Lung-gas composition and transfer analysis: O 2 and CO 2 diffusion coefficients and metabolic rates

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1 CHAPTER 3 Lung-gas composition and transfer analysis: O and CO diffusion coefficients and metabolic rates D.N. Ghista 1, K.M. Loh &D.Ng 3 1 School of Mechanical and Production Engineering, Nanyang Technological University, Singapore ; mdnghista@ntu,edu.sg School of Engineering Electronics, Nanyang Polytechnic, Singapore. 3 Department of Nuclear Medicine and PET, Singapore General Hospital, Singapore. Abstract The primary function of the lung is to i oxygenate the blood and thereby provide oxygen to the cells for metabolization purposes, and ii to remove the collected CO from the pulmonary blood. Herein, we will analyze the compositions of the inspired and expired air per breath, and from there compute the O consumption and CO production rates. Next, we derive expressions for diffusion coefficients D O and D CO in terms of the evaluated cardiac output, O and CO concentrations in arterial and venous blood, alveolar and blood O and CO partial pressures. We then take up a typical case study, and demonstrate the computation of D O and D CO, to represent the lung-performance capability to oxygenate the blood. 1 Introduction The lung-functional performance is characterized by i its ventilatory capacity, to bring air and hence O into the alveoli, and ii its capacity to transfer O and CO into and from the pulmonary capillary bed. Hence, the O and CO diffusion coefficients as well as the O consumption rate and the CO production rate represent the lung-performance indices. ISSN on-line WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press doi:10.495/ /03

2 78 Human Respiration Lung-air composition analysis and O consumption and CO production rates We carry out a mass-balance analysis, involving: i compositions of air breathed in and out ii consumption or losses of O,CO and H O. Table 1 provides clinical data on partial pressures and volumes of N,O,CO and H O of atmospheric air breathed in and expired out, one breath cycle. The monitored breathing rate BR 1 breaths/min, and we assume P H O at 37 C 47 mmhg. It can be noted that the expired air volume exceeds the inspired air volume for this particular breath cycle. The H O loss of 30.1ml ml contributes the major portion of this difference..1 Calculation of O consumption rate and CO production rate We now determine the O consumption rate and CO production rates from the inspired and expired gases. Assuming the patient breathes at 1 times per minute we he O Consumption Rate Inspired O Expired O ml/min Table 1: Inspired air composition and partial pressures. Atmospheric air Expired air Respiratory gases mmhg ml/% mmhg ml/% N % 74.5% O % 15.7% CO % 3.6% H O % 6.% Total % 100% WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

3 Lung-Gas Composition and Transfer Analysis 79 Dead space 150 ml Figure 1: Dead-space volume CO Production Rate Expired CO Inspired CO ml/min The amount of water vapor in the humidified expired air amounts to 6.% of the expired air compared to 0.49% of the dry inspired air corresponding to the partial-pressure ratio of water vapor in the expired air 47/760. The volume of the dry expired air ml 49.7 ml. Now, assume that out of 500 ml of inspired air, the dead-space air volume not taking part in the gas-transfer process is 150 ml and the alveolar air volume is 350 ml. We next compute the dead-space air volume composition.. Dead-space air composition The clinical data of expired air composition is: N 393.1ml O ml CO ml H O ml Total ml Now, the dead-space air will be made up of i a dry air portion from the inspired air assumed to be 141 ml, plus ii the water vapor taken up by the dry air WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

4 80 Human Respiration estimated to be 9 ml since the expired air portion of 141 ml will not he under gone O and CO transfer, its composition is the same as that of the inspired air: N 111 ml 78.55%, O 9.40 ml 0.84%, CO 0.06 ml 0.04%, H O 0.69 ml 0.49%. When this inspired air in the dead space of 141 ml is fully humidified, it will take up a further X ml of H O vapor, in the ratio of the partial-pressures, as: X X ml of H O vapor which is close to our estimate. So, by adding 9.9 ml of H O vapor to 0.69 ml of water vapor in the inspired air volume of 141 ml, the total water vapor in the dead-space air is 9.98 ml. The humified dead-space air composition will be: N ml 73.78% O 9.40 ml 19.55% CO 0.06 ml 0.04% H O 9.98 ml 6.63% Total ml.3 Alveolar-air composition and partial pressures We can now compute the alveolar air composition, by subtracting the dead-space air from the expired air. These values are tabulated in column 4 of the table below. Finally, we compute the partial pressure of O and CO as well as of N and H O, so that we can determine next the diffusion coefficients of O and CO based on the monitoring of arterial and venous blood concentrations. These values are tabulated in column 5 of the below table. Expired air Dead-space Alveolar air Alveolar-air partial ml air ml ml pressure mmhg N O CO H O Total WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

5 Lung-Gas Composition and Transfer Analysis 81 3 Lung gas-exchange model and parametric analysis 3.1 Expressions for D O and D CO The gas exchange between the alveolar air and pulmonary capillary blood is represented by the following O and CO conservation equations Fig. : Q VE C VE O Q AE CO AE + V O from the alveolar air to capillary blood DO ; P cap O PO AE, 1 Q AE CO AE + P O in which P cap O PO PRB O concentration of the preoxygenated blood Q VE C VE CO Q AE C AE CO V CO Q AE CCO AE P CO DCO ; P cap CO P VE CO, in which P cap CO PCO PRB CO concentration of the preoxygenated blood, wherein i Q AB and Q VB are arterial and venous blood flow-rates; Q AB Q VE at venous end, Q VB Q AE at arterial end ii PO al and P cap O are the alveolar and capillary O partial pressures iii PCO al and P cap CO are the alveolar and capillary CO partial pressure iv D O and D CO are the O and CO diffusion coefficients v P O erage of PO al P cap O over the capillary length; P CO erage of PCO al P cap O over the capillary length. Now we can equate the arterial and venous blood flow rates, as Q AB Q VB Q SV/EP CO/60, vi SV, EP and CO being the stroke volume, ejection period and cardiac output, respectively. Hence the above equations can be rewritten as: V O is the O transfer rate from alveolar air to capillary blood O consumption rate, V CO is the CO transfer rate from capillary blood to alveolar air. Figure : Schematic of blood-gas concentration in the pulmonary capillary. WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

6 8 Human Respiration From eqn. 1: Q VE C VE O From eqn. : wherein i ii QC VB O Q AB CO AB + P O QCO AB + P O DO DO ; P cap O P AE O P O D O QCVE O CO AE P O QCAB O CO VB P O. 3 Q VE C VE CO Q AE C AE Q, C VE O QC VE O CO P CO QCCO AE P CO DCO ; P cap CO P AE CO P VB CO DCO D CO QCVE CO CCO AB P CO. 4 and CO AE, CCO VE and CCO AE can be monitored because CO C AB and CCO AB and C AE and CCO AE C VB and C VB CO VE and C VE O O O CO D O and D CO eqns. 3 and 4 represent the lung gas-exchange parameters. Now from eqns. 3 and 4, if we want to evaluate the diffusion coefficients D O and D CO, we need to also express PO al, P cap O and PCO al, P cap CO in terms of monitorable quantities. In this regard, i Alveolar PO al can be expressed in terms of V the ventilation rate and V O the O consumption rate as Fig. 3: ] PO al k 1 1 e k V /V m/v O, 5 ii the O consump- CO VB. Equation where V m is the maximum ventilation rate and V O tion rate or absorption rate from the alveoli QCO AB 5 implies that as V /V m increases, the exponential term decreases, and P al O increases as in Fig. 3, and as V O increases P al O decreases as in Fig. 3. Alveolar P al CO can be expressed in terms of V and V O as in Fig. 4. P al CO k 3 e k 4 V /V m/v CO ], 6 where V CO the CO production rate or excretion rate from the blood QCCO VB CCO AB. This equation implies that as V /V m increases, PCO al decreases; also, as V CO increases the exponential term decreases, and hence PCO al increases. WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

7 Lung-Gas Composition and Transfer Analysis 83 Figure 3: Effect on alveolar P O of i alveolar ventilation, and ii rate of oxygen absorption from alveolar P O or O consumption rate from Guyton 1971, p. 476]. Figure 4: Effect on alveolar P CO of alveolar ventilation and rate of carbon dioxide excretion from the blood or CO production rate from Guyton 1971, p. 476]. iii Blood P O can be obtained in terms of blood CO, from the O disassociation curve providing concentrations in arterial and venous blood, is represented in Fig. 5 as: C O CO m 1 e k P O 5 P O m, or CO 1 e k 5PO, 7 WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

8 84 Human Respiration Figure 5: O dissociation curves, showing the total oxygen in each 100 ml of normal blood, the portion dissolved in the water of the blood from Guyton ], p. 485]. Figure 6: The carbon dioxide dissociation curve from Guyton ], p. 491]. iv where m and PO m are the maximum values of blood O partial pressure CO CO /CO m PO P O /PO m. Blood P CO can be obtained in terms of C CO, from the CO disassociation curve or CO concentration in arterial and venous blood can be represented WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

9 as per Fig. 6 as: Lung-Gas Composition and Transfer Analysis 85 C CO CO m 1 e k 6 P CO /PCO m or, C CO 1 e k 6 P CO /P m CO 3. Alveolar O and CO partial-pressure expressions 1 e k 6P CO. 8 Now, let us refer eqn. 4 for the PO al partial pressure curve Fig. 3, represented by the equation: PO al k 1 1 e k V /V / ] m V O k 1 1 e k ] V /V O, where V V V m 9 where V is the alveolar ventilation rate in liters/min, V m is the maximum ventilation rate 50 l/min and V O is the O consumption rate in liters/min. Herein, the coefficients k 1 and k can be determined by hing this equation match the Fig. 3 data. Note, in this equation, when V 0, PO al 0 from the equation, which satisfies the data. Now for V O 0.5 l/min, when V V V 0.5, PO al 140 mmhg. Hence, m 140 k 1 1 e k 0.5 ]] 0.5 k 1 1 e k. 10 Also, when V O 1l/min, V 0.3l/min, PO al 100 mmhg. Hence 100 k 1 1 e k 0.3 ]] 1 k 1 1 e 0.3k. 11 From eqns. 10 and 11, we get: k 11 e k k 1 1 e 0.3k 1 e k 1 e 0.3k e 0.3k e k or, e k + 140e 0.3k, so that k 4.18 min/l. 1 Upon substituting k 4.18 min/l into eqn. 10 we obtain: 140 k 1 1 e 4.18, so that k mmhg. 13 WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

10 86 Human Respiration Hence, the P al O curve can be represented by: O e 4.18 P al ] V /V O, 14 where, V O QCO AB CO VB and V V /50 l/min. Now, let us look at the PCO al expression: PCO al k 3 e k 4 V /V / m V ] CO k 3 e k 4 V /V ] CO We note from Fig. 4 that for V CO 0. l/min and V m 0., P al CO 1. Hence, from the above equation, we get: 1 k 3 e k 4 15 Also, for V O 0.8 l/min and V m 0., P al CO 6 mmhg. Hence 6 k 3 e k 4 From eqns. 15 and 16, we get: ] k 3 e k e k4 e e k 3 k ln 6 3 k 4, so that k Substituting k 4.46 into eqn. 16, we obtain: 6 k 3 e.46 4, k Hence, the P al CO curve can be represented as PCO al e.46 V /V / m V ] CO, 19 where V V / 50 l/min and V CO QC VB CO C AB CO. 3.3 Arterial and venous O and CO partial-pressure expressions We now need to express P AB O and PCO VB in terms of CO AB and CCO VB.. WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

11 Lung-Gas Composition and Transfer Analysis 87 So that let us look at the O disassociation curve, as shown in Fig. 5. C O C O max 1 e k / 5 ]] PO PO max, From Fig. 5, at P O Hence from eqn. 0: Also, P O Hence from eqn. 0: or, C O 1 e k 5P O, where CO C O, PO C P O. 0 O max P O max 40 mmhg 0.9 for normal venous blood, and 140 mmhg C O e 0.9k 5 k mmhg 0.68 for normal arterial blood, and 140 mmhg So, we take the erage value of k 5 : C O e 0.68k 5,or k k Then the O disassociation curve is given by: C O CO B 0. 1 e PO 140 ], 4 and P O ln CO B ] ] ln 0. CO B. 5 Finally, we look at the CO disassociation curve C CO C CO / max 1 e k 6 P CO P CO max, P CO or, CCO 1 e k 6 P CO max 1 e k 6PCO. 6 WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

12 88 Human Respiration Based on Fig. 6, when P CO 0 mmhg 140 mmhg 0.14, C CO , so that e 0.14k 6, k , 7 when PCO 70 mmhg 140 mmhg 0.5, C CO , so that 80 So, we take the erage value of k 6 : e 0.5k 6, k k Then the CO concentration is given from eqns. 6 9 by: PCO ] C CO CCO B e and P CO 9.7 ln C B CO ] Sequential procedure to compute D O and D CO 1 We first monitor: V t, V t, SV stroke volume, EPcardiac ejection period, CO VB, CO AB, CCO VB and CCO AB O and CO concentrations in pre oxygenated and post oxygenated blood. We substitute the values of CO AB CO VE and CO VB CO AE into eqn. 3, and the values of CCO AB CCO VE and CCO VB CCO AE into eqn We next determine: Q SV/ejection period, 3 V O t Q CO AB CO VB, 33 V CO t Q CCO VB CCO AB We then substitute the expressions for V O t and V CO t into the equations for PO al eqn. 14 and PCO al eqn We substitute the monitored values of CO VB CO AE and CCO VB CCO AE into eqns. 5 and 31, to obtain the values of P AE O O and P AE CO. 6 Now, in order to determine the values of the lung gas-exchange parameters D O and D CO, we substitute into eqns. 3 and 4 for Q from eqn. 3, PO al from eqn. 14, PCO al from eqn. 19, P VB from eqn. 6, and PCO VB from eqn. 31. WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

13 Lung-Gas Composition and Transfer Analysis Determining D O and D CO Figure 7 illustrates the variation of P O PO al P cap O length l of the capillary bed. Let l l / l m. Then we can express: P al O P AB O along the Then, P O PO max P O P O max f O l f O l dl P O max FO. 36 Based on data 3], since P O 1 mmhg for P O max 65 mmhg, we he F O We can similarly determine the erage value of P CO from Fig. 8 as: Let l l / l m. Then, we can represent Fig. 8 as: Then, P CO P CO max f O l. 37 P CO P O max 1 0 f CO l dl P CO max FCO. 38 Figure 7: Uptake of oxygen by the pulmonary capillary blood. The curve in this figure was constructed from data in Mihorn and Pulley: Biophys. J., 8: 337,1968. from Guyton 1971, p. 434.] WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

14 90 Human Respiration Figure 8: Diffusion of carbon dioxide from the pulmonary blood into the alveolus. The curve in this figure was constructed from data in Mihorn and Pulley: Biophys. J., 8: 337,1968. from Guyton 1971, p. 435.] Based on data 3], since P CO 0.5 mmhg for P CO max 5 mmhg, we he F CO 0.1. From the P O and P CO expressions, we can determine the O consumption and the CO production rates, as follows: D O Total O consumed P O D CO Total CO produced P CO 4 Case studies V O P O V CO P CO Q CO AB CO VB P O Q CCO VB C AB P CO CO A We monitor the partial pressures blood concentrations of O and CO as: C AE O C VB O 0.13, C VE O C VE CO C AB CO CO AB 0.18, CCO AE CCO VB 0.55, From eqn. 6, we obtain: ] ] PO VB ln 0. CO VB 9.7 ln mmhg. 41 WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

15 From eqn. 31, we obtain: P VB CO ln Lung-Gas Composition and Transfer Analysis C VB CO ] ] ln mmhg. 4 We now also monitor Q 5l/min, V 0.1 and V 5l/min. Then, from eqn. 33: so that from the above data, From eqn. 34: V O t Q C AB O CO VB, V O t ml O /min Consumption rate 43 V CO t Q CCO VB CCO AB Now, from eqn. 14. For V 0.1 and V O 0.5 l, we obtain PO al : PO al e 4.18 V 00 ml CO /min production rate. 44 /V O ], e /0.5]] mmhg. 45 From eqn. 19, for V 0.1 and V CO 0.0 l, we obtain PCO al : PO al e.19 V /V ] CO e.190.1/0.] mmhg. 46 Now, we can evaluate the diffusion coefficients: From eqns. 3, 36, 41, and 45: D O QCAB O P O C VB O ml O /min/mmhg. 47 WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

16 9 Human Respiration From eqn. 4: D CO QCVB CO CCO AB P CO ml CO /min/mmhg. 48 B Alternately, we derive data from: i the inspired and expired air analysis such as that carried out in Section.3: O consumption rate 83. ml/min, CO production rate 6.8 ml/min, PO al mmhg and PCO al mmhg and ii venous blood gas analysis: Then, as per eqn. 41, CO VB 0.13, CCO VB corresponding to CO VB 0.13 and, as per eqn. 4: ] PCO VB ln 0.8 CCO VB P VB O 31. mmhg, ln mmhg. 50 ] We obtain, from air-composition analysis, that V O t 83.3ml/min 51 Hence, and and V CO t 6.8ml/min. 5 D O V O P O mlo /min/mmhg, 53 D CO V CO P CO mlco /min/mmhg. 54 The advantage of this method B over A is that it does not require monitoring of the cardiac output, and is hence simpler to implement clinically. WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line

17 Lung-Gas Composition and Transfer Analysis 93 References 1] Guyton, A.C., M.D. Text Book of Medical Physiology, 8th edn, HBJ I.E. Saunders, ISBN , pp. 4 44, ] Comroe, J.H., Physiology of Respiration, nd edn, Year Book Medical Publishers Incorporated, ISBN , pp , ] Conney, D.O., Biomedical Engineering Principles, Marcel Dekker Inc., ISBN , pp , , WIT Transactions on State of the Art in Science and Engineering, Vol 4, 006 WIT Press ISSN on-line