Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support: a mathematical modeling approach

Objective To describe (1) the energy transfer from the ventilator to the lungs, (2) the match between venous-venous extracorporeal membrane oxygenation (ECMO) oxygen transfer and patient oxygen consumption (VO2), (3) carbon dioxide removal with ECMO, and (4) the potential effect of systemic venous oxygenation on pulmonary artery pressure. Methods Mathematical modeling approach with hypothetical scenarios using computer simulation. Results The transition from protective ventilation to ultraprotective ventilation in a patient with severe acute respiratory distress syndrome and a static respiratory compliance of 20mL/cm H2O reduced the energy transfer from the ventilator to the lungs from 35.3 to 2.6 joules/minute. A hypothetical patient, hyperdynamic and slightly anemic with VO2 = 200mL/minute, can reach an arterial oxygen saturation of 80%, while maintaining the match between the oxygen transfer by ECMO and the VO2 of the patient. Carbon dioxide is easily removed, and normal PaCO2 is easily reached. Venous blood oxygenation through the ECMO circuit may drive the PO2 stimulus of pulmonary hypoxic vasoconstriction to normal values. Conclusion Ultraprotective ventilation largely reduces the energy transfer from the ventilator to the lungs. Severe hypoxemia on venous-venous-ECMO support may occur despite the matching between the oxygen transfer by ECMO and the VO2 of the patient. The normal range of PaCO2 is easy to reach. Venous-venous-ECMO support potentially relieves hypoxic pulmonary vasoconstriction.


METHODS
In this supplementary material, we describe the principles of the mathematical modeling.

Energy transfer from the ventilator to the lungs
The basic principle of mechanical ventilation relies on active pulmonary inflation through the energy transfer from the mechanical ventilator to the lungs. In acute respiratory distress syndrome (ARDS) patients, the higher the pulmonary compliance and the lower the airway resistance, the lower will be ventilator-to-lungs energy transfer, and a high proportion of this low amount of energy transfer will be dissipated generating tidal volume (Vt) ( Figure 1S). Likewise, low pulmonary compliance with low airway resistance will be associated with a high amount of the ventilator-lungs energy transfer dissipated in alveoli overdistension and cyclic opening of respiratory units, a process determinant of the physical concept of lungs strain, in which the higher the strain, the more the lungs expand relatively to initial expiratory lung volumes. (1,2) The strain is partitioned in static strain, as generated by positive end-expiratory pressure (PEEP), and dynamic strain, as generated by Vt. The dynamic strain results in deformation of the lung structure, triggers inflammation, edema and hemorrhage formation. Furthermore, a high static strain is associated with slight inter-alveolus septum rupture and interstitial emphysema development. (2,3) Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support Figure 1S -Exploring the strain concept. Defining strain rate as tidal volume/ functional residual capacity ratio, in Panel A) (Normal lungs condition), the higher functional residual capacity promotes a lower strain rate than Panel B) (acute respiratory distress syndrome condition), considering the tidal volume is the same. In acute respiratory distress syndrome, the collapsed alveolus causes a lower pulmonary compliance than in normal condition; therefore the energy spent generating tidal volume in normal condition is also spent in mechanical alveoli hyperdistention (higher strain rate) as presented in acute respiratory distress syndrome condition. FRC -functional residual capacity; ARDS -acute respiratory distress syndrome. # Normally aerated alveoli; * collapsed alveoli; $ hyperdistended alveoli.
In order to quantify the amount of energy transferred from the ventilator to the lung, the mechanical power concept was used. (4) The mechanical power is derived from the equation of motion, and then computes the most important energy load components (Elastic and resistive, adding PEEP to the equation and neglecting inertial forces) of each respiratory cycle. (5) To quantify the total energy load transference per minute, the obtained value is multiplied by the respiratory rate. The calculation of energy load transferred from the mechanical ventilator to the lungs per minute is mathematically expressed as following: (4) Lungs compliance # # # Expiratio Inspiratio # # # Respiratory * # # $ $ Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support association of higher PEEPs (10 -15cmH2O) and lower plateau or driving pressures with lower mortality. (8,11) The impact of I:E ratio on any outcome is not well established.
Therefore, the marginal effect of each cited variable on the mechanical power was simulated in the supplementary results. As many ventilators offer the possibility of direct inspiratory time adjustment, the plot between inspiratory time normalized to a RR = 10BPM and mechanical power was also built.
In order to facilitate the understanding of PEEP interaction effect with driving pressure on the mechanical power, and RR interaction effect with driving pressure on the mechanical power, two graphs plotting driving pressure and mechanical power were built, one with different PEEP levels and other with different RRs.
The airway pressure release ventilation (APRV) is used in up to 11% of ELSO centers. (6) Our group has used the APRV in patients with alveolar hemorrhage or in patients with collapsed lung who is potentially recruitable. The APRV is also explored in the supplementary results section.

Arterial oxygenation and total amount of oxygen transfer
We did the mathematical modeling considering the venous-venous configuration with the drainage cannula introduced in the femoral vein, and the arterial cannula introduced in the jugular vein. The oxygenation, oxygen transfer, decarboxylation, and carbon dioxide transfer modeling were explored using the following model: Figure 2S -Patient and extracorporeal membrane oxygenation coupling scheme of the mathematical modeling. In this figure, the blood flows, oxygen and carbon dioxide blood contents used in the mathematical modeling are depicted. FiO2 -fraction of inspired oxygen; PalvO2 -oxygen alveolar partial pressure; RR -respiratory rate; Vt -tidal volume; ECMO -extracorporeal membrane oxygenation; CpreO2 -pre-extracorporeal membrane oxygenation oxygen blood content; CpostO2 -post-extracorporeal membrane oxygenation oxygen blood content; CpreCO2 -pre-extracorporeal membrane oxygenation carbon dioxide blood content; CpostCO2 -post-extracorporeal membrane oxygenation carbon dioxide blood content; QECMO -extracorporeal membrane oxygenation blood flow; QEshunt -extracorporeal membrane oxygenation parallel cava blood flow (cava blood flow without extracorporeal membrane oxygenation); QCO -cardiac output; CvO2 -venous oxygen blood content; CRAO2 -right atrium oxygen content; CvCO2 -venous carbon dioxide blood content; CRACO2 -right atrium carbon dioxide content; Qrecirc -extracorporeal membrane oxygenation recirculation blood flow; CpostCO2 -postextracorporeal membrane oxygenation carbon dioxide blood content; QLungs -pulmonary blood flow in matched aerated alveoli/capillaries; CcO2 -pulmonary alveolar capillary blood oxygen content; CcCO2pulmonary alveolar capillary carbon dioxide blood content; QLshunt -pulmonary blood flow in shunt regions; VO2 -peripheral compartment oxygen consumption; VCO2 -peripheral compartment carbon dioxide production. VCO2 -peripheral compartment carbon dioxide production.
CaCO2 -arterial carbon dioxide blood content, CvCO2 -venous carbon dioxide blood content, CpreCO2pre-ECMO carbon dioxide blood content, CpostCO2 -post-ECMO carbon dioxide blood content, CRACO2 -right atrium carbon dioxide content, and CcCO2 -pulmonary alveolar capillary carbon dioxide blood content.  Where Hb is the hemoglobin level in g/dL and QCO is the cardiac output only here is expressed in L/minute.

Standard formulas used
(2) ECMO recirculation term Qrecirc has been mathematically described as a function of the QECMO. (12)(13)(14) Using the optimized femoral-jugular configuration, with a QECMO of 5000mL, it is expected a recirculation rate up to 20%. (12) During low QECMO rate, the Qrecirc varies linearly according to the QECMO, (14) however, at higher QECMO rate this relation of Qrecirc with QECMO is non-linear (exponential function). (14) Intuitively, the QCO is also a modulator of the recirculation once keeping QECMO stable; the higher the QCO, the higher will be the right ventricular drainage of the cannulas region and smaller the recirculation rate due to the ventricular drainage of the arterial cannula blood output. In this way, an extremely low QCO is responsible for almost 100% of arterial cannula blood recirculation. (14) As a higher QECMO is directly associated with a higher Qrecirc, and a higher QCO is associated with a lower Qrecirc, the equation was fitted using recirculation rate against the QECMO/QCO ratio.
The recirculation rate equation was created as a single exponential function (basic equation: f(x) = a * (1 -b x )), where f(x) is the recirculation rate and x is the QECMO/QCO ratio. Fitting all the data exposed above resulted in the following equation: Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support The recirculating blood flow carries CPostO2, therefore there will be not oxygenation of this amount of blood passing through the oxygenator. Thus, the effective QECMO (the partition of QECMO which will really be oxygenated) will be defined as: The QECMOeff plotted against the QECMO (with QCO simulated in 2000, 5000, and 10,000mL) is shown below: Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support Figure 5S -Association of extracorporeal membrane oxygenation blood flow with the effective ECMO blood flow (QECMOeff) in three different cardiac output.
(3) Post-oxygenator oxygen content (CPostO2) The FiO2 in the oxygenator sweep gas flow was considered 100%. Considering the oxygenator as efficient, the hemoglobin will ever be 100% saturated by oxygen. The PPostO2 at higher QECMO is expected to be lower than PPostO2 at lower QECMO. (15) The PPostO2 at QECMO = 5500 -6000mL/minute is expected to be ≥ 150mmHg, (16) and a QECMO as low as 2000mL/minute the PPostO2 is expected to be around 400mmHg, with a nonlinear PPostO2 decrement with progressive higher QECMO. (12,15) In order to fit this non- The oxygen transfer is defined as following: (15) Q ECMO (mL/minute) Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support The CPreO2 as a separated fraction from the recirculated blood is considered equal to CVO2. The simulation of O2 transfer as function of QECMO leads to the main result of this modeling, and is presented in the results section of the main manuscript.
(5) Right atrial oxygen venous content (CRAO2) The coronary sinus blood flow (1mL/kg/minute) was considered negligible to an adult. The CRAO2 is the mean of the CPostO2 and CVO2 pondered according to the blood flows, that is: The CaO2 was calculated based on the pulmonary shunt as following: QS/QT already includes the physiological shunt, mainly constituted by the bronchial circulation. The CcO2 is calculated based on the oxygen alveolar partial pressure: Where Patm is the barometric pressure (~ 690mmHg), RQ is the respiratory quotient (~ 0.8), and PH 2O was considered as 47mmHg.
Considering the partial capillary pressure of oxygen equal to the PAO2, and hemoglobin oxygen saturation of 100%, CCO2 was defined as: Combining those three last equations, the CaO2 was calculated as following: SatO2 was calculated as following: After this point, the mathematical circuit reentries, closing the loop.

Looping and iteractions to reach the equilibrium
For starting equilibrium cycle of mathematical circuit ( Figure 7S), the arterial saturation at the time of ECMO initiation (64%) was used. For any new variable tested, the circuit was recycled until reaching the equilibrium. In order to assure the equilibrium of the SatO2, twenty iteractions (cycles) were tested (see the appendix to check the script).   Figure 7S -Visual inspection of each step described above closing the iteractions. Twenty cycles (iteractions) were considered to reach the equilibrium. ECMO -extracorporeal membrane oxygenation; VO2 -oxygen consumption; Hb -hemoglobin; QCO -cardiac output; PaCO2 -partial pressure of carbon dioxide; FiO2ventilator -Ventilator fraction of inspired oxygen; QECMO -extracorporeal membrane oxygenation blood flow; SatO2 -oxygen saturation; CaO2-arterial content of oxygen; P A O 2 -alveolar oxygen partial pressure; FiO2 -fraction of inspired oxygen; P atm -atmospheric pressure; P H2Opartial pressure of water; RQ -respiratory coefficient; CcO2 -pulmonary alveolar capillary blood oxygen content; Q S /Q T -shunt fraction; CpostO2 -postextracorporeal membrane oxygenation oxygen blood content; Q ECMOeff -ECMO effective blood flow; CvO2 -venous oxygen blood content; QEshunt -extracorporeal membrane oxygenation parallel cava blood flow (cava blood flow without extracorporeal membrane oxygenation); QCO -cardiac output; CRAO2 -right atrium oxygen content; CpostO2 -post-extracorporeal membrane oxygenation oxygen blood content; S v O 2 -Oxygen mixed venous saturation; Qrecirc -extracorporeal membrane oxygenation recirculation blood flow; P Post O 2 -Oxygen partial pressure post-ECMO.  Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support The behavior of equilibrated values was compatible with ECMO-supported patients at the bedside.

Arterial carbon dioxide and total amount of carbon dioxide transfer
The CO2 modeling was also based on femoral -jugular venous-venous configuration. The patient -ECMO coupling was considered as in the figure 2S.

Standard formulas used
Additionally to the 1.2.1 item, the blood CO2 content was calculated using the

Creating mathematical terms of the carbon dioxide modeling
(1) The recirculation term was used as described in the 1.2.2 item.
As described in the methods section, the patient had a PaCO2 = 62mmHg just before ECMO initiation.
(2) Initial PvCO2 was modeled using a predetermined VCO2, which will add CO2 (CCO2added) to the arterial blood resulting in the CvCO2 (mL of CO2 / 100mL of blood).
Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support The VCO2 will be carried out by blood in a minute by a volume of blood equivalent to QCO. Therefore the CCO2added was calculated as following: CCO2added = VCO2 / (QCO * 10) Taking into account the CaCO2, calculated as already described in the 1.3.1 item, the CvCO2 was calculated as following: Still using the Douglas formula rearranged, (17) the PvCO2 was calculated as following: PvCO2 = Plasma CO2 / (2.226 * s * (1 + 10 (ph-pk) )) The venous pH was considered as 0.02 lower than the arterial pH, as described in septic patients. (18) Testing the consistency of PvCO2, the following graphs were built.  This value of PaCO2 re-enters the loop until stabilization.
In order to test the model consistency and stability, as in the oxygenation model, Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support

Oxygen partial pressure responsible for pulmonary vasoconstriction inhibition (PstimulusO2).
The pulmonary pressure during ARDS results from many factors, such as hypoxemia, hypercapnia, pH, cardiac output, and lung parenchyma collapse. The oxygen partial pressure responsible for modulation of pulmonary arterial pressure results from the alveolar (PAO2) and venous oxygen partial pressure (PvO2) as following: (19) PstimulusO2 = PAO2 0.68 + PvO2 0.32 The QPshunt was taken into account considering the PAO2 in shunt regions as zero due to alveolar collapse or absence of fresh air. In this way, the final PstimulusO2 was

RESULTS
This section shows complementary analyses using the mathematical models.  Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support  Oxygen delivery, carbon dioxide removal, energy transfer to lungs and pulmonary hypertension behavior during venous-venous extracorporeal membrane oxygenation support