WO2023209657A1

WO2023209657A1 - Computer program that simulates in real time a system comprising a natural lung, an artificial lung comprising an oxygen-carbon dioxide exchange membrane

Info

Publication number: WO2023209657A1
Application number: PCT/IB2023/054425
Authority: WO
Inventors: Luciano Gattinoni; Peter Herrmann; Michael Quintel; Mattia BUSANA
Original assignee: Dimar S.P.A.
Priority date: 2022-04-29
Filing date: 2023-04-28
Publication date: 2023-11-02
Also published as: IT202200008615A1

Abstract

Method (100) for simulating an evolution over time of a system starting from input data measured and/or calculated at an initial time. The system comprises three components: a first component (A) comprising a natural lung (NL) and at least one flow generator for a ventilation, a second component (B) which is a tissue component constituting a body metabolism response, a third component (C) comprising an artificial lung that comprises an extracorporeal gas exchange system (ML). The method comprises a modelling part (200), a clinical part (300) and a simulator part (400). The modelling part (200) of the system simulates the system as a simulated system by means of three models (210, 220, 230) which are a first model (210) that simulates a behaviour of the first component (A), a second model (220) that simulates a behaviour of the second component (B) and a third model (230) that simulates a behaviour of the third component (C). The clinical part (300) introduces said input data into the simulator part (400), wherein said simulator part (400) simulates an evolution of said simulated system starting from said input data by means of the three models (210, 220, 230).

Description

COMPUTER PROGRAM THAT SIMULATES IN REAL TIME A SYSTEM

COMPRISING A NATURAL LUNG, AN ARTIFICIAL LUNG

COMPRISING AN OXYGEN-CARBON DIOXIDE EXCHANGE MEMBRANE

The present invention refers to a computer program that simulates in real time a system comprising a natural lung, an artificial lung comprising an oxygen-carbon dioxide exchange membrane.

To date, to assist hypercapnic patients or patients suffering from acute respiratory failure (ARDS), use is made of ventilator supports that help the patient in the gas exchange at the lung level and/or extracorporeal systems that facilitate the oxygenation of the blood and the removal of carbon dioxide CO₂ .

Pulmonary ventilators or Venturi or turbine flow generators are used to replace or augment spontaneous ventilation of a patient who has respiratory/ventilatory insufficiency owing to pathologies affecting the lung or chest pump.

Mechanical ventilation is mainly employed in acute respiratory failure. Less common is the use as a treatment aimed at controlling pathological conditions that chronically cause (such as diseases of the rib cage, neuromuscular diseases or COPD) a stable increase in carbon dioxide values and to remove excesses thereof from the body.

In such cases, it is preferred to proceed through non-invasive ventilations that help the patient in facilitating the gas exchange through flows provided at positive pressures that increase alveolar recruitment and therefore greater ease for the patient to remove excess CO₂ and increase oxygenation. Both pulmonary ventilators and flow generators facilitate the gas exchange at the patient's alveolar level.

The problem found in recent years caused by the pulmonary ventilators, is the so-called Ventilator- Induced Lung Injury (VILI), i.e. lacerations/ruptures of the lung tissues caused by excessive pressure or volume of the gases supplied by the pulmonary ventilators in cases of patients that are intubated and not spontaneously breathing. Such high volumes or pressures are the result of severely compromised lungs that are overstressed by mechanical ventilation to ensure a minimum of alveolar exchange at the physiological level.

To avoid such stresses to the lung, the patient may be subjected, when the clinical picture allows it, to NIV (Not Invasive Ventilation) leaving the patient spontaneously breathing or with a minimum of respiratory assistance and maintaining a positive pressure inside the lung below the safety threshold, and at the same time helping the partial removal of CO₂ through an extracorporeal circulation of blood (decapneization) in a gas-permeable membrane, thus assisting the activity of the natural lung. Decapneization means a therapy dedicated to removing CO₂ from the blood by means of a medical-surgical device referred to as a decapneization device, inserted into an extracorporeal blood circuit capable of selectively extracting carbon dioxide (CO₂) from blood by means of its passage through a filter membrane permeable to CO₂ . Inside the decapneization device a flow of air more or less enriched with oxygen flows and through a membrane permeable to gases only, the oxygen carbon dioxide O₂-CO₂ exchange takes place which is dictated by the difference in partial pressures of the individual gases in the two compartments.

The surface of the membrane is able to support an important removal of CO₂ thanks to the connection with a source of O₂ or of medical air that determines a pressure gradient of the individual gases within the fibres of the membrane itself.

Single-lumen or double-lumen catheters normally positioned in the femoral or jugular veins, i.e. in a lower or upper part of a patient's body respectively, are generally used for the removal and the reintroduction of blood into the patient's veins.

Intrapulmonary gas exchanges are the basis of the breathing mechanism that regulates both the diffusion of molecular oxygen (O₂) from the alveoli to the arterial blood (oxygenation), and the elimination of molecular carbon dioxide (CO₂). Gas exchanges depend, in addition to diffusion, also on the mechanical capacity of the respiratory system that allows the exchange of air in the lungs. Alterations in intrapulmonary gas exchanges may cause hypoxemia (insufficient oxygenation of the blood) or hypercapnia (inadequate elimination of CO₂ from the blood).

Generally in a patient in pathological conditions a dysfunction of both the exchange of O₂ and CO₂ occurs, but more commonly prevailing impairments of either of them occur. In some pulmonary and/or systemic pathologies, the elimination of CO₂ can only occur by resorting to mechanical ventilation, thus accepting the increased risk of possible complications. In fact, if after the failure of the non-invasive ventilation (NIV) techniques, the patient requires invasive ventilation methods, he will be faced with an increased probability of morbidity with an increased risk of mortality.

Extracorporeal gas exchange systems adapted to remove CO₂ are for example ECCO₂-R (extracorporeal CO₂ removers), ECC (Extracorporeal Circulation) system and systems referred to as ECMO (Extracorporeal Membrane Oxygenation) used in serious pathologies or lung transplants represent an additional opportunity for critical patients.

There is a clinical and experimental need to monitor a patient with acute respiratory failure treated with Extracorporeal Vein-Venous Membrane Oxygenation (ECMO) therapy. This therapy is increasingly used in Intensive Care and its intrinsic complexity poses great difficulties in clinical management, as the normal physiology of the patient is profoundly altered by the therapy itself.

Unfortunately it is difficult both to detect in real time and simultaneously data from the patient, and to simulate what happens in a system comprising natural lung, artificial lung and membrane once the physician changes a parameter of the system. Normally the CO₂ data in the blood is detected with blood gas analysis every 6 or 8 hours, therefore when the doctor changes a parameter of the system he must rely on his empirical knowledge and experience to predict what could happen to the system in 6 or 8 hours, without the possibility of simulating what will happen within a few hours.

In the article by Christopher John Joyce et at, "A mathematical model of CO, O and N exchange during venovenous extracorporeal membrane oxygenation", Intensive Care Medicine Experimental (2018) 6:25, DOI: 10.1186/S40635-018-0183-4 disadvantageously only a blood flow from a lower part of the body is considered, which is directed towards an artificial lung, furthermore not all input values are calculated or measured, but are estimated.

Aim of the present invention consists in realizing a method implemented by a computer that simulates in real time a system comprising a natural lung, an artificial lung comprising an oxygen-carbon dioxide exchange membrane and that predicts how the system evolves over time, for example what happens by varying a mechanical ventilation and/or settings of an artificial lung.

In accordance with the invention this aim is achieved with a simulation method implemented on a computer according to claim 1.

Another aim of the present invention consists in realizing a computer program that simulates in real time a system comprising a natural lung, an artificial lung comprising an oxygen-carbon dioxide exchange membrane and that predicts how the system evolves over time, for example what happens by varying a mechanical ventilation and/or settings of an artificial lung.

In accordance with the invention this other aim is achieved with a computer program according to claim 9.

Other features are provided in the dependent claims.

The features and advantages of the present invention will become more apparent from the following description, which is to be understood as non-limiting example, with reference to the appended schematic drawings, in which: figure 1 is a schematic view of a flowchart of the method implemented by a computer program according to the present invention that simulates in real time a system comprising a natural lung, an artificial lung comprising an oxygen-carbon dioxide exchange membrane and that predicts what happens by varying a mechanical ventilation and/or settings of an artificial lung; figure 2 is a diagram of the functioning of the natural lung; figure 3 is a diagram of two compartments of an ideal pulmonary unit, the first compartment is referred to as dead space, the second compartment is properly referred to as venous admixture (and also referred to as, albeit improperly, shunt); figure 4 is a diagram of a pulmonary membrane in which part of the blood flow that exits the device, which is the artificial lung, can re-enter the device, depending on the anatomical positioning of the cannulas and on the quantity of a blood flow pumped through an ECMO circuit Q_EC measured in L/min; figure 5 is a graph showing how the percentage saturation of haemoglobin varies as a function of the partial pressure of oxygen in the blood PO₂ measured in mmHg, two curves are represented, the former with dashed line is real and the latter with solid line is standard; figure 6 shows a flowchart of the method according to the present invention comprising a modelling part, a clinical part and a simulation part; fig. 7 shows a flowchart of the modelling part.

With reference to the cited figures there is described a computer program that simulates in real time a system comprising a natural lung, an artificial lung comprising an oxygen-carbon dioxide exchange membrane and that predicts what happens by varying a mechanical ventilation and/or settings of an artificial lung and which implements a simulation method 100.

The method 100 comprises a modelling part 200, a clinical part 300 and a simulator part 400.

The computer comprises at least one memory for storing the data and at least one processor for performing the actions of the program when the program that comprises instructions such that when the program is executed by the computer implement the method 100 for simulating the system.

Real time means that the method 100 receives input data at an initial time and through this input data it is able to simulate the evolution of the system at any future time chosen by taking a negligible time with respect to the time that would be required waiting for the evolution of the system, wherein this negligible time is the time it takes to the simulator to simulate the evolution of the system. The method simulates the evolution of the system over time means that the method simulates the evolution of the system at any future time starting from an initial time at which it receives input data.

The clinical part 300 of the program allows to enter patient data to measure or calculate key variables of clinical utility to monitor the evolution of the disease, for example advantageously avoiding performing invasive manoeuvres on the patient (e.g. pulmonary arterial catheter). For this, the ventilator configuration, a blood gas analysis (EGA) from the output of the membrane lung (ML), an arterial blood gas analysis EGA and the elimination of CO₂ from the natural and artificial lungs are necessary as input. Blood gas analysis provides measured data.

The obtained data can be sent to an interface of the second part of the program which is the simulator 400. The simulator 400, given the previously calculated variables, allows to simulate how the system evolves in real time, allowing the doctor to choose different management strategies and to observe the expected response without taking any action on the patient. The simulator 400 can also be used alone, i.e. without the clinical part 300, as an educational tool to explain the complex interaction between the patient, the Extracorporeal Vein-Venous Membrane (ECMO) and all the physiological variables involved.

Through this analytical and numerical/iterative approach the program that implements the method is able to predict the most clinically useful blood parameters at every point of the ECMO system - patient even in impossible/difficult to collect sites.

Advantageously the invention provides a greater degree of safety to the fragile ECMO patient and a deep understanding to the doctor of the patient's specific physiological needs, taking a step forward in a personalized approach to the patient in intensive care.

Advantageously, the method allows a learning of the mechanisms that regulate the gas exchanges and helps the doctor to understand and define the actual state of gas exchange of the patient and to predict in real time what happens when the parameters vary, for example what happens if the mechanical ventilation and/or the settings of the artificial lung are modified. As shown in figure 1 the method 100 and the program comprise a modelling part 200 comprising three patterns or blocks 210, 220, 230.

Each model 210, 220, 230 integrates the description and the effects of the mechanical ventilation in the natural lung NL and the description and the effects of adding an extracorporeal blood flow ML, another component, which connects the natural lung NL to the artificial lung, is a model of the body metabolism B.

Generally referring to figure 1 the simulated system provides a first component A comprising a natural lung NL and a ventilator or more generally a flow generator, a second component B comprising the body metabolism, i.e. a tissue component, a third component C which is the artificial lung and comprises a pulmonary membrane ML i.e. any extracorporeal gas exchange system such as ECMO or ECCO₂-R.

For each block 210, 220, 230 we will provide the classical equations that describe the gas exchange and we will add some original solutions to widely recognized but yet unsolved problems in clinical practice 300.

These solutions of the clinical part 300 and of the simulation part 400 act synergistically with each other and constitute the heart of the invention.

Referring in particular to figure 1 the first model or block 210 is the model of the component A of the system, otherwise referred to as natural lung NL, also comprising the ventilator.

Referring to figure 1 the system comprises a multiplicity of connections in blood flow communication between the three compartments A, B and C, these connections are also referred to hereinafter as lines. Said multiplicity of connections comprise a first input connection 1 that transmits a total blood flow Qt to the first compartment A; a second input connection 2 that transmits a portion of blood flow QL to the input of the natural lung NL; a third input connection 3 that transmits a blood flow Q_s to non-aerated parts of the natural lung (NL); a fourth connection 4 at the exit from the natural lung (NL) and which connects with the third connection 3 towards a fifth connection 5.

Said fifth connection 5 receives the flows from the third 3 and from the fourth connection 4 of the first compartment A and transmits the total blood flow Qt towards the second compartment B.

Referring in particular to figure 2, the natural lung is simulated through the simulation program and the first block 210 can be imagined as a single pulmonary unit, in which the gases are uniformly distributed over space and time. In other words, this ideal pulmonary unit 215 has a constant blood input 211 and output 212 (corresponding respectively to the lines 1 and 5 of figure 1) and a constant gas input 213 and output 214. This unit 215 is the heart of the gas exchange in the natural lung (NL hereinafter), i.e. in this unit 215 a certain quantity of O₂ is added by the gas to the blood that outflows from the lung and a certain quantity of CO₂ is eliminated from the blood by the gas. The diagram is presented in figure 2.

Figure 2 shows at input 211 a capillary blood flow Q_c (L/min) that enters the ideal alveolus 215. Q_c is equal to a total blood flow Qt (L/min) minus a deviated blood flow Q_s (L/min), which does not enter the ideal pulmonary unit 215. Qt is the volume of blood pumped by the heart in the unit of time (L/min).

Q_s is the volume of blood that flows into the nonaerated parts of the lung per unit of time (L/min).

At input 211 an O₂ content of venous blood entering the pulmonary unit 215 is C_v O₂ (mL/100mL) and a CO₂ content of venous blood entering the pulmonary unit is Cv CO₂ (mL/100mL). Note that C_v O₂ and C_v CO₂ are also the gas concentrations in the derived blood.

In figure 2 at output 212 an O₂ content of the capillary blood leaving the ideal alveolus 215 is C_c O₂ (mL/100mL), while a CO₂ content of the capillary blood leaving the ideal alveolus 215 is C_c CO₂ (mL/100mL).

In figures 1 and 2, VO₂ NL is the quantity of oxygen absorbed through the ideal pulmonary unit 215 in 1 min (mL/min) and VCO₂ NL is the quantity of carbon dioxide eliminated through the ideal pulmonary unit in 1 min (mL/min).

The constitutive equations describing the relationship between VO₂, VCO₂ and blood content at equilibrium are as follows:

The formulas are necessary to calculate the gas content in capillary, venous and arterial blood, respectively: C_c O₂ = 1.39•Hb (g/dL) +

0.00314 (ml/10OmL•mmHg)-PA O₂ (mmHg)

C_v O₂ = 1.39-Kb (g/dL)•S_v O₂ + 0.00314 (mL/10OmL•mmHg)-P_v O₂ (mmHg)

C_a O₂ = 1.39-Hb (g/dL)•S_a O₂ + 0.00314 (mL/100mL mmHg)-P_a O₂ (mmHg) where 1.39 is ml of O₂ that binds 1g of Hb (where Hb stands for haemoglobin) and 0.00314 is the solubility coefficient for O₂ in blood; PA O₂, Pv O₂ and P_a O₂ are the partial pressures of O₂ in the alveolus, in the venous blood and in the arterial blood, respectively; S_v O₂ and S_a O₂ are venous and arterial saturations. In the ideal compartment, i.e. in the capillary blood, the saturation is assumed to be 100%.

According to Douglas equation the CO₂ content is:

According to Kelman equation (Kelman GR. Digital computer procedure for the conversion of PCO₂ into blood CO₂ content. Respir Physiol. 1967 Aug;3 (1):111--5. doi: 10.2116/0034-5687 (67)90028-x. PMID: 6059098) the CO₂ content is:

Doxygenated and D_reduced are, respectively, the ratio

[CO₂]cells: [CO₂]plasma under completely reduced and completely saturated blood conditions, as determined experimentally by Van Slyke and Sendroy (1928). Since this ratio is based on experimentally determined values, it automatically takes into account the carbon dioxide carried in the form of carbamine compounds. The ratio of the intermediate saturations was calculated by linear interpolation.

where T is a temperature and Hct is the haematocrit content, i.e. a percentage of blood volume occupied by haematocrits: Hct = Hb * 3/100; in general pK= -logic(Ka), where Ka is an acid dissociation constant. In this case reference is made to the pK of carbonic acid (H₂CO₃), which is 6.1.

We remind that O₂ exchanged in the ideal pulmonary unit is equal to:

and that the eliminated CO₂ is equal to:

where F_i O₂ and FA O₂ are fractions (e.g.: 0.30, 0.40 etc.) of oxygen respectively in the inspired (index i) and alveolar (index A) gas from the natural lung NL.

It follows that, at equilibrium, the quantity of gas absorbed by the blood (for O₂) or eliminated from the blood (for CO₂) is equal to the quantity given/removed with ventilation, as shown here for the oxygen:

The ratio of VCO₂ to VO₂ is known as the respiratory quotient (R):

If R = 1, it means that one mole of CO₂ is produced for each mole of O₂ consumed.

From the above equation it is possible to derive that F_A O₂, i.e. the fraction of oxygen in the ideal pulmonary unit:

To the ideal pulmonary unit, which takes full account of the exchange of VO₂ and VCO₂, we can associate two other compartments: one ventilated and not perfused, and another perfused but not ventilated. The first is referred to as dead space and represents "wasted" ventilation, while the second is properly referred to as venous admixture and, albeit improperly, it is also referred to as shunt, and it represents "wasted" perfusion. The interaction of these two compartments with the ideal pulmonary unit is represented in figure 3.

In figure 3, VE is a minute ventilation at the input in the plant, VA is VE - VD where VA is the ventilation that reaches the ideal pulmonary unit 215, VD is the ventilation of the dead space that, being the not perfused compartment, does not participate in the gas exchange. Therefore, there is no difference in the composition of the gas entering and exiting the system (line 3 in figure 1).

In figure 3, F_i O₂ is a fraction of oxygen in the inspired gas; F_A O₂/P_A O₂ is a fraction pressure ratio of the oxygen present in the ideal pulmonary unit; F_A CO₂/P_A CO₂ is a fraction pressure ratio of carbon dioxide present in the ideal pulmonary unit; F_E O₂/P_E O₂ is a fraction-pressure ratio of the oxygen in the gas exiting the entire system, resulting from the gas mixture coming from the ideal pulmonary unit 215 (FA O₂) and from the dead space; F_E CO₂/P_E CO₂ is a fraction-pressure ratio of the carbon dioxide in the gas exiting the entire system, resulting from the gas mixture coming from the ideal pulmonary unit 215 (F_A CO₂) and from the dead space.

In figure 3, Qt is a total blood flow (cardiac output) entering the system, divided into Q_c which is a perfusion that reaches the ideal pulmonary unit and Q_s which is a venous admixture, which represents the quantity of blood perfusing the unventilated alveoli. The composition of the blood exiting this compartment is the same as the one of the blood entering the compartment and will mix with the blood coming from the ideal pulmonary unit.

The dead space, defined as the quantity of gas that does not participate in the gas exchange, was initially calculated by Bohr (Bohr dead space) according to the following:

it follows that

This is the fraction of gases that enters the system that reaches the ideal pulmonary unit 215, which is ventilated and perfused. Therefore, the complement to 1 of the above equation represents the fraction of gases that enters the ventilation system of the dead space compartment:

P_E CO₂ is calculated as the ratio of VCO₂NL to the total ventilation VE, while the alveolar partial pressure of carbon dioxide P_A CO₂ is the ratio of VCO₂NL to the ventilation that reaches the alveoli VA, i.e.:

The number 863 is a constant factor that converts the volumes from BTPS to STPD and transforms the fractions into partial pressures.

The dead space calculated with the Bohr equation is also referred to as physiological dead space. The greatest limit of this equation is that it requires P_A CO₂, i.e. the partial pressure of carbon dioxide in the ideal pulmonary unit 215. Since this value cannot be measured, the Bohr equation was modified by Enghoff using P_a CO₂ as a surrogate for P_A CO₂ . This modification would be precise if the P_a CO₂ value were a true surrogate of the partial pressure of CO₂ in the capillary of the ideal pulmonary unit 215 (P_c CO₂), which is in equilibrium with P_A CO₂ . Unfortunately, the composition of the blood exiting the system is also affected by the venous admixture compartment, which modifies the P_a CO₂ values. Therefore, the dead space calculated according to Enghoff's modification of the Bohr equation is falsely increased by any venous admixture effect.

The physiological dead space can be conceptually divided into two sections: the anatomical and the alveolar one.

For what concerns the alveolar dead space in the ideal pulmonary unit 215, ventilated and perfused, the following relationships can be derived:

Where VA_tot is the gas that enters the alveolar space, perfused or not perfused.

It follows that:

this ratio represents the fraction of ventilated and perfused alveolar ventilation with respect to the total alveolar ventilation. Consequently, the complement of this relationship represents the alveolar dead space:

where P_A CO₂ is usually replaced by the arterial CO₂ value (P_a CO₂).

As far as the anatomical dead space is concerned it reflects the anatomical structures, including the mechanical ventilation apparatus where gas exchange is physically impossible.

An approximate estimate of the anatomical dead space is 1 ml per 1 pound of body weight (1 kg = 2.2 pounds). For example, in a 70 kg man the anatomical dead space will be 70 kg x 2.2 ml/kg = 154 ml.

According to Bohr, the anatomical dead space can also be derived as follows:

it follows that:

therefore, the anatomical dead space will be:

Referring in particular to figure 1 the second model or second block 220, is the model of the tissue compartment B.

Referring to figure 1, the second block 220 simulates the behaviour of an ideal tissue that is continuously perfused with a blood flow (Qt), consumes a certain quantity of oxygen and produces a certain quantity of CO₂ . Under normal conditions, the quantity of O₂ consumed will be equal to the flow multiplied by the difference between arterial and mixed venous O₂ content, while the produced CO₂ will be equal to the oxygen consumed multiplied by the respiratory metabolic quotient.

Under normal conditions, the partial pressure of CO₂ in the ideal tissue is equal to the PCO₂ in the mixed venous blood, i.e. the CO₂ content exiting the ideal tissue compartment B is equal to the CO₂ content in the mixed venous blood.

When an artificial blood is added to the system, the CO₂ content leaving the tissue is different from the mixed venous blood content of CO₂ . This will require an estimate of CO₂ in the tissues. In addition, both O₂ consumption and CO₂ production, as well as blood flow, can be different in the upper and lower part of the body. This is absolutely relevant when the artificial lung C is added to the system, as the drainage cannula drains blood coming from specific districts of the body (usually the lower part of the body) whose gas composition is different from the weighted average concentration of gas found in the tissues.

To partially overcome this problem, we have divided the ideal tissue compartment into two sections: the upper one and the lower one.

We defined Q_up as the fraction of the cardiac output that perfuses the upper part 8 of the body and Q_low as the fraction that perfuses the lower part 9:

We defined f_c as the percentage difference in CO₂ and O₂ content between the upper and lower part of the body:

The model we adopted for the metabolic compartment is presented in figure 1 as second block 220.

The O₂ and CO₂ content of the ideal tissue was calculated as follows:

Said multiplicity of connections comprise a sixth connection 9 that transmits a blood flow Q_cath coming from the second compartment or component B. The blood flow Q_cath comes from the lower part 7 of the body and in part it could come from the upper part 8 of the body.

Said multiplicity of connections comprise a seventh connection 10 where the blood flow Qret of the upper part 8 of the body leaving the tissue component B flows and connects with the line 17 where the blood flow that arrives from the third component C flows reconnecting to the first connection 1 that enters the first component

A. Note that line 10 does not pass through the pulmonary membrane 230, in fact the blood flow coming from the lower part 7 of the body, comprising the blood flow coming from the femoral veins, has a higher pressure than the blood flow coming from the upper part 8 of the body, therefore the blood flow that comes from the upper part 8 of the body is sent directly to the natural lung NL 210 without passing through the lung membrane ML C, while only the blood flow that comes from the lower part

7 of the body ( and in part it could also come from the upper part 8 of the body) is directed towards the pulmonary membrane ML of the artificial lung C.

The blood flow of the lower body 7 has different values than the blood flow of the upper body 8.

Advantageously, by differentiating the two blood flows of the upper part 8 and the lower part 7 of the body, a greater precision of the simulation method 100 is obtained, also allowing the doctor a more correct clinical evaluation because it is based on data calculated or measured in real time and not on data estimated on the basis of samples taken once every 6 or

8 hours which do not differentiate the upper part 8 from the lower part 7 of the body as occurs in the state of the prior art.

The third model or third block 230 describes the artificial lung comprising the pulmonary membrane ML and its characteristics. The artificial lung, of which several models are available, is basically a device that allows the exchange of gases between two compartments: blood and gas. The material that separates them varies depending on the model and also the arrangement for the correspondence of blood and gas is different. The basic principles are the same as those adopted by the first block 210 for the natural lung NL: the material separating the two phases subrogates the alveolar- capillary barrier; the gas entering the gaseous compartment subrogates the total ventilation (VE_ML) and the flow entering the pulmonary membrane (Q_EC) through the line 12 subrogates the cardiac output.

The quantification of the gas composition in the pulmonary membrane ML and its correspondence with the blood encounters the same problems encountered with the natural lung NL. It is therefore convenient to approach the physiology of the pulmonary membrane ML like for the natural lung NL, defining in an analogous way to what has been done above for the natural lung an ideal compartment of the artificial lung to which the compartments of dead space and venous admixture of the artificial lung are added.

Ideal compartment: the ideal compartment in ML can be considered as a homogeneous section of the device in which all the exchange of oxygen and carbon dioxide takes place. Therefore, the correspondence of ideal blood flow and of the ideal gas flow is such that, at a given ideal P_oxy O₂ and ideal P_oxy CO₂, the observed O₂ input and CO₂ removal can be fully explained.

Venous admixture compartment: the venous admixture compartment is calculated as follows:

Dead space: Bohr dead space of ML:

Where is the average PCO₂ in output from the

ML (P_ECO₂).

The physiological dead space of ML is calculated as:

As schematically shown in figure 4 another complication is present in the ML with respect to the NL: part of the blood flow exiting the device can re- enter the device, depending on the anatomical positioning of the cannulas and on the quantity of Q_EC . By device is meant the artificial lung. Even the artificial lung works like the natural lung shown in figure 2, in fact it is exchanged VCO_{2 ML} and VO_{2 ML} which respectively represent the amount of CO₂ eliminated through the lung membrane ML and the amount of oxygen absorbed through the artificial lung unit ML, as shown more schematically in Figure 1.

Note that the blood flow coming from the lower body 7 is measured by hemo gas analysis, while the blood flow coming from the upper body 8 is measured by gas analysis. Advantageously, the gas analysis is carried out continuously and in real time.

Referring to figure 1 that shows the three blocks 210, 220, 230, consider: Qret is the blood flow of the upper part 8 of the body that goes directly from the tissue compartment B to the natural lung NL A following the line 10; Q_cath is the blood flow of the lower part 7 of the body from the tissue compartment B to the pulmonary membrane ML C following the line 9; Q_rc is the blood flow that recirculates through the pulmonary membrane ML following the line 11;

Q_ec is the extracorporeal blood flow that enters the pulmonary membrane C following the line 12, given by the sum of Q_cath and Q_rc;

Q_ec is the total blood flow that exits the artificial lung ML C following the line 16 and connecting to the lines 11 and 17;

Q_oxy is the portion of Q_ec that enters the ideal ML compartment C following the line 13 and exits therefrom following the line 15;

Q_sec is the portion of Q_ec that enters the shunt compartment of the ML following the line 14;

Said multiplicity of connections also comprise the lines 11-17 shown in figure 11.

Q_gas (VEML) is the total gas flowing in the ML;

VA_oxy is the gas flow that ventilates the ideal compartment ML;

VDML is the ventilation of the dead space of the ML;

Coxy O₂ and Coxy CO₂ respectively represent the O₂ and CO₂ content in the ideal ML compartment C;

C_in O₂ and C_in CO₂ respectively represent the O₂ and CO₂ content of the blood entering ML C; C_out O₂ and C_out CO₂ respectively represent the O₂ and CO₂ content of the blood exiting ML C; P_oxy O₂ and P_oxy CO₂ represent the partial pressure of O₂ and CO₂ in the ideal compartment of ML C;

P_in O₂ and P_in CO₂ represent the partial pressure of O₂ and of CO₂ respectively in the blood entering the ML;

P_out O₂ and Pout CO₂ represent the partial pressure of O₂ and of CO₂ in the blood exiting the pulmonary membrane ML C, respectively.

More generally the method 100 comprises said modelling part 200 of the system that simulates the system as a simulated system by means of three models 210, 220, 230 which are the first model 210 that simulates a behaviour of the first component A, the second model 220 that simulates a behaviour of the second component B, the third model 230 that simulates a behaviour of the third component C.

The first model 210 comprises a first input line 1 that transmits a total blood flow Qt towards a natural lung NL; said natural lung NL comprising an aerated part that exchanges oxygen and carbon dioxide with a fraction of the total blood flow Qt which is a blood flow QL entering from the second line 2 and exiting oxygenated QL from the third line 4, a non-aerated part of the natural lung that transmits a remaining fraction of the total blood flow Qt which is the non-oxygenated blood flow Q_s that is transmitted from the fourth line 3; a fifth output line 5 that transmits a total blood flow Qt that sums the oxygenated blood flow QL of the third line

4 and the non-oxygenated blood flow Q_s that is transmitted from the fourth line 3.

The second model 220 comprises a first input line

5 which is the fifth output line 5 of the first model 210; the tissue component that exchanges oxygen and carbon dioxide with the total input blood flow Qt through the first line 5; a second output line 10 that transmits a blood flow Qret from the tissue component towards the first input line 1 of the first component A; a third output line 9 that transmits a blood flow Q_cath from the tissue component towards the third component C.

The third model 230 comprises a first input line 12 that transmits a total blood flow Q_EC towards the artificial lung; said artificial lung ML comprising the extracorporeal gas exchange system ML that exchanges oxygen and carbon dioxide with a fraction of the total blood flow Qt which is a blood flow Q_OXY entering from the second line 13 and exiting oxygenated Q_OXY from the third line 15, a non-aerated part of the artificial lung that transmits a remaining fraction of the total blood flow Qt which is the non-oxygenated blood flow Q_sEC that is transmitted by the fourth line 14; a fifth output line 16 that transmits a total blood flow Q_EC that sums the oxygenated blood flow Q_OXY of the third line 15 and the non-oxygenated blood flow Q_sEC of the fourth line 14; a recirculation line 11 that transmits a fraction of the total blood flow QEC exiting the fifth output line 16 and referred to as recirculation blood flow QRC and which connects to the first line 12 and to the second output line 9 of the second component B by mixing the recirculation blood flow QRC with the blood flow Q_cath coming from the tissue component to form the input blood flow Q_EC of the first line 12 to the artificial lung C; a transmission line 17 that transmits a fraction of the remaining blood flow Q_EC from the fifth output line 16 deprived of the recirculation blood flow QRC towards the first input line 1 of the first component A. Said transmission line 17 connects to the second output line 10 of the second component B to mix the remaining blood flow of the transmission line 17 with the blood flow Qret from the tissue component.

The clinical part 300 of the method 100 and of the program implementing the method 100 is designed to provide the doctor with a complete picture of the patient's clinical status and its interaction with the pulmonary membrane ML, using a limited number of input variables that can be implemented automatically. The concepts on which the clinical section 300 is built are the same as those of the simulator part 400 in that the simulator 400 predicts how the output parameters change once the input parameters provided by the clinical part 300 are modified.

The structure of the clinical section 300 is exactly the same as the one of the simulator section 400: we have the same three compartments: A natural lung NL comprising the ventilator, B tissue compartment, C artificial lung comprising the pulmonary membrane ML, connected as described above and depicted in particular in figure 1.

The biggest difference between the simulator 400 and the clinical instrument 300 is that in the latter P_A CO₂, PA O₂ and the alveolar pH that must be estimated indirectly from the arterial blood gases, while they are calculated directly in the simulator 400. In the simulator 400, the most relevant variables of the ideal compartment, such as the venous admixture and the dead space, are calculated directly from the input variables used as input. In the clinical model 300 these technical characteristics are not primary variables to be estimated but are calculated. Therefore, the whole procedure must follow different steps.

The clinical part 300 needs initial measured input data to then calculate all input data needed for the simulation part 400.

More generally, the clinical part 300 provides in input for initial measured input data used by the clinical part 300 to calculate at least a part of input data for the simulation part 400. The remaining part of input data for the simulation part 400 are comprised among the initial input data of the clinical part 300. No input data is only estimated, but always either measured or calculated on the basis of measures. Note that calculating is different from estimating, because calculating implies having measured data to insert in a formula to calculate another data, while estimation means a probabilistic estimate affected by stochastic errors.

It is specified that the input data for the simulation part 400 are partly the initial input data also used by the clinical part 300 and partly are the data calculated by the clinical part 300 using the initial input data.

Referring to the arrows in figure 1 to establish pre- and post-pulmonary membrane ML C, the input variables for the clinical part 300 of the program and of the method are the initial input data and are divided in this manner.

General parameters: cardiac output (Qt), according to line 5 of figure

1; VCO_{2 ML} and VCO₂ NL, which represent respectively the quantity of CO₂ eliminated through the pulmonary membrane ML and the quantity of CO₂ eliminated through the natural lung NL;

Ventilator setting parameters for the natural lung NL, part A: F_i O₂, which represents the fraction of oxygen in the gas inspired by the natural lung (number between 0 and 1);

RR respiratory rate, the number of breaths taken per minute; V_t, current or tidal volume, a quantity of air moving in or out of the lungs for each respiratory cycle L;

Arterial BGA blood gas analysis parameters:

P_a CO₂ partial pressure of CO₂ in the arterial blood measured in mmHg;

P_a O₂ partial pressure of O₂ in the arterial blood measured in mmHg; pH_art is the pH of the arterial blood;

Hb_art represents the concentration of haemoglobin in the arterial blood in g/dL;

Post EGA ML parameters:

Pout CO₂ partial pressure of CO₂ in the blood flowing out of the ECMO circuit measured in mmHg;

Pout O₂ partial pressure of O₂ in the blood flowing out of the ECMO circuit measured in mmHg; pH_out is the pH of the blood flowing out of the ECMO circuit;

Hb_out is the concentration of haemoglobin in the blood flowing out of the ECMO circuit, measured in g/dL.

Pulmonary membrane ML setting parameters of part C: Q_ec the volume of blood pumped through the ECMO circuit, per unit of time in L/min;

Q_gas is the volume of gas flowing in the ECMO circuit, per unit of time in L/min; F_i O_{2 ML} fraction of oxygen in the gas flowing in the pulmonary membrane, is a number comprised between 0 and 1;

Pre ML parameters:

S_pre O₂ is the oxygen saturation in the haemoglobin in the blood flowing towards the ECMO circuit;

Tissue compartment parameters, part B:

ABE (v-a) is the difference in BE between tissue blood, i.e. venous blood not affected by extracorporeal circulation, and arterial blood; f_c is the percentage difference in O₂ and CO₂ content between the upper portion and the lower portion of the patient's body of the tissue compartment B; fq is the percentage fraction with respect to Qt between the upper and lower part of the body (e.g. percentage of Q_up with respect to Q_t, or percentage of Q_low with respect to Qt).

Consider that T=37°C is used in any formula that requires temperature.

The starting equations for the clinical part are:

where Hct is the haematocrit content that is measured by a blood test that indicates the percentage of the blood volume occupied by the haematocrits.

Oxygen saturations are available directly from BGA blood gas analysis.

The clinical part 300 recalculates with coherence formulas all other saturations not directly available (e.g.: venous saturation).

Saturations for the arterial and post-ML blood are calculated from the input variables, respectively (pH, PO₂ and PCO₂).

As is done in the simulator part 400, a correction is applied for the parameters: temperature, PCO₂ and pH. For this purpose we use the method proposed by Kelman (J. Appl. Fisio., 1966). Therefore, for each PO₂ value we calculate a new value of virtual PO₂ (virtual PO₂) which has no clinical and biological significance, but when inserted into the standard saturation curve depicted in figure 5 it provides the same value of oxygen saturation of haemoglobin SO₂ as the real curve with actual pH, PCO₂ and PO_2n.

Virtual PO₂ is calculated as:

This new value is inserted in place of the real PO₂ in the tree equations described above (Severinghaus, Kelman, Ruiz) and the saturation is obtained in the current condition of pH, PCO₂ and temperature:

Severinghaus:

Kelman:

After calculating the haemoglobin saturation values, it is possible to calculate the basic excess BE of arterial and post-ML blood according to Zander equation:

From the pressures to the arterial and post-ML contents:

Oxygen: C_a O₂ and C_out O₂ are calculated as follows:

Carbon dioxide: C_a CO₂ and C_out CO₂ are calculated according to Douglas or Kelman formulas: CO₂ content (according to Douglas eq.):

CO₂ content (according to Kelman eq.):

At this point the following variables are available for the arterial and post-ML blood: contents, pressures, pH, BE and saturations.

With regard to the oxygen in the pulmonary membrane ML an ideal compartment of pulmonary membrane ML is considered.

Since one of the input variables is the oxygen saturation of the blood entering the ML (pre ECMO saturation), we calculate the oxygen content C_in O₂ as follows.

First of all, a conversion of the saturation in input to pressure (P_in O₂) is performed with an inverse

Severinghaus equation:

It is important to point out that this calculation does not take into account the effect of pH, PCO₂ and temperature on the haemoglobin dissociation curve; in fact, the P_in O₂ calculated with inverse Severinghaus is the one that would be in equilibrium with the input saturation only if the pH was 7.40 and PCO₂ was 40 mmHg. This limit is, however, clinically irrelevant as it is limited to no more than 3-4 mmHg.

If the pH and PCO₂ of the input blood of the ML were known, however, it would be possible to add this additional correction.

In the second stage, the pre-ML blood oxygen content is calculated:

We can calculate now VO_{2 ML} and R_ML as follows:

It is assumed that the oxygen tension (P_oxy O₂) in the blood flowing in the ideal ML compartment C is equal to the oxygen tension in the gaseous compartment. So:

P_oxy O₂ is then converted into virtual P_oxy O₂ (using the pH and PCO₂ post-ML values for correction) in order to calculate the corresponding haemoglobin saturation (S_oxy O₂) with Severinghaus, Kelman or Ruiz equation.

At this point it is possible to calculate the oxygen content of this "ideal" compartment ML:

The shunt percentage of the compartment ML (line 14) with respect to Q_ec is then calculated with the usual shunt formula:

Therefore, it is possible to calculate the two components of the extracorporeal flow, namely the shunt flow (Q_sec) (line 14 of figure 1) and the flow that perfuses the ventilated part of the ML ( Q_oxy) (line 13 of figure 1):

With regard to the carbon dioxide in the pulmonary membrane ML, an ideal compartment of pulmonary membrane ML is considered. C_out CO₂ has already been calculated from the input variables.

Gin CO₂ is calculated as follows:

The CO₂ content in the ideal compartment ML is instead calculated as follows:

Physiological dead space in the compartment of the pulmonary membrane ML:

As is evident from the formula, it is not enough to have an initial setup provided by a flow generator/lung ventilator to calculate physiological dead space.

With regard to the tissue compartment B, the section of the model 220 where all O₂ is consumed and all CO₂ is produced. Therefore, the arterial carbon dioxide content will be modified as follows:

A catheter draining the blood from the tissue compartment B may be inserted into the upper or lower part of a patient's body, where the distribution of the total flow may be different. It is therefore convenient to divide the tissue compartment B into two sections: the upper metabolic 8 and the lower metabolic 7 compartment.

The direct flow to the upper compartment 8 will be defined as:

Considering this, then the flow to the lower compartment 7 will be:

The concentration of carbon dioxide in the two compartments 7, 8 is different, as the metabolism may be different. In fact, gravity affects the upper body differently than it does the lower body. Therefore, the oxygen saturation of the blood leaving the upper 8 or lower 7 compartment is different. The percentage difference in the fraction of oxygen saturation between the upper 8 and lower 7 compartment is an input variable and is referred to as f_c.

It should be noted that the documents of the prior art however do not differentiate the blood flow coming from the upper part 8 of the body from the blood flow coming from the lower part 7 of the body.

In this step we must estimate how much flow from the tissue compartment reaches the pulmonary membrane ML and how much will return directly to the right heart. The following variables are then defined:

Qret is the blood flow that from the upper part 8 of the body goes directly from the tissue compartment B to the natural lung NL A following the line 10 of figure 1; Q_cath is the blood flow that from the lower part 7 of the body goes from the tissue compartment B to the pulmonary membrane C along the line 9 of figure 1; Q_rc is the blood flow that recirculates through the pulmonary membrane C, shown along the line 11 of figure 1;

Q_ec is the extracorporeal blood flow (line 12 of figure 1) that enters the pulmonary membrane C, given by the sum of Q_cath and Q_rc;

Q_oxy is the portion of Q_ec that enters the ideal ML compartment C, see line 15 of figure 1; Q_sec is the portion of Q_ec that enters the shunt compartment of ML C, see line 14 of figure 1. In order to determine Q_cath, Q_rc and Qret it is necessary to understand how the blood exiting the tissue compartment B is drained in the extracorporeal circulation C. Two cases are possible:

Case 1: Q_cath ≤ Q_low

This occurs when Q_cath comes completely from the lower metabolic compartment 7;

Case 2: Q_cath > Q_low

This occurs when Q_cath receives blood from all the lower metabolic compartment 7 and from the upper one 8 (Q_{up cat} )•

When it is written that the blood flow of the lower part 7 of the body passes through line 9, it means that the blood flow of the lower part 7 of the body passes for the most part, but a part of the blood flow 8 of the upper part of the body can also pass in part as explained in the alternative case 2.

In line 10 of figure 1, on the other hand, only the blood flow of the upper part 8 of the body passes both in case 1 and in case 2.

In a first phase, we can assume that we are in the most complicated case, that is, in case 2. In this scenario, the following system of equations applies:

from which, we can simply calculate Q_{up cath} and Q_rc:

If Q_{up cat} > 0: then case 2 is confirmed, the two values of Q_{up cath} and Q_rc are accepted and Q_cath can be calculated:

If Q_{up cat} 0, then case 2 is impossible and we return to case 1.

In this scenario, i.e. in case 1, the following system of equations applies:

from which, we can simply calculate Q_cath and Q_rc:

In this step, regardless of the case, it is possible to calculate the recirculation rate as:

Advantageously to calculate the content of O₂ in the upper 8 and lower 7 tissue compartments, it is necessary to know the recirculation flow, the flow of the catheter and the oxygen content of the blood entering and exiting the artificial lung C, this allows to calculate, with "inverse" sequence, the oxygen content in the tissue section B and its division in the two metabolic compartments 7, 8. In fact:

it is then possible to calculate C_cath O₂, Cret O₂ and C_v O₂ . It is emphasized that these values are calculated and not estimated.

According to case 1:

As regards the role of carbon dioxide CO₂ in the upper 8 and lower 7 tissue compartments, it is possible to calculate C_cath CO₂, Cret CO₂ and C_v CO₂ .

In case 1:

In case 2:

The basic excess BE of the arterial and post-ML blood has already been calculated in the earlier steps using Zander equation.

According to Zander, no difference should be found between arterial and venous BE. However, we leave the possibility to set ABE as input by the user.

P_in CO₂ and pHin are calculated through the following iterative process in the pre-ML portion of figure 1. P_in CO₂ is calculated according to the inverse Zander equation as a function of the increase in the pHin levels (starting from 6.5 with a 0.0001 step).

The calculated P_in CO₂ is inserted into an equation of the CO₂ content (either Douglas or Kelman). The iteration is interrupted when the calculated CO₂ content is equal to the one given ( C_in CO₂).

For all remaining compartments (tissue and mixed venous blood), saturation of PO₂, PCO₂, pH and O₂ is derived from the content through the following iterative process .

A first step of the iterative process provides that PCO₂ is calculated according to the inverse Zander equation as a function of the increase in pH levels (starting from 6.5, with 0.0001 step), starting from an oxygen saturation of 100%.

An appropriate BE value is used for each compartment: tissue compartment: BE_tiss mixed venous compartment: BE_V

The calculated PCO₂ is inserted into an equation of the CO₂ content (either Douglas or Kelman). The iteration is interrupted when the calculated CO₂ content is equal to the one given (respectively C_tiss CO₂ and C_v CO₂).

A second step of the iterative process provides the PCCb/pH pair being used to calculate a PO₂/Sat pair through another iterative process. The PCO₂/pH pair is used to calculate PO₂ virtual as a function of the increase in PO₂ levels (starting from PO₂=1, e.g. with 0.01 step). PO₂ virtual is then inserted into Severinghaus, Kelman or Ruiz equation to calculate a new oxygen saturation value, then the O₂ content is calculated:

The process is interrupted when the calculated oxygen content is equal to the one given (respectively C_tiss O₂ and C_v O₂).

A third step of the iterative process provides that the PO₂/Sat pair obtained in the second step is inserted into the first step of the iterative process, to calculate a new PCO₂/pH pair, and so on.

The cycle of the three steps of the iterative process is interrupted when the last calculated pH is less than, for example, 0.00001 different from the previous one.

Alternatively it is possible to interrupt the cycle at different pH values.

As regards the first block 210 that models the natural lung NL.

At this point, the following calculations are possible:

As regards the oxygen in the natural lung NL, it is provided that PA O₂ is calculated by assuming P_A CO₂ = Pa CO₂:

Then, we determine the oxygen saturation (S_A O₂) and the pH (pH_A) of the capillary blood through iteration, in a similar way to what is done in the simulator 400.

Assuming P_A CO₂ = Pa CO₂, we calculate BE through Zander equation, starting from S_A O₂ = 1 and with increasing levels of pH_A (starting from pH 6.5 with increasing levels for example by a 0.0001 step), until the calculated BE is equal to the one given (BE_art):

At this point the process is interrupted and the pH value_A is accepted. The pH_A is used to calculate PO₂ virtual, along with PA O₂ and P_a CO₂ . P_virtual O₂ is then inserted into Severinghaus, Kelman, or Ruiz equation to calculate a new oxygen saturation value. If the calculated value of S_A O₂ differs from the previous one for example by more than 0.001, the cycle restarts from the beginning: the new saturation value is inserted into the initial Zander equation to calculate a new pH value, which will be used to find a new alveolar saturation value. The cycle continues until the last saturation value differs from the previous value less than, for example, 0.001. O₂ capillary is calculated as:

With regard to the natural shunt of the lung (line 3 of figure 1):

With regard to the carbon dioxide of the natural lung NL:

With regard to the dead space of the natural lung

NL:

Owing to the approximations indicated some of the values calculated so far may be slightly inaccurate. To solve this problem, we recalculated some of the variables.

We recalculated S_A O₂ and pH_A. TO do this, we repeated the cycle described above for the calculation of the oxygen in the natural lung NL, using the value P_A CO₂ obtained in the calculation of the carbon dioxide for the natural lung NL.

Thus, a new value C_A CO₂ can be obtained with the Kelman or Douglas equation.

We calculated a new C_a CO₂ according to the following formula:

A new iteration starts. The following equations are solved in this order:

If the calculated C_a CO₂ is different from the one calculated above in relation to the saturation calculation, see figure 5 for reference, the last value is used to restart the cycle. The iteration is interrupted when C_aCO₂ is different from the previous one, for example 0.01.

More generally and summarizing what the clinical part 300 of the method 100 does, the clinical part (300) comprises at least one iterative calculation step for determining all the simulated system data starting from the initial input data.

Said at least one iterative calculation step comprises a multiplicity of calculation cycles for determining an evolution over time of both the initial input data and of the data calculated by the clinical part 300.

Referring to the arrows in figure 1 to establish pre- and post-pulmonary membrane ML C, the input variables for the simulation part 400 of the program are divided in this way.

General parameters: cardiac output (Qt), according to line 5 of figure 1, (it is also used for clinical part 300);

Hb : haemoglobin concentration in the involved blood streams; BE_art is the basic excess of arterial blood. The basic excess is defined as the quantity of base/acid that must be added to the blood to obtain a pH of 7.40, providing a partial pressure of carbon dioxide PCO₂ of 40 mmHg and at a temperature of 37°C (mmmol/L).

Ventilator setting parameters for the natural lung NL, part A: F_i O₂, which represents the fraction of oxygen in the gas inspired by the natural lung (number between 0 and 1), (is also used for clinical part 300);

RR respiratory rate, the number of breaths taken per minute, (also used for clinical part 300); V_t, current or tidal volume, a quantity of air moving in or out of the lungs for each respiratory cycle L, (also used for clinical part 300);

V_{d phys}/V_t is the rate between V_d and Vd_Phys, where V_t is the total or tidal volume that represents the quantity of air moving in and out of the lungs at each respiratory cycle measured in L, while Vd_Phys is the total dead space, such as the volume of the ventilated but not perfused lung (measured in L).

Qs/Qt is the ratio between Q_s with respect to Qt, i.e. a number comprised between 0 and 1, where Q_s is the volume of blood flowing in the non-aerated parts of the lung, measured in unit of time, i.e. L/min.

VO₂ tot represents the quantity of oxygen absorbed both through the natural lung NL and through the pulmonary membrane ML in 1 minute (mL/min).

VCO₂ tot represents the quantity of carbon dioxide eliminated both through the natural lung NL and through the pulmonary membrane ML in 1 minute (mL/min).

Pulmonary membrane ML setting parameters of part C:

Q_ec the volume of blood pumped through the ECMO circuit, per unit of time in L/min; Q_gas is the volume of gas flowing in the ECMO circuit, per unit of time in L/min; F_i O_{2 ML} fraction of oxygen in the gas flowing in the pulmonary membrane, is a number comprised between 0 and 1;

VCO_{2 ML} is the quantity of CO₂ eliminated through the pulmonary membrane ML, (also used for clinical part 300);

QRC/Q_EC is the QRC ratio with respect to Q_EC, where QRC is the part of Q_EC that recirculates through the pulmonary membrane (L/min);

Q_sec/Q_EC is the ratio of Q_sec with respect to Q_EC, where Q_sec is the part of Q_EC that flows into the non- ventilated parts of the pulmonary membrane ML (L/min);

Post ML parameter:

BEpostML is the basic excess of blood flowing out of the pulmonary membrane (mmmol/L).

Tissue compartment parameters, part B:

ABE (v-a) is the difference in BE between tissue blood, i.e. venous blood not affected by extracorporeal circulation, and arterial blood, (also used for clinical part 300); f_c is the percentage difference in O₂ and CO₂ content between the upper portion and the lower portion of the patient's body of the tissue compartment B, (also used for clinical part 300); fq is the percentage fraction with respect to Qt between upper and lower part of the body (e.g. percentage of Q_up with respect to Qt, or percentage of Q_low with respect to Qt), (also used for clinical part 300).

Consider that T=37°C is used in each formula of the simulation part 400 requiring temperature, as also happens for clinical part 300. The clinical part 300 introduces said input data set forth above into the simulator part 400.

Said simulator part 400 simulates an evolution of said simulated system starting from said input data by means of the three models 210, 220, 230.

More generally and summarizing what the simulator part 400 does, the simulation part 400 comprises at least one iterative calculation step for determining all the simulated system data starting from the input data provided by the clinical part 300, wherein said at least one iterative calculation step of the simulation part 400 comprises a multiplicity of calculation cycles for determining an evolution over time of both the input data provided by the clinical part 300 and of the data calculated by the simulation part 400 starting from said input data provided by the clinical part 300.

More specifically, the following initial equations are used for the simulation part 400: for the blood flowing in the natural lung A, i.e. the shunt flow Q_s line 3 of figure 1 and total flow Qt:

for the blood flowing in the pulmonary membrane ML C, i.e. shunt flow Q_sec line 14 of figure 1, Q_oxy line 13 of figure 1 and recirculation flow Q_rc line 11 of figure 1:

The distribution of Qt from the tissue compartment B to the artificial lung ML C, i.e. Q_cath line 9 of figure 1, Qret line 10 of figure 1, Q_oxy line 17 of figure 1:

Other formulas used to calculate the other data needed for the simulation:

where FR is a respiratory rate, i.e. a number of respiratory acts performed in one minute (acts/min).

With regard to the simulation of the natural lung NL: ideal compartment A, the following passages are implemented in sequence.

The first step is the determination of the alveolar partial pressure of carbon dioxide P_ACO₂ :

The second step is the determination of the alveolar partial pressure of oxygen P_A O₂ according to the following formula:

Note that, in this initial step, we consider RNL =1 where RNL is the rate of VCO_2NL with respect to VO_2NL, which are respectively the quantity of CO₂ eliminated through the natural lung in 1 minute and the quantity of oxygen absorbed by the natural lung in 1 minute. P_B is a barometric pressure of 760 mmHg, while P_H20 is a saturated vapour pressure corresponding to 47 mmHg.

The third step consists in determining the saturation of the alveolar oxygen (Sat _aiv) and the alveolar pH (pH_alv) of the capillary blood by iteration as described below.

By definition, the PO 2 of capillary blood (P_c O₂) is equal to the ideal pulmonary unit PO₂. Since pH is a function of both basic excess and oxygen saturation, an iterative process is used to find the pH of the alveolar compartment. The reference equation is a modified Zander equation, which calculates BE:

Given a BE input (assumed equal to BE_art which is the basic excess of arterial blood), the iterative process starts calculating the alveolar pH assuming a 100% alveolar saturation. Increasing pH levels (starting from pH 6.5 with increasing levels e.g. by a 0.0001 step) are inserted into Zander equation until the calculated BE is equal to the given one. At this point the process is interrupted and the pH value is accepted. Then, the pH is used to calculate the actual alveolar saturation.

It is possible to calculate the saturation of haemoglobin as a function of P_AO₂ through Severinghaus, Kelman or Ruiz equation:

Severinghaus :

Ruiz :

It is important to point out that the equations listed above refer to a dissociation curve of the oxygen saturation measured at pH 7.40, PCO₂ of 40 mmHg and temperature T of 37°C. A correct correction is therefore preferable when referring to the clinical scenario, with different pH and PCO₂ values.

For this purpose we used a method proposed by Kelman (J Appl Fisio, 1966). The rationale behind the use of this equation is the generally accepted fact that all three factors alter the scale of the PO₂ axis, but do not alter the shape of the dissociation curve. It is thus possible to make the standard dissociation curve applicable to different temperatures and acid-base states by calculating a new value of virtual PO₂ (PO₂ virtual). This new value has no clinical and biological significance, but when inserted into the standard saturation curve it provides the same SO₂ value as the real curve with the actual pH, PCO₂ and PO₂ values (see figure 5).

This new value is inserted into place of the PO₂ real in the tree equations described above (Severinghaus, Kelman, Ruiz) and saturation is obtained under the actual conditions of pH, PCO₂ and temperature.

If the calculated alveolar saturation value differs from the previous one by more than for example 0.001, the cycle restarts from the beginning: the new saturation value is inserted into the initial Zander equation to calculate a new pH value, which will be used to find a new alveolar saturation value. The cycle continues until the last saturation value differs from the previous one by less than 0.001, for example.

More generally, this step of iterative calculation of said simulation part 400 can be summarized as comprising a first step that calculates the alveolar partial pressure of carbon dioxide P_ACO₂; a second step that calculates the alveolar partial pressure of oxygen P_A O₂; a third step saturation of alveolar oxygen (Sat alv) and of the alveolar pH (pHalv) of the capillary blood by means of an iterative process comprising calculating the alveolar pH starting from an alveolar saturation value and varying it step by step, using the alveolar pH calculated in the previous step and calculating the actual alveolar saturation under actual conditions of pH, PCO₂ and temperature.

For what concerns a calculation of the gas content, we divide the problem by carbon dioxide and oxygen. With regard to the carbon dioxide, we calculate the capillary content (C_c CO₂).

It is possible to choose which equation can be used between two different options: Douglas equation or Kelman equation. The input variables are: P_A CO₂, alveolar saturation and capillary blood pH.

For CO₂ Content (Douglas eq.):

For CO₂ Content (Kelman eq.):

For mixed venous blood (Cv CO₂):

For arterial blood (C_a CO₂):

For oxygen the capillary content (C_c O₂):

For what concerns the second tissue component B and for carbon dioxide the blood flow is divided into the upper and lower parts of the tissue compartment according to fq:

The CO₂ content in the ideal tissue is calculated as follows:

CO₂ concentrations in the upper (C_up CO₂) and lower part (C_low CO₂) of the body can be calculated according to the following system of equations:

It follows that:

At this point it is necessary to consider the characteristics of the input blood, which can originate in proportions different from the upper and lower metabolic compartments:

In case 1: Q cat

Q low this occurs when the catheter Q comes completely from the lower metabolic compartment;

In case 2: Q cat > Q low this occurs when Q_cath receives blood from all of the lower metabolic compartment and part of the upper one.

For what concerns case 1, since Q_cath comes only from the lower metabolic compartment, C_cath CO₂ in this case it is equal to C_low CO₂ .

In contrast, Qret is composed of blood coming from both the upper and lower metabolic compartments, and it is calculated as follows:

C_in CO₂, Coxy CO₂ and C_out CO₂ from the ML artificial lung are calculated through a matrix as follows:

For what concerns case 2, C_ret is only composed of blood coming from the upper metabolic compartment, so C_ret CO₂ = Cup CO₂ .

On the other hand, Q_cath is composed of blood coming from the upper and lower metabolic compartments and is calculated as follows:

C_in CO₂, Coxy CO₂ and C_out CO₂ from the artificial lung ML are calculated through a matrix as follows:

With regard to the ideal compartment of the pulmonary membrane ML. For further calculations it is useful to calculate P_oxy CO₂ and P_oxy O₂ . P_oxy CO₂ is calculated through an iterative process, assuming Sat_oxy =1 and BE_oxy = BE_art .

P_in CO₂ is calculated according to the inverse Zander equation as a function of the increase in pH levels (starting from 6.5, e.g. 0.0001 step):

Each calculated P_in CO₂ is inserted into an equation of the CO₂ content (either Douglas or Kelman). The iteration is interrupted when the calculated CO₂ calculated is equal to the one given (Coxy CO₂ .

Then, assuming RML =1, it is possible to calculate:

For what concerns the tissue compartment B, the problem is divided into two cases 1 and 2.

With regard to case 1 we can find C_a O₂, C_v O₂, C_Up O₂, C_low O₂, Cin O₂ and C_out O₂ by solving the following system of equations through a matrix:

Since Q_cath only comes from the lower metabolic compartment, C_cath O₂ in this case is equal to Ci_Ow O₂ .

As regards case 2, we can find C_a O₂, C_v O₂, C_Up O₂, Glow O₂, Cin O₂ and C_out O₂ by solving the following system of equations through a matrix:

In this case, Cret is only composed of blood coming from the upper metabolic compartment, so Cret O₂ = C_up O₂ .

Once this system of equations is solved, the O₂ content of the blood entering and exiting the natural lung NL and the artificial lung ML can be calculated. It is then possible to calculate the quantity of O₂ added to the blood from each lung with the following formulas:

consequently, since VCO₂ and VO₂ are available for both lungs, it is possible to calculate RNL and RML:

For accuracy purposes, it is now possible to recalculate P_A O₂ with the correct RNL:

It is possible to calculate BE according to Zander. No difference should be found between arterial and venous BE, however, we leave the possibility to set ABE as input by the user.

PO₂, PCO₂, pH and saturation can now be calculated for all compartments A, B and C.

For all compartments dealt with above (arterial, mixed venous, pre-ML, post-ML and tissue blood), PO₂, PCO₂, pH and O₂ saturation are derived from the contents through the following iterative process:

In the first stage, PCO 2 is calculated according to the inverse Zander equation as a function of the increase in pH levels (starting from 6.5, for example with 0.0001 step), starting from an oxygen saturation of 100%

An appropriate BE value is used for each compartment: Arterial: BE_art Mixed venous: BE_V Pre-ML, Post-ML and tissues: BE_tiss •

The calculated PCO₂ is inserted into an equation of the CO₂ content (either Douglas or Kelman). The iteration is interrupted when the calculated CO₂ content is equal to the one given (C_a CO₂, C_v CO₂, C_in CO₂, C_out CO₂ and C_tiss CO₂, respectively).

In the second stage, the PCO₂/pH pair is used to calculate a PO₂/Sat pair through another iterative process. The PCO₂/pH pair is used to calculate virtual PO₂ virtual (as discussed above) as a function of the increase in PO₂ levels (starting from P02=l, e.g. with step 0.01).

P_virtual O₂ is then inserted into Severinghaus, Kelman or Ruiz equation to calculate a new oxygen saturation value, then the O₂ content is calculated:

The process is interrupted when the calculated oxygen content is equal to the one given (respectively C_a O₂, C_v O₂, C_in O₂, C_out O₂ and C_tiss O₂)• In the third stage, the PO₂/Sat pair obtained in the second stage described above is inserted into the first stage, to calculate a new PCO₂/pH pair, and so on. The cycle is interrupted when the last calculated pH is lower than, for example, 0.00001 different from the previous one.

Alternatively, it is possible to choose other steps, with respect to those mentioned so far in the implementation example.

As far as the artificial lung dead space is concerned, it is possible to calculate it as the Bohr dead space of the ML:

where is the average PCO₂ in output from the ML

(PECO₂).

The physiological dead space of the ML is calculated as:

Alternatively and more generally it is possible to provide that in place of ECMO any extracorporeal gas exchange system such as ECCO₂-R is used.

Alternatively and more generally the pulmonary ventilator is a flow generator.

Alternatively and more generally initial input data for the clinical part 300 comprises: the total input blood flow Qt through the first line 5 of the first component (A); the quantity of CO₂ VCO_{2 ML} eliminated through the extracorporeal gas exchange system (ML); the quantity of CO₂ VCO₂ NL eliminated through the natural lung (NL); the oxygen fraction in the gas inspired by the natural lung (NL) F_i O₂; the respiratory rate RR of the natural lung (NL); the tidal volume V_t of the natural lung (NL); the partial pressure of CO₂ in an arterial blood P_s CO₂; the partial pressure of O₂ in the arterial blood P_a O₂; the pH of the arterial blood pH_art; the haemoglobin concentration in the arterial bloodH_art r the partial pressure of CO₂ in the blood P_out CO₂ flowing out of the third component (C); the partial pressure of O₂ Pout O₂ in the blood flowing out of the third component (C); the pH of the blood pH_out flowing out of the third component (C); the concentration of haemoglobin Hb_out in the blood flowing out of the third component (C). the blood flow Q_ec pumped through the third component (C); the volume of gas Q_gas flowing in the third component (C); the fraction of oxygen F_i O_{2 ML} in the gas flowing in the extracorporeal gas exchange system (ML); the oxygen saturation S_pre O₂ in the haemoglobin in the blood flowing towards the third component (C); a basic excess difference ABE (v-a) between tissue blood and arterial blood; a percentage difference f_c of O₂ and CO₂ content between an upper portion and a lower portion of the second component (B); a percentage fraction f_q with respect to Qt between the upper and lower part of the second component (B).

Alternatively and more generally input data for the simulation part 400 comprises: the total input blood flow Qt through the first line 5 of the first component (A); the concentration of haemoglobin Hb in the blood flows of the simulated system; the basic excess of the arterial blood BE_art; the oxygen fraction in the gas inspired by the natural lung (NL) F_i O₂; the respiratory rate RR of the natural lung (NL); the tidal volume V_t of the natural lung (NL); the rate Va _Phys/V_t between a tidal volume V_p and a total dead space volume V_{d Phys}; the Qs/Qt ratio between a volume of blood flowing in non-aerated parts of the natural lung Q_s with respect to the total blood flow Qt; the quantity of oxygen absorbed VO₂ tot both through the natural lung (NL) and through the extracorporeal gas exchange system (ML); the quantity of carbon dioxide eliminated VCO₂ tot both through the natural lung NL and through the extracorporeal gas exchange system (ML); the volume of blood Q_ec pumped through the third component (C); the volume of gas Q_gas flowing in the third component (C); the fraction of oxygen F_i O_{2 ML} in the gas flowing in the extracorporeal gas exchange system (ML); the quantity of CO₂ VCO_{2 ML} eliminated through the extracorporeal gas exchange system (ML); the QRC/Q_EC ratio of the recirculation blood flow QRC with respect to the blood flow Q_EC in the third component (C); the Q_sec/Q_EC ratio of the blood flow flowing in the non-ventilated parts of the extracorporeal gas exchange system (ML) Q_sec with respect to Q_EC; a basic excess difference ABE (v-a) between tissue blood and arterial blood; the percentage difference f_c of O₂ and CO₂ content between an upper portion and a lower portion of the second component (B); the percentage fraction f_q with respect to Qt between the upper and lower part of the second component (B).

Advantageously, the present invention implements the method 100 implemented by the computer which simulates in real time the system comprising the natural lung NL, the artificial lung ML comprising the oxygen carbon dioxide exchange membrane and which predicts how the system evolves over time, for example what happens by varying a mechanical ventilation and/or artificial lung settings.

Advantageously, method 100 does not estimate the values, but as input it has measurements or calculations made on measurements and as output method 100 has calculations performed on the input measurements and/or calculations obtained which are repeated iteratively.

The invention thus conceived is susceptible to many modifications and variants, all falling within the same inventive concept.

Claims

1. Method (100) for simulating an evolution over time of a system starting from input data measured and/or calculated at an initial time, wherein said system comprises three components: a first component (A) comprising a natural lung (NL) and at least one flow generator for a ventilation, a second component (B) which is a tissue component constituting a body metabolism response, a third component (C) comprising an artificial lung that comprises an extracorporeal gas exchange system (ML), wherein said method comprises a modelling part (200), a clinical part (300) and a simulator part (400), wherein said modelling part (200) of the system simulates the system as a simulated system by means of three models (210, 220, 230) that are a first model (210) that simulates a behaviour of the first component (A) and comprises a first input line (1) that transmits a total blood flow Qt towards a natural lung (NL), said natural lung (NL) comprising an aerated part that exchanges oxygen and carbon dioxide with a fraction of the total blood flow Qt which is a blood flow QL entering from the second line (2) and exiting oxygenated QL from the third line (4), a non-aerated part of the natural lung that transmits a remaining fraction of the total blood flow Qt which is the non-oxygenated blood flow Q_s that is transmitted from the fourth line (3), a fifth output line (5) that transmits a total blood flow Qt that sums the oxygenated blood flow QL of the third line (4) and the non-oxygenated blood flow Qs of the fourth line (3); a second model (220) that simulates the behaviour of the second component (B) and comprises a first input line (5) which is the fifth output line (5) of the first model (210), the tissue component that exchanges oxygen and carbon dioxide with the total input blood flow Qt through the first line (5), a second output line (10) that transmits a blood flow Qret from the tissue component of an upper part (8) of the body only towards the first input line (1) of the first component (A), a third output line (9) that transmits a blood flow Q_cath from the tissue component of a lower part (7) of the body or of the lower part (7) and of a part of the upper part (8) of the body towards the third component (C); a third model (230) that simulates the behaviour of the third component (C) and comprises a first input line (12) that transmits a total blood flow Q_EC towards the artificial lung, said artificial lung (ML) comprising the extracorporeal gas exchange system (ML) that exchanges oxygen and carbon dioxide with a fraction of the total blood flow Qt which is a blood flow Q_OXY entering from the second line (13) and exiting oxygenated Q_OXY from the third line (15), a non-aerated part of the artificial lung that transmits a remaining fraction of the total blood flow Qt which is the non-oxygenated blood flow Q_sEC that is transmitted from the fourth line (14), a fifth output line (16) that transmits a total blood flow Q_EC that sums the oxygenated blood flow Q_OXY of the third line (15) and the non-oxygenated blood flow QSEC of the fourth line (14), a recirculation line (11) which transmits a fraction of the total blood flow QEC exiting the fifth output line (16) and referred to as recirculation blood flow QRC and which connects to the first line (12) and to the second output line (9) of the second component (B) by mixing the recirculation blood flow QRC with the blood flow Q_cath coming from the tissue component to form the input blood flow Q_EC of the first line (12) to the artificial lung (C), a transmission line (17) that transmits a fraction of the remaining blood flow Q_EC from the fifth output line (16) deprived of the recirculation blood flow QRC towards the first input line (1) of the first component (A), wherein said transmission line (17) connects to the second output line (10) of the second component (B) to mix the remaining blood flow of the transmission line (17) with the blood flow Qret from the tissue component; wherein said clinical part (300) introduces said input data into the simulator part (400), wherein said simulator part (400) simulates an evolution of said simulated system starting from said input data by means of the three models (210, 220, 230).

2. Method (100) according to claim 1, characterized in that said input data comprises at least one configuration of said at least one flow generator, a blood gas analysis (BGA) from an output of the extracorporeal gas exchange system (ML), an arterial blood gas analysis (BGA_a) and data related to the removal of CO₂ from both the natural lung (NL) and the artificial lung.

3. Method (100) according to any one of claims 1 or 2, characterized in that said clinical part (300) provides at the input for initial input data used by the clinical part (300) to calculate at least a part of input data for the simulation part (400).

4. Method (100) according to claim 3, characterized in that the initial input data for the clinical part (300) comprises: the total input blood flow Qt through the first line (5) of the first component (A); a quantity of CO₂ VCO₂ ML eliminated through the extracorporeal gas exchange system (ML), a quantity of CO₂ VCO₂ NL eliminated through the natural lung (NL); an oxygen fraction in the gas inspired by the natural lung (NL) F_i O₂; a respiratory rate RR of the natural lung (NL); a tidal volume V_t of the natural lung (NL); a partial pressure of CO₂ in an arterial blood P_a CO₂; a partial pressure of O₂ in the arterial blood P_a O₂; a pH of the arterial blood pH_art; a haemoglobin concentration in the arterial blood H_art r a partial pressure of CO₂ in the blood P_out CO₂ flowing out of the third component (C); a partial pressure of O₂ Pout O₂ in the blood flowing out of the third component (C); a pH of the blood pH_out flowing out of the third component (C); a concentration of haemoglobin Hb_out in the blood flowing out of the third component (C). the blood flow Q_ec pumped through the third component (C); a volume of gas Q_gas flowing in the third component (C); a fraction of oxygen F_i O_{2 ML} in the gas flowing in the extracorporeal gas exchange system (ML); an oxygen saturation S_pre O₂ in the haemoglobin in the blood flowing towards the third component (C); a basic excess difference ABE (v-a) between tissue blood and arterial blood; a percentage difference f_c of O₂ and CO₂ content between an upper portion and a lower portion of the second component (B); a percentage fraction f_q with respect to Qt between the upper and lower part of the second component (B).

5. Method (100) according to any one of claims 3 or

4, characterized in that the clinical part (300) comprises at least one iterative calculation step for determining all the simulated system data starting from the initial input data, wherein said at least one iterative calculation step comprises a multiplicity of calculation cycles for determining an evolution over time of both the initial input data and of the data calculated by the clinical part (300).

6. Method (100) according to any one of claims 1-

5, characterized in that the simulation part (400) comprises at least one iterative calculation step for determining all the simulated system data starting from the input data provided by the clinical part (300), wherein said at least one iterative calculation step of the simulation part (400) comprises a multiplicity of calculation cycles for determining an evolution over time of both the input data provided by the clinical part (300) and of the data calculated by the simulation part (400) starting from said input data provided by the clinical part (300).

7. Method (100) according to claim 6, characterized in that said at least one iterative calculation step of said simulation part (400) comprises a first step that calculates the alveolar partial pressure of carbon dioxide P_ACO₂, a second step that calculates the alveolar partial pressure of oxygen P_A O₂, a third step of saturation of alveolar oxygen (Sat_alv) and of the alveolar pH (pH_alv) of capillary blood by means of an iterative process comprising calculating the alveolar pH starting from a value of alveolar saturation and varying it step by step, using the alveolar pH calculated in the previous step and calculating the effective alveolar saturation under actual pH, PCO₂ and temperature conditions.

8. Method (100) according to any one of claims 1- 7, characterized in that the input data for the simulation part (400) comprises: the total input blood flow Qt through the first line (5) of the first component (A); a concentration of haemoglobin Hb in the blood flows of the simulated system; a basic excess of the arterial blood BE_art; an oxygen fraction in the gas inspired by the natural lung (NL) F_i O₂; a respiratory rate RR of the natural lung (NL); a tidal volume V_t of the natural lung (NL); a rate V_{d Phys}/V_t between a tidal volume V_t and a total dead space volume V_{d Phys}; a Qs/Qt ratio between a volume of blood flowing in non-aerated parts of the natural lung Q_s with respect to the total blood flow Qt; a quantity of oxygen absorbed VO₂ tot both through the natural lung (NL) and through the extracorporeal gas exchange system (ML); a quantity of carbon dioxide eliminated VCO₂ tot both through the natural lung NL and through the extracorporeal gas exchange system (ML); a volume of blood Q_ec pumped through the third component (C); a volume of gas Q_gas flowing in the third component (C); a fraction of oxygen F_i O_{2 ML} in the gas flowing in the extracorporeal gas exchange system (ML); a quantity of CO₂ VCO₂ ML eliminated through the extracorporeal gas exchange system (ML) a QRC/Q_EC ratio of the recirculation blood flow QRC with respect to the blood flow Q_EC in the third component (C); a Q_sec/Q_EC ratio of the blood flow flowing in the non-ventilated parts of the extracorporeal gas exchange system (ML) Q_sec with respect to Q_EC; a basic excess difference ABE (v-a) between tissue blood and arterial blood; a percentage difference f_c of O₂ and CO₂ content between an upper portion and a lower portion of the second component (B); a percentage fraction f_q with respect to Qt between the upper and lower part of the second component (B).

9. Computer program loadable in a computer memory comprising instructions which, when the program is executed by the computer, implement a method (100) according to any one of claims 1-8.