Steady state transport function analysis of portions of the central circulation in man

COMPUTERS

AND

Steady

BIOMEDICAL

RESEARCH

9,291-305

State Transport of the Central

(1976)

Function Analysis of Portions Circulation in Man

R. JORDINSON, SUSAN ARNOLD, B. HINDE, G. F. MILLER, J. G. RODGER, AND A. H. KITCHIN Depurtmeni Unirvrsity

of Medicine, of Edinburgh;

Western General Hospital and Department of Applied Mathematics, and the Division of Numerical Analysis and Computing, National Physical Laboratory

Received July 25, 1975

I.

TNTR~DUCTION

Indicator dilution curves recorded downstream following single injections of indicators are widely used for measurementof cardiac output. For injections of a constant amount of the indicator the area under the primary time-concentration curve is inversely proportional to volume flow. The shape of the indicator dilution curve is influenced by suchtechnical factors asthe dead spaceof the sampling system, the speed of withdrawal of blood, and the dynamic frequency response of the concentration measuring instrument. Ideally such a curve describes what will happen to a population of blood cells or moleculesof blood plasma, starting simultaneously from a point in the bloodstream, in terms of their times of arrival at a sampling point downstream. This depends on variations in path length and in velocity along parallel paths, attachment of indicator to cells or plasma, intravascular turbulence and mixing, and the velocity profile in vessels.The curve forms a statistical frequency distribution of transit times of individual particles, from which can be calculated the shortest and longest transit times, the mean transit time, the degree of dispersion of transit times, the volume flow in the circulation. and the volume of blood in the systembetween the injection and sampling points. There are practical restrictions on the useof indicator dilution curve analysis. The indicator must be nontoxic, haemodynamically inert, rapidly miscible, and capable of accurate and continuous quantitation in the sampledblood stream. The injection must be rapid and complete. All the injected dye must travel directly to the common mixing chamber. After passagethrough the mixing chamber the dyed blood must not be further diluted by admixture of collateral blood flow. The injection and sampling sitesmust be separatedby a sufficient volume to permit satisfactory mixing of the indicator, and this is generally held to preclude study of flow from one chamber of the heart to an adjoining one. Assuming these conditions are met, we may record from a single sampling point (e.g., the brachial artery) dilution curves from one or more injection sites in the left or right heart. Alternatively, following a single right-sided injection, curves may Copyright ( 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

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be recorded from two sites in the left heart. The curves will diit‘er in contour. the appearance time and peak concentration in the longer tranxit being later and the curve more dispersed. Peak concentration becomes lower and recirculation tinall! interrupts the disappearance phase of the primary curve. Assuming a steady itate. however, the integral of the primary curve will he constant despite the changing contour. If we consider any pair of curves, the difference between them must represent the delay imposed by the passage of blood from one sampling site to the other. and this too will form a frequency distribution curve of transit times between these two points. Thus from curves recorded from different injection sites in the central circulation it might be possible to compute functions describing the behaviour of blood flow in the segments of circulation between the injection sites. This would allow detailed study of transport function in local segments of the central circulation. The mathematical problem of recovering the transport function from the pair of recorded time-concentration curves is that of deconvolution. A method based on Fourier Transform has been described and applied by Coulam rf c/l. (2). A different numerical approach based on the method of regularization is presented here which has proved more satisfactory in application than the Fourier technique. 2.

DISCUSSION

OF METHODS

Let us consider the relation between a pair of indicator dilution curves recorded at different sites in the circulation, following injection of indicator at some suitable point upstream of the first site. If the upstream and downstream curves are denoted respectively byf(t) and g(t) then, following Zierler (14), it can be shown that

The (physical) assumptions made in deriving [I] are as follows. During the period of inscription of the two curves the volume flow through the segment must be steady. The volume of the segment must be constant although, as with flow rate, variations of a relatively high frequency, e.g., pulse rate, will be unimportant (I). The movement of the indicator particles must represent the flow of whole blood; this includes the provisos that there is no net loss of indicator during its passage through the circulation, that adequate mixing is obtained, that the viscosity of indicator is not markedly different from that of blood and that the dyed blood traverses the same circulatory pathway as the whole blood. In the case of indocyanine-green the binding to albumen means that the flow measured will be that of plasma and, in so far as there is any inhomogeneity of flow between plasma and cells, an error may be introduced. This will of course apply to both curves. The RHS of [I] is known as the convolution integral and the function k(r) is the kernel of the integral. In physical terms k(t) represents a frequency distribution of

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STATE

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ANALYSIS

293

transit times for particles travelling from one sampling site to the other; for convenience we call it a transport function. Its vertical scale is arbitrary. We note, in passing, that if 2, J and k denote the Laplace transforms of g(t), ,f(t), and k(t) respectively, then the Laplace transform of [l] may be written

‘q(s) =f(s) R(s). In terms of control theory k(s) = g(s)/J;(s) is the transfer function corresponding to the portion of the circulation between the two sites. The main problem is to solve [I] for the function k(t) given the functionsf(t) and g(r) which are recorded in experiments. This problem is well known and a number of methods have been developed for its solution. In particular that of Coulam ef al. (2) is noteworthy. Briefly their method involves finding the Fourier series representations off(t) and g(t); the transfer function-the Fourier transform equivalent of E(.y)---is then computed and the transport function k(t) finally determined by further use of the Fourier series technique. A partial check on the validity of the computed solution can be made by substituting the computed k(t) in the integral on the RHS of Eq. [I] and comparing the result with the given values of g(t). A further refinement of these authors’ technique is that of including a special tail extrapolation on both upstream and downstream curves since parallel sampling of the curves means that at termination of sampling, more dye will have been recorded entering the circulation segment than leaving it, owing to the finite transit time of the dye particles. These authors have tested their procedures extensively on computer models and have found that it is necessary to calculate a large number of terms of the harmonic series in order to perform accurate convolution or deconvolution of curves. They have demonstrated the use of their system in animal preparations including the pulmonary circulation and the hind limb at rest and on exercise, and have shown a general linear relationship between mean transit time of the pulmonary transfer function and the dispersion or standard deviation. Where recirculation is a major problem, as it is in peripheral circulatory beds or in long transit curves in the central circulation, an approach such as this which includes recirculation in the calculations is clearly essential. The reasons for seeking a simpler approach to the problem in the present situation are these : (a) In short transit curves in the central circulation the problem of recirculation is much less severe. (b) Varying the injection site rather than the sampling site (as in (2)) offers less difficulty in the clinical situation, and in studies by Samet et al. (8) of pulmonary transit times in which both methods were employed. there seemed to be no significant difference between the two. (c) By using a single sampling system and densitometer, problems of matching catheter sampling systems and densitometer outputs are eliminated with resultant increased accuracy in the recording of curves.

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ET AI

Unfortunately when the method of Fourier series was applied to the experimental data obtained with the use of this approach, major difficulties were encountered. In the next section we briefly describe these difficulties and propose an explanation. Subsequently, in Section 4, we describe an alternative method, that of regularisation. which substantially resolves the difficulty and enables satisfactory results to be obtained. 3. THE RESULTS OF THE FOURIER SERIES METHOD

The formulae for this computation were taken from those given in (2, p. 880). The tail extrapolation technique was, however, not used here as distortion of the curves due to recirculation was removed. The essentialdifficulty lay in the fact that the shape of the function k(t) varied markedly as N was increased; This is illustrated in Fig. 1 in which (k)t is plotted for

N:40

FIG. 1. Deconvolution by the Fourier transform method: curves of k(l) plotted from the same input and output curves with increasing values of N.

STEADYSTATETRANSPORTFUNCTIONANALYSIS

295

various N. Inevitably the effect of increasing N is to diminish the integral of the square of the residual of Eq. [l]. However, the improved accuracy is achieved at the cost of an increasingly oscillatory solution k(t). Consequently it was often difficult to determine an N yielding both small residuals and a solution function k(t) of a physically acceptable form. These difficulties stem from the fact that the mathematical problem with which we have to deal is “ill-posed”; that is to say, we have an unstable situation in which a small change in the prescribed function g(t) may give rise to a large perturbation in the solution k(t). Conversely, substantially different shapes k(t), when convolved with the given input functionf( t), may lead to outputs which are very close to the g(t) actually measured. A detailed discussion of problems of this type is to be found in Twomey (13). In this paper the author argues strongly against the use of transform methods on the grounds that the fundamental equations are unstable and attempts at direct solution will generally give large oscillating errors. Our experience indeed confirms his criticisms. Twomey suggests the application of constraints to the problem to achieve acceptable results, and this idea is basic to the method of regularization which we describe in the next section. 4. AN ALGEBRAICSOLUTIONTOTHECONVOLUTIONINTEGRAL (THEMETHODOFREGULARIZATION)

This method solves [l] by an algebraic approach in which the integral is approximated by a sum using the trapezoidal rule. It is further assumed that the functions J”(r) and g(t), which are determined in experiment, have been processed to remove humps in the tail of each curve due to recirculation of indicator. Thus the initial and final values offand g are zero. When the above approximation is made, we obtain the system gi = h i’ kjfi-j, j=O

i=O, I ,..., n,

[31

where h is the interval of integration, tj = to +jh, kj = k(t,), fj =f(tj), g, = g(ti) and the prime on the summation indicates that the first term is halved. The starting point to is chosen so thatf, = go = 0 and the value of n is such that both f and g are zero beyond t = t,. Also, since the dye particles take a finite time in traversing the segment of the circulation between the sampling sites, the “output” function g is zero for some time interval to < t < t,, say, over whichfis nonzero. In matrix notation the system [3] becomes Fk=g

[41

where F is a lower triangular matrix and k and g are column vectors. The system cannot be solved by a direct method since the matrix F is singular, reflecting the fact that the original problem [I] is ill-posed. Although the instability is inherent in the

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problem itself, the difficulty may be effectively overcome by employing the method of regularization. This method. which furnishes smooth approximate solutions to integral equations of the first kind. was pioneered by Tikhonov (10). Phillips (6). and Twomey (12) and has subsequently received attention from many author\. Excellent expository accounts are to be found in Twomey (13) and Turchin ef u/. ( II 1. An essential feature of the regularization method, as applied to the discretized form of the problem, is that instead of seeking to satisfy Eq. [4] exactly we impose the weaker condition ijFk -‘q < e. [51 where ~1.!I denotes a euclidean norm and P is a small positive quantity; its value may, for example, be chosen to correspond with a known noise level in the data, g. From the infinity of vectors satisfying the inequality [5] we now seek to determine the vector k which minimises the expression H[k], where His a suitably chosen quadratic functional. It can be shown that the minimizing vector certainly exists and is unique if H is positive definite; for a rigorous account of the method under more general conditions on H the reader may consult Rib&e (7). The choice of functional is not critical but it should be such that the effect of constraining H[k] to be small is to exclude undesirable oscillations from li. The simplest choice, which we shall adopt here is H[k] = j;. kj’ = kj,‘. and corresponds to the so-called zero-order regularization. Some alternative choices are discussed by Twomey (13). It is readily shown that, provided e < I/gii, the minimal vector k satisfies Eq. [5] with the equality sign, i.e., ‘J-?t -RI’ = 6’. @I (For if an alleged minimal vector k gave IiFk - g,~ < c we could certainly reduce ~I/ir still further without violating [5].) By a well-known result in the calculus of variations, the problem of minimizing llk1j2 subject to the condition [6] is equivalent to that of minimizing where y(>O) is a Lagrange multiplier or regularization parameter. It would be possible to determine y uniquely so that [6] is satisfied. In practice, however, it is usually convenient to choose y rather than e and to observe the effect on the magnitudes of r and Ilk11 as y is varied. In this mode of operation the number e is not a preassigned tolerance but a measureof the quality of fit with a prescribed y. It should be noted that becauseof the “ill-posed” character of the problem no absolute measureof the accuracy of the solution is obtainable without further assumptions. The degree of successachieved may, however, be gauged from the size of e, indicating the overall accuracy of fit, and that of jikji which relates to smoothness.By varying y we may

STEADY

STATE

TRANSPORT

FUNCTION

ANALYSIS

297

alter the balance betweenthesetwo quantities. Ify ischosentoo small, the constraint on the size of llki[ will be insufficient and an oscillatory solution may result; while if y is made too large a smooth solution is obtained at the cost of an unduly large value of IjFk - ~11.The object should be to determine a value of y which achieves a satisfactory compromise between these two extremes. If an initial estimate of k, k* say, is available it is convenient to modify the above regularized formulation slightly. The system [4] may be rewritten in the form F(k - k*) = g - Fk”,

that is, as a system of equations for the corrections regularizing functional is then

[71 k,i -k*,.

H[h- - k*] = ,+ _ keel’,

The appropriate

PI

and the expression to be minimized becomes(cf. Eq. [7]) IlFk -811’ + yllk - k*il’.

[91

Provided that y > 0 the minimization problem possesses a unique solution, kY say, which can be computed by solving the systemof linear equations (F= F/icy’ - F=g) + y(k” - k*) = 0,

that is to say, (F= F + yl)kY = F=g + yk*.

DOI

It is natural to take k* = 0 initially. The resulting solution kY may then be used as the estimate k* for a further application of the process, and so on. The computer program allows for this type of iterative use. A modification permitting someeconomy in the number of equations to be solved, is desirable if the time ranges of the functionsf and g differ significantly. Suppose that gi = 0 for i = 0,l . . . . , s - 1, while g, # 0. Then clearly the first s equations are satisfied by taking kj = 0 forj = 0, 1, . . . s - 1. If theseequations are omitted the system [3] becomes h 2’ kjf i-j = gi, j=s

i=s,

s+ 1. . . . .( sfn.

PII

If we now replace i by i + s and j by j + s the equation assumesthe sameform as before, with gi and kj replaced by gi+, and kj+s respectively. More generally we may setk,=Oforj=O,l,. . . . , s-r-l,whererisanintegerintherangeO
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attributable to a combination of factors, namely the presenceof errors in the data and in the quadrature approximation, the sharpnessof the peak of Ii(r) in relation to the interval employed, and the smoothing or broadening effect of the regularization process. 5. APPLICATION

OF THE METHOD OF REGULARIZATION

As noted in the previous section there is somedegreeof choice in the values of the two main parameters, namely y and S. It is of obvious importance to see how sensitivethe function k(f) is to variation in thesevalues, by noting the corresponding changesin C, Ilg\l, and l\kll aswell as the statistical parameters of mean and standard deviation.

We consider a typical set of data obtained from measurementsof indicator in the right atrium (input,) and left atrium (output). In Fig. 2 we show k(t) plotted for values of y equal to I, IO, and 50, the calculations having been carried out for a fixed value of .Fequal to the recommended one as described in (b) below. The

FIG. 2. Functionk(t) plotted from the same input and output curves with increasing values of 7. r = 4.

STEADY

STATE TRANSPORT

TABLE EFFECT OF VARYING

y ON k(r)

FUNCTION

299

ANALYSIS

I (r = 4)-SEE

FIG.

2

e -

.-~-

___

--.-

.

One iteration

Two iterations

0.1 1.0 10.0 50.0

0.079 0.148 0.261 0.631

0.050 0.124 0.183 0.298

:P: tn

0.46 0.44 0.43

0.57 0.47 0.45 0.44

7.55 7.62 7.63

output

f7

754 7.62 7.62

1.09 1.22

0.96 1.30

22.24 22.24 22.24 22.24

Key: y = regularization parameter; r = number of time intervals before start of g; e = IlFk-g,: N residual error; m = mean transit time; jlk’j = norm of transport function; c = standard deviation.

corresponding values of the above parameters are shown in Table I; the graph of k(r) for y = 0.1 was not plotted becauseof the extreme scatter. The graphs bear out the remarks of the previous section, namely that as y increases this scatter, or oscillation, decreasesconsiderably. From the table we note that jlkjl itself does not vary greatly and this behaviour has been observed in other cases.For smaller values of y of the order 0.01 (not given here), ljkll shows a relative increase in value but this is accompanied by an even more marked oscillatory shape of the transport function. The residual error e, defined by e = l/F,4-g/j, increaseswith y, again confirming the previous remarks. The choice in the values of y seemstherefore to depend rather more on e than on llkil. Having made several runs with the above values of y with different sets of data we found that taking y = 10 seemsto provide a reasonable solution. The ratio I/e//: ilgjj is about lx, which compares with the experimental error. The residual error can be reduced still further by using the iteration procedure twice as suggestedearlier. The improvement in the value of e is rather more marked with the larger values of y; the visual improvement to the curve is not particularly noticeable and in general this refinement has not beenextensively used. Table I also shows the values of mean transit time and standard deviation. It is encouraging to note that these values are reasonably insensitive to the choice of )‘, although the oscillatory nature of the function k(t) for the smaller values of y did not permit calculation of theseparameters. (b) Variations

with s

In the last section reference was made to the reduction in the value of e which can be achieved by choosing a small nonzero value of r. In effect this meansthat k(t) instead of starting simultaneously with g(t)-the output function-as theory predicts, starts a few time intervals before g(t). Numerical tests on several setsof data

I 2

FIG. :‘=

IO.

3.

Function

3

0.414 0.223

4 5

0.261 0.271

6 7

0.269 0.270

k(r)

x

l,,

0.50

7.58

0.45 0.44

7.60 7.62

0.44 0.44

7.65 7.66

0.44

7.65

C’

plotted

from

the

same

input

and

output

curves

with

varying

values

ot

STEADYSTATETRANSPORTFUNCTIONANALYSIS

301

indicate that a reasonable value of e can usually be found by taking r = 3 or 4; use of a larger value produces no further significant reduction of e. The effect of such changes on the other parameters seems to be negligible-see Table II. If r is made smaller, thus forcing k(t) to start closer to its theoretical initial point, then e increases in value. The shape of the curve also changes, k(t) rising to a maximum over a shorter time interval so that a more pronounced peak is produced (Fig. 3). Such behaviour is in accordance with theory which predicts a very sharp increase in ii(t) whose initial behaviour resembles that of a step-function. It is interesting to note that this overlap of the transport function into the time interval preceding its theoretical starting point is present in the results of (2). It evidently stems from the difficulty of numerically representing the initial behaviour of a function like k(t). The pointwise representation adopted in the method of Sections 4 and 5 does have the advantage of providing some control over the extent of the overlap.

6. DESCRIPTION OF EXPERIMENTS (a) Recording of Indicator Dilution Curves Twenty patients undergoing cardiac catheterization for valvular heart disease were studied. They were sedated with diazepam 5 mg. A Waters X302 whole blood cuvette densitometer was used, connected directly to a polyethylene catheter 12 cm long and internal diameter 1.15 mm, inserted percutaneously into the right brachial artery. The dead space of the catheter-cuvette system was 0.6 ml. During inscription of curves blood was withdrawn by a Harvard infusion-withdrawal syringe, the flow-rate being 74 ml/min. In order to minimize errors in concentration measurement due to inadequate time for dye binding, the minimum appearance time was set at 5 set (Saunders et al. (9)). Injections of cardiogreen (indocyanine-green) of 5 mg contained in 2 ml water were made by two methods: (i) by hand injection and flushing through the injecting catheter, and (ii) using an ECG-triggered injector set to deliver the injection in early diastole at a flow rate of 10 ml/set. In these cases the volume of water containing the dye was increased to 4 ml. One or two curves during which extrasystoles occurred were rejected. Injections of dye were made (1) in the aortic root and (2) through a Brockenbrough transeptal catheter positioned in the mid left atrium. This catheter was then withdrawn and positioned in the right ventricular inflow tract and the third indicator dilution curve inscribed. Injections were performed serially within a space of 5 min. The output of the densitometer was recorded, using an ultraviolet recorder with 30 cm wide paper. The paper speed was 1.25 cmjsec and the sensitivity of the galvanometer such that concentration of 12 mg dye/liter of blood gave a deflection of

302

JOIIDINSON

ICI- ,\I.

approximately 15 cm. The injection times were signaled and recorded on the papei in two ways, (i) manually and (ii) when using the triggered injector, by recording the injection signal from the instrument on a separate channel. In analysing the curves obtained subsequently no significant difference was found between the two injection techniques. The condition of the steady-state flow is assumed over the period of inscription of the pair of curves, i.e., 2-3 min. As a check, cardiac output was calculated from both curves in all pairs. There was a mean difference of 0. I5 ‘;d with standard error of the mean of 0.43 “,J and coefficient of variation of 4.7:,:. The error involved in assuming steady-state flow must therefore be small. (b) Method of Analysis Distortion of the curves due to recirculation was removed. In the case of short transit curves where there was good separation between the primary and recirculation curves with concentration falling between them to less than IO per cent of the peak value, this was done by visual completion of the primary curve. Where this condition was not met however, as in the case of right-sided curves, the method of logarithmic extrapolation of the down-slope was used. Cardiac output was calculated, the densitometer being calibrated immediately after each study with dilutions of dye in whole blood. Appearance times of the curves were corrected for delay caused by the sampling system from a knowledge of the dead space and sampling flow-rate. Results Figure 4 shows, in a normal subject, a pair of indicator dilution curves recorded from right ventricular and left atria1 injections of dye. It also shows the calculated transport function between the two injection sites derived by the method described above. As already mentioned, the transport function can be regarded as a distribution of transit times and from this, the values for mean transit time, standard deviation, and dispersion can be calculated. Table III shows for groups of patients with aortic stenosis and mitral valve disease values for transit times of the pulmonary circulation and left heart and average values for dispersion of the transport functions. In aortic stenosis the transport function from left atrium to aortic root shows rapid clearance of dye. In mitral stenosis the principal abnormality is an increase in mean transit time from left atrium to aorta with correspondingly increased dispersion. Pulmonary transit time is also increased but only proportionately to the reduced pulmonary bIood flow. In mitral regurgitation the transfer function from left atrium to aorta is markedly impaired with prolonged mean transit time and increase in dispersion. Clearance of dye from the left atrium is grossly delayed, and thus the apparent pulmonary transit, which includes a portion of the left atrium, is greatly prolonged from this cause.

303

STEADY HATE TRANSPORT FUN("IION ANALYSIS TABLE MEAN TKANSIT

111

TIME (SEC) AND DISPERSION

Num- Right ventricle-Left atrium ber of ~ 11, (5 cases

(SEC) OF TKANSPON

Left atrium-Aorta ,,I -

Aortic stenosis Mitral stenosis Mitral incompetence

4 8 7

3.72 & 0.78 4.50+0.47 6.05 + 0.55

2.29 2~0.30 2.68kO.38 2.74 f- 0.48

FUNCI’IONS

I .90 + 0.16 4.23kO.45 6.64 F 0.44

CT ..___ 0.61 + 0.09 2.18t0.32 4.01 & 0.29

Cardiac output (,liters/min) __~~ 5.13 + 0.46 3.71 kO.42 3.27 k 0.29

LA

\

Fiti. 4. Input (left atria1 injection) and output (right ventricular injection) curves recorded in a normal subject with (below) transport function calculated by the method of regularisation.

304

.I( )I
t, I At

‘l-he potential clinical importance of the general principle of deconvolution 01 pairs of indicator dilution curves is that it offers :I method of estimating hetetxgeneity of flow within regional circulations such as the pulmonary (Knopp and Bassingthwaite, (4)) or coronary (Dobbs c’t N/. (3)) circulations: also in the case of the heart, it means that the effect of valvular stenosis or incompetence can be measured, as well as the detection of minor degrees of dysfunction in portions of the central circulation not revealed by pressure measurements or cardiac output determinations. With no more intervention than that already included in routine cardiac catheterization studies, serial recording of pairs of curves is feasible with little loss of accuracy as compared with simultaneous recording of curves from two catheter sampling sites. The numerical process of deconvoluting pairs of indicator dilution curves has been based on the method of regularization rather than the Fourier method. Our application of the latter method was not successfulfor the reasonsindicated earliet and we believe that the former method, which hasbeen successfullyapplied to many sets of data, is based on a more realistic approach to a numerically difficult (i.e.. “ill-posed”) problem-a fact which should be stressed. To be useful clinically the computational procedures should be minimised. Here it is intended to attempt digital recordings of the data for the upstream and downstream curves. These could then be used directly with the given computer program to produce the transport function in both tabular and graphical form soon after the conclusion of an investigation on a particular patient.

REFERENCES J. B. Circulatory transport and convolution integral. Mavo C’linic Prrw. 42, 137 (1967). 2. COULAM, C. M., WARNER, H. R., MARSHALL, H. W., AND BASSINGTHWAIGHTE, J. B. A steadystate transfer function analysis of parts of the circulatory system using indicator dilution techniques. Comput. Biomed. Res. 1, 124 (1967). 3. DOBBS, W. A., GREENLEAF, J. F., AND BAssINGTHwAIcit~TF. J. B. The transcoronary transport function, h(t), in dogs. Fed. Proc. 29, 951 (1970). 4. KNOPP, T. J. AND BASSINGTHWAIGHTE, J. B. Effect of flow on transpulmonary circulatory transport function. J. Appl. Physiol. 27, 36 (1969). 5. MICHOLES, K. K., WARNER, H. R., AND WOOD, E. H. A study of dispersion of an indicator in the circulation. Ann. N. Y. Acud. Sci. 115, 721 (1964). 6. PHILLIPS, D. L. A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Much. 9, 84-96 (1962). 7. RIBI~RE, G. Regularization d’ofirateurs. R~L’. Fran~aise Irlformat. Recherche Op&ationnelk 1. BASSINGTHWAIGHTE,

1, 57-79 8.

(1967).

P., BERNSTEIN, W. H., LOPEX, A., AND LEVINE, volume determination. Circulation 33, 847 (1966). SAMET,

S. Methodology

of true pulmonary blood

STEADY

STATE

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FUNCTION

ANALYSIS

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K. B., HOFFMAN, J. 1. E., NOBLE, M. 1. M., AND DOMENECH, R. J. A source of error in measuring flow with indocyanine green. J. Appl. Physiol. 28, 190 (1970). Zf). TIHONOV, A. N. On the solution ofincorrectly posed problems and the method of regularization. Soviet Math. Dokl. 4, 1035-1038 (1963). Translation from Russian. 11. TURCHIN, V. F., KOZLOV, V. P. AND MALKEVICH, M. S. Use of mathematical-statistics methods in solution of incorrectly posed problems. SocietPhys. USP, 13,681-702 (1971). 1-7. TWOMEY, S. On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. J. Assoc. Comput. Much. 10, 97-101 (1963). /3. TWOMEY, S. The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements. J. Franklin Inst. 279, 95-109 (1965). 14. ZIERLER, K. L. Circulation times and the theory of indicator-dilution methods for determining blood flow and volume. In “Handbook of Physiology Section 2 Circulation,” (W. F. Hamilton, and P. Dow, Eds.) Vol. I, Ch. 18 pp. 585-615. American Physiological Society. Washington, 1962. Y. SAUNDERS,