An animated, non-compensating pulmonary model for teaching ventilation and perfusion relationships to medical students

Math/ Compul. Modeiling, Vol.1I,pp.823-827,1988

08957177/88$3.00+0.00 PergamonPressplc

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AN ANtMATFXJ NON-COhfPENSATlNG PULMONARY MODEL FOR TFACHING VENTILATION AND PFRFVSION RELATIONSHIPS TO MEDICAL STUDE3a-S

Bruce L. Johns,Alan D. Scott,and David J.Thuente Indiana University -- Purdue University at Fort Wayne, Center for Medical Education and Dept. of Mathematical Sciences, 2101 Coliseum Blvd. East, Fort Wayne, IN. 46805.

Abstract. Medical students, in their study of gas exchange within the lung, are typically challenged by the complexities and interactions resulting from the non-uniform ventilation, diffusion, blood flow maldistribution. and the mechanisms of blood gas transport. The model uses simple animated graphics to show how a non-compensating, alveolus / capillary lung model would respond to changes in systemic and environmental parameters. The alveolus acts like a well-mixed compartment and has a non-cyclical bulk-flow input that is determined by frequency and depth of respiration. The resulting volume and partial pressures of the gas compartment are obtained by integrating the difference between this inflow and the outflow caused by simple diffusion into the blood compartment. Diffusion of oxygen from alveolus to blood is influenced by the mean-capillary oxygen partial pressure that is obtained by expressing Fick's first law of diffusion in terms of blood O2 concentration and integrating between pulmonary artery and end-capillary values of concentration and time. This procedure also provides the end-capillary oxygen concentration which varies as diffusion characteristics and blood flow influence gas exchange. The student can observe the effect on arterial oxygen by changing frequency or depth of breathing, dead space volume, diffusion coefficient, rate of blood flow. and the hemoglobin content of blood. After perturbing the system from steady-state, the student can try to return to normal values by altering blood flow, rate and depth of breathing, or metabolism. Clinical interventions, such as O2 supplementation or transfusion, may also be used. Students observe the clinical effects of diffusion impairment, increased dead space, venous admixture, hyper- and hypoventilation. altered cardiac output, anemia, and high altitude. Our goal is to facilitate the medical student's understanding of a complex system and to emphasize the benefits and limitations of physiologic compensations and clinical interventions. Keywords.

Oxygen, Bulk flov. Diffusion, Pulmonary capillary, Hemoglobin.

INTRODUCI-ION

This model describes the interaction between the alveolar gas phase and the vascular network of capillaries -- the process called gas exchange. The systemic arterial partial pressure of oxygen (P02) can be increased or decreased by altering the characteristics of ventilation, diffusion, perfusion, and blood. The student acts as the respiratory control center whose function is to maintain a normal arterial PO2 (100 mmHg). They are asked to alter one of the model's parameters (Table 1). allow time for some altered state to develop, and then use a physiologic or therapeutic measure to return the model to a normal level of arterial oxygen. GENERALO-

After using this model the student will: 1) take a simplified vascular-lung system and predict the consequences of manual or environmentally altered system characteristics

on arterial, venous, and alveolar oxygen partial pressures; and 2) develop an intuitive feeling for how the system parts and the environment interact and make a physiologically relevant corrective response to return arterial oxygen to normal. GENJZRAL CHARACPFZJSGCS

OFTHE

MODEL

Model A.5.mnptions

The model (Fig. 1) represents gas exchange in a non-compensating (no reflex) ideal lung having a well-mixed, single alveolus resulting in a constant and uniform alveolar P02. A single capillary passes by the alveolus such that there is a constant capillary dimension and ,hence, constant diffusion properties at the blood-gas barrier. A single pathway allows uniformity of red blood cell (RBC) residence times. The RBC's are assumed to have normal hemoglobin (Hb) saturation chemistry, an instantaneous chemical combination of oxygen and Hb. and no interactive effect of oxygen vith carbon dioxide or hydrogen ions.

x24

Oxygen consumption by the tissues of the body (F3) is obtained as the difference between arterial (Ca) and Venous (Cv) oxygen concentrations times total blood flow (Q). Since oxygen consumption is a system parameter, the model uses Eq. (4) to calculate systemic venous (or pulmonary arterial) blood O2 concentration. This concentration will directly influence diffusion since a decrease in concentration will increase the alveolarto-pulmonary capillary partial pressure gradient. Hemoglobin Eiffeds

.2;

i

residence

Eie.

50 time -

1. Alveolar _ . and the Dlffuslon

The O2 binding characteristic of hemoglobin in RBC's plays a dominant role in affecting diffusion across the alveolar septum since hemoglobin represents an O2 sink which increases O2 exchange rate and transport at moderate P02's. There is no simple expression for 02-hemoglobin interactions as a function of PO2 like there is with Henry's Law of solubility. Therefore, the exchange of gas between alveolus and capillary is much more complicated than simple inert gas diffusion and the resultant O2 solubility in blood since the reaction of oxygen with RBC's introduces nonlinearities. Figure 2 (Roughton, 1964) shows the non-linear relationship between blood plasma POP, X saturation (I sat) of concentration hemoglobin, and blood oxygen hemoglobin ([02-Hb]). Considering due to the 5 billion RBC's / cc, and the presence of four oxygen sites on each hemoglobin molecule,

75

set

of

Oqqcn

Basic Model

The model is designed around flows of 02 across boundaries of 3 compartments:

____~__ ‘ool-----

air --> alveolus Fl = f * (VT - VII) * (FIO2 - FA02)

(1) I

The flow of oxygen into the alveolar compartment (Fl) is dependent on the depth (VT) and rate (f) of respiration, the amount of dead space (i.e., VII = the volume unavailable for gas exchange), and the differential of inspired (FIO2) and expired (FA02) oxygen fractions. alveolus --> blood F2 = DL * (PA02 - PC')

Oxygen leaves (F2) by simple diffusion as determined by the trans-membrane pressure gradient (alveolar (PA02) and capillary (PC') mean partial pressures), and the conductance of the alveolar membrane (DL). An important dependent variable for the model is the volume of o2 in the alveolus (VA02). obtained by the integrated difference between the inflow (Fl) and outflow (F2) of oxygen: t2 (Fl - F2) * dt

VA02 = VA02 t I

(3)

t1

Division of alveolar oxygen volume (VA02) by total alveolar volume (VA) provides a value for PA0 which is the alveolar driving force for diffusion. blood --> tissue F3 = (Ca - Cv) * Q

60

(2)

(4)

po2

SO

100

120

1 100

hHQ)

it is understandable that the reversible association of oxygen and Hb occurs in incremental steps yet results in the smooth curve illustrated. Since there are four sites, each RBC shows 25% increments in saturation. Normally RBC's operate between 75% saturation (venous blood at PO2 = 40 mmHg) and nearly saturated (arterial blood at PO2 = 100 mmHg). For the model a curve fit to this relationship was obtained to fit the relationship over normal and sub-normal PO2 ranges (Roughton, 1964): PO2 > 40 mmHg

Proc. 6th Int. Conf. on Mathematical

1%

1

1

Xsat

w

100 - Xsat

2.7 * log (PO21

3.88

(5)

where r - mean capillary residence time of a red cell. Since mean transit time equals compartment volume (Vc) divided by washthrough flow (cardiac output, Q), or 1

PO2 < 40 mmlig

-=

-

1=

r (10)

VC

Q

Xsat log

825

Modellip

substituting this into Eq. (8) yields

I 100 - Xsat 2.21 * log (P02) - 2.85

(6)

CC

dC

DL =-

In addition to this non-linear relationship, the linear solubility of oxygen in plasma follows Henry's Law. The amount of oxygen in solution in blood is not normally significant. However, as alveolar PO2 rises above 150 mmHg, the significant increase in oxygen transport is a direct result of soluble oxygen. Moancapillary PO* Diffusion by Fick's Law (flow F2) depends on a mean capillary partial pressure that occurs at some point along the capillary as the Hb molecules pick up their final O2 molecule. This makes the vascular compartment spatially dependent when compared with the well-mixed alveolar compartment. Therefore, the mean diffusional gradient can only be determined precisely if the 02 gradient from gas to blood is known at every point along the pulmonary capillary. It is necessary to look at all momentary rises in O2 saturation as the incremental volumes of blood pass from the pulmonary arterial end to the end-capillary location as Bohr did in 1909 (Briscoe, 1980; Comroe, et al, 1962; Wagner, 1977). Considering O2 in any small element of capillary volume, the change in total O2 in time dt must equal the amount that diffused in: Vc * d I

P2-Hbl

+ LO21

I

=

(PA02 - PC) * DL * dt

(71

where [02-Hb] = concentration of 02 bound to Hb (ml O2 bound / ml blood), [02] = concentration of dissolved O2 and PC = PO2 in the small element. It is assumed that these two blood compartments are in equilibrium and that total concentration, Cc' - [02-Hb] + Rearranging terms and integrating along P21. the pulmonary capillary from the beginning venous (pulmonary artery or systemic venous) to end-capillary (Ca) values of concentration and residence time, one obtains CS.

TC dc

DL vc

= - PC)

J

dt

(81

0

Based on the assumption that diffusing capacity (DL) and capillary blood volume (Vc) are constant along the capillary, the right side of this relationship integrates to the expression DL -*T VC

(9)

(PA02 - PC)

(111

Q

This integral is complicated by the fact that the O2 partial pressure of any small element is dependent on O2 concentration of O2 on hemoglobin based on the hemoglobin saturation curve. For operation of the model. integration of Eq. (11) until area = DL / 2 * Q provides a concentration (Cc) that corresponds to the mean capillary partial pressure for oxygen (PC). Likewise, integration to the full value of DL / Q provides an estimate of end capillary PO2 (Pa), representing the equilibration between alveolus and capillary blood. Figure 1 shows what happens to the saturation of blood as the corpuscles pass from the venous to the arterial end. Venous blood contacts the maximum PO2 gradient so gas exchange is highest at the venous end of the capillary. This process continues along the capillary, though at a diminishing rate, since the blood PO2 is rising while alveolar PO2 is constant. This suggests that there may be an interaction between diffusion and blood flow. An example of exercise is included showing that the increase in blood flow (cardiac output) will decrease residence time of a particular element of blood. The increase in velocity will reduce the contact time with the alveolus and reduce the effectiveness of diffusion at a fixed membrane conductance CDL). Fortunately, in a normal lung there is sufficient margin of safety since the system is over-designed for normal operation. In other words, normal equilibrium of blood with alveolar PO2 's occurs within the first third of the capillary (normal residence time is estimated at 0.75 set). This means that even in exercise, when RBC's must travel more capillary distance before equilibrium is reached, there is still sufficient time for equilibration between the alveolar gas and the RBC at the end of the capillary. As flow increases further, or as membrane conductance decreases (an increase in membrane resistance), equilibrium will not occur and there will be a disparity between alveolar and arterial (end-capillary) P02's. This is referred to as the Alveolar - arterial difference. Ciic&tion and Tissue EXfeds Finally, the vascular - tissue component of the model has an interesting requirement in order to function physiologically. A change in arterial oxygen content does not appear instantly in the venous return, nor does the

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Proc. 6th Int. Confi on Mathematical

content change instantly and totally once the change does appear. Therefore, the model has an appearance time (lag) of 5 seconds; ie, all arterial changes take at least 5 seconds before any effect will be observed in venous blood. Then, once the change appears there is an exponential effect with a time constant (7 ) of 10 seconds. This effect represents the myriad of transient times in the many vascular circuits. The effect of an instantaneous arterial change appears in the venous blood according to the following: Time (set) 10 20 30 40

time constants

% of max. response

1 2 3 4

63 86 95 98

Thus, considering lag and exponential effects, it takes 35 seconds for an arterial PO2 step response to show 95% of its effect in venous P02. This time constant and transport delay situation adds stability to the physiological system but can, at times, make for some interesting mathematical problems that must be considered in this unique modeling situation. Parametersand Physiologic Ranges The students are given a protocol to facilitate their achieving the general and specific program objectives. The following comments refer to the ranges of operation open to the student for experimentation. Because the students are operating as the respiratory controller, they are challenged to make incremental physiologic or therapeutic corrections to keep the model near a normal arterial P02. The graphics allow the student to focus on changes in venous, alveolar, and arterial partial pressures. Hardcopy of all model parameters and variables is available at any time. Barometric Pressure -- Pbar (mmHg). Higher elevations cause barometric pressure and inspired oxygen partial pressure to decrease: Place

Elevation (feet)

Denver

5.000 15.000 29,028

Everest

Pbar (mmHg) 640 429 253

Insp. O2 @mHg) 134 SO 52

Since the model makes an instantaneous change in barometric pressure, pressures lower than 550 mmHg are to be done in 100 mmHg increments at 30 second intervals. Oxygen Fraction Inspired -- FI02. Normally, as barometric pressure drops the fraction of O2 remains constant (0.2093). However, in cases of asphyxia in a closed space or because of displacement of oxygen, the fraction of O2 can drop precipitously. For inhaled fractions below 0.15. steps of 0.025 at 30 second intervals are used. Ventilation Rate -- f (breaths / min). On the high side of normal, respiratory rates can go to a maximum of 40-50 / min under very heavy exercise. More moderate values of 24-30

Modelling

/ min represent typical high rates. Tidal Volume -- VT (ml). The volume inhaled per breath is normally about 500 ml. In exercise, tidal volumes may reach 50-60X of vital capacity (or 2.5-3 L in an average man). As tidal volume decreases (approaching dead space volume), alveolar ventilation technically approaches zero. Cardiac Output -- Q (ml / qin). This model presently operates as though left heart and right heart outputs are equal and there are no shunts of either anatomic or physiologic origin. Viable range of flows would be from 1,000 ml / min (circulatory failure) to a maximum of 25,000 ml / min (maximal exercise). To decrease flow, use 1,000 ml / min decrements in flow at 30 second intervals. For large increases in flow, 2,000 ml / min increments are made at 30 second intervals with the expectation that other physiologic parameters will require alteration once 10,000 ml / min is reached. Dead Space -- VD (ml). Parts of the lung that are ventilated but not perfused can occur for anatomic or physiologic reasons. With age, dead space continues to increase from 25% of tidal volume at age 20 years to 40% of tidal volume at 60 years old. Emphysema may increase dead space because of distention of the alveolar sacs. Use of a 20" long by 3/4" ID snorkel increases dead space by 150 ml. Lung diffusing Capacity -- DL (ml O2 / min * mmHg). The diffusing capacity may be increased by increases in cardiac output as seen in exercise, since more oxygen would be transferred / unit of time at a constant O2 gradient. This represents recruitment of exchange surface area. Normally, the alveolar septum may increase in thickness due to interstitial edema or fluid buildup (as in pneumonia), but it doesn't usually decrease thickness to facilitate gas transport. Decreases in diffusing capacity to 5-7 ml / min * mmHg are not unusual in diseased lungs. These levels are attained by decrementing DL in steps of 5 ml / min * mmHg at 30 second intervals. Cxygen Consumption -- 02use (ml / min). Increased work loads to 2,000 kg * m / min in well-trained athletes can increase O2 consumption to 4,500 ml / min. representing a 15-fold increase. Hemoglobin Cont. -- [Hb] (mg / ml). Hemoglobin concentration may increase to values of 0.234 mg / ml for acclimatized people at 5.350 meters (17,550 feet) elevation. Anemia decreases hemoglobin. Hemoglobin 02 Sites -- Hb sites. The number of effective carrying sites can decrease from four as carbon monoxide completely binds in place of oxygen. By selecting fractional decreases in site numbers the effect of CO may be mimicked. Sp&fic Objectives After working with the model, the student will be able to describe the effect of the

821

following situations on alveolar, arterial. and venous oxygen partial pressures, and will develop a strategy for correcting arterial PO2 to normal: 1. Sudden airplane cabin depressurization at 15,000 feet (Pbar = 429 mmHg). 2. Asphyxia in a sealed compartment (FI02 = 0.10). 3. Snorkeling (VD = 300 ml). 4. Pneumonia (DL = 12 ml / min * mmHg). 5. Increased cardiac output (Q = 12,000 ml / min). 6. Heart failure (Q = 2,500 ml / min). 7. Anemia ([Hb] = .l mg / ml). 8. Altitude acclimatization ([Hb] = .2 mg / ml). 9. Carbon monoxide poisoning (Hb sites = 3.5). 10. Hypoventilation - drug overdose of CNS depressant (VT = 250 ml). 11. Hyperventilation (f = 15 & VT = 600 ml; thus MV = 9,000 ml / min). 12. Increased level of exercise (02use = 900 ml / min). 13. Increased respiratory frequency from excitement (f - 18 / min). CONCLUSIONS Using this model, the student is able to gain experience learning about the interactions between the environment and the oxygen compartments of the body. The student is challenged to think as a controller in the

circuit and use physiologic and therapeutic measures to maintain a constant arterial oxygen partial pressure. During operation of the model attention is focused on the lifegiving arterial blood, though total access to model parameters and variables is available to those who desire it. Finally, the model can focus on the alveolus / capillary interface where ventilation and perfusion meet in a highly nonlinear process, and can allow the student to dissect the effects of changes in individual system parameters. REFERENCES Gas transfer in Briscoe, W. A. (1980). In J. B. West (Ed.), diseased lungs. Pulmonary Gas m, Vol. 2. Academic Press, pp. 298-302. Comroe, J. H.. R. E. Forster, A. B. Dubois, W. A. Briscoe. and E. Carlsen (Eds.) (19621, ULune, 2nd ed, pp. 353-356. Transport of Roughton, F. J. W. (1964). oxygen and carbon dioxide. In W. 0. Fenn and H. Rahn (Eds.), -ok of Pm . Vol. I. Section 3: Res, American Physiological Society, pp. 782784. Wagner, P. D. (1977). Diffusion and chemical in pulmonary reaction gas exchange, . -Reviews, 57~274-284, 1977.

ters and Control Alterable Parameters and Their Control Values: Pbar FI02 f VT Q VD DL 02use [Hbl Hb sites

= = = = = = = = =

Barometric pressure ----------Fraction of inspired O2 ------Respiratory rate -------------Tidal volume __________________ Cardiac Output ---------------Pulmonary dead space ---------Lung O2 diffusing capacity ---Tissue oxygen consumption (F3)Hemoglobin concentration -----# of available sites on Hb ----

760 mmHg 0.2093 12 / min 508 ml 6103 ml / min 148 ml 27 ml O2 / min * mmHg 298 ml / min 0.15 mg / ml blood 4

Unchangeable parameters: = 5

Lag

circulation appearance time -- 5 seconds exponential time constant ----- 10 seconds

System Variables: Fl F2 MV PA02 Pa02 Pv02 PC PC FA02 [02-Hb] CC' cc Ca vc T Tc T

= = = = = = = = = = = = = =

Oxygen flow into the alveolus ------------Oxygen diffusion, alveolus to blood ------Minute ventilation of air ----------------Alveolar oxygen partial pressure ---------arterial oxygen partial pressure ---------venous oxygen partial pressure -----------pulmonary capillary 02 partial pressure --mean capillary oxygen partial pressure ---Oxygen fraction in alveolar gas Blood concentration of O2 due to Hb ------Total O2 concentration in blood ----------Mean capillary O2 concentration ----------End-capillary O2 concentration -----------Total capillary blood volume -------------Mean of capillary residence times --------Maximum time of RBC in capillary ---------Systemic circulation time-constant --------

ml / min ml / min ml / min mmHg mmHg mmHg mmHg mmHg ml / ml / ml / ml / ml set set set

ml ml ml ml