Blood Oxygenators and Artificial Lungs

Abstract

This chapter is aimed at discussing the main engineering aspects involved in the design of artificial lungs, defining the limits of the currently available devices and understanding the challenges for further developments. To that end, the first part of the chapter provides a short overview of the functions of the respiratory system, that allows defining the medical requirements for the artificial devices and the goals that must be achieved in their design; information on the historical development of the artificial lungs is also included. Subsequently, the physical and chemical fundamentals of gas solubility in blood and gas transport through the membranes widely used in artificial lungs are presented. These fundamentals provide the basis for the engineering analysis of the artificial lung and assessment of its performance. The chapter is concluded with a survey of the state of the art of the clinically used devices with indications for possible further improvements.

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1 Introduction

In healthy humans, lungs are the main respiratory organs that provide oxygen to and remove carbon dioxide from the blood. The lungs perform two main functions: ventilation, i.e., the rhythmic movement of air inspiration and expiration, and respiration, i.e., the real gas exchange between air and blood that occurs in the alveoli. Both lung functions may be impaired by many pathologies that modify the capacity of the respiratory system to ensure air flow and gas exchange; when such pathologies become severe, artificial devices to support the lung functions may be required either temporarily or permanently. Furthermore, artificial devices able to replace the lung function are required in cardiac surgery with extracorporeal circulation, when the blood flux is diverted from the heart–lung compartment. Hereafter, the term artificial lung will be used to refer to all devices aimed at providing oxygen and carbon dioxide exchange to replace or to support the function of the natural lungs.

This chapter is aimed at discussing the main engineering aspects involved in the design of artificial lungs, defining the limits of the currently available devices, and understanding the challenges for further developments. To that end, the first part of this chapter provides a short overview of the functions of the respiratory system that allows to define the medical requirements for the artificial devices and the goals that must be achieved in their design; information on the historical development of the artificial lungs is also included. Subsequently, the physical and chemical fundamentals of gas solubility in blood and gas transport through the membranes widely used in artificial lungs are presented. These fundamentals provide the basis for the engineering analysis of the artificial lung and assessment of its performance. This chapter is concluded with a survey of the state of the art of the clinically used devices with indications for possible further improvements.

2 Structure and Function of Respiratory System

This and the following section were authored by Felice Eugenio Agrò, Marialuisa Vennari, and Maria Benedetto—University School of Medicine “Campus Bio-Medico” of Rome, Italy.

The respiratory system includes the upper airway (mouth, nose, nasal cavity, pharynx, and larynx), lower airway (trachea, bronchi, bronchioles, and alveoli), and respiratory pump (rib cage, intercostal muscles, diaphragm, and accessory muscles). Bronchioles and alveoli compose the lungs bound by visceral pleura. Visceral pleura reflects in the parietal pleura, on the inner face of the chest, connecting lungs to respiratory pump.

The main function of the respiratory system is connected to the aerobic metabolism of cells, which requires oxygen and produces carbon dioxide. In detail, the respiratory system guarantees the elimination of the $\mathrm {CO_{2}}$ and the uptake of the O$_{2}$ in two phases: ventilation and respiration. Ventilation involves the respiratory pump and airway conducts, till bronchioles, while respiration involves alveoli and blood, through the respiratory membrane (external respiration) as well as blood and tissue through the capillary membrane (internal respiration).

Ventilation consists of a mechanic and rhythmic process in which air movement in (inspiration) and out (expiration) of lungs is permitted. Air flux is due to a transpleural pressure gradient, which determines convection movements. At rest, during inspiration, the contraction of diaphragm and intercostal muscles leads to the expansion of chest cavity and lungs, which, due to a reduction of intrapleural pressure, determines air inflow. During expiration, the relaxation of muscles leads to elastic return of the chest and lungs, with an increase in intrapleural pressure, which causes the outflow. Through pressure variation at each respiratory act, the movement of 500 ml of air is realized (tidal volume). With respect to this volume, only 350 ml participates in the real gas exchange: in fact, 150 ml remain in the conducting airway (anatomic dead space¹) that does not take part in gas exchange. The rhythmic function of ventilation has a rate of about 15 acts/min, determining a volume rate of about 7.5 l/min (total ventilation = tidal volume $\times$ respiratory rate). Considering the dead space, the alveolar ventilation (the total volume of air arriving to alveoli in a minute) is about 5.25 l/min.

Respiration is realized in the alveoli that are the real functional parts of the whole airway. Alveoli have a mean diameter of 200–300 µm and represent the distal part of the lungs in which air is conducted by a terminal airway ramification (terminal bronchioles). It has been estimated that in a mean adult, the respiratory zone of the lungs has a surface of about 100–150 $\mathrm {m^{2}}$ and contains a volume of 2.5–3 l of air. The effectiveness of the large surface of the respiratory zone in providing gas exchange is further increased by the characteristics of the alveolar capillaries and alveolar membrane (blood–gas interface). Alveolar capillaries are large enough to permit the passage of a single red cell (7–10 µm), leading to a continuous sheet of blood flowing over the alveolar membrane and allowing a sufficient contact for gas exchange even in a short time (0.75 s per red cell, 0.25 on exertion). Moreover, the alveolar membrane is extremely thin (1.5 µm), enhancing the gas exchange rate between the alveolar gas and blood. As previously reported, both $\mathrm {O_{2}}$ and $\mathrm {CO_{2}}$ must be exchanged between the alveolar gas and blood to balance the gas consumption and production due to the cell aerobic metabolism. Actually, $\mathrm {O_{2}}$ diffuses from alveoli ($\mathrm {O_{2}}$ pressure of 150 mmHg) to venous blood ($\mathrm {O_{2}}$ pressure of about 40 mmHg), while $\mathrm {CO_{2}}$ diffuses from venous blood ($\mathrm {CO_{2}}$ pressure about 45 mmHg) to alveoli ($\mathrm {CO_{2}}$ pressure of 40 mmHg). It is worth noting that since the ratio of $\mathrm {CO_{2}}$ production to $\mathrm {O_{2}}$ consumption by tissues (respiratory exchange ratio) is about 0.8, an adequate respiratory function requires the same ratio to apply to the rate of $\mathrm {CO_{2}}$ elimination and $\mathrm {O_{2}}$ uptake in lungs.

As discussed later in detail (Sect. 6.7), gas exchange efficiency depends on ventilation and perfusion. Different parts of the lungs are not equally perfused nor ventilated; in particular, the cardiac output is distributed in the different parts of the lungs according to the transmural vessel pressure (the difference between the capillary and alveolar pressures). Actually, in the upper part of the lung (apical lung), the pulmonary arterial pressure (i.e., the pressure of the blood entering the pulmonary capillary) is lower than the alveolar pressure, this tends to narrow the cross section of capillaries, and only minimal blood flux is permitted; as a consequence, this region is only minimally participating to gas exchange. In the middle and basal lung, arterial pulmonary pressure is higher than alveolar pressure and a higher blood flux is permitted: in the middle zone, the arterial pressure is higher, but the venous pressure (i.e., the pressure of the blood at the end of the pulmonary capillary) is lower than alveolar pressure and the blood flux is determined by arterial–alveolar pressure difference; on the other hand, in the basal zone, both the arterial and venous pulmonary pressures are higher than alveolar pressure and the blood flux is determined by the arterial–venous pressure difference. The middle zone is the ideal zone where ventilation/perfusion rate is near to 1. Actually, while arterial oxygen concentration is largely affected by the ventilation–perfusion ratio, the $\mathrm {CO_{2}}$ concentration is mainly dependent on ventilation. As a consequence, an increase in alveolar ventilation may correct $\mathrm {CO_{2}}$ elimination, while it does not surely correct hypoxemia caused by alterations of the ventilation–perfusion ratio.

3 Extracorporeal Gas Exchange Devices in Clinical Practice

Extracorporeal devices able to provide gas exchange are routinely used intraoperatively in cardiac surgery with extracorporeal circulation. In this case, usually a membrane oxygenator is included as an essential part of the cardiopulmonary pump needed to replace the cardiac and pulmonary function; cardiopulmonary bypass (CPB) is generally applied for few hours, but, in some cases, a prolonged support may be needed. In these cases, extracorporeal membrane oxygenation (ECMO) is used as life support or lung assist, to provide continuous support, typically for a period of the order of days to weeks.

The use of ECMO to supplement the insufficiencies or the failure of the respiratory system is less frequent, but the interest in this procedure has been increasing in recent years. In fact, in clinical practice, the two functions of the respiratory system (ventilation and respiration) may be affected by many pathologies: both primitive lung diseases—e.g., pneumonia, chronic obstructive pulmonary disease (COPD), fibrosis, and acute respiratory distress syndrome (ARDS)—and secondary pulmonary involvements—e.g., cardiogenic edema, neurological impairment, and chest alterations—may modify the capacity of the respiratory system to ensure sufficient air flux and gas exchange. When the decrease of this capacity is life-threatening, the functions of the respiratory system should be artificially supported. In this case, mechanical ventilation (MV) and/or extracorporeal oxygenators (EOs) may be indicated. Historically, ECMO has been used with benefits in neonates and children with reversible cardiorespiratory failure. In recent years, evidence supporting ECMO use in adults has emerged, with increased survival rate compared to the optimization of standard therapy both in severe respiratory failure and cardiac failure. This evidence has been underlined during H1-N1 epidemic, in CESAR [37] trials on patients with ARDS, and in reports on patients with cardiac arrest, cardiogenic shock, and who failed weaning from CPB (Extracorporeal Life Support Organization, ELSO, registry).

Hereafter, the focus will be put on ECMOs. Indeed, ECMO seems to present some advantages compared to standard therapy based on mechanical ventilation: in particular, the ventilator-induced lung injury (barotrauma or volutrauma) is avoided, and the lungs are allowed to rest and acute damage recovery.

Two main types of ECMO may be distinguished: veno-venous (VV) and veno-arterial (VA).² VA ECMO supports both pulmonary and cardiac functions, while VV ECMO provides only respiratory support. The main indications to ECMO use are acute cardiac failure (AV ECMO) and acute respiratory failure (VV ECMO) with high mortality risk, despite optimal conventional therapy. In particular, according to ELSO guidelines for ECMO centers, the main indications for ECMO use in adults are as follows:

• acute respiratory failure with a ratio of arterial oxygen pressure³ (mmHg) to the fraction of oxygen in the inspired air (PaO2/FiO2) $< 150$ on FiO2 $> 90\,\%$ and/or Murray score⁴ 2–3 (ECMO suggested) or with a PaO2/FiO2 $< 80$ on FiO2 $> 90\,\%$ and Murray score 3–4 (ECMO indicated);

• $\mathrm {CO_{2}}$ retention with PaCO2 $> 80$ mmHg or inability to achieve safe inflation pressures;

• severe air leak syndromes (pneumothorax, broncho-pleural fistula);

• refractory cardiogenic shock;

• septic shock with severe cardiac dysfunction (indication is some center);

• failure to wean from cardiopulmonary bypass;

• cardiac arrest;

• as a bridge to the placement of a ventricular assist device (VAD), to cardiac transplantation, and during the recovery of revascularization in myocardial infarction, myocarditis, and postcardiotomy. In case of respiratory failure, generally, there are no absolute contraindications to ECMO use. Relative contraindications are conditions with known poor outcome despite ECMO (MV $> 7$ days with high pressure) and specific patient conditions (e.g., severe obesity and comorbidities).

Recently, ECMO use has been suggested in case of chronic pulmonary diseases such as COPD. The use of systems for specific $\mathrm {CO_{2}}$ elimination may reduce the incidence of intubation in case of COPD exacerbation in association with noninvasive ventilation (NIV). These results may be encouraging to further experience ECMO use in chronic respiratory failure.

In pediatric cases, ECMO is indicated within the first week of MV at high pressure or when a shock refractory to standard treatment developed, the weaning from CPB failed, after a successful cardiopulmonary reanimation the patient is still unstable, or a severe cardiac failure of any etiology develops. Contraindications depend on age, comorbidities, and the presence of contraindication to anticoagulation.

4 General Remarks on Blood Oxygenator Design

Before going into details on the study of blood oxygenators, it is beneficial to briefly discuss the requirements for these devices and the goals to achieve in their design. Thus, it will be easier to understand the technological development of blood oxygenators, the present-day applications, and the work that is still to be done to obtain better devices for wider application fields.

A device capable of providing the complete respiratory function, as in a cardiopulmonary bypass for an open-heart surgery, will be considered: such device must transfer up to $250\,\text {Nml/min}$ (11 mmol/min) of oxygen, in order to meet the basal metabolic requirement of an adult patient; in the past, the gas exchange requirement was reduced by lowering the patient’s body temperature, but nowadays, there is a trend toward normothermic perfusion. At the same time, the device has to remove about 200 Nml/min (9 mmol/min) of carbon dioxide. The $\mathrm {CO_{2}}$-to-$\mathrm {O_{2}}$ rate ratio must equal the respiratory ratio (0.8), and a fine control of the amount of carbon dioxide removed is required to avoid both hyper- or hypocapnia.

Fig. 6.1

Scheme of operating conditions for an oxygenator used in cardiopulmonary bypass

Typical operating conditions are summarized in Fig. 6.1. The whole blood flow rate (5 l/min) is pumped through the device, with a roller or centrifugal pump; the inlet blood has $\mathrm {O_{2}}$ and $\mathrm {CO_{2}}$ partial pressures of about $40\,\text {mmHg}$ (corresponding to about $6.6\,\text {mM}$, see Sect. 6.6.1.1) and $45\,\text {mmHg}$ (corresponding to about $12.7\,\text {mM}$, see Sect. 6.6.1.2), respectively; the arterialized blood has to attain $\mathrm {O_{2}}$ and $\mathrm {CO_{2}}$ partial pressures of about 100–150 mmHg (8.7–8.9 mM) and 30–40 mmHg (8.5–11 mM), respectively.

The inlet gas is usually pure oxygen or an oxygen-rich mixture, which provides a large driving force for oxygen transfer to the blood. On the other hand, the gas flow rate controls the carbon dioxide concentration in the sweep gas and, therefore, the effectiveness of its removal from blood: the higher the gas flow rate, the lower of the carbon dioxide pressure in the sweep gas and the higher the driving force for carbon dioxide removal [39]. Therefore, the carbon dioxide transfer rate depends directly on the gas flow rate. More specifically, since $\mathrm {CO_{2}}$ partial pressure in the outlet gas cannot exceed 40–45 mmHg in order to ensure a sufficient driving force for its removal, a minimum gas flow rate of about 3.6 l/min is required.⁵ Usually, the actual gas flow rate ranges from 5 to $10\,\text {l/min}$.

Regardless of the type of device used, the gas transfer rate is given by:

$$ \text {exchange surface}\times \text {mass transfer coefficient}\times \text {driving force} $$

or

$$ \text {volume}\times \frac{\text {exchange surface}}{\text {volume}}\times \text {mass transfer coefficient}\times \text {driving force} $$

From a clinical point of view, in order to minimize the transfusion of donor blood or plasma expander solution, it is desirable to minimize the oxygenator priming volume as well as the volume of the extracorporeal circuit; therefore, it is of paramount importance that the oxygenator be designed so as to have a high exchange surface area to volume ratio. On the other hand, the exchange surface area should be kept as low as possible, still meeting the gas transfer rate requirements. Indeed, blood exposure to exogenous surfaces causes the activation of the complement and coagulation cascade, which impair the biocompatibility of the device; furthermore, for membrane oxygenators, a reduction of the membrane area will also reduce the cost of the device. Therefore, a fundamental target to be pursued in the design of blood oxygenators is to obtain high mass transfer coefficients and exploit the maximum driving force available.

As for the mass transfer coefficient, this parameter is mainly determined by the blood-side gas transport resistance and, in membrane oxygenators, also by the mass transfer resistance of the membrane itself; therefore, the new technological developments should be aimed both at optimizing blood fluid dynamics in the oxygenators, in order to improve gas transport in the blood layer, and producing membranes with high gas permeability.

The maximum available driving force for gas transfer is about $650\,\text {mmHg}$ for oxygen⁶ and $40\,\text {mmHg}$ for carbon dioxide. In order to mimic the respiratory exchange ratio and best exploit the maximum available driving force for both gases, the $\mathrm {CO_{2}}$-to-$\mathrm {O_{2}}$ mass transfer coefficient ratio should approach $0.8\times 650/40\simeq 13$. For lower mass transfer coefficient ratios, carbon dioxide removal controls the exchange surface required; on the other hand, for higher mass transfer coefficient ratios, the $\mathrm {CO_{2}}$ removal rate must be controlled by proper choice of the gas flow rate and/or carbon dioxide concentration in the inlet gas.

5 Development of Blood Oxygenators: History and Current Solutions

The birth of blood oxygenators is strictly related to the requirements set by the cardiopulmonary bypass (CPB) procedure in open-heart surgery; therefore, the early devices developed were integrated systems including both a blood pump and an oxygenator.

The first devices used between 1930 and 1960 in open-heart surgery provided $\mathrm {O_{2}}$ and $\mathrm {CO_{2}}$ exchange by direct contact of blood with a gas phase. Two methods were used to obtain a large gas–blood contact area: to contact the gas phase with a thin blood film flowing on a solid surface (film oxygenators) or to disperse small gas bubbles into venous blood (bubble oxygenators) [40].

In film oxygenators, a thin blood film is formed on stationary or rotating surfaces; gas exchange occurs through the film surface directly exposed to a gas phase with high oxygen and low carbon dioxide partial pressure. In the first successful CPB operation, a screen oxygenator was used. Such device consisted of a series of upright wire mesh screens with the venous blood flowing from the top by gravity and forming a thin film on the screens; arterialized blood was collected at the bottom of the screens to be returned to the patient. Screen oxygenators were included in a commercial device for CPB (Mayo-Gibbon pump oxygenator), but such apparatus was bulky and required a large blood and saline solution priming volume.

Fig. 6.2

Scheme of a rotating disk oxygenator

In the mid-fifties, rotating disk oxygenators were also introduced in the clinical practice. In this type of oxygenators (see Fig. 6.2), a blood film is created on vertical disks rotating on an horizontal axis and dipping into a pool of venous blood; in the upper part of the disks, the blood film is exposed to an oxygen-rich atmosphere, thus resulting in a good oxygenation capacity. In spite of the large priming volume required and assembling and sterilization issues, rotating disk oxygenators were largely used in the clinical practice until the seventies: this was mainly due to the perception that blood trauma caused by this type of device is generally quite low [40].

Almost in the same period (1950s), the idea to obtain a large gas–blood contact area and efficient gas transfer by bubbling an oxygen-rich gas phase into venous blood was explored. In spite of the apparent simplicity of this approach, which also allows to operate with a low priming volume, several issues had to be solved before a bubble oxygenator suitable for clinical application could be obtained. First of all, gas bubble dispersion in blood required a defoaming system to remove gaseous emboli before returning the arterialized blood to the patient; in this respect, the size of gas bubbles is crucial: while smaller gas bubbles, with high surface-to-volume ratio, result in a larger specific surface area and more efficient gas exchange, they are less prone to rise spontaneously to the surface and are more difficult to remove. Furthermore, the need for an adequate ratio between the oxygen transfer flow rate (which is kinetically limited by the exchange surface area) and carbon dioxide transfer flow rate (which is limited by $\mathrm {CO_{2}}$ accumulation in the gas phase) had to be compromised to achieve an efficient device.

Fig. 6.3

Scheme of the De Wall bubble oxygenator

In 1955, a helical reservoir pump oxygenator (De Wall oxygenator [41]) was used for the first time in an intracardiac surgical operation: in such oxygenator (see Fig. 6.3), oxygen is mixed with venous blood in a vertical cylinder where gas exchange occurs; on top of the mixing tube, the two fluid phases enter a silicon-coated chamber, where gas bubbles coalesce and some debubbling occurs; finally, debubbling is pushed further in a helical tubular reservoir, in which the gas bubbles float upward while blood flows downwards. De Wall bubble oxygenator gained a large acceptance, being used in 90 % of open-heart operations in 1976 [40]; indeed, such device was cost-effective and easy to assemble and sterilize. The popularity of this device was also supported by the introduction of a single-use, presterilized, and prepacked plastic version [41], together with different systems to improve defoaming (see for example [42]). Even if a high oxygenation efficiency can be achieved with bubble oxygenators, two important shortcomings limit their use: firstly, the need for a thorough removal of gas bubbles in order to reduce the risk of embolism; secondly, the blood damage due to mechanical stress and the direct contact with gas and solid surfaces. In particular, blood damage issues limit the duration of sessions with direct contact oxygenators and made them unsuitable for long-term therapeutic support, as required for patients with ARDS or newborns with neonatal respiratory distress syndrome.

A significant improvement to the state of the art was obtained with the introduction of membrane oxygenators, in which a gas permeable membrane is interposed between the gas phase and blood, thus avoiding the need of a defoaming unit and significantly reducing blood damage. Indeed, membrane oxygenators mimic the natural lung configuration, where gas exchange occurs through the alveolar membrane. Actually, the first idea to use membranes for blood oxygenation dates back to the mid-forties, when Kolff noted that during hemodialysis sessions, blood oxygenation occurred in parallel with detoxification, due to gas exchange with the oxygen-saturated dialysis solution [43]; nevertheless, more than twenty years were necessary to develop effective membrane oxygenators for clinical use and to challenge the dominant position held by bubble oxygenators [44]. In the early developments, the focus was put on producing reliable membranes with high permeability to both $\mathrm {O_{2}}$ and $\mathrm {CO_{2}}$. Polyethylene membranes showed a low $\mathrm {CO_{2}}$ permeability (only five times greater than oxygen permeability, compared to a permeability ratio in the lung membrane exceeding 20): as a result, the membrane surface area was controlled by the required $\mathrm {CO_{2}}$ transfer rate. A significantly better performance was obtained in the sixties with silicone rubber membranes, which showed higher permeabilities and a more favorable $\mathrm {CO_{2}}$ to $\mathrm {O_{2}}$ transfer ratio; on the other hand, with the improvement of membrane performance, oxygen diffusion through the blood film became a significant resistance to the gas transfer.

The development of membrane oxygenators is then the result of advancements on membrane materials, leading to improved permeability and selectivity, and optimization of fluid dynamics of the devices, which allows to reduce both the membrane surface area and the priming volume. Spiral coil (Kolobow oxygenators) as well as plate and screen configurations were firstly used; the hollow fiber configuration, already adopted in hemodialysis, was transferred to blood oxygenators in the middle of seventies [45–47].

The next major advance came with the introduction of microporous hydrophobic membranes with a pore size below 1 µm: in these membranes, O$_{2}$ and CO$_{2}$ diffusion does not occur through the membrane material but in the gas-filled pores. The use of highly hydrophobic materials prevents plasma leakage through the membrane pores: at the beginning of eighties, the first commercial hollow fiber oxygenator used silicone-coated microporous polypropylene membranes; more recently, poly(4-methyl-1-pentene) (commonly called with the trademark TPX) membranes showed a very good performance [48]. Although microporous membranes exhibit high gas transfer rates, in long-term use they undergo a progressive alteration of surface properties due to protein and lipid adsorption, the wetting of membrane pores, and the plasma infiltration and leakage through the pores; as a consequence, the membrane permeance markedly reduces over time. Composite hollow fiber membranes, including a non-porous polymer layer (“skin”) on the microporous membrane surface, that prevent the blood infiltration into the pores and control the long-term performance have been developed in the last years.

Blood oxygenators currently used in clinical practice are membrane oxygenators, in most cases based on microporous hollow fiber membranes. The fibers (diameter 200–300 µm) are wound or bundled in a hard shell, with a fiber packing density of 40–60 % and a large surface area-to-volume ratio (1–2.5 $\text {m}^{2}$, with a priming volume in the range of 100–350 $\text {ml}$). The gas usually flows inside the lumen of the fibers, while blood flows outside, through the fiber bundle (extraluminal flow). The opposite flow pattern, with blood flowing inside the fibers (intraluminal flow), is also possible, but less frequent; indeed, extraluminal flow results in a lower resistance to gas transfer through the blood film, because the flow past the fibers induces a secondary flow that enhances mixing; furthermore, since the extraluminal side has a wider cross section, also the resistance to blood flow and pressure drop are reduced in extraluminal flow. A passive secondary flow may be also obtained by putting obstacles in the blood path, creating undulation or texturing the membrane surface or using a specific flow geometry (e.g., helical coil). It is important to note that turbulence and secondary flow increase the pressure drop and the shear rate experienced by red cells; therefore, also the shear-induced hemolytic damage is increased.

Fig. 6.4

Flow pattern in Affinity^® NT blood oxygenator, including a heat exchanger

Fig. 6.5

Picture of the Maquet Quadrox^®. Flow pattern highlighted. Reproduced from [50], with permission

In extraluminal flow, the angle between the fibers and blood flow affects the mass transfer coefficient [49]. Good results have been obtained with blood flowing perpendicularly to the hollow fiber axis, but different flow patterns are found in the different commercial devices: blood flows radially through the fiber bundle in the Medtronic Affinity^® NT oxygenator (Fig. 6.4) or perpendicularly to the gas pathway in the Maquet Quadrox^® (Fig. 6.5). Silicone oxygenators with non-porous silicone sheets have poorer performance in terms of gas exchange efficiency, but are clinically used for long-term support. Many of the modern devices are integrated with a rigid reservoir and a heat exchanger [45].

Configurations and performance of some membrane oxygenators used in the clinical practice are summarized in Table 6.1.

Table 6.1 Properties of hollow fiber membrane blood oxygenators

6 Fundamentals of Gas Exchange

The knowledge and quantitative description of gas solubility and transport in blood are required in order to understand gas exchange both in healthy lungs and in blood oxygenators. This section offers a brief overview of these fundamental aspects with specific focus on respiratory gases, i.e., $\mathrm {O_{2}}$ and $\mathrm {CO_{2}}$.

6.1 Gas Solubility in Blood

The dissolution of both $\mathrm {O_{2}}$ and $\mathrm {CO_{2}}$ in blood follows a combined physical and chemical mechanism: both gases are absorbed into plasma as molecular species and then take part in chemical reactions in the liquid phase, which enhance their solubility. More specifically, molecular $\mathrm {O_{2}}$ binds to hemoglobin, while $\mathrm {CO_{2}}$ dissociates leading to the formation of carbonate and bicarbonate ions⁷; furthermore, $\mathrm {CO_{2}}$ and $\mathrm {O_{2}}$ chemical equilibria are coupled via the pH effect with oxyhemoglobin dissociation and CO$_{2}$ binding to hemoglobin.

Since the early works of Adair [51], several studies have been reported in the literature and extensive reviews have also been published [52, 53]; here, the main results and theoretical models useful for the design of blood oxygenators are reported.

6.1.1 Oxygen Solubility

Molecular oxygen (physical) solubility in plasma is described by Henry’s law⁸

$$\begin{aligned} p_{\mathrm {O_{2}}}=H_{\mathrm {O_{2}}}^{\prime }\, c_{\mathrm {O_{2}},m} \end{aligned}$$

(6.1)

where $p_{\mathrm {O_{2}}}$ is the oxygen partial pressure in the gas phase, $c_{\mathrm {O_{2}},m}$ is the concentration of free, molecular oxygen in the liquid phase, and $H_{\mathrm {O_{2}}}^{\prime }$ is the Henry constant of molecular oxygen in plasma. The value of $H_{\mathrm {O_{2}}}^{\prime }$ ($1\cdot 10^{6}\,\text {atm mol}^{-1}\text {cm}^{3}$) denotes a very low solubility of oxygen in plasma: indeed, the physical solubility alone is insufficient to ensure sufficient oxygen transport to fulfill the tissue metabolic requirements, which explains why the presence of an oxygen carrier in blood is necessary.⁹

Oxygen transport is facilitated by its binding to hemoglobin inside red blood cells; in this tetrameric protein, each amino acid chain contains a heme group which binds to an oxygen molecule and a terminal amino group which can bind to a CO$_{2}$ molecule. As for oxygen–hemoglobin binding, Adair [51] suggested to account for four binding reactions between the partially oxygenated hemoglobin, indicated as $\mathrm {Hb(O_{2})}_{n-1}$, and the oxygen molecule. The general form of such reactions is as follows:

$$ \mathrm {Hb(O_{2})}_{n-1}+\mathrm {O_{2}}\rightleftarrows \mathrm {Hb(O_{2})}_{n}\qquad n=1,\ldots ,4 $$

and the corresponding equilibrium constants are as follows:

(6.2)

Accounting for these reactions, a fractional saturation of hemoglobin, defined as

$$ S_{\%}=\frac{\text {oxygen molecules bound to hemoglobin}}{\text {maximum oxygen molecules bound to hemoglobin}} $$

can be evaluated as:

(6.3)

where represents the hemoglobin concentration¹⁰; typical values of the binding equilibrium constants for human blood are reported in Table 6.2: the cooperativity of heme groups for $\mathrm {O_{2}}$ binding, i.e., the increase hemoglobin affinity for oxygen after binding the first $\mathrm {O_{2}}$ molecule, results in a sigmoid saturation curve; as for many engineering calculations, such a curve can be described by the simplified Hill equation [55]:

Table 6.2 Equilibrium constant for oxygen-hemoglobin-binding reaction (${\text{ mmHg }^{-1}}$) [54]

$$\begin{aligned} S_{\%}=\frac{\left( p_{\mathrm {O_{2}}}/p_{50}\right) ^{n}}{1+\left( p_{\mathrm {O_{2}}}/p_{50}\right) ^{n}} \end{aligned}$$

(6.4)

which is based of the following apparent reaction

$$\begin{aligned} \mathrm {Hb+}n\mathrm {O_{2}}\rightleftarrows \mathrm {Hb(O_{2})}_{n} \end{aligned}$$

(6.5)

In Eq. 6.4, $p_{50}$ is the oxygen partial pressure at which 50 % of hemoglobin oxygen-binding sites are saturated (the higher the $p_{50}$ value, the lower the hemoglobin affinity for oxygen), while n is defined by Eq. 6.5. A value for n of 2.7 was found to fit well data for normal human blood in the saturation range of 20–98 %, while a value of about $27\,\text {mmHg}$ is usually assumed for $p_{50}$; however, it is well known that $p_{50}$ is affected by temperature, $\text{ CO }_{2}$ concentration, pH, and ratio of diphosphoglycerate¹¹ (DPG) to hemoglobin concentrations. More specifically, $p_{50}$ is a decreasing function of temperature, CO$_{2}$ and DPG concentration, and an increasing function of pH. Several correlations have been proposed to describe the dependence of $p_{50}$ on the above-listed variables; as an example, those reported by Samaja et al. [56, 57] are reported in Table 6.3.

Table 6.3 Dependence of $p_{50}$ on several physiological variables ($p_{50}$, $p_{\mathrm {CO_{2}}}$, mmHg; T, K) [56, 57]

Finally, the total oxygen concentration in blood is given by:

(6.6)

Fig. 6.6

Oxygen concentration in blood as a function of $\mathrm {O_{2}}$ partial pressure

Figure 6.6 reports the total oxygen concentration in blood as a function of oxygen partial pressure in the gas phase: it can be seen that more than 99 % of the dissolved oxygen is present as oxyhemoglobin complex; in other words, the capacity of blood to transport oxygen to peripheral tissues is strongly affected by its hemoglobin content. In Fig. 6.6, the regions of the plot corresponding to oxygen pressures in venous and arterial blood are marked: it is evident that the effect of a change in $p_{\mathrm {O_{2}}}$ on oxygen solubility is much stronger in venous rather than in arterial blood. Furthermore, it is worth noting that the $p_{\mathrm {O_{2}}}$ range from 30 to 50 mmHg, which corresponds to the oxygen level that must be maintained in the peripheral tissues, is also characterized by a strong dependence of oxygen solubility on $p_{\mathrm {O_{2}}}$: this feature allows to have a significant $\mathrm {O_{2}}$ exchange rate between blood and tissues even with small $p_{\mathrm {O_{2}}}$ differences.

By differentiating Eq. (6.6), the following expression is obtained:

(6.7)

where $\mathscr {K}=4H_{\mathrm {O_{2}}}^{\prime }\left( \partial S_{\%}/\partial p_{\mathrm {O_{2}}}\right)$. In the physiological blood oxygen pressure range (40–95 mmHg), a linearized form of Eq. 6.6 with a constant value of about $12.1\cdot 10^{6}\,\text {mol}^{-1}\text {cm}^{3}$ for $\mathscr {K}$ provides a reasonable approximation of the oxygen solubility curve; therefore, in this partial pressure range, an apparent Henry’s constant for $\mathrm {O_{2}}$ in blood can be used to describe $\mathrm {O_{2}}$ solubility

(6.8)

6.1.2 Carbon Dioxide Solubility

While almost all oxygen dissolved in blood is bound to hemoglobin inside red cells, the majority of carbon dioxide is found in plasma (2/3) and red cells (1/3) as bicarbonate ion and only less than 5 % is bound to hemoglobin. Table 6.4 reports some typical values of the carbon dioxide concentration in arterial and venous blood. Distribution of $\mathrm {CO_{2}}$ among different chemical species, namely molecular carbon dioxide, bicarbonate, and carbonate ions, occurs according to the following hydrolysis reactions:

Table 6.4 $\text {CO}_{2}$ content in arterial and venous blood (concentrations are referred to the blood volume; $\mathrm {Htc}=45\,\%$) [58]

$$ \mathrm {CO_{2}}+\mathrm {H_{2}O}\rightleftarrows \mathrm {H_{2}CO_{3}}\qquad K_{h}= 1.7\cdot 10^{-3}{}\quad $$

$$ \mathrm {H_{2}CO_{3}}\rightleftarrows \mathrm {HCO_{3}^{-}+H^{+}}\qquad K_{a1}= 2.5\cdot 10^{-4}\,\text {M} $$

$$ \mathrm {HCO_{3}^{-}\rightleftarrows CO_{3}^{2-}+H^{+}}\qquad K_{a2}= 5.6\cdot 10^{-11}\,\text {M} $$

By accounting for the above-listed reactions, the total carbon dioxide concentration can be expressed as:

$$\begin{aligned} c_{\mathrm {CO_{2}}}=c_{\mathrm {CO_{2}},m}+c_{\mathrm {HCO{}_{3}^{-}}}+c_{\mathrm {CO{}_{3}^{2-}}}=c_{\mathrm {CO_{2}},m}\left( 1+\frac{K_{h}K_{a1}}{10^{-\mathrm {pH}}}+\frac{K_{h}K_{a1}K_{a2}}{10^{-2\,\mathrm {pH}}}\right) \end{aligned}$$

(6.9)

In Eq. 6.9, $\mathrm {CO_{2}}$ in the form of carbonic acid or carbamino compounds (bound to hemoglobin) was neglected. At the physiological blood pH (about 7.3), the ratio of $\mathrm {HCO_{3}^{-}}$ to $\mathrm {CO_{2}}$ concentrations is about 93 %.

Assuming that free $\mathrm {CO_{2}}$ solubility is described by the Henry’s law, the total $\mathrm {CO_{2}}$ concentration in the liquid phase can be expressed as a function of its partial pressure in the gas as

$$\begin{aligned} c_{\mathrm {CO_{2}}}=\frac{p_{\mathrm {CO_{2}}}}{H_{\mathrm {CO_{2}}}^{\prime }}\left( 1+\frac{K_{h}K_{a1}}{10^{-\mathrm {pH}}}+\frac{K_{h}K_{a1}K_{a2}}{10^{-2\mathrm {pH}}}\right) \end{aligned}$$

(6.10)

Therefore, the solubility of $\mathrm {CO_{2}}$ in blood can be expressed by referring to a pH-dependent apparent Henry’s constant, $\mathscr {H}_{\mathrm {CO_{2}}}$, which can be calculated as follows

$$\begin{aligned} \frac{1}{\mathscr {H}_{\mathrm {CO_{2}}}}=\frac{c_{\mathrm {CO_{2}}}}{p_{\mathrm {CO_{2}}}}=\frac{1}{H_{\mathrm {CO_{2}}}^{\prime }}\left( 1+\frac{K_{h}K_{a1}}{10^{-\mathrm {pH}}}+\frac{K_{h}K_{a1}K_{a2}}{10^{-2\mathrm {pH}}}\right) \end{aligned}$$

(6.11)

It is worth noting that blood pH depends on $\mathrm {CO_{2}}$ concentration and venous blood has a slightly but significantly lower pH than arterial blood. As a consequence, a nonlinear relation between $c_{\mathrm {CO_{2}}}$ and $p_{\mathrm {CO_{2}}}$ is actually observed. However, as previously shown for $\mathrm {O_{2}}$ solubility (see Eq. 6.8), in the relevant range of physiological conditions for gas exchange, the $\mathrm {CO_{2}}$ solubility curve can be linearized and an approximately constant value for $\mathscr {H}_{\mathrm {CO_{2}}}$ assumed, that is:

$$\begin{aligned} \mathscr {H}_{\mathrm {CO_{2}}}=\frac{\varDelta p_{\mathrm {CO_{2}}}}{\varDelta c_{\mathrm {CO_{2}}}}= 5\cdot 10^{-3}\,\text {mmHg mol}^{-1}\,\text {l} \end{aligned}$$

(6.12)

6.2 Gas Transport in Blood

The analysis of gas solubility presented in Sect. 6.6.1 showed that dissolved $\mathrm {O_{2}}$ and $\mathrm {CO_{2}}$ are present in blood both as free molecular and chemically bound species. This aspect affects also the transport of gases in blood, especially with regard to diffusive transport.

This section deals with the analysis of gas transport in blood. The contents of the first part of this section apply equally to oxygen and carbon dioxide transport; therefore, for the sake of simplicity, $\mathrm {O_{2}}$ or $\mathrm {CO_{2}}$ will not be included in subscripts and equations of general validity will be presented. Rather, subscripts m and b will be used to refer to free and chemically bound fractions of the dissolved gas, respectively. In the second part of the section, the analysis will be separately focused on $\mathrm {O_{2}}$ or $\mathrm {CO_{2}}$ and different equations will be introduced for the two gases.

With the above-described notation, the total concentration of a gas in blood is as follows:

$$\begin{aligned} c=c_{m}+c_{b} \end{aligned}$$

(6.13)

and the steady-state gas balance equation for gas transport may be written as:

$$\begin{aligned} \mathbf {v}\cdot \nabla \left( c_{m}+c_{b}\right) =\nabla \cdot \left( \mathscr {D}_{m}\nabla c_{m}+\mathscr {D}_{b}\nabla c_{b}\right) \end{aligned}$$

(6.14)

where the left-hand and right-hand side terms account for convective and diffusive transport of all species, respectively.

By assuming that the binding reactions are fast enough to ensure local equilibrium conditions, it is possible to write

$$\begin{aligned} \nabla c_{b}=H^{\prime }\frac{\partial c_{b}}{\partial p}\nabla c_{m} \end{aligned}$$

(6.15)

Therefore, the continuity equation (Eq. 6.14) may be rewritten as

$$\begin{aligned} \mathbf {v}\cdot \left( 1+H^{\prime }\frac{\partial c_{b}}{\partial p}\right) \nabla c_{m}=\nabla \cdot \left[ \mathscr {D}_{m}+\mathscr {D}_{b}H^{\prime }\frac{\partial c_{b}}{\partial p}\right] \nabla c_{m} \end{aligned}$$

(6.16)

The bracketed term in Eq. (6.16) is usually referred to as the facilitated diffusion coefficient, $\mathscr {D}_{f}$, that is,

$$\begin{aligned} \mathscr {D}_{f}=\mathscr {D}_{m}\left[ 1+\frac{\mathscr {D}_{b}}{\mathscr {D}_{m}}H^{\prime }\frac{\partial c_{b}}{\partial p}\right] \end{aligned}$$

(6.17)

Equation 6.17 shows that chemical binding results in an apparent enhancement of the molecular diffusivity of the dissolved gas; the augmentation factor (bracketed term in Eq. 6.17) quantifies the importance of facilitated diffusion and depends on the ratio of free to bound species diffusivity as well as on binding equilibrium conditions.

Finally, if a constant value can be assumed for $\partial c_{b}/\partial p$, Eq. 6.14 can be rearranged in terms of $c_{m}$ only:

$$\begin{aligned} \mathbf {v}\cdot \nabla c_{m}=\mathscr {D}_{ eff }\nabla ^{2}c_{m} \end{aligned}$$

(6.18)

where $\mathscr {D}_{ eff }$ is the effective diffusion coefficient, which is defined as

$$\begin{aligned} \mathscr {D}_{ eff }=\mathscr {D}_{m}\frac{1+{\displaystyle \frac{\mathscr {D}_{b}}{\mathscr {D}_{m}}}H^{\prime }{\displaystyle \frac{\partial c_{b}}{\partial p}}}{1+H^{\prime }{\displaystyle \frac{\partial c_{b}}{\partial p}}}=\frac{\mathscr {D}_{f}}{1+H^{\prime }{\displaystyle \frac{\partial c_{b}}{\partial p}}} \end{aligned}$$

(6.19)

6.2.1 Oxygen Transport

As reported in Sect. 6.6.1.1, dissolved oxygen is present in blood as a free molecular species and as oxygenated hemoglobin, which can be considered as oxygenated heme groups. Based on this consideration, Eq. 6.13 can be written for $\mathrm {O_{2}}$ as

$$\begin{aligned} c_{\mathrm {O_{2}}}=c_{\mathrm {O_{2}},m}+c_{\mathrm {HbO_{2}}} \end{aligned}$$

(6.20)

where $c_{\mathrm {HbO_{2}}}=4S_{\%}c_{\mathrm {Hb}}$. According to Eqs. 6.13, 6.17, and 6.20, the $\mathrm {O_{2}}$ facilitated diffusion coefficient can be written as

$$\begin{aligned} \mathscr {D}_{\mathrm {O_{2}},f}=\mathscr {D}_{\mathrm {O_{2}},m}\left[ 1+\frac{\mathscr {D}_{\mathrm {HbO_{2}}}}{\mathscr {D}_{\mathrm {O_{2}}}}\frac{\partial c_{\mathrm {HbO_{2}}}}{\partial c_{\mathrm {O_{2}},m}}\right] \end{aligned}$$

(6.21)

Since $\mathscr {D}_{\mathrm {HbO_{2}}}/\mathscr {D}_{\mathrm {O_{2}}}\ll 1$ due to the high hemoglobin molecular weight and concentration in red blood cells, the augmentation factor is significant only for very low oxygen partial pressures and can be neglected in physiological conditions (in which oxygen partial pressure is in the range of 40–95 $\text {mmHg}$). As a consequence, it can be assumed that $\mathscr {D}_{\mathrm {O_{2}},f}\simeq \mathscr {D}_{\mathrm {O_{2}},m}$ and the effective $\mathrm {O_{2}}$ diffusivity is given by

$$\begin{aligned} \mathscr {D}_{\mathrm {O_{2}}, eff }=\frac{\mathscr {D}_{\mathrm {O_{2}},m}}{1+H_{\mathrm {O_{2}}}^{\prime }{\displaystyle \frac{\partial c_{\mathrm {HbO_{2}}}}{\partial p_{\mathrm {O_{2}}}}}}=\frac{\mathscr {D}_{\mathrm {O_{2}},m}}{1+\mathscr {K}c_{\mathrm {Hb}}} \end{aligned}$$

(6.22)

6.2.2 Carbon Dioxide Transport

As already pointed out, several species originate from chemical binding of dissolved $\mathrm {CO_{2}}$ in blood (see Sect. 6.6.1.2). According to the notation introduced, the total concentration of these species¹² will be denoted as $c_{\mathrm {CO_{2}},b}$ and $\mathrm {CO_{2}}$ facilitated diffusion coefficient can be expressed as

$$\begin{aligned} \mathscr {D}_{\mathrm {CO_{2}},f}=\mathscr {D}_{\mathrm {CO_{2}},m}\left[ 1+\frac{\mathscr {D}_{\mathrm {\mathrm {CO_{2}}},b}}{\mathscr {D}_{\mathrm {CO_{2}},m}}\frac{\partial c_{\mathrm {\mathrm {CO_{2}}},b}}{\partial c_{\mathrm {CO_{2}},m}}\right] \simeq \mathscr {D}_{\mathrm {CO_{2}},m}\left[ 1+\frac{\mathscr {D}_{\mathrm {\mathrm {CO_{2}}},b}}{\mathscr {D}_{\mathrm {CO_{2}}}}\frac{H'_{\mathrm {CO_{2}}}}{\mathscr {H}{}_{\mathrm {CO_{2}}}}\right] \end{aligned}$$

(6.23)

while $\mathrm {CO_{2}}$ effective diffusivity is given by

$$\begin{aligned} \mathscr {D}_{\mathrm {CO_{2}}, eff }=\frac{\mathscr {D}_{\mathrm {CO_{2}},f}}{1+{\displaystyle \frac{H_{\mathrm {CO_{2}}}^{\prime }}{\mathscr {H}{}_{\mathrm {CO_{2}}}}}} \end{aligned}$$

(6.24)

7 Gas Exchange Between Capillary Blood and Alveolar Air

Before presenting and discussing mathematical models of blood oxygenators, a simple analysis of gas transfer in alveoli is reported in this section. Such analysis is aimed at understanding the results that blood oxygenators should obtain and why their performance is currently far from that of healthy lungs.

Fig. 6.7

Gas exchange between air in the alveolar sac and blood in the alveolar capillary

Let us consider gas exchange between blood in the alveolar capillary and air in the alveolar sac, as schematically represented in Fig. 6.7. Blood enters the venous end of the alveolar capillary with a gas (O$_{2}$ and CO$_{2}$, subscript omitted) pressure $p_{v}$; gas exchange occurs between blood and the alveolar sac, where the gas partial pressure is $p_{alv}$, as in the exhaled air. At steady state, the gas balance on an infinitesimal capillary segment along the axial direction z gives

$$\begin{aligned} v\frac{\mathrm {d}c}{\mathrm {d}z}=-\frac{4}{d}N_{tm} \end{aligned}$$

(6.25)

where v is the blood velocity, d is the capillary diameter, and $N_{tm}$ is the transmembrane gas flux from blood to alveolar air. The flux $N_{tm}$ can be written as

$$\begin{aligned} N_{tm}=K_{c}\left( c-c^{*}\right) \end{aligned}$$

(6.26)

where $K_{c}$ is the overall mass transfer coefficient through the respiratory membrane and $c^{*}$ is the gas concentration in blood in equilibrium with the air in the alveolar sac.

By substituting Eq. 6.26 in Eq. 6.25, the following equation is obtained

$$\begin{aligned} \frac{\mathrm {d}c}{\mathrm {d}z}=-\frac{1}{ Pe _{tm}}\left( c-c^{*}\right) \end{aligned}$$

(6.27)

where the transmembrane Peclet number defined as

$$\begin{aligned} Pe_{tm}=\frac{vd}{4K_{c}} \end{aligned}$$

(6.28)

was introduced. It is worth noting that $Pe _{tm}$ may be equally defined as

$$\begin{aligned} Pe_{tm}=\frac{{\displaystyle \frac{1}{aK_{c}}}}{{\displaystyle \frac{L}{v}}}=\frac{Q_{B}}{AK_{c}} \end{aligned}$$

(6.29)

where $a=4/d$ is the specific exchange area per unit capillary volume, L is the capillary length, $Q_{B}$ is the blood volumetric flow rate, and A is the total exchange area. Equation 6.29 highlights the physical meaning of $Pe _{tm}$, which is the ratio of the transmembrane diffusion time to the blood residence time in the capillary.

Equation 6.27 can be integrated with the boundary condition given by the known gas concentration at the venous end of the capillary, $c_{B,v}$; the solution obtained can be cast in the following dimensionless form

$$\begin{aligned} \tilde{c}(\tilde{z})=1-\exp (-\tilde{z}) \end{aligned}$$

(6.30)

where

$$\begin{aligned} \tilde{z}=\frac{z}{L\, Pe_{tm} }\ \ \ ;\ \ \ \tilde{c}=\frac{c-c_{v}}{c^{*}-c_{v}} \end{aligned}$$

(6.31)

Fig. 6.8

Dimensionless gas concentration in blood along the alveolar capillary (Eq. 6.30)

Figure 6.8 shows the plot of the dimensionless gas concentration in blood along the alveolar capillary (Eq. 6.30). From this dimensionless plot and accounting for the gas solubility in blood (see Sect. 6.6.1), it possible to determine the partial pressure profiles of oxygen and carbon dioxide along the capillary, which are shown in Fig. 6.9.

Equation 6.30 shows that the dimensionless gas concentration at the outlet ($z=L$, arterial end) of the alveolar capillary is $1-\exp (-1/ Pe _{tm})$; therefore, for low $Pe _{tm}$ values, blood leaves the capillary with a gas concentration close to equilibrium with the alveolar air ($\tilde{c}\simeq 1$); in this case, it can be easily shown that the overall rate of gas exchange approaches the limiting value $vA\left( c^{*}-c_{v}\right)$.

Fig. 6.9

Oxygen and carbon dioxide pressure in blood along the alveolar capillary. Oxygen pressure is obtained with $p_{v}= 40\,\text {mmHg}$, $p_{alv}= 105\,\text {mmHg}$, and $c_{{\text {Hb}}}= 2.2\,\text {mM}$; carbon dioxide concentration is obtained with $p_{v}= 45\,\text {mmHg}$ and $p_{alv}= 40\,\text {mmHg}$

Healthy human lungs operate with a low $Q_{B}/A$ ratio ($5\,\text {l/min}$ of blood are spread over a surface of 100–150 $\text {m}^{2}$); furthermore, the respiratory membrane offers a very low resistance to gas transfer, so that the overall mass transport coefficient $K_{c}$ depends only on the blood-side resistance and has a very high value. Due to both the low $Q_{B}/A$ ratio and high $K_{c}$ value, $Pe_{tm}$ is low in the characteristic operating conditions of human lungs, so that gas pressure at the arterial end of the capillary and in the alveolar sac is virtually equal.

It is worth noting that the compositions of alveolar and inspired air are different, in fact, accounting for gas mass balance over the whole blood and alveolar compartments:

$$\begin{aligned} Q_{B}c_{v}+\frac{V_{i}}{\mathbb {R}T}p_{i}=Q_{B}c_{a}+\frac{V_{i}}{\mathbb {R}T}p_{alv} \end{aligned}$$

(6.32)

where $Q_{B}$ is the volume blood flow rate and $V_{i}$ the volume flow rate of air entering the alveoli. The above equation may be rewritten as:

$$\begin{aligned} c_{a}-c_{v}=\frac{p_{a}-p_{v}}{\mathscr {H}}=\frac{V_{i}}{Q_{B}\mathbb {R}T}\left( p_{i}-p_{alv}\right) \end{aligned}$$

(6.33)

where, according to Eqs. 6.8 and 6.12, $\mathscr {H}$ is set equal to the ratio $\left( p_{a}-p_{v}\right) /\left( c_{a}-c_{v}\right)$; by further assuming that $p_{a}\sim p_{alv}$, we get:

$$\begin{aligned} p_{a}=\frac{{\displaystyle \frac{V_{i}\mathscr {H}}{Q_{B}\mathbb {R}T}}p_{i}+p_{v}}{{\displaystyle \frac{V_{i}\mathcal{\mathscr {H}}}{Q_{B}\mathbb {R}T}}+1}\ \ \ or\ \ \ \frac{p_{a}-p_{v}}{p_{i}-p_{v}}=\dfrac{\frac{V_{i}\mathcal{\mathscr {H}}}{Q_{B}\mathbb {R}T}}{\frac{V_{i}\mathscr {H}}{Q_{B}\mathbb {R}T}+1} \end{aligned}$$

(6.34)

Equation 6.34 underlines the influence of the ventilation–perfusion ratio $V_{i}/Q_{B}$ on the difference between inspired and alveolar (or arterial) gas partial pressure. In healthy lungs at rest, $V_{i}/Q_{B}\sim 1$; therefore, as for blood oxygenation, using a mean $\mathscr {H}_{\mathrm {O_{2}}}$ value of $27.8\,\text {mmHg mmole}^{-1}\,\text {l}$, we have $V_{i}\mathscr {H}_{\mathrm {O_{2}}}/\left( Q_{B}\mathbb {R}T\right) \sim 1$ and $\left( p_{a,\mathrm {O_{2}}}-p_{v,\mathrm {O_{2}}}\right) /\left( p_{i,\mathrm {O_{2}}}-p_{v,\mathrm {O_{2}}}\right) \simeq 0.5$.

8 Gas Exchange in Membrane Oxygenator

Most of the oxygenators used in current clinical practice are based on hollow fiber membranes, with high exchange surface-to-volume ratio. Oxygen and carbon dioxide exchange occurs through the membrane, avoiding the direct contact between blood and sweep gas. From this point of view, such artificial devices try to mimic the natural operation of the lung; however, from a quantitative point of view, parameters such as total surface area, contact time, or membrane permeability are still not comparable (see Table 6.5). As a consequence, the performance of the currently available artificial devices is sufficient only for the support of the basal metabolic needs of patients.

Table 6.5 Comparison of some operating variables in human lungs and membrane oxygenators

The gas exchange rate may be written as:

$$\begin{aligned} F_{tm}=K_{p}A\left( p^{G}-p^{*}\right) \end{aligned}$$

(6.35)

where $K_{p}$ is the overall mass transport coefficient, A is the exchange surface area, $p^{*}$ is the partial pressure of the gas in equilibrium with the blood, and $p^{G}$ is the partial pressure in the sweep gas phase.

In order to determine the overall mass transport coefficient, the analysis reported in Chap. 2 must be extended to include also the mass transport resistance due to the membrane; we obtain:

$$\begin{aligned} \frac{1}{K_{p}}=\frac{1}{k_{p}^{G}}+\frac{\delta }{\mathscr {P}}+\frac{1}{k_{p}^{B}} \end{aligned}$$

(6.36)

where $\mathscr {P}$ is the membrane permeability (see Chap. 3), $\delta$ is the membrane thickness, and $k_{p}^{G}$ and $k_{p}^{B}$ are the mass transport coefficients (referred to partial pressure as driving force) in the gas phase and blood phase, respectively.¹³

Usually, the resistance in the gas phase is negligible; as a consequence, the overall mass transfer coefficient is independent of gas-phase flow rate. In such a condition, reducing the gas-phase mass transfer resistance has a negligible effect on the transmembrane gas flux; rather, in order to increase the gas transfer rate per unit surface of the device, all efforts should be aimed at improving the membrane permeance or reducing the resistance in the blood boundary layer.

Another way of increasing the transmembrane gas flow is by increasing the partial pressure driving force for gas exchange. This can be done quite easily for oxygen, by using enriched air or even pure oxygen (properly humidified) as gas phase: in this way, a driving force up to 500–600 mm Hg can be obtained. On the other hand, this is not as easily done for carbon dioxide, for which the maximum attainable driving force is 40–45 mm Hg, when a $\mathrm {CO_{2}}$-free gas phase is fed to the device. Considering that oxygen and carbon dioxide exchange rates must be in the metabolic ratio (1:0.8), the mass transfer coefficient must be higher for carbon dioxide than for oxygen: more specifically, in order to exploit the maximum driving forces available, the CO$_{2}$ to O$_{2}$ mass transport coefficient ratio should be about 10. Such a result can be only be obtained by relying on the membrane resistance properties, since mass transport in blood phase is not significantly different for the two gases.

The following part of this section provides information on how to evaluate the mass transfer resistance of both the blood boundary layer and different types of membranes used in blood oxygenators.

8.1 Membranes for Gas Oxygenators

8.1.1 Dense Membranes

Polymeric membranes made of thin polymer sheets are often used for gas exchange in artificial lung, mainly for long-term extracorporeal life support.

Gas permeation through a polymer layer is usually described by solution–diffusion model (see Chap. 3), which leads to the following expression for the membrane permeance:

$$\begin{aligned} \mathscr {P}=\alpha _{m}\mathscr {D}^{m} \end{aligned}$$

(6.37)

where $\alpha _{m}$ is the gas partition coefficient in the membrane polymer.

The properties of some polymeric materials are reported in Table 6.6. Good results are obtained with 100–200 µm-thick silicone membranes, which allow to obtain mass transfer coefficients of the order of $0.05\,\text {mol min}^{-1}\,\text {m}^{-2}\,\text {atm}^{-1}$ and exhibit excellent properties in terms of biocompatibility.

Table 6.6 Permeability of various polymeric materials [59]

It is worth noting that while dense membranes have a lower permeability than the porous ones, so that a higher surface area and priming volume are required, they greatly reduce the risk of blood leakage or, conversely, gas entrainment into the blood; these properties as well as their longevity make them still the ideal candidate for long-term support.

8.1.2 Porous Membranes

Most of the membranes used in blood oxygenators are of the hollow fiber porous type, with fiber diameters of 200–400 µm and porous wall thickness of 20–50 µm; the fiber wall is highly porous (30–50 %) with pore size below 0.1 µm.

Hydrophobic polymers are used to prevent blood penetration into the membrane pores, which thus remain filled with gas. If $\theta$ is the contact angle between gas and blood on the polymer, the pressure required to fill the pores is $4\sigma \cos \theta /d_{P}$, where $\sigma$ is the gas–blood surface tension and $d_{P}$ is the pore diameter: For polymers that are not wetted by blood ($\theta \simeq \pi$) and small pore diameter, pore wetting is easily prevented, possibly by applying a small gas overpressure.

In empty pores, respiratory gases diffuse relatively fast and the permeability can be estimated by accounting for the membrane wall porosity ($\varepsilon _{w})$ and tortuosity factor ($\tau$):

$$\begin{aligned} \mathscr {P}_{dry}=\frac{\varepsilon _{w}}{\tau }\frac{\mathscr {D}^{p}}{\mathbb {R}T} \end{aligned}$$

(6.38)

Due to the small size of membrane pores, gas diffusion is likely to occur in the Knudsen regime (see Chap. 3) and diffusivity is given by $\mathscr {D}^{p}=2d_{p}/3\,\sqrt{2\mathbb {R}T/\pi M}$, where M is the diffusing gas molecular mass; therefore, we get:

$$\begin{aligned} \mathscr {P}=\frac{2\varepsilon _{w}}{3\tau }d_{p}\sqrt{\frac{2}{\pi M\mathbb {R}T}} \end{aligned}$$

(6.39)

According to Eq. 6.39, the ratio $\mathscr {P}_{dry} \delta$ (membrane permeance) can be as high as $50\,\text {mol min}^{-1}\,\text {m}^{-2}\,\text {atm}^{-1}$ ($2.5 \times 10^{-2}\,\text {ml cm}^{-2}\text {s}^{-1}\,\text {cmHg}^{-1})$; slightly lower values are obtained for carbon dioxide. With such high values, blood-side resistance controls the gas transfer process and $K_{p}\simeq k_{p}^{B}$.

On the other hand, the performance of porous membranes is limited by pore wetting issues: in fact, mainly when the membrane is used for prolonged support, adsorption of plasma components renders the membrane surface hydrophilic; in these conditions, plasma can enter and fill the pores. Therefore, the permeating gas has to diffuse in a plasma layer ($\mathscr {D}^{p}=\mathscr {D}$, where $\mathscr {D}$ is the diffusivity in plasma) and the permeability reduces to:

$$\begin{aligned} \mathscr {P}=\frac{\varepsilon _{w}\alpha _{B}}{\tau }\mathscr {D} \end{aligned}$$

(6.40)

where $\alpha _{B}$ is the gas partition coefficient in blood. Pore wetting can cause a reduction of permeability up to 5 orders of magnitude compared to dry membranes, with a consequent degradation of the device performance.

8.1.3 Composite Membrane

In order to prevent wetting and infiltration into the microporous wall and improve the long-term performance of membrane oxygenators, composite hollow fibers are used, which incorporate a thin layer of non-porous polymer on the fiber surface. Composite fibers are produced either by coating a previously manufactured microporous membrane or by a one-step process in which both the porous and dense layers are formed at the same time.

The non-porous “skin” prevents plasma infiltration in the pores even during prolonged applications, but offers a further resistance to gas transport across the membrane. The non-porous skin may be considered as an additional membrane, and a term accounting for its mass transfer resistance should be included in Eq. 6.36. To that end, the permeance of the dense layer can be calculated as in Eq. 6.37. However, the non-porous skin is usually very thin, so that the gas exchange performance of composite and porous membranes is in general comparable.

8.2 Gas Transport in the Blood Film

The blood film near the membrane surface offers in general the controlling mass transfer resistance when a porous membrane is used. Therefore, improvement of blood fluid dynamics and mass transport coefficient is among the major aims of the current research on blood oxygenators.

As discussed in Chap. 2, the mass transport coefficient depends on the thickness of the boundary layer formed near the solid surface, which in turn depends on the blood velocity field. A decrease of the boundary layer thickness may be achieved by increasing the blood flow rate or reducing the cross-sectional area available to flow, with an appropriate device design.

Several studies have been carried out to evaluate the mass transport coefficient in blood oxygenators. A theoretical analysis based on the solution of the equation of motion in laminar flow is suitable only for simple geometries and flow conditions such as in intraluminal blood flow; differently, more complex flow patterns such as extraluminal flow require a different approach, which can involve experimental testing of commercial devices or 3D computational fluid dynamic simulations. Moreover, chemical binding of $\mathrm {O_{2}}$ and $\mathrm {CO_{2}}$ in blood strongly affects their transfer rates and should be accounted for.

The following approach is generally considered: firstly, the mass transport coefficient in a non-reacting system is determined (e.g., by experiments carried out by using water or simulated blood as liquid phase); then, the effect of binding reactions is accounted for.

As already reported in Chap. 2, the mass transport coefficient in a non-reacting system, $k_{p}^{B0}$, is usually described by correlations of the form:

$$\begin{aligned} Sh=\frac{k_{p}^{B0}\ell }{\alpha _{B}\mathscr {D}_{B}}=aRe^{b}Sc^{1/3} \end{aligned}$$

(6.41)

Equation 6.41 shows that the mass transport coefficient depends on both the physical properties (Sc) and fluid dynamics of the system (Re). In Eq. 6.41, $\ell$ is a characteristic length for blood flow: for intraluminal flow, $\ell =d_{f}$, while for extraluminal flow, $\ell$ is the equivalent diameter given by $\varepsilon _{f}d_{fe}/\left( 1-\varepsilon _{f}\right)$; $d_{f}$ and $d_{fe}$ are the inner and outer diameters of the hollow fibers, respectively, and $\varepsilon _{f}$ is the void fraction of the fiber bundle. Some correlations for the blood-side mass transport coefficient in different flow configurations are summarized in Table 6.7.

Table 6.7 Correlations for the blood-side mass transfer coefficient

Two different approaches have been suggested in the literature to account for the enhancement of the gas transfer rate due to the binding reactions.¹⁴

A first approach [65, 66] uses the same relations derived for non-reacting system, but accounts for facilitated and effective diffusion as defined in Sect. 6.6.2. Specifically, it was proven that the effective diffusivity should be used in calculating the Schmidt number for blood, whereas the Sherwood number is based on the facilitated diffusion. Following this approach, the ratio of the oxygen transport coefficient in blood to its transport coefficient in a non-reacting system is given by (the augmentation factor in the facilitated diffusivity is negligible for oxygen):

$$\begin{aligned} \frac{k_{B}}{k_{B}^{0}}=\left[ 1+\dfrac{4c_{\mathrm {Hb}}\left( \mathrm {d}S_{\%}/\mathrm {d}p_{\mathrm {O_{2}}}\right) }{\alpha }\right] ^{1/3} \end{aligned}$$

(6.42)

In a second approach, the effect of binding reaction on the mass transport coefficient is quantified by multiplying the mass transport coefficient determined for a non-reacting system, $k_{B}^{0}$, by an enhancement factor E:

$$ k{}_{B}=k_{B}^{0}E $$

As for oxygen, the enhancement factor can be evaluated from experimental data [67] or from a theoretical analysis based on the film model [68].

Relations to evaluate the enhancement factor are reported in Table 6.8.

Table 6.8 Enhancement factor for oxygen mass transport coefficient

9 Membrane Oxygenator Modeling

A reliable mathematical model can be considered as the basis for a rational design of a blood oxygenator, as for any other device; furthermore, mathematical models can be a valuable tool to get insight into the working principles of oxygenators, by helping in the definition of the critical design parameters and of their effect on the performance, also without having to carry out extensive, costly, and complex experimental campaigns.

Though more sophisticated and detailed models can be developed at the price of a greater mathematical and numerical complexity, in this section we present simplified models based on reasonable hypotheses. By providing a fair trade-off between simplicity and reliability of analysis, such models allow to focus only on the fundamental aspects of the oxygenation process, rather than on involved mathematical solutions.

The model presented here is based on the following assumptions: (a) steady-state conditions; (b) the oxygen–hemoglobin reaction is always at equilibrium; (c) the linearized form of the oxygen saturation curve holds; (d) gas transport in both bulk phases is purely convective; (e) the active volume (i.e., the space where blood and gas are indirectly contacted through the membrane and gas exchange occurs) of the device can be considered as a porous pseudo-continuous medium, through which blood and gas flow; and (f) the gas transfer rate between gas and blood is given as $k_{p}^{B}a\left( p^{G}-p^{*}\right)$, where a is the membrane surface area per unit total volume.

Under the above-listed hypotheses, the oxygen balance in blood may be written as:

$$\begin{aligned} \mathbf {v}_{B}\cdot \nabla \left( c_{\mathrm {O_{2}},m}+c_{\mathrm {HbO_{2}}}\right) =K_{p,\mathrm {O_{2}}}a\left( p_{\mathrm {O_{2}}}^{G}-p_{\mathrm {O_{2}}}^{*}\right) \end{aligned}$$

(6.43)

where $\mathbf {v}_{B}$ is the blood superficial velocity and a is the membrane surface area per unit total volume. The above equation may be rearranged in the form

$$\begin{aligned} \mathbf {v}_{B}\cdot \frac{1+\mathscr {K}c_{\mathrm {Hb}}}{H_{\mathrm {O_{2}}}^{\prime }}\nabla p_{\mathrm {O_{2}}}^{*}=\mathbf {v}_{B}\cdot \frac{\nabla p_{\mathrm {O_{2}}}}{\mathscr {H}_{\mathrm {O_{2}}}}=K_{p,\mathrm {O_{2}}}a\left( p_{\mathrm {O_{2}}}^{G}-p_{\mathrm {O_{2}}}^{*}\right) \end{aligned}$$

(6.44)

As for the gas phase, the oxygen balance is written as

$$\begin{aligned} \frac{\mathbf {v}_{G}}{\mathbb {R}T}\cdot \nabla p_{\mathrm {O_{2}}}^{G}=-K_{p,\mathrm {O_{2}}}a\left( p_{\mathrm {O_{2}}}^{G}-p_{\mathrm {O_{2}}}^{*}\right) \end{aligned}$$

(6.45)

where $\mathbf {v}_{G}$ is the gas superficial velocity.

In the following sections, this general model is solved for two common flow patterns used in blood oxygenators.

Fig. 6.10

Schematic geometry of countercurrent flow oxygenators

9.1 Countercurrent Blood Oxygenator

In countercurrent flow, blood and gas flow in opposite sense along the same direction and exchange oxygen and carbon dioxide across the membrane (see Fig. 6.10). This flow pattern is found in hollow fiber or flat-sheet oxygenators, and the simple analysis presented here applies equally to both types of device.

Let x be the direction of flow, and then both gas and blood have velocities directed along x; furthermore, both blood and gas compositions vary with x, due to mass transfer across the membrane. The oxygen balance equations on an infinitesimal segment of the device along the x direction can be written as

$$\begin{aligned} \frac{v_{B}}{\mathscr {H}_{\mathrm {O_{2}}}}\frac{\mathrm {d}p_{\mathrm {O_{2}}}^{*}}{\mathrm {d}x}=K_{p,\mathrm {O_{2}}}a\left( p_{\mathrm {O_{2}}}^{G}-p_{\mathrm {O_{2}}}^{*}\right) \end{aligned}$$

(6.46)

$$\begin{aligned} \frac{v_{G}}{\mathbb {R}T}\frac{\mathrm {d}p_{\mathrm {O_{2}}}^{G}}{\mathrm {d}x}=-K_{p,\mathrm {O_{2}}}a\left( p_{\mathrm {O_{2}}}^{G}-p_{\mathrm {O_{2}}}^{*}\right) \end{aligned}$$

(6.47)

The above set of differential equations can be solved with the boundary conditions:

$$ {\left\{ \begin{array}{ll} x=0\qquad &{} p_{\mathrm {O_{2}}}^{*}=p_{v}\\ x=L\qquad &{} p_{\mathrm {O_{2}}}^{G}=p_{in,\mathrm {O_{2}}}^{G} \end{array}\right. } $$

where $p_{v}$ is the oxygen pressure in the inlet venous blood and $p_{in,\mathrm {O_{2}}}^{G}$ is the oxygen partial pressure in the inlet gas. By some simple mathematical rearrangements and integration via variable separation, we get:

$$\begin{aligned} \ln \frac{p_{out,\mathrm {O_{2}}}^{G}-p_{v}}{p_{in,\mathrm {O_{2}}}^{G}-p_{a}}=K_{p,\mathrm {O_{2}}}aL\left( \frac{\mathbb {R}T}{v_{G}}-\frac{\mathscr {H}_{\mathrm {O_{2}}}}{v_{B}}\right) \end{aligned}$$

(6.48)

or

$$\begin{aligned} \ln \frac{p_{out,\mathrm {O_{2}}}^{G}-p_{v}}{p_{in,\mathrm {O_{2}}}^{G}-p_{a}}=K_{p}A\left( \frac{\mathbb {R}T}{V_{G}}-\frac{\mathscr {H}_{\mathrm {O}_{2}}}{Q_{B}}\right) \end{aligned}$$

(6.49)

where $p_{a}$ is the oxygen pressure in the arterial blood, $p_{in}^{G}$ and $p_{out}^{G}$ are the oxygen pressure in the inlet and outlet gas, respectively, A is the total membrane area, and $V_{G}$ and $Q_{B}$ are the gas and blood volumetric flow rates, respectively.

An oxygen balance over the whole device allows to obtain the following expression for the overall oxygen transfer rate in the device

$$\begin{aligned} F_{tm,{\mathrm {O}_{2}}}=\frac{V_{G}}{\mathbb {R}T}\left( p_{in,\mathrm {O_{2}}}^{G}-p_{out,\mathrm {O_{2}}}^{G}\right) =\frac{Q_{B}}{\mathscr {H}_{\mathrm {O_{2}}}}\left( p_{a}-p_{v}\right) \end{aligned}$$

(6.50)

By substituting Eq. 6.50 in Eq. 6.49, the following expression is obtained

$$\begin{aligned} F_{tm,{\mathrm {O}_{2}}}=K_{p{,\mathrm {O}_{2}}}A\frac{\left( p_{out}^{G}-p_{v}\right) -\left( p_{in}^{G}-p_{a}\right) }{\ln \dfrac{p_{out}^{G}-p_{v}}{p_{in}^{G}-p_{a}}} \end{aligned}$$

(6.51)

$$\begin{aligned} F_{tm,{\mathrm {O}_{2}}}=K_{p{,\mathrm {O}_{2}}}A\,\varDelta p_{LM,\mathrm {O}_{2}} \end{aligned}$$

(6.52)

In Eq. 6.52, the log mean partial pressure difference, $\varDelta p_{LM,\mathrm {O}_{2}}$, was introduced. It is worth noting that Eq. 6.51 is valid also when it is possible to assume that the partial pressure of oxygen in the gas stream does not vary significantly across the device (i.e., $p_{in,\mathrm {O_{2}}}^{G}\simeq p_{out,\mathrm {O_{2}}}^{G}=p_{\mathrm {O_{2}}}^{G}$); such condition holds if $V_{G}/\mathbb {R}T\gg Q_{B}/\mathscr {H}_{\mathrm {O_{2}}}$. Furthermore, Eq. 6.51 applies also to cocurrent flow, i.e., when gas and blood flow in the same sense inside the oxygenator; however, such flow pattern is not used in clinical devices because of its lower efficiency compared to countercurrent flow.

Fig. 6.11

Schematic geometry of radial flow oxygenators; a perspective view, b cross-sectional view with flow pattern, c cross-sectional view with sizes

9.2 Hollow Fiber Oxygenator with Radial Blood Flow

Another flow pattern used in membrane oxygenators, blood flows radially (direction r) across bundles of woven hollow fibers, while gas flows in the fiber lumen in the axial (z) direction as schematically described in Fig. 6.11 (see also Fig. 6.4). In such configuration, the velocity of blood is not constant, but varies along the flow direction as follows:

$$\begin{aligned} v_{B}(r)=\frac{Q_{B}}{2\pi Lr} \end{aligned}$$

(6.53)

The local oxygen balances in the blood and gas sides are as follows:

$$\begin{aligned} \frac{Q_{B}}{2\pi L\mathscr {H}_{\mathrm {O_{2}}}}\frac{1}{r}\frac{\partial p_{\mathrm {O_{2}}}^{*}}{\partial r}=K_{p,\mathrm {O_{2}}}a\left( p_{\mathrm {O_{2}}}^{G}-p_{\mathrm {O_{2}}}^{*}\right) \end{aligned}$$

(6.54)

$$\begin{aligned} \frac{v_{G}}{\mathbb {R}T}\frac{\partial p_{\mathrm {O_{2}}}^{G}}{\partial z}=-K_{p,\mathrm {O_{2}}}a\left( p_{\mathrm {O_{2}}}^{G}-p_{\mathrm {O_{2}}}^{*}\right) \end{aligned}$$

(6.55)

with the boundary conditions:

$$ {\left\{ \begin{array}{ll} r=R_{0}\qquad &{} p=p_{v}\\ z=0\qquad &{} p^{G}=p_{in}^{G} \end{array}\right. } $$

where $R_{0}$ is the inner radius of the active space of the device, i.e., where blood is fed in the external space of the fiber bundle.

It is worth noting that, in Eqs. 6.54 and 6.55, $p_{\mathrm {O_{2}}}^{G}$ and $p_{\mathrm {O_{2}}}^{*}$ are functions of both r and z. The above set of differential equations can be solved numerically. A closed solution is obtained if the oxygen partial pressure in the gas phase can be considered as constant; in this case, it can be shown that Eq. 6.52 holds also for radial flow devices.

9.3 Response of the Membrane Oxygenator to Different Operating Conditions

Equation 6.51 may be rewritten in a more useful form, to analyse the response of a blood oxygenator to changes in the operating conditions, such as gas flow rate or inlet gas composition. To this aim, it is useful to define two-dimensionless groups

$$\begin{aligned} R=\frac{K_{p}A}{Q_{B}/\mathscr {H}}\qquad Y=\frac{V_{G}/(\mathbb {R}T)}{Q_{B}/\mathscr {H}} \end{aligned}$$

(6.56)

The first one, R, accounts for the ratio of the mass transfer rate to blood convection rate: a low value of R means that the gas transfer is slow compared to the rate at which blood passes through the device; therefore, when R is low, gas transfer tends to be ineffective. The second parameter, Y, accounts for the gas to blood flow rate ratio. Equations 6.50 and 6.51 may be rearranged to get:

$$\begin{aligned} \frac{F_{tm}}{{\displaystyle \frac{Q_{B}}{\mathscr {H}}}\left( p_{in}^{G}-p_{v}\right) }=\frac{p_{a}-p_{v}}{p_{in}^{G}-p_{v}}=\frac{Y\left[ 1-\exp \left[ -R\left( 1-Y\right) /Y\right] \right] }{1-Y\exp \left[ -R\left( 1-Y\right) /Y\right] } \end{aligned}$$

(6.57)

Fig. 6.12

Effectiveness factor of the gas transfer in the oxygenator as a function of the two operating parameters R and Y

The ratio $\left( p_{a}-p_{v}\right) /\left( p_{in}^{G}-p_{v}\right)$ represents an effectiveness factor for the gas transfer in the device, i.e., the ratio between the gas rate effectively transferred to the blood (proportional to the difference between the gas pressure in the arterial and venous blood) and the maximum gas transfer rate using a sweep gas at $p_{in}^{G}$ (proportional to the difference between the pressure in the sweep gas and the gas pressure in the venous blood); $\left( p_{a}-p_{v}\right) /\left( p_{in}^{G}-p_{v}\right) =1$ corresponds to the maximum transfer rate achievable in the device, with an outlet blood in equilibrium with the inlet gas; on the other hand, when $\left( p_{a}-p_{v}\right) /\left( p_{in}^{G}-p_{v}\right) =0$, gas transfer does not occur.

Figure 6.12 reports the plot of $\left( p_{a}-p_{v}\right) /\left( p_{in}^{G}-p_{v}\right)$ as a function of R and Y. From the plot, it is evident that for low Y values—i.e., for low values of the ratio between gas and blood flow rates—the gas transfer rate to blood increases approximately linearly with Y, but becomes relatively constant once Y is sufficiently high; for $Y\rightarrow \infty$, the ratio $\left( p_{a}-p_{v}\right) /\left( p_{in}^{G}-p_{v}\right)$ approaches the limiting value of $1-e^{-R}$. Therefore, if the goal is to ensure the maximum possible rate of gas exchange, we need to operate above the minimum gas flow rate that corresponds to a gas transfer rate independent of the sweep gas flow; on the other hand, operating at lower Y values, it is possible to control the gas exchange rate by manipulating the sweep gas flow rate.

10 Current Research and Perspectives

The artificial lungs used today in the clinical practice are extracorporeal devices, mainly used in open-heart surgery to replace the heart and lung functions; in this case, the device is required to ensure gas exchange for some hours. The patient is anticoagulated with heparin to prevent thrombosis within the extracorporeal circuit and potential formation of thromboemboli; the whole bypass circuit is often coated with heparin to prevent clotting and reduce the amount of systemic anticoagulant required.

Current research is devoted to finding solutions also for different applications, such as providing support in acute respiratory failures (such as ALI or ARDS), treating chronic respiratory diseases (such as COPD), and bridging to organ transplantation or allowing the natural lungs to recover; solutions for efficient and safe longer-term support (from several days to months) are then required. The traditional approach based on mechanical ventilation has some major drawbacks: the positive airway pressures and volume excursions associated with mechanical ventilation can cause further damage to the lung tissue, including barotrauma (high airway pressures), volutrauma (lung distension), and parenchymal damage due to the toxic levels of oxygen required for effective mechanical ventilation. Despite advances in supportive care, the mortality rate in patients with the acute respiratory distress syndrome (ARDS) is widely considered to have remained high and generally in excess of 50 %. Therefore, the idea is to provide breathing support independent of the lungs, using respiratory assist devices as alternatives or adjuvants to mechanical ventilators for patients with failing lungs.

Lung assists are usually classified in (a) extracorporeal devices, not too different from those currently used in heart surgery, (b) paracorporeal or wearable devices that can be attached directly to the patients, and (c) intracorporeal or implantable devices that can be implemented with the intravenous or intratoracic configuration.

This section presents an overview of the recent developments in the field of artificial lungs.

10.1 Extracorporeal Lung Assist (ECLA)

Extracorporeal lung assist is aimed at allowing the lungs to rest and recovery or bridging the patient to lung transplantation. In this framework, it is interesting to define two specific therapeutic approaches [69]

• total ECLA when both blood oxygenation and $\mathrm {CO_{2}}$ removal are required;

• partial ECLA which aims principally to $\mathrm {CO_{2}}$ removal, possibly with mild oxygenation to support natural lungs or low-pressure ventilation.

ECMO is suitable for total ECLA, providing both extracorporeal oxygenation and carbon dioxide removal. It is important to recognize that oxygenation is controlled by hemoglobin saturation¹⁵ and effective oxygenation can be obtained only with high blood flow rates (at least 50–60 ml/min per kg of body weight). In ECMO, a high blood flow rate (70–80 % of the cardiac output) is diverted to a pump-driven external circuit including a membrane oxygenator and a heat exchanger to control the body temperature. The system is then similar to a cardiopulmonary bypass circuit, which demands a continuous bedside management of trained staff. A large membrane surface area (a square meter of hollow fiber membrane) is currently required to provide adequate gas exchange; furthermore, respiratory support may be required also for relatively long periods of time; therefore, in extracorporeal circuits, blood/biomaterial contact is extensive and the inflammatory or thrombogenic complications are exacerbated; finally, the membrane device is required to maintain its performance for a long period and resist to plasma wetting, which causes a decrease of permeability.

Research on extracorporeal devices for prolonged use is now focused on the enhancement of gas exchange and, therefore, on the reduction of the surface area of the membrane; improvement of the biocompatibility of the materials used is also a major target. In parallel, efforts to simplify the circuit device system and reduce the need for intensive monitoring are carried out. In order to increase the gas exchange effectiveness, improvements of fluid dynamics have been considered, which involve optimization of the design of the gas exchanger or active mixing of blood.

In contrast to the high blood flow required for blood oxygenation, the removal of metabolic $\mathrm {CO_{2}}$ can be obtained also by treating a low blood flow rate (less than 25 % of the cardiac output) with a less invasive procedure. This is the basis for implementing a low flow technique for extracorporeal $\mathrm {CO_{2}}$ removal (ECCO2R), while oxygenation remains a function of natural breathing. Blood flow in ECCO2R circuit can be pump-driven veno-venous or pump-less arteriovenous.

Pump-driven veno-venous ECCO2R devices use a blood flow rate depending on the clinical demands (see Table 6.9 for an example) up to a high-flow ECMO, if a suitable catheter is used.

Table 6.9 Features of the iLA active^® system by Novalang

A very small membrane area (0.3 m$^{2}$) is used in the decap^® system (Hemodec, Salerno, Italy): in this device, a recirculation loop of ultrafiltrate produced by a hemofilter—in series with the membrane oxygenator—increases the flow rate in the membrane module and enhances carbon dioxide removal (see Fig. 6.13). In a similar system (Decapsmart, Medica, Medolla, Italy), the $\mathrm {CO_{2}}$ removal is combined to renal replacement therapy (RRT) in the patient with multiple organ failure needing both respiratory and renal support (see Fig. 6.14) [70].

Fig. 6.13

Scheme of the decap^® system

Fig. 6.14

Scheme of the decapsmart^® system

ALung Technologies, Inc. commercializes the Hemolung^® Respiratory Assist System, a dialysis-like alternative or supplement to mechanical ventilation originally developed at the University of Pittsburgh. The primary component of the Hemolung^® system is a cartridge which houses a cylindrical bundle of hollow fiber membranes; the fibers are positioned around a spinning core, which simultaneously drives blood flow (centrifugally) through the cartridge and $\mathrm {CO_{2}}$ transfer from blood to the oxygen sweep gas flowing under negative pressure through the fiber lumen (see Fig. 6.15). The pump-driven flow past the membranes markedly increases the gas exchange efficiency, allowing for a significant $\mathrm {CO_{2}}$ removal (30 to 40 %) at a relatively low blood flow in the range of 300 to 500 ml/min. Such a blood flow rate, similar to that used in hemodialysis, can be obtained with a single double-lumen venous catheter. A priming volume of 300 ml and minimal heparinization are required [72, 73].

Fig. 6.15

Hemolung^® cartridge. Left cross-sectional drawing (reproduced from [71], under the CC BY 2.0 license—http://creativecommons.org/licenses/by/2.0/). Right picture of the filled cartridge (© 2013 ALung Technologies, Inc.)

Arteriovenous carbon dioxide removal () is carried out with a membrane gas exchange device connected directly from arterial to venous circulation, without a blood pump; a fraction of the cardiac output (10–30 %), dictated by the arterial to venous pressure difference and hydraulic resistance of the device, is diverted to the membrane unit, which operates with a high ventilation ratio. In this way, the device provides sufficient gas exchange to achieve nearly total removal of $\mathrm {CO_{2}}$ and provides for about 10 % of the $\mathrm {O_{2}}$ requirement. Oxygenation is then maintained by simple diffusion across the patient’s alveoli and/or reduced mechanical ventilation. Even if a standard membrane unit can be used, it is of paramount importance to minimize the pressure drop in the device: computational fluid dynamics offers a major help to proper design the membrane module with pressure drops of few mmHg. An example of these devices is the interventional lung assist (iLA) of Novalung (GmgH, Hechineng, Germany), which uses a membrane “ventilator” with arteriovenous femoral connection.

10.2 Paracorporeal or Intrathoracic Devices

Extensive research is devoted to develop totally artificial lungs able to fully support basal O$_{2}$ and CO$_{2}$ transfer requirements for long-term support in chronic respiratory failure. The ultimate goal is to obtain an implantable device that can be placed inside body cavities; nevertheless, the present-day prototypes are implemented and tested in a paracorporeal configuration, with the device placed externally to the patient’s chest.

Fig. 6.16

In-series and in-parallel application of paracorporeal oxygenators

The fundamental idea is to attach the artificial lung directly to the pulmonary circulation, by utilizing the right heart as the blood pump. In-series (with both outflow and inflow cannula connected to the pulmonary artery) and in-parallel (with outflow cannula connected to the pulmonary artery and the inflow cannula connected to the left atrium) applications are possible (see Fig. 6.16). In the former case, the whole cardiac output is diverted to the gas exchange device and the natural lungs act as embolic filter; however, the load on heart increases because of the hydraulic resistance offered by the device. In the case of in-parallel application, only a fraction of the cardiac output is diverted to the natural lung and receives respiratory support; the blood flow rate fed to the device depends on the relative hydraulic resistance of the artificial lung compared to the natural lung. This configuration requires a lower load for the heart. Recently, a new cannula design (Wang-Zwische double-lumen cannula, or W-Z DLC) based on the double-lumen cannula used in neonatal and pediatric veno-venous ECMO was also used. This new cannula replaces the in-series and in-parallel anastomoses with a single access including two pathways: a drainage pathway and an infusion pathway. The cannula is placed percutaneously through the internal jugular vein; the drainage lumen is open to both the superior and inferior vena cava, while the infusion lumen is open to the right atrium. Blood from systemic circulation flows through the superior and inferior vena cava into the drainage lumen to the artificial lung device. Blood is oxygenated and returned via the infusion lumen into the right atrium. This oxygenated blood is then pumped through the native circulation and pulmonary bed, thus receiving the full metabolic and filtering capacities of the native lungs [74].

It is important to consider that virtually all standard devices for heart–lung machines with hollow fiber membranes could be used for paracorporeal applications, but important technical improvements are required to obtain efficient and safe devices. Firstly, while pressure drop across the fiber bed is only of secondary importance for heart–lung machines equipped with external pumps, this parameter should be kept as low as possible when the right ventricle is used to drive blood circulation into the device. As an example, BioLung^® from MC3 Inc. (Ann Arbor, MI, USA) utilizes radial flow through a concentrically wound hollow fiber fabric; computational fluid dynamics has been utilized to optimize the device, and the pressure drop has been reduced to 5–10 mmHg with a blood flow rate of 4–6 $\text {l/min}$ and a membrane area of 1.5–2 $\text {m}^{2}$ [75].

In order to reduce the required surface area of membrane and the size of the devices, active mixing with a rapidly rotating disk is introduced in chronic artificial lung (CAL), developed by the University of Maryland [76]; mixing improves gas exchange and the centrifugal motion enables to pump the blood and to reduce the impact on the right heart in the in-series configuration.

Haemair Ltd. (Swansea, UK) patented a prototype of a portable lung, to be used by conscious, mobile patients [77]. Such system is based on a compact gas exchange device with high membrane surface area (5–20 $\text {m}^{2}$), still lower than the surface of natural lung but much larger than conventional oxygenators. The large surface area is required since the device is designed to use air instead of oxygen (thus avoiding the need of an oxygen supply like a bulky oxygen cylinder); furthermore, the idea to use the device for long-term conscious, mobile patient increases the oxygen demand above the basal requirement. A control system for sensing the patient demand for oxygen and adjust the blood and/or air flow is included: more specifically, the sensor detects the pulse rate of the patient, which is related to the oxygen demand.

The medium-term development of the devices presented here is aimed at obtaining small, implantable mass transfer devices, with blood flow driven by the natural circulation and supported by a small air pump placed outside the body. In the long-term, the gas exchange device should be included within a complete prosthetic lung that will employ no electrical or mechanical parts: the natural lungs should be removed and replaced with an elastic air sac placed in the pleural cavity; the natural breathing action should expand and contract the air sac, drawing air through the mass exchange apparatus.

10.3 Intravenous Devices

Intravenous gas exchange represents an attractive modality to support the respiratory function in ARDS. The fundamental idea is to transfer oxygen to and to remove carbon dioxide from venous blood with a bundle of hollow fiber inserted through a peripheral vein and placed in the vena cava, without requiring extracorporeal circulation. Obviously, the fiber bundle must be compact to be safely placed in the vena cava, which has a diameter of 1.5–3 $\text {cm}$. This clearly introduces very strong constraint on the maximum allowable membrane area, so that the device can only support the gas exchange function, without replacing it. Generally, a respiratory support at 50 % of the basal requirement is considered as an appropriate target for these devices.

The first prototype of intravascular artificial lungs dates back to the 1980s with the IVOX [78] at CardiPulmonics, Inc. (Salt Lake City, UT, USA). Ivox consisted of a bundle of crimped hollow fiber membranes connected to a dual-lumen gas conduit which led outside the body to a console for providing gas flow through the fibers. The crimped fibers helped to minimize fiber clumping and to disturb blood flow improving the gas exchange permeance. The device was tested in clinical trials with a membrane area of 0.2–0.5 $\text {m}^{2}$ and providing for about 30 % of basal gas exchange requirements; the trial demonstrated the possibility to insert a fiber bundle in the vena cava for a prolonged periods without hemodynamic complications or thrombus formation; however, the extent of respiratory support was considered insufficient.

The McGowan Institute for Regenerative Medicine at the University of Pittsburgh has been active in the development of a respiratory support catheter usually referred to as intravenous membrane oxygenator (IMO) or Hattler catheter [79]: like IVOX, IMO consists of a bundle of hollow fibers, but a central polyurethane balloon which rhythmically (300 beats/min) inflates and deflates and provides blood mixing, thus enhancing the gas transfer coefficient (see Fig. 6.17). Flow velocity profiles visualized in the laboratory have shown that balloon pulsation also disrupts the layer of fluid near to the vessel wall, where the shear stress is higher, thus reducing the hemolytic damage. Tests for and $\mathrm {CO_{2}}$ exchange in cow both ex-vivo and in vivo show that the balloon pulsation results in a significant increase in gas exchange and allows to meet the design target.

Fig. 6.17

Scheme of the IMO device with the hollow fiber bundle and the central balloon used to improve blood-side gas transport (reproduced from [80], with permission)

10.4 Microfluidic Devices

Upon recognizing that the main limitations of ECMOs (even in the case of recent commercial equipment, such as the Maquet Quadrox) are ultimately to be ascribed to limited surface-to-volume ratio of the membrane module, research has been oriented toward scaling down the membrane exchange module to the microfluidic domain, i.e., shrinking down the characteristic dimension of blood- and gas-hosting channels from the current values of order 500 µm to channels of characteristic dimension of order 10–20 µm. Likewise, microfabrication techniques allow to reduce the thickness of the membrane separating blood and gas from the current value of order 50–100 µm to membranes as thin as 1–10 µm.

Such reduction of characteristic lengths makes it sensible to predict that ECMO devices exploiting this technology should eventually overcome the limitations of current hollow fiber technology by increasing the surface-to-area ratio up to two orders of magnitude and decreasing membrane resistance by an order of magnitude or even more, while maintaining the same or even decreasing the priming volume. Furthermore, because of the increased efficiency and lower flow rates involved by downsizing the characteristic cross-sectional length scales, atmospheric gas feeding and lower pressure drop can be afforded, thus allowing blood flow to be driven by heart pumping and the membrane exchanger to be fed by room air.

A quantitative assessment of the potential benefit of exploiting microfluidic technology in ECMO has been recently put forward by Potkay, who set up a simple transport model to estimate oxygen exchange rate per unit surface in a microfluidic device where the blood and the gas mixture are arranged in a multichannel cross-flow configuration [50, 81]. A conceptual scheme of the portable device together with the flow configuration for the membrane module is depicted in Fig. 6.18. This model provides a useful prediction of the average rated flow, $Q_{B}/A$, defined as the maximum blood flow rate for a given surface area of the membrane exchanger that allows inlet blood at oxygen saturation of 70 % to be collected at saturation of 95 % at the module outlet. Note that these values are fixed by physiological constraints. Quantitatively, Potkay proposes

$$\begin{aligned} \frac{A}{Q_{B}}=\alpha _{B}\frac{1}{K_{p,\mathrm {O_{2}}}}\log \frac{p_{in,\mathrm {O_{2}}}^{*}-p_{\mathrm {O_{2}}}^{G}}{p_{out,\mathrm {O_{2}}}^{*}-p_{\mathrm {O_{2}}}^{G}} \end{aligned}$$

(6.58)

The model expressed by Eq. 6.58 allows to estimate the potential limits of ECMO exploiting the microfabrication/microfluidic approach. For instance, in the case where the characteristic length of the channel cross-section is set to 10 µm and assuming that the membrane resistance to oxygen transport can be neglected, the predicted rated flow per square meter membrane area approaches 27 $\mathrm {l\, min^{-1}}$ [81], a value that overcomes by a factor five the best currently available commercial ECMOs. However, there are still many technological and even theoretical issues that prevent the achievement of this theoretical limit. Among the first category, one can single out the necessity for an accurate design of the microfluidic channel network, which should minimize pressure drop, thus allowing natural blood pumping through the device.

Fig. 6.18

Conceptual scheme of a portable lung support device, together with the flow configuration for the membrane module as proposed by [50] (reproduced with permission)

In this regard, natural systems still provide the most useful source for inspiration, such as in Murray’s seminal work carried out in the mid-twenties of the last century, which established the minimum work principle in physiological flows [82, 83]. Murray’s principle fixes a rule for the diameter ratio and the angle that should be satisfied when a larger channel branches into two smaller channels. In this respect, the possibility of integrated computer-assisted design of the channel network joint with soft lithography techniques could provide the practical chance to test these ideas at length scales that closely mimic those of natural vascular systems. Note, however, that an ideal microfluidic ECMO should contain several thousands of branched channel networks running in parallel in order to achieve suitable exchange surface area, an occurrence that makes the practical implementation of device microfabrication considerably more troublesome than the simple configuration devised by Potkay. Further practical challenges come from biocompatibility issues associated with the materials that come into contact with blood. Materials that are currently used to prevent thrombogenesys and platelet activation (e.g., heparin and PMEA) allow device lifetimes that are measured out in weeks.

Among the practical challenges so far described, which prevent ECMO from being a completely clinically successful technique, one is likely to be met in the near future, i.e., that of membrane thickness. New fabrication techniques such as initiated chemical vapor deposition (iCVD) are indeed continuously being proposed and tested, which are pushing membrane thickness below the single micrometer scale. To this end, it is worth remarking how iCVD has been shown to be compatible with branching channel network geometries [84]. Figure 6.19 shows an example of application of this technique to a branched channel membrane geometry.

Fig. 6.19

Example of a branched channel membrane produced with the iCVD technique. Reproduced from [84], with permission

Beyond the constructive issues concisely addressed above, altogether difficult hindrances to be overcome are to be expected even on the theoretical modeling of microscale ECMOs. In this regard, the most peculiar aspect associated with scaling down the device is given by the fact that the 10 µm limit sought for the characteristic dimension of the channel cross-section is comparable to the size of red blood cells, an issue that brings into play new phenomena both for the hydrodynamics and the mass transport processes that take place in the exchanger. Specifically, as regards the blood microhydrodynamics, the rheological behavior of blood is yet to be completely understood, and constitutive relationships such as the Bullik power law model [85] or other approaches [86] used to characterize flow regimes in hollow fiber exchangers must yet be validated at these length scales. In turn, 3 different blood rheologies as well as modified flow regimes should also be expected to have a significant impact on mass transport in the blood phase, and therefore, the validity of model predictions on overall mass transfer coefficients tested in hollow fiber modules for both oxygen [86] and carbon dioxide [66] should not be taken for granted. Because in microchannels of 10- µm cross-sectional dimension, red blood cells can be expected to flow “one at a time”, it appears sensible to assume that the oxygen uptake and carbon dioxide discharge will be described by a time-periodic model, where the frequency with which a given portion of the membrane surface is visited upon by the streaming red blood cells introduces a new timescale that is altogether absent in larger hollow fiber modules. These observations make it clear that the rational design of microfluidic-assisted ECMOs passing the benchmark test of clinical practice will be the result of the synergetic cooperation between researchers of many different branches of science, experimentalists, and theoreticians alike.

Acronyms

AV:

Arteriovenous

:

Arteriovenous carbon dioxide removal

ARDS:

Acute respiratory distress syndrome

CPB:

Cardiopulmonary bypass

DPG:

Diphosphoglycerate

ECCO2R:

Extracorporeal $\mathrm {CO_{2}}$ removal

ECMO:

Extracorporeal membrane oxygenator

ELSO:

Extracorporeal Life Support Organization

EL:

Extraluminal (flow)

iCVD:

Initiated chemical vapor deposition

IL:

Intraluminal (flow)

MV:

Mechanical ventilation

PEEP:

Positive end-expiratory pressure

PMEA:

Poly(2-methoxyethylacrylate)

PMP:

Polymethylpentene

PP:

Polypropylene

VV:

Veno-venous

Symbols

A :

Total exchange area

a :

Specific exchange area per unit volume

c :

Concentration in blood

d :

Capillary diameter

$d_{f}$ :

Fiber

$d_{p}$ :

Pore diameter

$\mathscr {D}$ :

Diffusivity

E :

Enhancement factor

$F_{B}$ :

Blood volumetric flow rate

$Gz$ :

Graetz number

H :

Henry’s constant (liquid composition as molar fraction)

$H^{\prime }$ :

Henry’s constant (liquid composition as molar concentration)

$\mathrm {Htc}$ :

Hematocrit

$\mathscr {H}$ :

Apparent Henry’s constant

$K_{1},\dots ,K_{4}$ :

Oxygen–heme-binding constants

$K_{a1},K_{a2}$ :

First and second carbonic acid dissociation constants

$K_{c}$ :

Overall mass transfer coefficient (concentration driving force)

$K_{h}$ :

Carbon dioxide hydration equilibrium constant

$K_{p}$ :

Overall mass transfer coefficient (pressure driving force)

$k_{p}$ :

Gas-phase mass transfer coefficient (pressure driving force)

$\mathscr {K}$ :

Constant defined in Eq. 6.7

L :

Length (capillary or channel)

$\ell$ :

Characteristic length

$N_{tm}$ :

Transmembrane gas flux

$Q_{B}$ :

Blood flow rate

$Pe _{tm}$ :

Transmembrane Peclet number

$Pr$ :

Prandtl number

$p_{50}$ :

Oxygen partial pressure at 50 % of hemoglobin saturation

p :

Partial pressure

$\mathscr {P}$ :

Membrane permeance

R :

Dimensionless parameter defined in Eq. 6.56

$R_{0}$ :

Inner radius

$\mathfrak {\mathbb {R}}$ :

Gas constant

$Re$ :

Reynolds number

$S_{\%}$ :

Oxygen fractional saturation

T :

Temperature

$V_{i}$ :

Volumetric flow rate of inspired air

Y :

Dimensionless parameter defined in Eq. 6.56

v :

Blood velocity

$\alpha _{B}$ :

Gas partition coefficient in blood

$\alpha _{m}$ :

Gas partition coefficient in membrane

$\delta$ :

Thickness

$\varepsilon _{f}$ :

Void fraction of the fiber bundle

$\varepsilon _{w}$ :

Membrane wall porosity

$\theta$ :

Contact angle

$\sigma$ :

Gas–blood surface tension

$\tau$ :

Tortuosity factor

Subscripts

a :

Arterial blood

$alv$ :

Alveolar air

b :

Chemically bound form of gas dissolved in blood

$\mathrm {CO_{2}}$ :

Carbon dioxide

$\mathrm {Hb}$ :

Hemoglobin

i :

Inspired air

in :

Inlet

m :

Molecular (free) form of gas dissolved in blood

$\mathrm {O_{2}}$ :

Oxygen

out :

Outlet

$\mathrm { RBC }$ :

Red blood cell

v :

Venous blood

Superscripts

*:

In equilibrium conditions

$\mathrm {0}$ :

In absence of chemical reactions

B :

In blood

G :

In gas phase

i :

At the interface

m :

In the membrane

p :

In the pores of the membrane

Overscripts

$\sim$ Dimensionless