slope of this triangle is the change in cardiac output per the gradient for venous return, or venous conductance (G VR ). Based on a P MSF of 8 mmHg, this value is estimated to be 10 mL/kg/min per 1 mmHg. Multiplying the base of this triangle (i.e., P MSF -P RA ) by the slope of its hypotenuse (G VR ) gives the height of this triangle, which is also total circulatory flow, Q circulatory .
Equation 4 is solved for P RA Setting equation 3 equal to equation 5, we can reduce the equation to Q circulatory as follows: Because the inverse of conductance, G, is resistance, this equation can be written as: Accordingly, in this model the shared apex of the two triangles (i.e., the operating point, which defines Q circulatory ) is a function of the total base of the two triangles (i.e., P MSF less P PC ) and the inverse of the slopes of their respective hypotenuses (i.e., R VR and R cardiac ).

FIGURE 5
The R VR and the venous return curve. P MSF is constant, but the resistance to venous return changes. The shallow curve is a higher resistance (↑ R VR ) the steeper curve is a lower resistance (↓ R VR ). At the same P RA , lower resistance and higher resistance generate increased and decreased flow (L/min), respectively. V S and C are the stressed volume and average compliance, respectively, of the systemic vasculature. The flattening of the venous return curve is where the great veins collapse; this creates a maximal venous return in each state.

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More concretely, if R VR and R cardiac remain constant, increased P MSF and/or decreased pressure surrounding the heart (P PC ) raise the height of their shared apex (Figure 8). A concomitant decrease in R cardiac (i.e., increased slope of the Starling-Sarnoff curve) or R VR (i.e., increased slope of the venous return curve) would further elevate their shared apex (Figure 8).
In this model, P RA plays no role in cardiac output because the operating point (i.e., the shared apex) is the dependent variable; Q circulatory and P RA both fall out from this equilibrium (Feihl and Broccard, 2009a). The equations above could have equally been solved for P RA instead of Q circulatory ; P RA , nevertheless, would still be dependent upon P MSF , P PC , R cardiac and R VR .
To further develop this model with an emphasis on heart-lung interaction, the determinants of P PC are included. Doing so reveals additional, clinically-relevant independent variables when placing an ARDS patient in the prone position. The P PC is the x-intercept of the hypotenuse defined by R cardiac (i.e., the cardiac function curve) (Magder, 2004;Feihl and Broccard, 2009a). As originally hypothesized by Guyton (Feihl and Broccard, 2009a) and demonstrated by Marini and colleagues (Marini et al., 1981), increasing P PC initiates a parallel, right-shift of the cardiac function curve. Consequently, increased P PC decreases the shared apex (i.e., the operating point) and Q circulatory is diminished but only if there is no simultaneous change in P MSF , R cardiac or R VR .
Given that the P PC is a summation of: 1.) pleural pressure (P PL ), 2.) pressure added by mechanical ventilation (i.e., estimated as the mean airway pressure, P AW , multiplied by the ratio of the chest wall to respiratory system elastances, E CW /E RS ) (Gattinoni et al., 2004) and 3.) the elastic recoil pressure of the pericardium (P PC EL ) (Cabrera et al., 1989), we can expand equation 10 above.
Accordingly, increased pleural (e.g., thoracic supports) and/or elastic recoil pressure from the pericardium (e.g., right ventricular dilatation in acute cor pulmonale), raise the pressure surrounding the heart, P PC . Furthermore, elevated P AW (e.g., increasing positive end-expiratory pressure, PEEP) or a stiffened chest wall (e.g., prone position increases the E CW /E RS ratio) both amplify P PC ; from equation 11, we see that increasing P PC reduces Q circulatory but only if P MSF , R cardiac and R VR are constant. It should not escape the reader's attention that including P PC in this model is a crucial link between cardiac and respiratory physiologies.

Cardiac limitation
While the proposed model is meant to illuminate the clinicallyrelevant independent variables determining Q circulatory , equation 11 has important caveats (Magder, 2012). The most important is that it is predicated upon the intersection of two hypotenuses; in vivo, both the venous return and cardiac function curves have portions that flatten out. When the operating point falls upon the flat portion of  Figure 2, the x-axis is right atrial pressure in millimeters of mercury (mmHg) and y-axis is total blood flow in liters per minute (L/min). The P MSF is approximately 8 mmHg at the x-intercept of the venous return curve (in blue). The Starling-Sarnoff (or cardiac function) curve is in red in a normal, upright position; its x-intercept is the pressure around the right atrium, the pericardial pressure (P PC ). In this model, the dependent variable is the operating point, at the intersection of the venous return and cardiac function curves at equilibrium. Accordingly, both the x-(i.e., P RA ) and y-(i.e., Q circulatory ) coordinates defined by the operating point are also dependent variables. (B) How the P RA and Q circulatory are determined by the system. Normal cardiac function but diminished P MSF1 (e.g., volume loss, venodilation) results in operating point 1 (OP 1 ), diminished P RA and Q circulatory . Normal cardiac function with increased P MSF2 (e.g., volume expansion, decreased venous capacitance from adrenergic agents) causes OP 2 (i.e., increased P RA and Q circulatory ). Reduced P MSF and diminished cardiac function (e.g., acute cor pulmonale with tricuspid regurgitation) leads to OP 3 . Elevated P MSF with reduced cardiac function leads to OP 4 . Both Q circulatory and P RA are dependent variables in this system. The independent variables are reflected in the position and slopes of the venous return and cardiac function curves.
Frontiers in Physiology frontiersin.org the cardiac function curve, Q circulatory depends only upon the independent variables of cardiac function: P PC , the right atrial pressure at which the cardiac function curve begins to plateau, P RAplat and the R cardiac (Figure 9). Q circulatory P RA plat − P PC R cardiac Fundamentally, this equation relays that Q circulatory is no longer determined by peripheral factors when the operating point is above the P RAplat . Changing P MSF or R VR only alter P RA with fixed Q circulatory .

Venous limitation
In a manner similar to cardiac function, the venous return curve also flattens when the P RA falls below venous collapse pressure, P CRIT (Magder, 2012). This is the formation of a Starling resistor when the great veins enter the thorax and is observed with ultrasound as collapse of the great veins. This phenomenon is also termed 'waterfall' physiology because the pressure below P CRIT has no bearing on flow, just as the height of a waterfall does not mediate its flow (Permutt and Riley, 1963). Consequently, when the operating point lies to the left of P CRIT (i.e., on the "flat portion" of the venous return curve) Q circulatory becomes independent of cardiac function or P PC ; the independent variables are P CRIT , P MSF and R VR (Figure 9). Q circulatory P MSF − P CRIT R VR (13)

FIGURE 7
Simplified geometrical model. This model borrows from the Guyton diagram where the red line represents cardiac function and the blue line venous return. Two right triangles are formed as described in the text; the operating point is the apex of the two right triangles. Note that the slope (change in flow per unit pressure) is conductance, G. The inverse of conductance is resistance. As in previous figures, P PC is pericardial pressure, P MSF is mean systemic filling pressure, R cardiac and R VR are cardiac and venous resistance, respectively. Q circulatory is blood flow of the system with right atrial pressure (P RA ) in millimeters of mercury (mmHg) on the x-axis and blood flow in liters per minute (L/min) on the y-axis.

FIGURE 8
The independent and dependent variables of the geometric model. (A) The effect of changing the independent variables, P PC and P MSF on the dependent variable (operating point, OP). OP 1 depicts baseline conditions, its x-(P RA ) and y-(Q circulatory ) coordinates are shown. A solitary increase in P MSF (e.g., volume infusion) results in OP 2 , that is, increased P RA and Q circulatory . A selective decrease in P PC (e.g., spontaneous inspiration) leads to OP 3 which increases Q circulatory , but decreases P RA . If P MSF rises and P PC falls, the result is OP 4 , increased Q circulatory at a slightly reduced P RA relative to baseline.
(B) The effect of changing the independent variables, R VR and R cardiac on the dependent variable (operating point, OP). A selective decrease in venous resistance (e.g., shunting blood away from the splanchnic circulation) leads to OP 2 (i.e., both P RA and Q circulatory rise). A selective decrease in cardiac resistance (i.e., improving cardiac function by, for example, reducing pulmonary vascular resistance) leads to OP 3 (i.e., P RA falls, while Q circulatory rises). Reducing venous and cardiac resistance together leads to OP 4 (i.e., little P RA change with large Q circulatory augmentation).

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In other words, when venous limited, reducing R cardiac (i.e., improving cardiac function) or changing P PC has no bearing on Q circulatory ; only changing P MSF , R VR or P CRIT might alter total flow.

Implications for the prone position
With a Guyton-based circulatory model proposed above, anticipating the change in Q circulatory follows the independent variables of the system: P MSF , P PC , R cardiac and R VR . At present there are three key studies that have elucidated interactions between the circulation and prone position in ARDS (Vieillard-Baron et al., 2007;Jozwiak et al., 2013;Lai et al., 2021). Much of the discussion below is taken from these investigations.

Mean systemic filling pressure
Recently, Lai and colleagues studied the effect of prone position on the determinants of venous return (Lai et al., 2021). They measured P MSF by extrapolating to zero flow a series of P RA -cardiac output pairings in response to increasing airway pressure. Though this method overestimated P MSF in a porcine model (Berger et al., 2016), this observation was restricted to euvolemic conditions which are less likely in ARDS patients in the ICU. Nevertheless, considering the discussion on P MSF measurement above, a retrospective calculation of P MSA would be of great interest given that the average P MSF measured by Lai et al. was clinically quite high, especially in the prone position. Irrespective of absolute values, Lai and colleagues observed that P MSF increased significantly from the semi-recumbent to prone position; they hypothesized that this was due to increased intra-abdominal pressure (IAP). However, the baseline value and change in IAP had no bearing on P MSF behavior. This is unsurprising given what is known about the mechanisms by which PEEP increase P MSF . Initially, it was also hypothesized that IAP mediated P MSF augmentation with PEEP application and/or stiffening of the chest wall (i.e., akin to prone position) in early canine models . However, IAP had no role in raising P MSF , instead, adrenergic reflexes (i.e., changing vascular capacitance) and redistribution of blood volume from the central to peripheral circulation were the main drivers of P MSF rise Fessler, 1995;Fessler, 1997). Accordingly, central blood volume, adrenergic reserve and exogenous vasoactive agents all undoubtedly mediate the change in P MSF upon pronation, rather than IAP. Parenthetically, this could also explain hemodynamic differences noted between elective surgical and critically-ill ARDS patients when prone position is employed (Edgcombe et al., 2008). The latter are more likely to be on vasoactive agents and volumeloaded, while the former more likely euvolemic; as well, anesthetic agents may blunt reflexive changes in vascular capacitance which would limit P MSF rise in the operating room. As described above, P MSF is directly related to Q circulatory when the patient is not cardiac limited and without concurrent changes in P PC , R cardiac or R VR .

Pericardial pressure
There are no known direct measurements of P PC in humans with ARDS placed in the prone position. Yet, inferences can be made given the mathematical approximation of P PC presented above. The prone position increases the elastance (i.e., stiffness) of the chest wall (E CW ) (Pelosi et al., 1998). To the extent that pronation also decreases the elastance (i.e., improves compliance) of the lungs by alveolar recruitment, the E CW relative to the elastance of the

FIGURE 9
Operating point positions in the geometrical model. (A) reveals cardiac limitation where the operating point (OP) is on the flat portion of the cardiac function curve. Change in P MSF or R VR change only P RA and not Q circulatory (OP 1 versus OP 2 ). (B) is when the system is neither venous nor cardiac limited. Q circulatory is changed by P MSF , P PC , R cardiac and R VR (see Figure 5). (C) shows venous limitation or 'waterfall' physiology. Changes in cardiac function alter only P RA and not Q circulatory (OP 1 versus OP 2 ).
Frontiers in Physiology frontiersin.org respiratory system (i.e., the lungs and the chest wall together, E RS ) rises. Multiplying the mean airway pressure generated by mechanical ventilation by the E CW /E RS ratio approximates P PC augmentation when a patient is passive with the ventilator. For example, if the mean airway pressure is 10 mmHg with an E CW /E RS ratio of 0.3 in the supine position, then 3 mmHg is added to the P PC . If mean airway pressure remains constant and prone position increases the E CW /E RS ratio to 0.5, then 5 mmHg is added to the P PC . Additionally, pericardial restraint could play an important role determining P PC , especially if there is comorbid acute cor pulmonale (ACP). Typically, when right atrial volume is low (i.e., estimated by a transmural pressure below 5 mmHg (Hamilton et al., 1994)), there is little recoil pressure generated by the pericardium around it. As atrial volume increases beyond this, the pericardial sac is engaged and moves up its volume-pressure relationship. This leads to an increasingly large elastic recoil pressure from the pericardium, which raises the P PC . Elevated P PC , therefore, restricts right ventricular filling and 'protects' from overdistention; however, this blunts Q circulatory by narrowing the P MSF -P PC gradient.
In the setting of ACP, often seen in moderate-to-severe ARDS (Guérin and Matthay, 2016;Mekontso Dessap et al., 2016), pericardial recoil may play an important role upon prone position. With ACP, co-existent right atrial distension elevates P PC by pericardial recoil; this is especially true with P RA above 10-12 mmHg (Hamilton et al., 1994). While prone position is expected to further increase P PC (i.e., by increasing P PL ), to the extent that the elevated P PL shrinks cardiac volume, P PC may remain constant, or even fall, as the elastic recoil pressure imparted by the pericardium is reduced. More simply, the rising P PL experienced by the pericardial space is offset by falling recoil pressure of the pericardium. This was originally observed in models of continuous positive airway pressure in heart failure (Huberfeld et al., 1995). Were this to occur upon prone position in a patient with ACP, P PC would remain constant or fall. Taken with the effect of prone position on P MSF noted above, the P MSF -P PC gradient would be maintained (or enhanced) and so too would Q circulatory if R cardiac and R VR remain constant.
Finally, some have argued for the execution of prone position with thoracoabdominal supports that allow the abdomen to hang freely (Chiumello et al., 2006). These supports are typically placed mid-sternum and below the pelvis. Chiumello and colleagues compared these supports to the abdomen flush with the bed in prone ARDS patients (Chiumello et al., 2006). They found that the supports accentuated local pressure without any benefit to gas exchange while diminishing stroke volume. Given support placement directly at the sternum, it is possible that P PC is accentuated, reducing the P MSF -P PC gradient and Q circulatory barring a concomitant decrease in R cardiac or R VR .

Cardiac resistance
While not a commonly-employed term within the sphere of clinical hemodynamics, 'cardiac resistance' (R cardiac ) is analogous to R VR . Graphically and mathematically, R cardiac is simply the inverse slope of the cardiac function curve. A decrease in R cardiac (i.e., a steeper slope of the cardiac function curve) represents improved cardiac function and raises the operating point (i.e., Q circulatory ) unless the system is venous limited. In an elegant ultrasonographic study, Vieillard-Baron and colleagues illuminated the salubrious effects on the RV prompted by prone position (Vieillard-Baron et al., 2007). In 21 patients with P a O 2 /F i O 2 ratio of less than 100 mmHg and ACP defined as RV enlargement and septal dyskinesia, 18 h of prone position led to a significant reduction in heart rate and increase in cardiac output. Furthermore, RV end-diastolic area fell while LV end-diastolic area increased and tricuspid regurgitation was reduced. Taken together, the rise in cardiac output with diminished RV size strongly implies reduced R cardiac as a mechanism of improved Q circulatory , at least in patients with ACP. The mechanism for this improvement (detailed at the outset of this review and by others (Repessé et al., 2016)) was reduced pulmonary vascular impedance to flow facilitating RV ejection (Jardin and Vieillard-Baron, 2003;Vieillard-Baron et al., 2007) which improves stroke volume and cardiac output for any given P RA .
Recent studies also imply reduced R cardiac . Ruste and colleagues investigated the hemodynamic effects of prone position in over 100 patients (Ruste et al., 2018). 25% of prone sessions led to significantly increased cardiac output, while 23% had a significant decrease; the remainder showed no change. Importantly, of those sessions where cardiac output rose, 56% had no change or a decrease in global end-diastolic volume (GEDV) measured by transpulmonary thermodilution. Rising cardiac output without an increase in end-diastolic volume infers reduced R cardiac . Importantly, static GEDV with prone position could signify a shrinking RV end diastolic volume with enlarging LV end diastolic volume consistent with the reduced RV-to-LV end-diastolic area ratio observed with echocardiography by Vieillard- Baron et al. (Vieillard-Baron et al., 2007). Finally, Boesing and colleagues recently published on different PEEP titration strategies and their interaction with prone position (Boesing et al., 2022). In this study, esophageal pressure (P ES ) was used as a surrogate for P PL . Curiously, the PEEP titration strategy that led to the greatest increase in cardiac output from supine to prone was associated with the smallest rise in transmural P RA (i.e., P RA less P ES ), in other words, the least preload augmentation. Similar to the observations by Ruste and colleagues, this finding suggests, but does not prove, enhanced cardiac function (i.e., reduced R cardiac ).

Resistance to venous return
In the study of Lai and colleagues (Lai et al., 2021), the R VR was calculated from semi-recumbent to prone position in ARDS patients. In total, R VR increased in the vast majority, though there were a few with stable or slightly diminished R VR . Like P MSF , the change in R VR was not related to IAP and like P MSF , this is unsurprising given the foundational work of Takata and Robotham (Takata et al., 1990). In their original model, Takata and Robotham proposed that the relationship between great vein pressure and IAP would behave analogously to West zones in the lung. That is, if the IAP is much greater than inferior vena cava (IVC) pressure (i.e., zone 2), then venous return is impaired when the abdomen is pressurized by diaphragmatic descent and, in theory, prone position. However, if IAP is much less than IVC pressure (i.e., zone 3), then increased IAP generated by diaphragmatic descent (or prone position) enhances venous return. Their initial Frontiers in Physiology frontiersin.org work confirmed this model, however, they later found that the model held even with an open abdomen and evisceration, that is, constant IAP (Takata and Robotham, 1992). Thus, the ambient pressure of import was more likely focal subcostal, crural, or intrahepatic pressure, rather than general IAP. This was observed by Decramer and colleagues (Decramer et al., 1984) and explored further by Brienza et al. in a porcine model (Brienza et al., 1995) and Jellinek et al. in humans (Jellinek et al., 2000). Accordingly, diaphragmatic shape-matching between the liver and upper abdomen, active versus passive diaphragm displacement, intrahepatic compliance (e.g., intrinsic liver disease) and the use of focal thoracoabdominal supports, among other factors might affect hepatic pressure (P hepatic ) upon pronation. Diminished venous pressure (e.g., hypovolemia, venodilation) relative to P hepatic might increase R VR . By contrast, elevated venous pressure (e.g., high blood volume, low venous capacitance) relative to P hepatic might blunt a rise in R VR with prone positioning. Another possible mechanism for increased R VR with prone position follows that of P MSF . As described above, reflex sympathetic tone is a key mediator of increased P MSF . However, when alpha agonists act upon veins to increase the V S , resistance necessarily rises. This is because change in volume is proportional to the second power of vessel diameter but resistance is related to the fourth power. More concretely, if the diameter of a vein falls by 20% from its baseline, its volume is diminished by 36% (i.e., this reduces its capacitance, increases P MSF ) but its resistance rises by 244% (Rothe, 1993). Because the splanchnic circulation is a crucial reservoir for venous blood, the rise in resistance in response to V S recruitment can be offset by beta-agonism (Green, 1977) in the hepatic veins, or redistribution of blood flow to short time constant vascular beds, as noted above (Magder, 2016) (Figure 4). Nevertheless, hepatosplanchnic blood flow during prone position in ARDS changes little (Hering et al., 2002;Matejovic et al., 2002). Interestingly, one study found decreased renal blood flow (Hering et al., 2001)-a fast time-constant bed; diversion of blood in this manner contributes to increased R VR . A final, potential mechanism for R VR augmentation with prone position lies in the superior vena cava (SVC). Fessler found that the rise in total R VR following PEEP application was predominantly due to the veins draining into the SVC rather than the IVC (Fessler et al., 1992). Because P PL is the pressure that surrounds SVC and prone tends to raise P PL for any given P AW (see equation 11 above), it is possible that mechanical compression of the SVC contributes to R VR (Lansdorp et al., 2014;Berger et al., 2016). Regardless of the mechanism, R VR is a critical determinant of Q circulatory (Pinsky, 2021).

Knowing the limits
Taking the above into consideration, a key factor when predicting the hemodynamic response to prone position is the location of the operating point whilst semi-recumbent; is the operating point 'cardiac limited', 'venous limited' or 'unlimited' (Figure 6) (Magder, 2012)? Knowing this focuses the clinician on the independent variables most likely affecting Q circulatory . For instance, if the operating point is cardiac limited (Figure 6) we see that changes in P MSF and R VR play no role, while changes in cardiac characteristics (e.g., R cardiac ) mediate Q circulatory . Of course, this depends on how close the operating point is to the P RA at which the cardiac function curve flattens out, but this is, nevertheless, a

FIGURE 10
The geometrical model applied to representative data from Lai et al. (A) The effect of prone position on a preload responsive patient. At baseline, P PC is estimated by assuming a mean airway pressure of 15 mmHg, an E CW /E RS ratio of 0.2 and a pleural pressure (P PL ) at functional residual capacity of-2.5 mmHg. With prone position, the P MSF rises much more than P PC . There is an increase in R VR and an assumed decrease in R cardiac due to reduced pulmonary vascular resistance. The operating point with prone position (OP prone ) leads to an increase in total blood flow (Q CIRC ) and increased right atrial pressure (P RA ). By this model, P RA does not determine Q CIRC ; both P RA and Q CIRC are determined by P MSF , P PC , R cardiac and R VR . (B) Prone position in a preload unresponsive patient at baseline. P PC in prone is estimated by assuming a mean airway pressure of 15 mmHg, and E CW /E RS ratio of 0.5 and a P PL at functional residual capacity of-2.5 mmHg. With cardiac limitation, only a significant change in R cardiac would increase Q CIRC .
Frontiers in Physiology frontiersin.org reasonable clinical heuristic. Jozwiak and colleagues studied 18 ARDS patients with elevated right ventricular-to-left ventricular end-diastolic areas (RVEDA/LVEDA), but without ACP (Jozwiak et al., 2013). Prior to prone position, the change in cardiac output in response to a passive leg raise was evaluated. By the model above, 'cardiac limitation' is detected when a patient is preload unresponsive. In this state, only improved cardiac function during pronation (i.e., reduced R cardiac ) increases Q circulatory ; changes in P MSF and R VR shift the operating point along the x-axis, but not the y-axis. In other words, P RA changes but not blood flow. Jozwiak and colleagues found that in 'cardiac limited' patients, prone position significantly reduced pulmonary vascular resistance and the RVEDA/LVEDA which should diminish R cardiac and improve Q circulatory . However, these patients were also found to have depressed left ventricular ejection fraction. Furthermore, in the face of prone position, systemic afterload increased; total R cardiac , therefore, did not improve. When patients are not 'cardiac limited,' the operating point may be either 'unlimited' or 'venous limited.' In the study of Jozwiak and colleagues, imaging of the great veins was not reported, but those patients who were preload responsive were unlikely to have great vein collapse (i.e., venous 'waterfall') given that their average, baseline P RA was relatively high (i.e., 15 mmHg) with increased RVEDA/LVEDA ratios. Thus, based on equation 11 above, the change in Q circulatory was probably subject to all of: P MSF , P PC , R cardiac and R VR . Given what we know from Lai and colleagues, prone position likely increased P MSF ; P PC may have increased less than the rise in P PL because of reduced pericardial restraint and R cardiac fell due to diminished pulmonary vascular resistance. Each of these effects raise Q circulatory , presumably offsetting heightened R VR with prone ( Figure 10).
It is also possible for preload responsive patients to be 'venous limited' as described by equation 13 above. When the operating point lies on the flat portion of the venous return curve (i.e., below P CRIT ) then R cardiac ceases to affect Q circulatory . Said another way, blood flow is determined solely by peripheral venous factors. When 'venous limited', volume status is likely a crucial determinant of the hemodynamic response to prone position based on the model of Takata and Robotham described above (Takata et al., 1990). A zone 3 abdomen might have a stable or enhanced P MSF relative to P CRIT and blunt any increase in R VR (i.e., stable or increased Q circulatory ), while a zone 2 abdomen would diminish P MSF relative to P CRIT and favour elevated R VR (i.e., stable or reduced Q circulatory ). There is little data on 'venous limited' ARDS patients being placed in prone. In the study by Lai and colleagues, there were four 'preload responsive' patients who had no change (n = 3) or a decrease (n = 1) in Q ciculatory when placed in prone position. These patients may have been venous limited, but this data was not collected. Given that at low trans-mural pressure, the great veins are very compliant (Bodson and Vieillard-Baron, 2012), generation of a hemodynamically-significant Starling resistor, i.e., 'venous limitation,' should lead to great vein collapse throughout most of the respiratory cycle. In a patient passive with the ventilator, collapse is an inspiratory event for the SVC and expiatory event for the IVC. Collecting this data with ultrasound before and after pronation could help delineate this hemodynamic phenotype.
Finally, it is possible to be both venous and cardiac limited simultaneously, in other words, the operating point is on both the flat portion of the venous return and cardiac function curves concurrently. This might happen in states of high P CRIT (e.g., high PEEP, high subdiaphragmatic pressure) coupled with depressed cardiac function. In the setting of ARDS, this could be a syndrome of alveolar over-distension (Jardin and Vieillard-Baron, 2003). Prone position in such a patient might reduce Q circulatory , especially if the patient is hypovolemic. Managing this hemodynamic phenotype might involve PEEP titration to reduce P CRIT and enhance cardiac function as this could move the operating point onto steep portions of the venous return and cardiac function curves.

Conclusion
At equilibrium, the intersection of venous return and cardiac function generates the hemodynamic operating point. The operating point and both of its coordinates (i.e., P RA and Q circulatory ) are dependent variables. The independent variables of the system are the P MSF , resistance to venous return, cardiac function and the pressure surrounding the right atrium. These are not new principles; however, clinical physiology can be muddied in terms of how dependent and independent variables are discussed. A simplified geometrical model was presented to clarify the mechanisms of blood flow at equilibrium founded on Guyton's model of the circulation; this focuses the clinician on how interventions in the ICU (e.g., prone position) might affect hemodynamics. Recent mechanistic investigations into the circulatory consequences of prone position have been reported. These findings were incorporated into the simplified geometrical model with emphasis on the link between cardiac and respiratory physiologies. The pericardial pressure is one nexus binding the heart and the lungs; so too are changes in cardiac function from pulmonary vascular recruitment. Measuring 'preload responsiveness' locates the system's operating point; this helps predict the hemodynamic response to any intervention in the ICU, including the decision to prone a patient with ARDS.

Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

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