 Research
 Open Access
 Published:
A mathematical model of CO_{2}, O_{2} and N_{2} exchange during venovenous extracorporeal membrane oxygenation
Intensive Care Medicine Experimental volume 6, Article number: 25 (2018)
Abstract
Background
Venovenous extracorporeal membrane oxygenation (vvECMO) is an effective treatment for severe respiratory failure. The interaction between the cardiorespiratory system and the oxygenator can be explored with mathematical models. Understanding the physiology will help the clinician optimise therapy. As others have examined O_{2} exchange, the main focus of this study was on CO_{2} exchange.
Methods
A model of the cardiorespiratory system during vvECMO was developed, incorporating O_{2}, CO_{2} and N_{2} exchange in both the lung and the oxygenator. We modelled lungs with shunt fractions varying from 0 to 1, covering the plausible range from normal lung to severe acute respiratory distress syndrome. The effects on P_{a}CO_{2} of varying the input parameters for the cardiorespiratory system and for the oxygenator were examined.
Results
P_{a}CO_{2} increased as the shunt fraction in the lung and metabolic CO_{2} production rose. Changes in haemoglobin and F_{I}O_{2} had minimal effect on P_{a}CO_{2}. The effect of cardiac output on P_{a}CO_{2} was variable, depending on the shunt fraction in the lung.
P_{a}CO_{2} decreased as extracorporeal circuit blood flow was increased, but the changes were relatively small in the range used clinically for vvECMO of > 2 l/min. P_{a}CO_{2} decreased as gas flow to the oxygenator rose and increased with recirculation. The oxygen fraction of gas flow to the oxygenator had minimal effect on P_{a}CO_{2}.
Conclusions
This mathematical model of gas exchange during vvECMO found that the main determinants of P_{a}CO_{2} during vvECMO were pulmonary shunt fraction, metabolic CO_{2} production, gas flow to the oxygenator and extracorporeal circuit recirculation.
Background
The use of extracorporeal respiratory support to manage severe respiratory failure is increasing [1]. When oxygenation is adequate and the main problem is hypercapnia, extracorporeal CO_{2} removal may be used, but if oxygenation is inadequate then venovenous extracorporeal membrane oxygenation (vvECMO) is required. Management of these patients requires a thorough understanding of vvECMO physiology, which is a complex physiological interaction between the cardiorespiratory system and the vvECMO circuit [2]. The main goal of vvECMO is to avoid hypoxia, but with consideration of minimising the important adverse effects associated with hyperoxia [3], hypercapnia and hypocapnia [4], all of which may occur during vvECMO. Understanding the factors that determine O_{2} exchange and CO_{2} exchange during vvECMO and safely manipulating the complex physiology is vital.
Clinical studies have examined O_{2} and CO_{2} exchange during vvECMO for respiratory failure [5]. Circuit blood flow was the main determinant of arterial oxygenation, while the sweep gas flow through the oxygenator was the main determinant of CO_{2} elimination. While clinical studies are important in exploring vvECMO physiology, mathematical modelling is an essential tool to extend understanding of physiological systems [6]. It enables study of dynamic scenarios that cannot be examined in clinical studies due to ethical considerations, the inability to isolate the effects of changing a single parameter due to reflex responses and the logistics of clinical studies in a limited number of patients.
Hollow fibre membrane oxygenators have become the standard of care for vvECMO [7]. Mathematical models of O_{2} and CO_{2} exchange in these devices have been developed [8, 9], and provide useful information about how they behave. However, these models only examine oxygenator function, and do not consider its interaction with the cardiorespiratory system, so their clinical utility is limited. The determinants of oxygenation during vvECMO have been explored with mathematical models that incorporate both an oxygenator and the cardiorespiratory physiology and their findings fit well with clinical studies [10, 11], but these models have not incorporated CO_{2} exchange.
The purpose of this work was to develop a new model of the cardiorespiratory system during vvECMO, which incorporated O_{2}, CO_{2} and N_{2} exchange, in both the oxygenator and the lung. It was used to assess the effects of a various physiological parameters and vvECMO circuit settings on vvECMO physiology.
Methods
Model description
A simple model of the cardiorespiratory system during vvECMO (Fig. 1) was implemented using Matlab R2017a (Mathworks, USA). A more detailed description of the model, including the mathematical equations underlying it, is given in the Additional file 1.
A threecompartment model of the lung, based on Riley and Cournard’s model [12], was used. One compartment was “shunt”, with perfusion but no ventilation, and did not participate in gas exchange. The second compartment was the “ideal lung” compartment, in which pulmonary gas exchange occurred. Diffusion equilibrium was assumed. The third compartment was “dead space”, with ventilation but no perfusion, and did not participate in gas exchange. Kelman’s subroutines [13,14,15] were used to calculate contents from partial pressures. The equations for gas exchange were based on those used by West and Wagner [16]. The model examines the system at equilibrium. Because of the presence of the oxygenator, N_{2} exchange across the lung was not assumed to be zero. Expired minute ventilation (\( {\dot{V}}_E \)) and expired alveolar ventilation (\( {\dot{V}}_A \)) were expressed at body temperature and pressure saturated (BTPS), and contents of gases dissolved in blood were expressed as standard temperature and pressure dry (STPD).
The oxygenator was modelled in a similar way to the lung, except that the shunt fraction and the dead space were set to zero, and recirculation was incorporated. Diffusion equilibrium was assumed in the “ideal oxygenator” compartment. Sweep gas flow to the oxygenator (\( {\dot{V}}_{\mathrm{SWEEP}} \)) was expressed at ambient temperature (24 °C) and pressure (760 mmHg) dry (ATPD).
Input parameters required by the model are total cardiac output in l/min (\( {\dot{Q}}_T \)), pulmonary shunt fraction \( \left(\frac{{\dot{Q}}_S}{{\dot{Q}}_T}\right) \), inspired oxygen fraction to the lung (F_{I}O_{2}), blood flow to the extracorporeal circuit in l/min (\( {\dot{Q}}_{EC}\Big) \), fraction of \( {\dot{Q}}_{EC} \) that is recirculated to the extracorporeal circuit \( \left(\frac{{\dot{Q}}_{RC}}{{\dot{Q}}_{EC}}\right) \), shunt fraction of the extracorporeal circuit \( \left(\frac{{\dot{Q}}_{s\ EC}}{{\dot{Q}}_{EC}}\right) \), oxygen fraction of gas flowing to the oxygenator (F_{OXY}O_{2}), sweep gas flow to the oxygenator in l/min ATPD (\( {\dot{V}}_{\mathrm{SWEEP}} \)), haemoglobin in g/dl (Hb), body temperature in degrees Celsius (temp), a factor that accounts for shifts in the O_{2} dissociation curve due to changes in 2,3 diphosphoglycerate (DP50), oxygen consumption in ml STPD/min (\( \dot{V}{\mathrm{O}}_2\Big) \) and respiratory quotient (RQ). The base excess was assumed to be zero. Haematocrit (Hct) was calculated as 3 × Hb / 100.
The program uses initial trial values of P_{pa}O_{2}, P_{pa}CO_{2} and P_{pa}N_{2} (gas partial pressures in pulmonary arterial blood). The principle of conservation of mass for O_{2}, CO_{2} and N_{2} allows the partial pressures and contents in each of the domains of the model to be calculated sequentially, including calculated estimates of P_{pa}O_{2}, P_{pa}CO_{2} and P_{pa}N_{2}, that follow from the trial values. The mathematical problem is considered solved when trial values of P_{pa}O_{2}, P_{pa}CO_{2} and P_{pa}N_{2} have been found that generate calculated estimates that are all within 0.001 mmHg of the trial values. The problem is essentially that of finding the root in three dimensions of a system of nonlinear equations which is solved using the false position method, in three dimensions.
To examine the physiological interactions between vvECMO settings and the cardiorespiratory system during vvECMO, the lung in the model was set up to mimic a patient with varying severity of acute respiratory distress syndrome (ARDS). The approach taken was based on Gattonini’s concept of the ARDS lung consisting of a “baby lung” with near normal specific compliance and dependent areas of atelectasis the size of which corresponds to the shunt fraction [17]. Contemporary approaches to mechanical ventilation limit the distending pressure the lung is exposed to, so \( {\dot{V}}_A \) falls, as the amount of atelectasis and corresponding pulmonary shunt fraction increases. This was modelled by keeping the ventilation perfusion ratio of the baby lung \( \left(\frac{{\dot{V}}_A}{{\dot{Q}}_L}\right) \) at a constant value, mimicking the clinical situation where \( {\dot{V}}_A \) falls and \( \frac{V_D}{V_T} \) rises with worsening ARDS. To achieve this, \( {\dot{V}}_D \) was maintained at a constant value, and \( {\dot{V}}_E \) varied to achieve the desired value of \( \frac{{\dot{V}}_A}{{\dot{Q}}_L} \).
The \( {\dot{V}}_E \) required to give a P_{a}CO_{2} of 40 mmHg, when the lung had no shunt and \( \frac{V_D}{V_T} \) was 0.3, was initially calculated. In the standard ARDS lung, this value of \( {\dot{V}}_E \) was 6.157 l/min, \( {\dot{V}}_D \) was 1.847 l/min and the corresponding ventilationperfusion ratio of perfused lung \( \left(\frac{{\dot{V}}_A}{{\dot{Q}}_L}\right) \) was 0.7183. In keeping with the baby lung concept, \( \frac{{\dot{V}}_A}{{\dot{Q}}_L} \) was maintained constant at 0.7183, while \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) was varied between 0.5 and 1. \( {\dot{V}}_D \) was maintained constant at 1.847 l/min allowing the \( \frac{V_D}{V_T} \) and \( {\dot{V}}_E \) for any given scenario to be calculated. Total atelectasis of the lung is common during lung protective ventilation on vvECMO for severe ARDS, and this is represented in the model when \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) is 1.0.
Where the model would require \( {C}_{\overline{v}}{\mathrm{O}}_2 \) < 0 for a solution, this was considered physiological untenable and the program returned an error warning and no results were output. Results are presented with the upper and lower values on the x axis scale set to encompass the x axis values tested, so missing data are due to an untenable result.
Scenarios examined with the model
First, a validation was performed by comparing the findings of the model about oxygenation to those of Zanella’s model [11]. Input parameters were set to those in Zanella’s paper, and \( {\dot{V}}_{\mathrm{SWEEP}} \) adjusted to obtain P_{a}CO_{2} of 40 mmHg. The same output parameters were examined, enabling validation by reproducing the graphs from Zanella’s paper.
For all subsequent modelling, unless otherwise stated, F_{I}O_{2} 1.0, F_{OXY}O_{2} 1.0, \( {\dot{Q}}_T \) 6 l/min, \( \dot{V}{\mathrm{O}}_2 \) 250 ml/min STPD, Hb 10 g/dl, DP50 0, \( \frac{{\dot{Q}}_{s\ EC}}{{\dot{Q}}_{EC}} \) 0, \( \frac{{\dot{Q}}_{RC}}{{\dot{Q}}_{EC}} \) 0, temp 37 °C, RQ 0.8 and\( {\dot{V}}_D \) 1.847 l/min were used as input parameters. \( {\dot{Q}}_{EC} \), \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) and \( {\dot{V}}_{\mathrm{SWEEP}} \) were set and varied as required for the scenario being simulated. The derived parameters \( \frac{{\dot{V}}_A}{{\dot{Q}}_L} \) and \( {\dot{V}}_D \) were maintained constant at 0.7183 and 1.847 l/min. The output of the program for each scenario was written to a .csv file (these have been combined into the Additional file 2).
Second, the effect on oxygenation of different approaches to managing \( {\dot{V}}_{\mathrm{SWEEP}} \) was investigated. The standard ARDS lung was used with \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) ranging from 0.5 to 1.0. Three ways in which \( {\dot{V}}_{\mathrm{SWEEP}} \) could be managed as \( {\dot{Q}}_{EC} \) was varied were examined: (a) \( {\dot{V}}_{\mathrm{SWEEP}} \) adjusted to maintain arterial P_{a}CO_{2} = 40 mmHg; (b) \( {\dot{V}}_{\mathrm{SWEEP}} \) fixed at 2.5, 5, 10 and 15 l/min; and (c) \( {\dot{V}}_{\mathrm{SWEEP}} \) adjusted to maintain \( \frac{{\dot{V}}_{\mathrm{SWEEP}}\ }{{\dot{Q}}_{EC}} \) constant.
Third, the factors that affect P_{a}CO_{2} in the patient on vvECMO were studied using the standard ARDS lung.
Finally, the sensitivity of the output from the model to the choice of parameters set in the model lung was examined. The standard ARDS lung model was chosen to mimic a lung where the shunt fraction parallels the development of atelectactic lung, which is perfused but unventilated. \( {\dot{V}}_A \) is proportional to the amount of nonatelectatic lung.
The other extreme is a lung in which \( {\dot{V}}_A \) remains constant regardless of \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) (constant \( {\dot{V}}_A \) model). \( \frac{{\dot{V}}_A}{{\dot{Q}}_L}=\frac{{\dot{V}}_A}{{\dot{Q}}_T\left(1\frac{{\dot{Q}}_S}{{\dot{Q}}_T}\right)} \). The constant \( {\dot{V}}_A \) model was modelled with \( {\dot{V}}_A \) maintained constant at 4.310 l/min. This was the \( {\dot{V}}_A \) that gave P_{a}CO_{2} of 40 mmHg when the lung had no shunt. The effect on P_{a}CO_{2} of changing \( {\dot{Q}}_{EC} \), over a range of \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \), was examined with both the standard ARDS lung and the constant \( {\dot{V}}_A \) model, and the results of the two models compared.
Results
First, when the validation was performed with input parameters similar to Zanella’s model [11], the model produced almost identical graphical output (see the Additional file 3). This provided independent verification of Zanella’s model, and validation of the more complex mathematical approach that we have taken.
Second, when the effect on oxygenation of different approaches to managing \( {\dot{V}}_{\mathrm{SWEEP}} \) was investigated, the way in which \( {\dot{V}}_{\mathrm{SWEEP}} \) was managed as \( {\dot{Q}}_{EC} \) varied had minimal effect on oxygenation (Fig. 2 and Additional file 4: Figure S1). As \( {\dot{Q}}_{EC} \) increased, S_{a}O_{2} and \( {\mathrm{S}}_{\overline{\mathrm{v}}}{\mathrm{O}}_2 \) rose. The higher the \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \), the higher the \( {\dot{Q}}_{EC} \) to achieve “acceptable” values of S_{a}O_{2} > 85% and \( {\mathrm{S}}_{\overline{\mathrm{v}}}{\mathrm{O}}_2 \) > 60%. For all these scenarios, at any given \( {\dot{Q}}_{EC} \) and \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \), the output S_{a}O_{2} and \( {\mathrm{S}}_{\overline{\mathrm{v}}}{\mathrm{O}}_2 \) did not differ by more than 1.5 (% oxygen saturation) from the S_{a}O_{2} and \( {\mathrm{S}}_{\overline{\mathrm{v}}}{\mathrm{O}}_2 \) output in the simulation that held P_{a}CO_{2} constant at 40 mmHg. Output C_{a}O_{2} did not differ by more than 0.2 (ml STPD/100 ml blood). The biggest difference in P_{a}O_{2} was 19 mmHg, which occurred at high \( {\dot{Q}}_{EC} \) when P_{a}O_{2} was > 500 mmHg.
Third, the factors with important effects on P_{a}CO_{2} were \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \), \( {\dot{Q}}_{EC} \), \( {\dot{V}}_{\mathrm{SWEEP}} \), \( \dot{V}{\mathrm{CO}}_2 \) and \( \frac{{\dot{Q}}_{RC}}{{\dot{Q}}_{EC}} \). Higher \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) increased P_{a}CO_{2} (Fig. 3). Increases in \( {\dot{Q}}_{EC} \) reduced P_{a}CO_{2}, but for \( {\dot{Q}}_{EC} \) > 2 l/min, the changes were relatively small (Fig. 3), though they became larger with high \( \dot{V}{\mathrm{CO}}_2 \) (Fig. 4). Increases in \( {\dot{V}}_{\mathrm{SWEEP}} \) reduced P_{a}CO_{2} (Fig. 5). The \( {\dot{V}}_{\mathrm{SWEEP}} \) required to maintain P_{a}CO_{2} at 40 mmHg fell as \( {\dot{Q}}_{EC} \) increased, but for \( {\dot{Q}}_{EC} \) > 2 l/min, the changes were relatively small (Fig. 6). There was a linear increase in P_{a}CO_{2} with \( \dot{V}{\mathrm{CO}}_2 \) (Fig. 4). P_{a}CO_{2} rose and S_{a}O_{2} fell as recirculation increased (Fig. 7). At high \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \), when the recirculation was high, S_{a}O_{2} fell to physiological untenable levels (Additional file 5: Figure S2).
Changes in Hb, F_{I}O_{2} and F_{OXY}O_{2}, had minimal effect on P_{a}CO_{2}, despite the important effects these parameters have on oxygenation (Additional file 6: Figure S3, Additional file 7: Figure S4, and Additional file 8: Figure S5). The effect of \( {\dot{Q}}_T \) on P_{a}CO_{2} was variable, depending on the shunt fraction (Additional file 9: Figure S6).
Finally, the sensitivity of the model to the parameters set in the model lung was studied. With both the constant \( {\dot{V}}_A \) lung model and the standard ARDS lung model, increases in \( {\dot{Q}}_{EC} \) produced a fall in P_{a}CO_{2}, but for \( {\dot{Q}}_{EC} \) > 2 l/min, the changes were relatively small. The results for \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) of 0, and for \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) of 1 were identical, but the curves for \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) between these values were in different positions (Additional file 10: Figure S7).
Discussion
Previous mathematical models of gas exchange during vvECMO have only examined O_{2} exchange [10, 11], or made unrealistic assumptions about CO_{2} exchange [18]. They are unable to assess the determinants of CO_{2} exchange. To overcome this limitation, a mathematical model of gas exchange during vvECMO, that incorporated O_{2}, CO_{2} and N_{2} exchange, was developed. The effect of input physiological parameters and vvECMO circuit settings on oxygenation was compared to the findings of Zanella’s model [11], which only included O_{2} exchange. Our model closely reproduced the outputs of Zanella’s model and can be reviewed in the Additional files. This provides validation of the more complex mathematical approach presented here. This validation is limited to the effects on oxygenation, as Zanella did not examine CO_{2} exchange. Zanella’s simpler approach would be preferred if only O_{2} exchange was of interest, but it cannot be used if CO_{2} exchange is to be examined.
The major advantage of this model over previous models is the incorporation of CO_{2} exchange, which allows the effect of input physiological parameters and vvECMO circuit settings on P_{a}CO_{2} to be assessed. The key finding of our paper is that providing \( {\dot{Q}}_{EC} \) ≥ 2 l/min, the major determinants of P_{a}CO_{2} are \( \dot{V}{\mathrm{CO}}_2 \), \( {\dot{V}}_{\mathrm{SWEEP}} \) and \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T}. \) This provides guidance to the clinician who wishes to control P_{a}CO_{2}. \( {\dot{V}}_{\mathrm{SWEEP}} \) can be adjusted by changing the sweep gas flow rate on the oxygenator, and \( \dot{V}{\mathrm{CO}}_2 \) can be modified with sedation, paralysis and temperature control. \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) is more dependent on the underlying lung pathology. Manipulation of Hb, F_{I}O_{2}, F_{OXY}O_{2} and \( {\dot{Q}}_T \) will have minimal effect on P_{a}CO_{2}.
While ultraprotective ventilation strategies have been recommended for patients on vvECMO [19], practices vary widely between centres [20]. The effect of different ventilation strategies was not studied with the model. Unless \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) is 1, differences in minute ventilation that modify \( {\dot{V}}_A \) will directly affect P_{a}CO_{2}. Differences in positive end expiratory pressure may affect lung recruitment and alter \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \).
Schmidt et al. observed in vivo the determinants of oxygenation and decarboxylation during vvECMO in ten adult patients with respiratory failure [5]. When \( {\dot{Q}}_{EC} \) was reduced from a baseline of 5.8 ± 0.8 l/min by 40% to 2.4 ± 0.3 l/min, P_{a}O_{2} and S_{a}O_{2} fell, but there was no change in P_{a}CO_{2}. When maintaining \( {\dot{Q}}_{EC} \) at baseline values, reducing F_{OXY}O_{2} from 1.0 to 0.4 resulted in a fall in P_{a}O_{2} and S_{a}O_{2}, but no change in P_{a}CO_{2}. In contrast, when the sweep gas flow rate reduced from 10 to 2 l/min, P_{a}O_{2} did not change, and P_{a}CO_{2} rose. In the three patients given a red blood cell transfusion, there was a rise in P_{a}O_{2} (no data were presented for P_{a}CO_{2}). These findings are consistent with the predictions of the physiology made by the mathematical model, providing clinical validation of the model.
The model has several limitations. Gas exchange in the oxygenator was modelled as perfusion limited, which is not always the case with a real oxygenator. When the surface area of a membrane lung is low and blood flow is high, CO_{2} elimination is limited by diffusion. As the surface area increases, diffusion limitation becomes less important and CO_{2} elimination becomes predominantly perfusion limited [21]. O_{2} uptake increases linearly with blood flow in the Quadrox oxygenator, up to the rated maximum \( {\dot{Q}}_{EC} \) of 7 l/min [9], though this is not the case for all oxygenators. The assumption of predominantly perfusion limited gas exchange is unlikely to result in major inaccuracy during vvECMO, providing a suitable oxygenator is used. Oxygenator performance deteriorates with time, due to deposition of protein and clot formation on the surface exposed to blood, and water accumulation on the surface exposed to gas [22]. With longterm use, the findings of the model may become less applicable as the oxygenator performance deteriorates. Input parameters of the model were treated as independent variables, which is not true in vivo. For example, cardiac output changes in response to anaemia, hypercapnia or hypoxia. \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) changes in response to P_{pa}O_{2} which affects pulmonary vascular tone. Despite these limitations, the findings of the model are in keeping with in vivo data [5].
The model does not include a dead space compartment in the oxygenator. Castagna et al. studied oxygenators after several days of vvECMO (4.0 days, IQR 2.0–8) and found \( \frac{{\dot{V}}_D}{{\dot{V}}_T} \) of 47.8 ± 15.3 (mean ± sd) [22]. This finding fits the clinical observation that the sweep gas flow rate required to maintain normocapnia often gradually increases during an ECMO run. As dead space develops, the same gas transfer across the oxygenator can be maintained by increasing \( {\dot{V}}_{\mathrm{SWEEP}} \).
The findings presented in this paper relate to the ARDS patient with high \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \), that requires \( {\dot{Q}}_{EC} \) ≥ 2 l/min to support oxygenation. During extracorporeal CO_{2} removal, \( {\dot{Q}}_{EC} \) is usually < 2 l/min, and \( {\dot{Q}}_{EC} \) becomes a major determinant of CO_{2} elimination. Gas exchange across the membranes used for extracorporeal CO_{2} removal is often limited by incomplete diffusion [21], and this may be exacerbated by the high gas sweep to blood flow ratios that are used to facilitate CO_{2} removal. A range of techniques have been tried to enhance CO_{2} removal including using carbonic anhydrase [23], acidification of blood or dialysate [24] and electrodialysis [25]. Further modelling of the factors affecting gas exchange during extracorporeal CO_{2} removal may prove enlightening.
Consider a scenario where a patient that was stable on ECMO becomes febrile and hypoxic. As the metabolic rate increased with the fever, the sweep gas was increased to maintain normal P_{a}CO_{2}. To correct the hypoxia, it is planned to increase \( {\dot{Q}}_{EC} \) from 2.5 to 5 l/min. What should be done with \( {\dot{V}}_{\mathrm{SWEEP}} \) to maintain normal P_{a}CO_{2}? The model predicts that with normal \( \dot{V}{\mathrm{CO}}_2 \), changes in \( {\dot{Q}}_{EC} \) would have minimal effect on P_{a}CO_{2} providing \( {\dot{Q}}_{EC} \) is > 2 l/m, so no change in \( {\dot{V}}_{\mathrm{SWEEP}} \) would be required. This is consistent with standard teaching on managing ECMO. However, in this scenario, \( \dot{V}{\mathrm{CO}}_2 \) is elevated, and the model predicts that increasing \( {\dot{Q}}_{EC} \) will increase CO_{2} elimination, so the clinician will need to reduce \( {\dot{V}}_{\mathrm{SWEEP}} \) to maintain a normal P_{a}CO_{2}. Another scenario where the model may be of assistance to the clinician is when a patient is anaemic on ECMO and requires blood transfusion to maintain adequate oxygenation. What should be done with \( {\dot{V}}_{\mathrm{SWEEP}} \) to maintain normal P_{a}CO_{2}? Despite the importance of Hb in CO_{2} transport in blood, the model predicts that during vvECMO, even large changes in Hb will have minimal effect on P_{a}CO_{2}. No change in \( {\dot{V}}_{\mathrm{SWEEP}} \) is required. The emphasis of this paper has been on CO_{2} exchange during vvECMO, but the model is equally applicable to O_{2} exchange. It supports clinical decision making by improving the clinician’s understanding of vvECMO physiology.
Conclusion
A mathematical model of gas exchange during vvECMO, incorporating O_{2}, CO_{2} and N_{2} exchange, was developed. The results of the model were consistent with previous mathematical models that only examined O_{2} exchange and predict the findings of clinical studies. The main determinants of P_{a}CO_{2} during vvECMO were lung \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \), metabolic CO_{2} production, \( {\dot{V}}_{\mathrm{SWEEP}} \) and extracorporeal circuit recirculation.
Abbreviations
 \( \dot{V}{\mathrm{O}}_2 \) :

O_{2} consumption in ml STPD/min
 \( \dot{V}{\mathrm{CO}}_2 \) :

CO_{2} production in ml STPD/min
 C_{a}O_{2} :

Arterial oxygen content in ml/dl blood STPD
 S_{a}O_{2} :

Arterial oxygen saturation
 P_{a}CO_{2} :

Arterial partial pressure of CO_{2} in mm Hg
 P_{a}O_{2} :

Arterial partial pressure of O_{2} in mm Hg
 \( {\dot{Q}}_{\mathrm{EC}} \) :

Blood flow to the extracorporeal circuit in l/min
 \( {\dot{Q}}_{\mathrm{T}} \) :

Cardiac output in l/min
 \( {\dot{Q}}_{\mathrm{P}} \) :

Blood flow to the lung compartment in l/min
 \( {\dot{Q}}_{\mathrm{S}} \) :

Pulmonary shunt blood flow in l/min
 \( \frac{V_D}{V_T} \) :

Dead space to tidal volume ratio
 \( {\dot{V}}_D \) :

Dead space ventilation of the lung in l/min BTPS
 \( {\dot{V}}_A \) :

Expired alveolar ventilation of the lung in l/min BTPS
 \( {\dot{V}}_E \) :

Expired minute ventilation of the lung in l/min BTPS
 \( \frac{{\dot{V}}_A}{{\dot{Q}}_L} \) :

Expired ventilation perfusion ratio of the lung
 \( \frac{{\dot{Q}}_{RC}}{{\dot{Q}}_{EC}} \) :

Fraction of \( {\dot{Q}}_{EC} \) that is recirculated to the extracorporeal circuit
 F_{I}O_{2} :

Inspired oxygen fraction of the lung
 F _{OXY}O_{2} :

Inspired oxygen fraction of the oxygenator
 _{CvO2} :

Mixed venous oxygen content in ml/dl blood STPD
 \( {S}_{\overline{v}}{O}_2 \) :

Mixed venous oxygen saturation
 P_{pa}CO_{2} :

Partial pressure of CO_{2} in the pulmonary artery
 P_{pa}N_{2} :

Partial pressure of N_{2} in the pulmonary artery
 P_{pa}O_{2} :

Partial pressure of O_{2} in the pulmonary artery
 \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \) :

Pulmonary shunt fraction
 \( \frac{{\dot{V}}_D}{{\dot{V}}_T} \) :

Ratio of dead space ventilation to total ventilation
 \( \frac{{\dot{Q}}_{s\ EC}}{{\dot{Q}}_{EC}} \) :

Shunt fraction of the extracorporeal circuit
 \( {\dot{V}}_{\mathrm{SWEEP}} \) :

Sweep gas flow rate to the oxygenator in l/min ATPD
 ARDS:

Acute respiratory distress syndrome
 ATPD:

Ambient temperature and pressure dry
 BTPS:

Body temperature and pressure saturated
 DP50:

Factor that accounts for shifts in the O_{2} dissociation curve due to changes in 2,3 diphosphoglycerate
 Hb:

Haemoglobin in g/dl
 Hct:

Haematocrit
 STPD:

Standard temperature and pressure dry
 temp:

Body temperature in °C
 vvECMO:

Venovenous extracorporeal membrane oxygenation
References
 1.
Thiagarajan RR, Barbaro RP, Rycus PT, McMullan DM, Conrad SA, Fortenberry JD, Paden ML (2017) Extracorporeal life support organization registry international report 2016. ASAIO J 63:60–67
 2.
Brogan TV, Lequier L, Lorusso R, MacLaren G, Peek G (2017) Physiology of extracorporeal life support. In: Bartlett RH, Conrad SA (eds) Extracorporeal life support: the ELSO red book, 5th edn. Extracorporeal Life Support Organization, Ann Arbor
 3.
Helmerhorst HJ, Schultz MJ, van der Voort PH, de Jonge E, van Westerloo DJ (2015) Benchtobedside review: the effects of hyperoxia during critical illness. Crit Care 19:284. https://doi.org/10.1186/s1305401509964
 4.
Eastwood GM, Young PJ, Bellomo R (2014) The impact of oxygen and carbon dioxide management on outcome after cardiac arrest. Curr Opin Crit Care 20:266–272
 5.
Schmidt M, Tachon G, Devilliers C, Muller G, Hekimian G, Brechot N, Merceron S, Luyt CE, Trouillet JL, Chastre J, Leprince P, Combes A (2013) Blood oxygenation and decarboxylation determinants during venovenous ECMO for respiratory failure in adults. Intensive Care Med 39:838–846
 6.
Glynn P, Unudurthi SD, Hund TJ (2014) Mathematical modeling of physiological systems: an essential tool for discovery. Life Sci 111:1–5
 7.
Yeager T, Roy S (2017) Evolution of gas permeable membranes for extracorporeal membrane oxygenation. Artif Organs 41:700–709
 8.
Hany Hazfiza M, Ahmad Khairi Abdul W, Fathiah Mohamed Z (2017) Mathematical modelling of carbon dioxide exchange in hollow fiber membrane oxygenator. IOP Conference Series: Materials Science and Engineering 210. https://doi.org/10.1088/1757899X/210/1/012003
 9.
Tabesh H, Amoabediny G, Poorkhalil A, Khachab A, Kashefi A, Mottaghy K (2012) A theoretical model for evaluation of the design of a hollowfiber membrane oxygenator. J Artif Organs 15:347–356
 10.
Spinelli E, Bartlett RH (2014) Relationship between hemoglobin concentration and extracorporeal blood flow as determinants of oxygen delivery during venovenous extracorporeal membrane oxygenation: a mathematical model. ASAIO J 60:688–693
 11.
Zanella A, Salerno D, Scaravilli V, Giani M, Castagna L, Magni F, Carlesso E, Cadringher P, Bombino M, Grasselli G, Patroniti N, Pesenti A (2016) A mathematical model of oxygenation during venovenous extracorporeal membrane oxygenation support. J Crit Care 36:178–186
 12.
Riley RL, Cournand A (1949) Ideal alveolar air and the analysis of ventilationperfusion relationships in the lungs. J Appl Physiol 1:825–847
 13.
Kelman GR (1966) Digital computer subroutine for the conversion of oxygen tension into saturation. J Appl Physiol 21:1375–1376
 14.
Kelman GR (1966) Calculation of certain indices of cardiopulmonary function, using a digital computer. Respir Physiol 1:335–343
 15.
Kelman GR (1967) Digital computer procedure for the conversion of PCO2 into blood CO2 content. Respir Physiol 3:111–115
 16.
West JB, Wagner PD (1977) Pulmonary gas exchange. In: West JB (ed) Bioengineering Aspects of the Lung. edn. Marcel Decker, New York
 17.
Gattinoni L, Marini JJ, Pesenti A, Quintel M, Mancebo J, Brochard L (2016) The "baby lung" became an adult. Intensive Care Med 42:663–673
 18.
Seear M, Anderson B, Hall R, Hui H (1995) Mathematical model of oxygen transport: a teaching aid for normal physiology adaptable to extracorporeal oxygenation circuits. Am J Phys 268:S32–S39
 19.
Schmidt M, Pellegrino V, Combes A, Scheinkestel C, Cooper DJ, Hodgson C (2014) Mechanical ventilation during extracorporeal membrane oxygenation. Crit Care 18:203
 20.
Marhong JD, Telesnicki T, Munshi L, Del Sorbo L, Detsky M, Fan E (2014) Mechanical ventilation during extracorporeal membrane oxygenation. An international survey. Ann Am Thorac Soc 11:956–961
 21.
Karagiannidis C, Strassmann S, Brodie D, Ritter P, Larsson A, Borchardt R, Windisch W (2017) Impact of membrane lung surface area and blood flow on extracorporeal CO2 removal during severe respiratory acidosis. Intensive Care Med Exp 5:34
 22.
Castagna L, Zanella A, Scaravilli V, Magni F, Deab SA, Introna M, Mojoli F, Grasselli G, Pesenti A, Patroniti N (2015) Effects on membrane lung gas exchange of an intermittent high gas flow recruitment maneuver: preliminary data in venovenous ECMO patients. J Artif Organs 18:213–219
 23.
Arazawa DT, Kimmel JD, Finn MC, Federspiel WJ (2015) Acidic sweep gas with carbonic anhydrase coated hollow fiber membranes synergistically accelerates CO2 removal from blood. Acta Biomater 25:143–149
 24.
Zanella A, Mangili P, Giani M, Redaelli S, Scaravilli V, Castagna L, Sosio S, Pirrone F, Albertini M, Patroniti N, Pesenti A (2014) Extracorporeal carbon dioxide removal through ventilation of acidified dialysate: an experimental study. J Heart Lung Transplant 33:536–541
 25.
Zanella A, Castagna L, Abd El Aziz El Sayed Deab S, Scaravilli V, Ferlicca D, Magni F, Giani M, Salerno D, Casati M, Pesenti A (2016) Extracorporeal CO2 removal by respiratory Electrodialysis: an in vitro study. ASAIO J 62: 143–149
Availability of data and materials
The datasets supporting the conclusions of this article are included within the article and its additional files. See the file “Explanation of supplementary material.docx” for details.
Author information
Affiliations
Contributions
CJ and DC developed the mathematical model. CJ did the programming, ran the simulations, formatted the results and wrote the paper. CJ, DC and KS revised the manuscript and intellectual content. All authors gave their approval to the final version of the manuscript to be published. All authors read and approved the final manuscript.
Corresponding author
Correspondence to Christopher John Joyce.
Ethics declarations
Authors’ information
CJ is Director of Intensive Care at Princess Alexandra Hospital, and Associate Professor at University of Queensland. Qualifications are MB, ChB, PhD, FCICM, FANZCA. DC is Senior Staff Specialist in Intensive Care at Princess Alexandra Hospital, Adjunct Professor at Queensland University of Technology, and Associate Professor at University of Queensland. Qualifications are MB, BS, PhD, FCICM, FANZCA. KS is Senior Staff Specialist in Intensive Care at The Prince Charles Hospital. Qualifications are MB, BS, PhD, FCICM.
Ethics approval and consent to participate
“Not applicable”
Consent for publication
“Not applicable”
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Additional files
Additional file 1:
Description of the ECMO model and the mathematical equations on which it is based. (PDF 205 kb)
Additional file 2:
This file contains the output of the program for the scenarios modelled. Each worksheet contains the output for a single scenario. (XLSX 236 kb)
Additional file 3:
This file contains the output of the program for the validation against Zanella’s model. Each worksheet contains the output of the program and graphs generated from this output, corresponding to one of Zanella’s figures. (XLSX 155 kb)
Additional file 4:
Figure S1. Shows the “Effect on \( {S}_{\overline{v}}{O}_2 \) of different approaches to managing \( {\dot{V}}_{SWEEP} \)”. (PDF 77 kb)
Additional file 5:
Figure S2. Shows the “Effect of recirculation at high \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \)”. (PDF 67 kb)
Additional file 6:
Figure S3. Shows the “Effect on P_{a}CO_{2} of Hb”. (PDF 74 kb)
Additional file 7:
Figure S4. Shows the “Effect of F_{I}O_{2} on P_{a}CO_{2}”. (PDF 74 kb)
Additional file 8:
Figure S5. Shows the “Effect of F_{OXY}O_{2} on P_{a}CO_{2}”. (PDF 76 kb)
Additional file 9:
Figure S6. Shows the “Effect of changes in \( {\dot{Q}}_T \) on P_{a}CO_{2}, over a range of \( \frac{{\dot{Q}}_S}{{\dot{Q}}_T} \)”. (PDF 72 kb)
Additional file 10:
Figure S7. Shows the “Sensitivity to which lung model is used”. (PDF 88 kb)
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Joyce, C.J., Shekar, K. & Cook, D.A. A mathematical model of CO_{2}, O_{2} and N_{2} exchange during venovenous extracorporeal membrane oxygenation. ICMx 6, 25 (2018) doi:10.1186/s4063501801834
Received
Accepted
Published
DOI
Keywords
 Extracorporeal membrane oxygenation
 ECMO
 Venovenous ECMO
 Extracorporeal CO_{2} removal
 Gas exchange
 Mathematical model
 Respiratory failure
 Acute respiratory distress syndrome