Mathematical Model for the Quantification of Pulmonary and Intrapulmonary Arteriovenous Anastomoses Blood Flow

Abstract

Blood flow quantification through intrapulmonary arteriovenous anastomoses (IPAVA) is vital because IPAVA blood flow (QIPAVA) prevents gas exchange and can provide a pathway for emboli to bypass the pulmonary microcirculation and provoke myocardial infarction, transient ischemic attack (TIA), or stroke. Some current techniques allow only an approximate estimate and present technical evaluation difficulties that can compromise the result. To overcome the limitations and obtain an accurate detection of QIPAVA, it is necessary to isolate the fundamental parameters of gas exchange associated with chemical-physical laws that regulate them and develop a specific algebraic model. Therefore, literal equations were elaborated and some experimental data, obtained through noninvasive and easy-to-use methods, were entered. These equations quantified the pulmonary blood flow that permits gas exchange and the one that does not (QIPAVA or generally shunt). Because they are accurate and based on the law of conservation of mass, the use of these equations will allow a general advance in understanding the consequences of QIPAVA on physiological and pathophysiological processes.

1. Introduction

The pulmonary circulation supports the entire cardiac output (Q_t) with high flow maintained at low intravascular pulmonary arterial pressure [1]. Additionally, pulmonary vascular resistance (PVR) is low in healthy adult lungs, enabling delivery of full cardiac output by the thin-walled right heart with relatively low right ventricular pressure compared to the left ventricular pressure generated by the thick-walled left heart that is required to overcome systemic pressures [2]. This is because the lungs have an inherent protective mechanism against high PVR through the recruitment of capillaries and, to a lesser extent, the distension of the elastic vessels present in the circuit [2, 3]. In fact, under conditions of rest, most of the capillary bed is not recruited [2, 4]; and during exercise, in healthy subjects, when the cardiac output increases up to six times, the pulmonary artery pressure increases only moderately [2, 5].

However, the pulmonary vasculature is a dynamic system that responds rapidly to vasoactive mediators [2]. For example, vasoconstriction due to smooth muscle contraction is caused by hypoxia [1, 6, 7, 8] and circulating vasoconstrictive mediators [2]. Given this significant vasoreactivity, it is important to verify the distribution of pulmonary blood flow within the lung with respect to a given cardiac output because it is only the blood that passes through the pulmonary microcirculatory area that determines the real gas exchange through the pulmonary epithelial tissue and the blood, which does not cross it, passes through alternative routes that do not allow gas exchange.

“Physiologically, pulmonary arteriovenous shunting is commonly defined as the passage of blood through the lungs without taking part in gas exchange” (Genovesi, et al.) [9]. There are two types of shunts, namely, extrapulmonary and intrapulmonary shunts. They are normally present in cardiorespiratory diseases.

Thus, cardiac and pulmonary blood flow that passes through a patent foramen ovale and/or large-diameter pulmonary arteriovenous malformations (PAVMs), which act as a right-to-left shunt in the lungs, are considered shunts.

Another possible route for pulmonary blood flow is through intrapulmonary arteriovenous anastomoses (IPAVA), whose presence is also in healthy humans. IPAVA in healthy lungs is indisputable [10] and has been known to exist in human lungs for over 60 years [10], but there is an ongoing dispute over their functional role and relevance.

In particular, in healthy subjects, IPAVA are typically closed or poorly perfused under resting conditions but can open during moderate- to high-intensity exercise in normoxia [10, 11, 12]. In hypoxia, these pathways are perfused under resting conditions depending on the severity of hypoxemia [13, 14], and there is greater blood flow through these pathways with exercise and hypoxia [10, 11, 12, 14, 15, 16, 17, 18]. Therefore, IPAVA are dynamic pulmonary pathways through which blood flows, and that fraction of pulmonary blood flow is variably present depending on the conditions to which the subjects are exposed.

Approximately 30% of healthy, young, asymptomatic subjects have IPAVA open at rest in normoxia; but breathing 100% O₂ and assuming an upright body position, blood flow through IPAVA persists in only 5% of these subjects [19].

IPAVA open in almost all healthy subjects, in normoxia during moderate- and high-intensity exercise [11, 12], and in hypoxia are already perfused at rest [13] and have even greater perfusion with exercise [12].

The transient patency of these IPAVA [14, 20] appears to safeguard small capillaries when there are high flows and pressures in the pulmonary artery, especially at high-intensity exercise and under hypoxia [11, 12].

On the other hand, IPAVA may pose a health risk, as they are a potential route for thrombi and other particles to bypass the sieve of the pulmonary capillary network and embolize the systemic arterial circulation [14]. In fact, “if an embolus were to flow through an IPAVA it could end up in the heart or brain and result in myocardial infarction, transient ischemic attack (TIA), or stroke” [10].

Consequentially, it is important to accurately quantify the pulmonary blood flow that exchanges gas with respect to the total cardiac output and the blood flow related to the shunts, which not only prevent gas exchange but can also allow the emboli to bypass the pulmonary microcirculation and cause serious complications [10].

IPAVA, in healthy subjects, have a large diameter, that is, >50 μm [21], and as already highlighted, have a dynamic regulation. They can bypass the pulmonary capillary circulation, and the blood flow flowing in them goes directly to the left side of the heart.

Currently, IPAVA are approximately estimated with echocardiographic methods (transthoracic saline contrast echocardiography [TTSCE]) [10, 11, 12, 15] and with technetium-99 m-labeled macroaggregated albumin (99mTc-MAA) [10, 16, 17, 22]. However, both techniques have important limitations [18], but they made it possible to detect blood flow through IPAVA in healthy humans in normoxia during exercise [11, 14] and, at rest and during exercise, in hypoxia [12, 13]. Furthermore, the simultaneous use of the two techniques, TTSCE and 99mTc-MAA, led to consistent data [18].

While the measurement of cardiac output is currently carried out through various methods, such as indicator dilution techniques, the Fick method, arterial pulse contour analysis, ultrasound, and bioimpedance [23], and specifically during exercise, cardiac output can be measured by open-circuit acetylene uptake [24, 25].

Therefore, for the reasons listed above, it is necessary to develop a specific algebraic-analytical model to precisely quantify the actual pulmonary blood flow and shunt blood flow. This mathematical model must be a perfect representation of the real system, to overcome all technical problems and provide, in each subject, a precise quantification of blood flow through the pulmonary microcirculatory area that determines the real gas exchange (Qp), even without knowing the measure of cardiac output (Q_t) and intrapulmonary arteriovenous anastomoses blood flow (Q_IPAVA).

To elaborate the mathematical model, first, the fundamental parameters of gas exchange were identified. Then, the main chemical-physical laws involved were enucleated, and the theory was interpreted by formulating several literal equations.

In particular, to apply the law of conservation of mass, it was first necessary to precisely identify the masses leaving the blood of the pulmonary microcirculation during expiration and to equate them with the exhaled ${\mathrm{CO}}_2$ mass. Therefore, the two forms of ${\mathrm{CO}}_2$ circulating in the blood have been identified, namely, ${\mathrm{CO}}_2$ dissolved and bicarbonate (${\mathrm{HCO}}_3^{-}$).

Subsequently, to validate the mathematical model, the experimental data available in the relevant literature [25] were entered into the equations. The human physiological data in this experiment [25] included blood gas data, metabolic data, and noninvasive measures of cardiac output (open-circuit acetylene uptake) from healthy young men.

I chose Jonk’s work [25] because the measured parameters, required by the equations, in his experiment had the characteristic of having been detected simultaneously, and Jonk’s experimental model faithfully reproduces the structure of the equations, namely, what is lost between venous and arterial blood and, at the same time, what is found in exhaled ${\mathrm{VCO}}_2$ (conservation of mass).

Since only data from healthy subjects were analyzed in this study, the considered shunts are intrapulmonary arteriovenous anastomoses (IPAVA).

2. Methods and materials

2.1 Modeling method: introductory note to the equations

The measured physiological parameters taken into consideration in this study, that are involved in respiratory gas exchanges, are HCO₃⁻, ${\mathrm{PCO}}_2$, (two blood parameters), ${\mathrm{VCO}}_2$, and cardiac output (Qt).

It is necessary to consider the HCO₃⁻ concentration to quantify the total ${\mathrm{CO}}_2$ content in blood as bicarbonate is transformed into ${\mathrm{CO}}_2$ because of the reaction—CO₂ + H₂O ↔ H⁺ + HCO₃⁻— catalyzed by carbonic anhydrase present in as many as 16 different isoforms [26] in various organ systems, including the lungs.

The formulation phase started with the use of Lavoisier’s Law on Conservation of Mass. This law states that “nothing is created and nothing is destroyed, but everything is transformed”; thus, by this law, ${\mathrm{VCO}}_2$ must represent the mass that is lost each minute in terms of ${\mathrm{PCO}}_2$ and $\left[{\mathrm{HCO}}_3^{-}\right]$ (the other form of ${\mathrm{CO}}_2$) between venous and arterial blood through the lungs.

Because ${\mathrm{VCO}}_2$ is expressed in L/min, ${\mathrm{PCO}}_2$ is expressed in mmHg, [HCO₃⁻] is expressed in mmol/L, and Q_t in L/min, to perform the calculation, it is necessary to align all the dimensions and the units of physical quantities of all parameters involved, thus ensuring, through dimensional analysis, the same dimensions (dimensional homogeneity) when equality is established in the equations.

Therefore, new physiological parameters that were symbolized with the Greek letters $\mathrm{\Phi }$ (uppercase phi) and $\varphi$ (lowercase phi) were defined, and several equations were formulated.

In particular, the letter $\mathrm{\Phi }$ defines the total molar flow rate of ${\mathrm{CO}}_2$, whereas $\varphi$ better defines the partial flow components, which constitute the total molar flow rate of ${\mathrm{CO}}_2$.

Specifically:

$\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{e}\right)}$ = total molar flow rate per minute of expired ${\mathrm{CO}}_2$ (measured with a metabolic cart), which is expressed in units of $\frac{\text{mmol}}{min}$ and calculated as follows:

$${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}=\frac{{\mathrm{VCO}}_2}{\mathrm{Vm}{\mathrm{CO}}_2}=\frac{{\mathrm{VCO}}_2}{22.26},$$E1

where ${\mathrm{VCO}}_2$=$\frac{\mathrm{mL}{\mathrm{CO}}_2}{min}$ and $\mathrm{Vm}{\mathrm{CO}}_2$= 22.26 mL/mmol.

$\mathrm{Vm}{\mathrm{CO}}_2$ = molar volume at STP (volume occupied by a mole of ${\mathrm{CO}}_2$ to Standard T^o and P).

$\mathrm{Vm}{\mathrm{CO}}_2$ = 22.26 L/mol. If we refer to 1 millimole, it occupies 22.26 mL and its dimensions are mL/mmol.

$\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$ = total molar flow rate per minute of ${\mathrm{CO}}_2$ left by pulmonary blood in the form of ${\mathrm{CO}}_2$ and ${\mathrm{HCO}}_3^{-}$ (measured with a blood gas analyzer), calculated using the measure of cardiac output (Q_t). It is expressed in $\frac{\text{mmol}}{min}$ and consists of two components, as given below:

$$\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}=\varphi {\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}+\varphi {\mathrm{HCO}}_{3\left(\mathrm{Qt}\right)}^{-}$$E2

where

1. $\varphi {\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$= molar flow rate of only ${\mathrm{CO}}_2$ left by pulmonary blood, calculated using Q_t. It is expressed in $\frac{\text{mmol}}{min}$.

2. $\varphi {{\mathrm{HCO}}_3^{-}}_{\left(\mathrm{Qt}\right)}=$ molar flow rate of only ${\mathrm{HCO}}_3^{-}$ left by pulmonary blood, calculated using Q_t and expressed in $\frac{\text{mmol}}{min}$.

2.2 $\text{Calculation of}{\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}$

$$\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}=\left(∆{\left[{\mathrm{CO}}_2\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}+∆{\left[{\mathrm{HCO}}_3^{-}\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}\right)\mathrm{x}\mathrm{Qt}$$E3

where $∆{\left[{\mathrm{CO}}_2\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}$ refers to the difference in concentration of ${\mathrm{CO}}_2$ dissolved in venous and arterial blood, and $∆{\left[{\mathrm{HCO}}_3^{-}\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}$ refers to the difference in concentration of ${\mathrm{HCO}}_3^{-}$ present in venous and arterial blood.

Proof of Eq. (3).

From the law of conservation of mass (Antoine Lavoisier), we must hypothesize that the ${\mathit{CO}}_2$ exhaled per minute must be exactly equal to the ${\mathit{CO}}_2$content lost per minute between venous and arterial blood, that is,

$$\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{e}\right)}=\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}.$$E4

However, to obtain $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$ we need to use Eq. (2).

To obtain $\varphi {\mathrm{CO}}_{2\left(\mathrm{Qt}\right)},$

$$\varphi {\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}=\left[\left({\mathrm{PCO}}_{2\mathrm{v}}-{\mathrm{PCO}}_{2\mathrm{a}}\right)\mathrm{K}\right]\times \mathrm{Qt}$$E5

${\mathrm{PCO}}_{2\mathrm{v}}$ = pressure of ${\mathrm{CO}}_2$ in venous blood, expressed in mmHg.

${\mathrm{PCO}}_{2\mathrm{a}}$ = pressure of ${\mathrm{CO}}_2$ in the arterial blood, expressed in mmHg.

$\mathrm{K}$ = solubility constant of ${\mathrm{CO}}_2$. At 37°C in plasma, with a value of 0.03 mmol/L · mmHg.

Q_t = cardiac output expressed in $\frac{\mathrm{L}}{min.}$

The component relating to $\left[\left({\mathrm{PCO}}_{2\mathrm{v}}-{\mathrm{PCO}}_{2\mathrm{a}}\right)\mathrm{K}\right]=∆{\left[{\mathrm{CO}}_2\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}$ represents the concentration gradient of ${\mathrm{CO}}_2$ eliminated through expiration between the venous and arterial blood (application of Henry’s law). When the concentration gradient $∆{\left[{\mathrm{CO}}_2\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}$, expressed in $\frac{\text{mmol}}{\mathrm{L}}$, is multiplied by Q_t expressed in $\frac{\mathrm{L}}{min}$, we obtain $\varphi {\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$= $\frac{\text{mmol}{\mathrm{CO}}_2}{min}$.

This flow corresponds to the portion of ${\mathrm{CO}}_2$ removed as such from the blood, between venous and arterial blood, during exhalation.

To obtain $\varphi {\mathrm{HCO}}_{3\left(\mathrm{Qt}\right)}^{-}$,

$$\varphi {\mathrm{HCO}}_{3\left(\mathrm{Qt}\right)}^{-}=\left({\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{v}}-{\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{a}}\right)\times \mathrm{Qt}$$E6

${\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{v}}$ = concentration of ${\mathrm{HCO}}_3^{-}$ in venous blood, expressed in $\frac{\text{mmol}}{\mathrm{L}}$,

${\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{a}}$ = concentration of ${\mathrm{HCO}}_3^{-}$ in the arterial blood, expressed in $\frac{\text{mmol}}{\mathrm{L}}$,

The concentration gradient of bicarbonate between venous and arterial blood, $∆{\left[{\mathrm{HCO}}_3^{-}\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}$, multiplied by Q_t, expressed in$\frac{\mathrm{L}}{min}$, yields $\varphi {\mathrm{HCO}}_{3\left(\mathrm{Qt}\right)}^{-}=\frac{\text{mmol}{\mathrm{HCO}}_3^{-}}{min}$.

This flow corresponds to the portion of ${\mathrm{HCO}}_3^{-}$ removed from the blood, between venous and arterial blood, during exhalation.

Therefore, the complete equation is as follows:

$$\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}=\left\{\left[\left({\mathrm{PCO}}_{2\mathrm{v}}-{\mathrm{PCO}}_{2\mathrm{a}}\right)\mathrm{K}\right]\times \mathrm{Qt}\right\}+\left({\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{v}}-{\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{a}}\right)\times \mathrm{Qt},$$E7

reworkable in Eq. (3), namely, $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}=\left(∆{\left[{\mathrm{CO}}_2\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}+∆{\left[{\mathrm{HCO}}_3^{-}\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}\right)\mathrm{x}\mathrm{Qt}.$ ${\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}$, according to Lavoisier’s law, must be equal to ${\mathit{\Phi CO}}_{2\left(e\right)}$, but if the required experimental data [25] are entered into Eq. (3), we obtain an inequation, namely, ${\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}>{\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right).}$

This is because the cardiac output (Q_t) is greater than the actual pulmonary blood flow that exchanges with the outside.

In fact, to verify if the blood flow, related to the pulmonary microcirculation, which is the real flow of blood that exchanges with the outside, coincides with the cardiac output, it will be necessary to verify if there is a difference between the flow of ${\mathrm{CO}}_2$ leaving the blood, calculated with the cardiac output, ${\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}$ and the measured flow of the exhaled ${\mathrm{CO}}_2$, ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$, as described below:

$$∆{\mathrm{\Phi CO}}_2={\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}-{\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}.$$E8

If the difference between the two flows, $∆{\mathrm{\Phi CO}}_2$, is equal to zero, then the pulmonary blood flow, related to pulmonary microcirculation (Qp) is equal to the cardiac output (Qt).

If $∆{\mathrm{\Phi CO}}_2>0$, then it is necessary to calculate the pulmonary blood flow (Q_p), introducing the parameter $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qp}\right)}$.

2.3 $\text{Calculation of}{\mathrm{\Phi CO}}_{2\left(\mathrm{Qp}\right)}$

If we substitute the value of Qp in Eq. (3) instead of the Q_t value, we obtain

$$\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qp}\right)}=\left(∆{\left[{\mathrm{CO}}_2\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}+∆{\left[{\mathrm{HCO}}_3^{-}\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}\right)\mathrm{x}\mathrm{Qp}$$E9

2.4 $\text{Identity between}{\mathrm{\Phi CO}}_{2\left(\mathrm{Qp}\right)}\text{and}\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{e}\right)}$

To calculate the pulmonary blood flow in pulmonary microcirculation (Qp), it will be necessary to equate $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qp}\right)}\text{to}\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{e}\right)}$ (application of the conservation of mass) as follows:

$$\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qp}\right)}=\left(∆{\left[{\mathrm{CO}}_2\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}+∆{\left[{\mathrm{HCO}}_3^{-}\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}\right)\mathrm{x}{\mathrm{Qp}}_{}={\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$$E10

where Qp is unknown.

2.5 Calculation of Q_p

$\mathrm{Qp}$ can be derived from Eq. (10) by reworking it as follows:

$$\mathrm{Qp}=\frac{\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{e}\right)}}{∆{\left[{\mathrm{HCO}}_3^{-}\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}+∆{\left[{\mathrm{CO}}_2\right]}_{\left(\mathrm{v}-\mathrm{a}\right)}}$$E11

Qp = pulmonary blood flow in pulmonary microcirculation, expressed in L/min

2.6 Calculation of Q_IPAVA

If Qp is lower than Qt, it means that a part of the cardiac output does not pass through the pulmonary microcirculation, and therefore the difference between cardiac output (Qt) and the pulmonary blood flow in pulmonary microcirculation (Qp) identifies and quantifies, in L/min, the blood flow passing through a shunt (referred to as Q_IPAVA for this study).

To calculate the blood flow in Q_IPAVA, the following operation will need to be performed:

$${\mathrm{Q}}_{\text{IPAVA}}=\mathrm{Qt}–\mathrm{Qp}$$E12

Q_IPAVA is expressed in L/min.

Proof of Eq. (12)

Pulmonary perfusion (Q) is the blood flow, expressed in L/min, through the pulmonary circulation and corresponds to the blood flow of cardiac output (Q_t), that is,

$${\mathrm{Q}}_{\mathrm{t}}=\mathrm{Q}$$E13

However, not always all pulmonary perfusion, determined by cardiac output, exchanges gas with the outside, and therefore pulmonary perfusion, (Q), consists of two flows:

• The pulmonary blood flow that exchanges gas with the outside, namely, $\mathrm{Qp}.$

• The pulmonary blood flow that does not take part in gas exchange, that is, Q_IPAVA.

Therefore,

$${\mathrm{Q}}_{\mathrm{t}}=\mathrm{Qp}+{\mathrm{Q}}_{\text{IPAVA}}$$E14

reworkable in Eq. (12).

2.7 Evidence that Q_IPAVA negatively affects gas exchange efficiency

$$\frac{\text{QIPAVA}}{\mathrm{Qt}}=1—\frac{{\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}}{{\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}}$$E15

Proof of Eq. (15)

Dividing both sides of Eq. (12), namely, ${\mathrm{Q}}_{\text{IPAVA}}={\mathrm{Q}}_{\mathrm{t}}-\mathrm{Qp}$, by Q_t, we obtain

$$\frac{\text{QIPAVA}}{\mathrm{Qt}}=1—\frac{\mathrm{Qp}}{\mathrm{Qt}}$$E16

and, taking into account that

$$\frac{\mathrm{Qp}}{\mathrm{Qt}}=\frac{{\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}}{{\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}}$$E17

by substituting for a $\frac{\mathrm{Qp}}{\mathrm{Qt}}$ the ratio $\frac{{\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}}{{\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}}$, we obtain Eq. (15)

Eq. (15) represents the direct evidence that Q_IPAVA affects the pulmonary gas exchange efficiency with its negative contribution. If we multiply the $\frac{\text{QIPAVA}}{\mathrm{Qt}}$ ratio by 100 (expressing it as a percentage), we obtain the percentage of IPAVA blood flow (Q_IPAVA) with respect to cardiac output (Q_t):

$$\frac{\text{QIPAVA}}{\mathrm{Qt}}\mathrm{x}100=\%{\mathrm{Q}}_{\text{IPAVA}}$$E18

Please note that Eq. (12) abd Eq. (15) are valid for healthy subjects, because in the presence of cardiopulmonary pathologies such as intracardiac shunt (atrial septal defect or PFO-patent foramen ovale) and large diameter pulmonary arteriovenous malformations, the blood flow, which does not exchange with the external environment, can take different routes than IPAVA.

However, since IPAVA are dynamic pathways, whose opening is inducible by hypoxemia, several clinical conditions that cause embolic insults in subjects with diseases associated with concomitant arterial hypoxemic conditions can be explained through the opening of IPAVA at rest [10, 14]. Therefore, Eq. (12) and Eq. (15) are also valid for these subjects in these particular clinical conditions.

2.8 Two more ways to calculate Q_p

It is possible to calculate Qp also using the measurement of cardiac output (Qt) in the following ways:

$$\mathrm{Qp}=\mathrm{Qt}-\left(\mathrm{Qt}\times \frac{\mathrm{\Delta \Phi }{\mathrm{CO}}_2}{{\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}}\right)$$E19

$$\mathrm{Qp}=\left(\frac{\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{e}\right)}}{{\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}}\right)\times \mathrm{Qt}$$E20

Note: Eq. (11), Eq. (19), and Eq. (20) equations to calculate $\mathsf{Q}\mathrm{p}$ always lead to the same result.

2.9 Application of experimental data on equations

Some experimental data presented in a prior work [25] were input into the equations because the parameters required by the equations in this experiment had the characteristic of having been detected simultaneously (namely, venous and arterial ${\mathrm{PCO}}_2$, venous and arterial bicarbonate concentration, Qt and ${\mathrm{VCO}}_2$).

In fact, this experimental model faithfully reproduces the structure of the equations that represent what is lost between venous and arterial blood and, at the same time, what is found in exhaled ${\mathrm{VCO}}_2.$

The experimental data refer to the following physiological parameters: ${\mathrm{VCO}}_2,\mathrm{Qt},{\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{v}},{\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{a}},{\mathrm{PCO}}_{2\mathrm{v}},\text{and}{\mathrm{PCO}}_{2\mathrm{a}}$, measured in different physiological conditions to which the subjects were subjected: *normoxia–placebo, **hypoxia –placebo, in conditions of rest and, during steady state, at moderate (50% of VO_2max) and heavy exercise (≥ 90% of VO_2max).

Note: In Jonk’s experiment [25], placebo refers to the experimental condition without acetazolamide.

*normoxia FIO₂ = 0.2093

**hypoxia FIO₂ = 0.125

2.10 Instruments and methods of measurement in the experiment

Q_t was measured at rest and, during steady state, at 50% and ≥ 90% of ${\mathrm{VO}}_{2max}$ using the noninvasive open-circuit acetylene uptake method [24, 25].

Blood gas was analyzed by a Blood Gas Analyzer (IL Synthesis 45 analyzer).

${\mathrm{VCO}}_2$ was measured with a TrueOne 2400 Parvo Medics Metabolic Cart.

Venous blood was drawn through a catheter placed in the left femoral vein pointing distally. Arterial blood was drawn through a catheter placed in the radial artery of the nondominant arm.

The measured arterial and femoral venous data, $\mathrm{P}{\mathrm{O}}_2,\mathrm{P}{\mathrm{CO}}_2$, and pH, were all corrected to body temperature, together with Hb, Hct, and standard P₅₀ (see calculation of in vivo P₅₀).

The corrected blood data are presented in [25].

The Jonk’s study “was approved by the Human Research Protection Program at the University of California, San Diego, and conducted in accordance with the Declaration of Helsinki. Each subject was informed of potential risks and discomforts and signed an informed consent prior to participation. A self-reporting medical history was obtained that was followed up by a physical examination (which included three-lead electrocardiogram, spirometry, and venous blood draw for haemoglobin and haematocrit assessment) to exclude participants with evidence of cardiac, pulmonary and haematological abnormalities” [25].

3. Results

3.1 Introductory note

The application of the mathematical model started from the input of some experimental data [25] into Eq. (3) to obtain $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$. The input data were related to five parameters:

• $\mathrm{Qt}$(measured using the noninvasive open-circuit acetylene uptake method),

• ${\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{v}},{\left[{\mathrm{HCO}}_3^{-}\right]}_{\mathrm{a}},{\mathrm{PCO}}_{2\mathrm{v}},{\mathrm{PCO}}_{2\mathrm{a}}$ (measured with blood gas analyzer).

${\mathrm{VCO}}_2$ (measured with metabolic cart) was transformed into ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$, as described in Eq. (1).

These experimental data [25] were obtained from subjects subjected to three different physiological conditions (at rest, moderate exercise, and heavy exercise) in normoxia-placebo and hypoxia-placebo conditions.

3.2 Comparison between ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$ and $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$

The results of the experimental data elaboration are shown in Figure 1, where the two flows, ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$ and $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$, are observed to be quite similar but not identical, as indicated by the law on mass conservation. This is because the cardiac output (Q_t) is an excessive multiplication factor (see Eq. (3)) for obtaining a flow $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$ identical to the expired ${\mathrm{CO}}_2$ (${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$). Therefore, $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$is always greater than ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$, except in two cases (normoxia-placebo/rest and hypoxia-placebo/moderate) in which ${\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}$is less than its respective ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$. This is an impossible situation because of the underestimated cardiac output (Q_t) value during the measurement.

Figure 1.

Comparison between ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$ and $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$ in different conditions. ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$ and $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$ are quite similar, but $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$is always greater than ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$. Only in normoxia-placebo/rest and hypoxia-placebo/moderate was $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$ less than ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$ because the Q_t measurement was underestimated.

3.3 Calculation of pulmonary flow of blood ($\mathrm{Qp}$)

Analyzing the data presented in Figure 1, in order for the two flows to be equal and to respect the law on the conservation of mass, it is necessary to calculate Qp.

$\mathrm{Qp}$ was calculated with Eq. (11) (also Eq. (19) and Eq. (20) lead to the same result) for each physiological condition (rest, moderate exercise, heavy exercise) both in normoxia and in hypoxia states. Then, by entering the calculated values of $\mathrm{Qp}$ into Eq. (9), instead of the measured Q_t values, as required by Eq. (3), we obtain the values of $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qp}\right)}$that coincide with the values of ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$. By mathematically relating the two total flows, $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qp}\right)}$ and ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$, we obtain an identity function (n = 6, y = x, R² = 1), as shown in Figure 2.

Figure 2.

${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$ in relation to $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qp}\right)}$, calculated with $\mathrm{Qp}$ values, is an identity function.

3.4 Cardiac output (Qt) compared to pulmonary blood flow ($\mathrm{Qp}$)

Figure 3A, B show the behavior of $\mathsf{Q}\mathrm{p}$, with respect to its own measured Q_t, under the various conditions (normoxia-placebo and hypoxia-placebo) and physiological states (rest, moderate and heavy exercise) examined. We can notice that Q_t was always greater than $\mathsf{Q}\mathrm{p}$, except in two conditions because of the underestimated Q_t and marked with the red arrow.

Figure 3.

Cardiac output (Q_t) compared to calculated pulmonary blood flow ($\mathrm{Qp}$) in normoxia-placebo (A) and hypoxia-placebo (B) under different physiological conditions (rest, moderate, and heavy exercise). (A) In normoxia, Q_t > $\mathrm{Qp}$ under all physiological conditions, except in the rest-placebo (reported with the arrow) because of the underestimated Q_t. (B) In hypoxia Q_t > $\mathrm{Qp}$ in all the physiological conditions, except in moderate-placebo (reported with the arrow) because of the underestimated Q_t, but at high intensity of exercise Q_t » Q$\mathrm{p}$.

Although $\mathrm{Qp}$ is less than its own Q_t, it is only minimally reduced under normoxia-placebo, whereas the reductions are very evident, especially under hypoxia-placebo, at heavy exercise.

In fact, in Figure 3A (normoxia-placebo), we can see that $\mathrm{Qp}$ decreased slightly, reaching its maximum decrease during normoxia under heavy exercise (97.56% of Q_t).

In contrast, in Figure 3B (hypoxia-placebo), we can see that during heavy exercise, $\mathrm{Qp}$ was considerably reduced (85.25% of Q_t), while at rest-hypoxia, the reduction in $\mathrm{Qp}$ was limited (93.51% of Q_t).

3.5 Cardiac output (Q_t), pulmonary blood flow (Q_p), and Q_IPAVA during heavy exercise in normoxia and hypoxia

Where Qt was greater than Qp, it was possible to calculate the shunt flow (Q_IPAVA) with Eq. (12).

Figure 4 shows, in placebo, the effect of hypoxia on Q_t, $\mathrm{Qp}$, and Q_IPAVA, compared to normoxia. A single physiological condition was examined, that of heavy exercise (≥ 90% of ${\mathrm{VO}}_{2max}$), because at rest (in normoxia), and at moderate exercise (in hypoxia), the two values of Q_t, being underestimated, did not allow an investigation.

Figure 4.

Comparison of Q_t, $\mathrm{Qp},$ and Q_IPAVA in normoxia and hypoxia during heavy exercise (≥ 90% of ${\mathrm{VO}}_{2max}$) in placebo. In normoxia, $\mathrm{Qp}$ represents 97.56% of Q_t, and Q_IPAVA represents 2.44% of Q_t; whereas in hypoxia, $\mathrm{Qp}$ represents 85.25% of Q_t, and Q_IPAVA represents 14.75% of Q_t, equal to 3.48 L/min.

The results showed that under heavy exercise, hypoxia considerably reduced $\mathrm{Qp}$ (85.25%) compared to its own Q_t, and Q_IPAVA increased up to 14.75% of Q_t, equal to 3.48 L/min; whereas in normoxia$,\mathrm{Qp}$ decreased only slightly (97.56% of Q_t), and Q_IPAVA represented 2.44% of Q_t, equal to 0.58 L/min.

3.6 Relationship between $\frac{\text{QIPAVA}}{\mathrm{Qt}}$ and $1—\frac{{\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}}{{\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}}$ (Eq. (15))

Applying Eq. (15) to the experimental data, we notice that it is an identity function. Unfortunately, two physiological conditions (normoxia-placebo/rest and hypoxia-placebo/moderate) were excluded from the data processing, as the measured value of Q_t was underestimated. This identity relationship is the direct demonstration that the percentage of Q_IPAVA affects pulmonary gas exchange efficiency with its negative contribution (see Figure 5).

Figure 5.

The relation between 1-$\frac{{\mathrm{\Phi CO}}_{2\left(e\right)}}{{\mathrm{\Phi CO}}_{2\left(\mathrm{Qt}\right)}}$ and $\frac{\text{QIPAVA}}{\mathrm{Qt}}$ (expressed as a percentage) is an identity function (n = 4, y = x, R² = 1). Two physiological conditions (normoxia-placebo/rest and hypoxia-placebo/moderate) were excluded from the data processing, as the measured value of Q_t was underestimated. This relationship is the direct demonstration that the percentage of Q_IPAVA affects its negative contribution to pulmonary gas exchange efficiency.

4. Discussion

In healthy humans, pulmonary blood flow $\left(\mathrm{Qp}\right)$ is currently thought to support the entire cardiac output (Q_t), and both are considered identical [24]; when Q_t is measured, this measure is also referred to as $\mathrm{Qp}$.

According to the results of this study, instead, it is necessary to identify cardiac output only with pulmonary perfusion (Q) and not with $\mathrm{Qp}$. In fact, from the results obtained by applying Eq. (3) to experimental data [25], in which Q_t is used as a multiplicative factor, $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qt}\right)}$ is always greater than ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$, which is the real flow of ${\mathrm{CO}}_2$exhaled from the lungs (see Figure 1). This result is because of the Q_t value, which is greater than Q$\mathrm{p}$ (see Figure 3).

However, this is an impossible situation because, according to the law on mass conservation, the ${\mathrm{CO}}_2$ flow left by pulmonary blood must be identical to expired ${\mathit{CO}}_2$. Therefore, we can deduce that if we use the measure of Q_t as a parameter to assess $\mathrm{Qp}$, we carry out a relatively high overestimation of it.

In contrast, if we calculate $\mathrm{Qp}$ with Eq. (11), we obtain $\mathrm{\Phi }{\mathrm{CO}}_{2\left(\mathrm{Qp}\right)}$ as identical to ${\mathrm{\Phi CO}}_{2\left(\mathrm{e}\right)}$ (see Figure 2). Therefore, only using $\mathrm{Qp}$, the law of the conservation of mass can be satisfied for the excretion of ${\mathrm{CO}}_2$ from human lungs, and all the ${\mathrm{CO}}_2$ content that is lost between venous and arterial blood in terms of ${\mathrm{HCO}}_3^{-}$ and ${\mathrm{CO}}_2$, multiplied by $\mathrm{Qp}$, can be found in the expired ${\mathrm{VCO}}_2$.

Furthermore, using Eq. (11) for calculating $\mathrm{Qp}$ made it possible to immediately identify errors in the measurement of cardiac output (see the two conditions mentioned in Figures 1 and 3); and this aspect is relevant, as the measurements of Q_t are sometimes incorrect, but there is no awareness of it.

4.1 Definition of pulmonary blood flow ($\mathrm{Qp}$)

From this study, we can define pulmonary blood flow ($\mathrm{Qp}$) as the actual blood flow, expressed in liters, which passes through the pulmonary microcirculatory area each minute and allows the effective exchange of respiratory gases through the pulmonary epithelium. Therefore, it cannot be quantified with the cardiac output value because, although minimal, there may be a quantitative difference between cardiac output and pulmonary blood flow ($\mathrm{Qp})$.

Cardiac output (Q_t) is exactly identical to the pulmonary blood flow ($\mathrm{Qp})$ only when all the blood flow starting from the right ventricle of the heart passes through the pulmonary microcirculation area; and consequently, the ${\mathrm{CO}}_2$ flow leaving the pulmonary blood, consisting of bicarbonate and ${\mathrm{CO}}_2$ flows (measured with a blood gas analyzer), is identical to the ${\mathrm{CO}}_2$ flow exhaled, measured externally with a metabolic cart. This can happen in healthy subjects when IPAVA are closed, at rest, and when there are no malformations, such as patent foramen ovale and/or large diameter pulmonary arteriovenous malformations (PAVMs).

Consequently, with the equation that calculates $\mathrm{Qp}$, we can evaluate the pulmonary blood flow, which represents the pulmonary perfusion (Q) net of shunt. In fact, the part of pulmonary perfusion, represented by shunt, is not able to eliminate ${\mathrm{CO}}_2$ towards the external environment; therefore, the component of ${\mathrm{CO}}_2$ present in shunt is not present in the $\mathrm{V}{\mathrm{CO}}_2$ measured with the metabolic cart.

4.2 Pulmonary flow of blood ($\mathrm{Qp}$) and vasoreactivity

$\mathrm{Qp}$ is extremely variable because the pulmonary vasculature is a dynamic system that responds rapidly to vasoactive mediators [2]. In this study, it varies compared to its Q_t (see Figure 3) according to the physiological state in which the subjects are (rest, moderate, or heavy exercise) and the conditions to which they are subjected (normoxia, hypoxia); but to ascertain whether there is a reduction (a loss of efficiency) in pulmonary flow with respect to cardiac output, it is necessary to refer to the $\frac{\mathrm{Qp}}{\mathrm{Qt}}$ ratio.

The $\frac{\mathrm{Qp}}{\mathrm{Qt}}$ ratio represents the efficiency of pulmonary blood flow with respect to cardiac output and should be correlated with pulmonary arteriolar resistance. The more $\mathrm{Qp}$ is less than Q_t, the greater the resistance offered by the vessels and the lower the efficiency of pulmonary flow with respect to cardiac output. Therefore, the use of the measure of $\mathrm{Qt}$to quantify $\mathrm{Qp}$, besides being inaccurate, does not allow the real evaluation of loss of efficiency of pulmonary blood flow especially when the subjects are in different physiological states and/or changed environmental conditions (e.g., at rest, in exercise, in normoxia, in hypoxia) or in the presence of cardiovascular diseases.

Particularly, if the subjects were in normoxia-placebo, $\mathrm{Qp}$ decreased slightly compared to its Q_t (see Figure 3A) in the different physiological states, reaching the maximum reduction with heavy exercise (97.56%). Therefore, in healthy subjects under normal conditions (normoxia), we verified that the value of $\mathrm{Qp}$ was very close to that of Q_t. The physiological mechanisms underlying these results are well described by various authors [2, 3, 5] when they state that the lungs react to the increase in cardiac output during exercise, with an in-built protective mechanism against the increase in pulmonary vascular resistance through the recruitment of capillaries and the distension of the elastic vessels present in the circuit.

In contrast, the effect of hypoxia alone significantly reduced $\mathrm{Qp}$ (85.25% under heavy exercise) compared to its own Q_t (Figures 3B and 4). This is caused by hypoxic pulmonary vasoconstriction, which intervenes in hypoxia, as already observed by several authors [1, 6, 7, 8]. Pulmonary vasoconstriction increases the resistance of the pulmonary arterioles, and consequently, we observed a decrease in Qp.

4.3 IPAVA blood flow (Q_IPAVA)

With Eq. (12), it was possible to quantify Q_IPAVA with a simple, precise method and easy to perform.

Since the pulmonary blood flow Qp represents the only portion that exchanges with the external environment, when the cardiac output is greater than Qp, the part of the flow that does not exchange ${\mathrm{CO}}_2$ with the external environment, namely, Q_IPAVA, is that one which flows through the vessels referable to intrapulmonary arteriovenous anastomoses (IPAVA).

With Eq. (15), it was possible to verify that Q_IPAVA prevents gas exchange and affects with its negative contribution to the efficiency of pulmonary gas exchange (see Figure 5).

Therefore, as verified by various authors through the TTSCE and 99mTc-MAA techniques mentioned above [10, 11, 12, 13, 14, 15, 16, 17, 18], in this study, the presence of Q_IPAVA was confirmed in healthy humans in conditions of both normoxia (during exercise) and hypoxia (at rest and during exercise). Unfortunately, as already pointed out, in normoxia at rest, the Q_t measure was underestimated; therefore, it was not possible to verify, in this study, if IPAVA were open during this condition, although it should be emphasized that various authors [13, 14] have verified that in healthy humans, in normoxia at rest, IPAVA are closed, but only in the vast majority of cases and not in all subjects [19]. In normoxia, during exercise at 50% VO_2max, instead, we could see a Q_IPAVA of 1.56% of Q_t; whereas at high intensity ≥90% of VO_2max, we quantified a Q_IPAVA of 2.44% of Q_t, equal to Q_IPAVA of 0.58 L/min.

Under normoxia, during exercise, Q_IPAVA was rather limited; under hypoxia, especially during heavy exercise, the Q_IPAVA was very high. In fact, we calculated a Q_IPAVA of 3.48 L/min equal to a Q_IPAVA of 14.75% of Q_t.

At rest-hypoxia, instead, Q_IPAVA increased up to 6.49% of Q_t, equal to Q_IPAVA of 0.37 L/min, and these data are congruent with what has been verified by other authors [13, 14] with the abovementioned methods.

4.4 Limitations of the mathematical model

The model allows us to understand the path taken by the pulmonary blood flow (Qp), which exchanges ${\mathrm{CO}}_2$ with the outside, that is, the pulmonary microcirculatory area, and to quantify it with precision in L/min and/or in % of cardiac output.

Logically, the equations that measure Qp provide a precise measurement as a function of the entered data. Therefore, only the data, which reflect the true pulmonary venous and arterial blood concentrations of ${\mathrm{CO}}_2$ and bicarbonate, will provide a very accurate Qp value.

Regarding the pulmonary blood flow that runs through the shunt and therefore does not exchange ${\mathrm{CO}}_2$ with the outside, the model cannot discriminate whether the flow is going through intrapulmonary and/or extrapulmonary shunts but only succeeds in precisely quantifying the shunt in L/min and/or in % of cardiac output.

In this work, we refer only to IPAVA because they are present in healthy subjects, and the experimental data entered refer to these subjects.

To discriminate the pathways traveled by the blood in any cardiopulmonary malformation, this diagnostic method should be integrated with other diagnostic methods (e.g., TTSCE).

5. Conclusion

This mathematical model, being based on the law of conservation of mass, is made up of literal equations that represent the real system of gas exchange, providing an accurate measurement, in each subject, of pulmonary blood flow through the pulmonary microcirculatory area, Qp, and intrapulmonary arteriovenous anastomoses blood flow, Q_IPAVA, (in general shunt), it being understood that the data entered into the equations reflect the real values present in the mixed venous blood of the pulmonary artery and in the arterial blood at the end of the pulmonary capillaries, as well as the measurement of cardiac output. Only in this way an accurate measurement of Qp and Q_IPAVA can be obtained.

The data entered into the equations were obtained through noninvasive and easy-to-use methods.

The use of these equations will allow a general advance in understanding the consequences of Q_IPAVA on physiological and pathophysiological processes. In fact, it will be useful in experimental research and hopefully applicable in the clinical setting especially in people with cardiorespiratory pathologies.

In particular, with a targeted study, it will be possible to better characterize the various pathologies, where the presence of respiratory and metabolic pathologies should cause a specific variation of Qp and Q_IPAVA, strictly depending on the type of disease.

Additional information

Parts of this chapter were previously published in a preprint by the same author: Vanni R. Mathematical Model for the Quantification of Pulmonary and Intrapulmonary Arteriovenous Anastomoses Blood Flow [Internet]. Research Square Platform LLC; 2023. Available from: http://dx.doi.org/10.21203/rs.3.rs-1629034/v3