Upsala J Med Sci 78: 169-180, 1973 The Adequacy and Compatibility of Compartmental Models of Electrolyte Exchange in the Dog’s Heart G. ARTURSON,l T. GROTH,2 G. GROTTE,3 P. MALMBERG,4 R. SAMUELSSON5 and U. SJOSTRAND5 From rhe ‘Burn Center, rhe 3Department of Pediatric Surgery, rhe 4Deparrment of Clinical Physiology, University Hospital; the Uppsala University Duta Center and the 5Department of Physiology and Medical Biophysics, Uppsula Uniuersity, Uppsala, Sweden ABSTRACT The exchange of electrolytes (Na, K, SO,) and albumin in the dog’s heart was studied by using tracer technique in a modified heart-lung preparation. The data were analysed in order to investigate if the system could be characterized kinetically in terms of a few distinct compartments, and to what extent tracer measurements in systemic plasma and cardiac lymph provide information about the capillary and cellular exchange of electrolytes in the dog’s heart. The compartmental models used for sodium and potassium were easily conformed so a s to be compatible with the tracer data and it was also possible to make them consistent with independent relevant data from the literature. However, it was not possible to construct adequate models in the sense that the model parameters were well deter- mined by the data. This was not due to inaccurate data but to the insufficiency of this type of data to identify adequate models of the required complexity. Therefore, detailed quanti- tative information on the electrolyte transport across cellular membranes cannot be obtained in experimental studies of the type presented in this investigation. The excitability of the heart muscle is heavily depend- ent on the extra- and intracellular concentrations of sodium and potassium ions. Analysis of the rela- tions between changes of these concentrations and the corresponding changes of excitability and con- tractility would be considerably simplified if the heart could be characterized kinetically in terms of a few distinct exchanging compartments. This type of description necessarily involves simplifying assump- tions concerning the number and interconnections of compartments, the relation of measurements in the body fluids to compartments etc., assumptions which limit the possibilities of drawing detailed conclusions. However, models of this kind provide simple concepts, which may be used for the descrip- tion of the essential features of the system studied. For the complete identification of a multicompart- ment system, it is necessary t o have data from each compartment (27). This condition is often not f u l - filled, which means that some aspects of the model will be less well determined. It is therefore of primary importance t o test a proposed model not only for compatibility with the experimental data available, but also for the more strict condition of adequacy, which besides compatibility, requires that the para- meters of the model are well determined by the data. Compartmental analysis has been used in previous investigations of electrolyte transport in the dog’s heart. Transcellular sodium and potassium exchange have been calculated and correlated to changes of cardiac frequency (8, 9, 16, 29). However, the models used in these studies were not tested for adequacy, a circumstance which limits the reliability of the results. In the present communication, the experimental data were gathered from a simultaneous study of 24Na, 42K, 1311-albumin and 3 5 S 0 4 in systemic plasma and heart lymph. The compartmental analysis was performed, with due consideration to the requirement of model adequacy, in order t o investigate t o what extent this type of data provide information about capillary and cellular exchange of electrolytes in the dog’s heart. MATERIAL A N D EXPERIMENTAL PROCEDURE T h e experiments were performed o n modified heart-lung preparations (HLP), prepared according to Areskog (1). in four young, anaesthetized Vorsteh dogs, weighing about 20 kg. N e m b u t a P (30 mg/kg) was used as the anaesthetic agent. T h e fall in body temperature during the experiment was diminished by heating the operation table, irradiating the dog with a heat lamp and regulating the temperature of the reservoir and tubing system in the HLP. Evaporation from the open thorax was diminished by means of a plastic foil cover and the HLP was continuously supplied with 5 . 5 % Upsala I M e d Sci 78 -170 G. Arturson et al. H e a r t lung p r e p a r a r a t i o n ( d o g l ) m m Hg 1 cm H20 1 I I c 5 I 9 12 I I I I I I I I 1 I u H - v Y -_ m . z N VI m N I 1 I 1 0 50 100 150 200 250 300 350 min. t Radioactivity i n systemic blood ( c o n t i n u o s infusion) Fig. I . Experimental results for dog 1. glucose solution ( I ml/min), which was added to the reservoir. The heart rate (HR), the arterial systemic pressure (BP), the central venous pressure (CVP) and the temperature of the venous blood were measured. The cardiac output was meas- ured continuously by means of an ultrasonic detector located on the tubing on the arterial side. The tracers 24Na, 42K, 35S04 and 1311-albumin were added Gimultaneously to the blood reservoir. The 1311-albumin had opsala J M e d Sci 78 been separated one hour previously, according to the method of Bill et al. ( 6 ) , in order to remove free 1311. 3 5 S 0 4 and lalI- albumin were given as a single injection. Together with the initial addition of 24Na and 42K, continuous infusions of these isotopes were started in order to keep the systemic plasma concentration as constant as possible. Cardiac lymph was collected in heparinized plastic tubes as described previ- ously (2). Blood samples were taken from the reservoir and Electrolyte exchange kinetics in the dog's heart 171 H e a r t lung preparation (dog21 \ , I - K J 0 5 0 100 153 200 250 303 350 100 min , R a d i o a c t i v i t y in systemic b l o o d ( c o n t i n u o u s Infuslon1 Fig. 2. Experimental results for dog 2. were centrifuged immediately in order to separate the plasma and the red blood cells. The experiments were discontinued when the systolic systemic prsssure fell below 40 mmHg. The concentrations of sodium and potassium in cardiac lymph and systemic blood plasma were determined with an Eppendorf flame photometer. The activities of Z4Na, 42K and 1311-albumin in systemic plasma, red blood cells and cardiac ymph were measured by a gamma spectrometer (cf. 26). 3 6 S 0 , in the systemic plasma and the cardiac lymph was measured in a Beckman CPM-2CO scintillation counter (cf. 18) after the separation of "SO, from the haemoglobin in a SephadexB C-50 column (cf. 19). The mean plasma concentration of the various isotopes was calculated as the mean of the values in all samples of systemic plasma and the separate concentrations (per cent) in the plasma, red blood cells and lymph samples were related Upsnla J M e d Sci 78 172 G . Arturson et al. t o this mean concentration. In Figs. 1 , 2 and 4 the radioactive data are calculated as successive means of three consecutive measurements. The magnitudes of the errors in the experi- mental data were estimated from the errors of the radioactive measurements and the errors in the weighing procedurzs (cf. 26). During the first 15-20 min of the experimental period the errors were calculated to be about f 5 46 (S.D.) or even slightly larger, whereafter they decreased. EXPERIMENTAL RESULTS The experimental results from two dogs are pre- sented in Figs. 1 and 2. The results from the other two dogs have been given in a previous communi- cation (3). The cardiac output and the arterial blood pressure gradually decreased, whereas the central venous pressure was fairly constant in the range of 0-10 cm H,O during the experiments. The heart rate decreased somewhat, probably due to a slight decrease in body temperature during the experimental period. The lymph flow rate usually increased at the end of the experimental period. The Na and K concentra- tions in the systemic plasma and the lymph were practically constant throughout the experiments. These results indicate that the ionic fluxes in the heart muscle could be considered to be in steady state. For unknown reasons, in one dog the activity of 24Na in the cardiac lymph increased t o values ex- ceeding that of the systemic plasma (Fig. 2). These data were therefore excluded from the subsequent computational analyses. As can be seen in Figs. 1 and 2 , the 42K and 1311- albumin activities were fairly constant in the red blood cells throughout the experiments, whereas the 24Na activity gradually increased. The initially high relative activity of the isotopes 24Na, 42K and 1311- albumin in the packed red blood cells may be ex- plained by an unavoidable amount of trapped plasma. The red blood cell compartment was con- sidered to be in negligibly slow exchange with plasma and was therefore not taken into account during the subsequent compartmental analysis. is localized in the interface between the compart- ments and the mixing of tracer is assumed to be very rapid within each compartment. The flux of tracer from one compartment to another is taken as being proportional to the concentration of tracer in the compartment which the tracer is leaving. Thus the rate constants are lumped parameters, including the exchange of tracer by way of bulk flow, diffusion and other possible transport processes which can be described in this way. The various mechanisms of the transport of substances cannot therefore be resolved in this kind of analysis. The physical models generate kinetic equations, forming a system of linear differential equations of the first order. The equations are formulated in terms of the concentration of tracer C , (counts min-l g-l) in the various compartments ( i ) of volume Vi (ml) and the rate constants ki, (mi-'), describing the transport of substance from compartment j t o compartment i. The system of equations was inte- grated numerically by the use of a fourth-order Runge-Kutta method (see e.g. (5)). This approach has the apparent advantage over analytical methods (see 8) that cases with arbitrary plasma functions can be analysed, and not only cases in which the plasma concentration is constant with respect to time. The parameters (rate constants ki, and volume ratios YJV,) were estimated by conforming the models in a least-squares sense to the experimental tracer-concentration data from plasma and lymph samples, using a method due to Powell ( 2 2 ) . The models were tested for compatibility with the tracer data, and the parameter estimates and the quantities derived from them were also tested for consistency with available independent information about the biological system studied. However, the aim of the analysis was not only to find models which were compatible with the experimental data but rather to define adequate models for the descrip- tion of the transport of the various substances, i.e. to find compartmental models of minimal complexity, which, besides Compatibility, also define parameters which are reliably determined by the data. The estimation situation was investigated by drawing likelihood contours. The region enclosed by the THEORY A N D COMPUTATIONAL METHODS sum-of-squares contour S=S(O) Various types of compartmental models have been used (cf. Fig. 3), all of which are based on the assump- tion of a steady state, i.e. that the compartments are constant in time with respect to composition and volume. Furthermore, the resistance to tracer flow I P s(e) Q s(6) 1 + ~ where 6 is the best estimate of the parameters 8 , gives an approximate 100 (1 - a ) yo confidence region, n - P ) { n - p Upsala J M e d Sci 78 Electrolyte exchange kinetics in the dog’s heart 173 1 k21 k12 , I ki2 2 . k 2 1 _1 I Compartmental M o d e l s L Model l a 1 k i o r - I L - 1 M o d e l lb Model 5 1 = F u n c t i o n a l p l a s m a Compartment 2 - F u n c t i o n a l i n t e r s t i t i a l c o m p a r t m e n t F u n c t i o n a l i n t e r s t i t i a l Compartment Il 3. F u n c t i o n a l l y m p h c o m p a r t m e n t 5: F u n c t i o n a l i n t r a c e l l u l a r c o m p a r t m e n t n 1 = F u n c t i o n a l i n t r a c e l l u l a r c o m p a r t m e n t I o r Fig. 3. Compartmental models used in the analysls. if the model is reasonably linear in 8 within the region of interest (4, 7). F,(p, n - p ) is the c( significance level of the F distribution with p and n - p degrees of freedom (p=number of parameters, n=number of data The kinetic equations for model l a are and the concentrating effect on the tracer due to evaporation during the experiment was simulated by an equivalent inflow ( k , , ) of tracer to this compart- ment. points). dC; _ _ dt - - k,, C; + k , , vz Vl C i + k , , C ; and MODEL APPROACHES AND RESULTS 1311-albumin dCi As 1311-albumin can be regarded as having a purely _ _ ~ (k12 + k,,) Ci + k,, v; C ; dt v 2 extracellular distribution (14), the 1311-albumin tracer data were analysed in terms of an open two- compartment model (model 1 a in Fig. 3). The plasma compartment was not regarded as infinite The parameters kzl, klz, koz and Vz/Vl were estimated from the experimental plasma and lymph- tracer data (representing the plasma and interstitial Upsala J M e d Sci 78 \ 174 G . Arturson et al. 100 1311-a1bumln ........ __.._- ...... ....... 100 E 1311- a l b u m i n ........ ...... .- ........ .. I P 50 - tl 50 m S y s t e m i c p l a s m a Cardiac l y m p h - M o d e l l a - 0 " 0 50 100 150 200 m i n % I I J C * S y s t e m i c p l a s m a x Cardiac l y m p h - M o d e l l a 0 50 100 150 200 m i n er S y s t e m i c p l a s m a f o r c i n g functmn A C a r d i a c l y m p h - M o d e l l b and m o d e l 3 0 200 mm 50 100 150 S y s t e m i c p l a s m a , , 1 r d i a c lym; - M o d e l 2a 0 0 0 50 100 150 200 m i n D i i " t S y s t e m i c p l a s m a I C a r d i a c l y m p h - M o d e l 2 a 0 50 100 150 200 m i n u C a l c u l a t e d " c a p i l l a r y p l a s m a " forcing f""Ctl0" Cardiac Lymph , 4 Mode(: a n d model; 0 50 100 150 2 0 0 mi- , Fig. 4 . Illustration of the results of conforming various models to the experimental data. Figs. A-D and F refer to dog 2 and Fig. E t o dog 1. Upsala J M e d Sci 78 l 3 l I - a t b Dog 2 M o d e l l a Ko2[mm -11 A d 1 V2/V1=0055 2 V2/V1=O060 1 V 2 / V 1 = 0 0 7 0 5 V 2 / V 1 = 0 075 kl2=0 0009min-1 Electrolyte exchange kinetics in the dog's heart 175 SOL ; D o g 2 M o d e l l a 0 015 kl2[mnn-1] B 0 021 0 020 0 019 0 018 0 017 k21[min-1] 0 0009 0 0010 00011 0 0012 0 0 016 Z L N a . D o g 1 Model 3 0 0 1 3 1 1 l k 0 2 = 0 1 1 2 2k02=0115 3 k02=0117 0 0 3 9 i Lk02'0119 1 5 k o 2 = 0 122 0035 0 0050 0 5 0 7 0 9 1 1 1 3 1 5 1 7 ZLNa ;Dog 1 M o d e l l b O Z o : 1 0 05 0 10 0 15 0 20 00055 00060 00065 0 0070 2 L N a , Dog 1 M o d e l 3 w i t h c o n s t r a i n t s V C Z l X V , . c , r l L * C c 0 21 0 0 5 S k ~ 2 S O 6 020 - 019 . 018 - 017 - 016 ' I 011 015 016 017 018 019 0 2 0 L 2 K , D o g 2 M o d e l 3 Fig. 5 Illustration of the multi-dimensional, 95 % confidence regions for the estimated model parameters by representative cross-sections. Upsala J M e d Sci 78 176 G , Arturson et al. Table I. Parameter estimates f o r 131Z-albumin using model l a and model 2a (dog 2 ) in the analysis of dogs I, 2 , 3 and 4 Dog 2 Dog 1 Dog 3 Dog 4 Dog2a Model Model Model Model Model Model Parameters l a l a 2a l a l a l a k,, (min-l) k,, (min-l) k,, (min-l) .k,,(min-') k,, (min-l) ko3 b i n - ' ) TLAG ( m i 4 V2l v, V2I v 3 Derived .quantities CPICI 0 0.0001 0 0.003 - 2 0.019 - 0.6 0.0010 0.0010 0 0.001 1 0.0010 0.0001 0.0009 0.0043 0 0.019 - 0.005 0.028 - 0.030 - - - 20 0 7 0.065 0.044 0.016 - 1.21 - 0 0.0008 0 0.018 - 26 0.050 - 1,2*0:2 1.0 0.9 1 . 1 0 0.0002 0 0.017 - 20 0.014 - 1.2 Not corrected for evaporation, k,, = 0. a compartments). The influence of evaporation and other factors on the size of the plasma pool was estimated from data for stable sodium in plasma. In one dog (dog 2) there was a systematic decrease in the plasma-pool size, which was corrected for by a constant inflow of tracer (k,,=O.OOl min-I); in the other dogs, the estimated pool size fluctuated irreg- ularly but was regarded as approximately constant for the analysis (cf. k,, in Table I and the comments below). In order to get a good fit in the interstitial Compartment, it was also necessary t o introduce a time-lag parameter, T L A G . T L A G was chosen from the experimentai data by visual extrapolation. Curvilinear interpolation of adjacent lymph values was used to calculate Iymph values at times corre- sponding to the plasma-sample times. The parameter estimates are given in Table I and the corresponding simulated curves for dog 2 are shown in Fig. 4 A , compared with the experimental data. Despite the fact that the deviations in some regions are greater than the analytical errors (see page 172), model 1 a was accepted as a satisfactory description of the data, in view of the combined effects of biologically but mainly methodologicaiiy uncontrollable errors. The parameter confidence for model l a and dog 2 is illustrated in Fig. 5 A . An alternative model (model 2a in Fig. 3) was used in an attempt to describe the data by the introduction of a special lymph compartment con- nected to the interstitial compartment, instead of Upsala J M e d Sci 78 the time-lag parameter of model l a . The kinetic equations for this model are: and d C j _ _ - - k,, C i i k,, 3 C,* dt v 3 (3) (4) Only dog 2, which was judged to be the best ex- periment, was analysed by this model and the para- meters k,,, k,,, k,,, k,, and V , / V , were estimated from a least-squares fit of compartments 1 and 3 to the experimental plasma- and lymph-tracer data. By introducing the plausible but still questionable con- straint C: ( t ) - C,* ( t ) , t 20, it was also possible to estimate V,/ V, and V,/ V3 separately. The goodness of fit is illustrated in Fig. 4 B (dog 2) and the least- squares estimates of the parameters are given in Table I. As can be seen from Fig. 4 B , there is a need for still more complex models in order to describe the experimental data within the limits of the analytical errors, for example, by adding a series of small compartments t o explain the time lag. However, this work did not seem justified in view of the avail- able data, as the time lag must be strongly influenced by the drainage of cardiac lymph. From the parameter estimates in Table I, it fol- lows, by the use of equations for mass transport and the steady-state assumption, that the concentration of unlabelled substance is the same in the plasma (C,) and the interstitial (C,) compartments. The amount of albumin brought back to the plasma per minute (influx rate) is of the order of 4 7 % of the amount transported from the plasma per minute (efflux rate), as calculated for dog 2. These conclu- sions are not critically dependent on the choice of model l a or 2a for this case. The correction for evaporation in dog 2 is, however, rather critical for the estimation of the parameters k , , and VJV,, as can be seen in the last column of Table I. This effect may be the explanation of the discrepancies between dog 2 and dogs 1, 3 and 4 concerning V,, and this is also the reason for the choice of dog 2 as the best experiment. 36SOa In accordance with studies of the distribution of various ions and carbohydrate molecules in the Electrolyte exchange kinetics in the dog's heart 177 Table 11. Parameter estimates for 3 5 S 0 4 using model l a and model 2a (dog 2 ) in the analysis of dogs 2, 3 and 4 Dog 2 D o g 3 D o g 4 parameters Model l a Model 2a Model l a Model l a 0.0010 0.0010 0.0062 0.0085 0.041 0.047 0.012 - 0.01 1 - 0.066 0.12 0.15 - 6.25 12 - - 0 0.0035 0.043 0.024 - - 0.05 3 - 0 0.0010 0.006 0.052 - - 0.02 1 3 - 1 .o 0.8 1 .o 1.2 mammalian heart by Page & Solomon (21) and Page (20), and in the frog heart by Danielson (lo), SO, can be regarded as having a purely extracellular distribu- tion. Model l a was therefore used for the analysis of the 35s04 data. The least-squares estimates of the parameters are given in Table I1 for dogs 2, 3 and 4 and the corresponding curves for dog 2 are shown in Fig. 4C. The parameter estimates for dogs 3 and 4 are influenced by the same uncertainty concerning evaporation as in the case of lSII-albumin. The good- ness of fit was acceptable for model l a but was not improved by using the more complex model 2a, as can be seen in Fig. 4D. The 95% confidence region for the estimated parameters is shown in Fig. 5B. From the values for dog 2 in Table 11, it can be seen that k , , is larger than the corresponding value for I-albumin. Independent lymph-flow data can be used to estimate the volumes of the functional plasma and interstitial pools. The functional plasma volume, V,, is of the same size, 42-54 ml, but the functional interstitial volume, V,, is larger (5 ml, compared with 3.5 ml for the corresponding value for 1311-albumin). It should be stressed that V , here signifies the part of the interstitial compartment drained by the cannula. The estimates of pool volumes are also subject t o large numerical errors. Furthermore, it can also be concluded from the figures for dog 2 that the influx rate of a5S04 to plasma from the interstitial space approaches 100 % of the corresponding efflux rate from plasma. 12 - 732853 In this case, and also in the case of 42K, the electro- lyte was infused continuously, in order to maintain a constant plasma level. The reason for this was primarily to simplify the mathematical analysis (cf. 8); however, it was not possible t o maintain the plasma level within + 5 % (cf. 8) and, instead of analytical methods, numerical methods had t o be used, with the advantage of accepting an arbitrary function for the plasma concentration of tracer. data for dog 2 were invalidated for some unknown measuring and/or technical reason (see experimental results). The data from dogs 1 , 3 and 4 were analysed. Using model 3 (cf. 9), the kinetic equations of which are The dC,* Vl dt vz v, __ = - (k,, + k,, + k,,,) Cz* + k,, - C ; + k,, y4 C ; ( 6 ) d C ; _ _ - - k 2 , C ; + k , , 5 C , * dt v, (7) a quite satisfactory fit was obtained, as illustrated for dog 1 in Fig. 4E. In this model the disappearance of the tracer from the interstitial fluids via the lymph system and other routes is described by a special rate constant ( k o z ) . This parameter cannot be resolved (see eqs. (6) and (7)), but only estimated in addition to k , , (intersti- tium to plasma). Since the measured plasma con- centration is used as a governing function, C: ( t ) , the other parameters k , , and Vl/Vz can also only be estimated as the product k , , ( Vl/ V,). The parameter estimates are given in Table 111 for dogs 1 , 3 and 4 and the 95% confidence region is shown in Fig. 5C for dog 1. As can be seen from this figure, the parameters k42 and k , , are unreliabfy determined by the tracer data, and, in view of the apparent discrepancy of the conclusions about the concentration of stable Na in interstitial (C,) and intracellular space (C,) with common knowledge (C,- 14Cc, cf. (28), (13)), model 3 is not an adequate model for the data. By introducing the con- straints Vc-4V,(cf. 8) and CI-14CC, i.e. k , , = 3.5k4,, it was also possible to fit the experimental tracer data, thus making model 3 compatible with these data. This model is, however, still not ade- quate, as can be seen from Fig. 5D (see also Table 111). The exchange between the interstitial and intra- cellular compartments (k,,, k 4 , ) is also in this case unreliably determined, while the transcapillary ex- Upsala J M e d Sci 78 178 G . Arturson et aI. Table 111. Parameter estimates for 24Na using model I b and 3 f o r the analysis of dogs I , 3 and 4 Table IV. Parameter estimatesfor 4aK using model 3 in the analysis of dogs I , 2, 3 and 4 Dog 1 Model 3 with Model Model con- Parameters lb 3 straints Dog 3 D o g 4 Model Model lb lb k,,x Vl/Vz(min-1)0.13 0.32 0.17 k l z + k,, (min-l) 0.14 0.34 0.18 k,, (min-l) - 1.45 0.60 kza (min-l) - 1.05 2.10 TLAG b i n ) 6 6 6 v4/ v, - 4 frozen 4 frozen 0.09 0.16 0.11 0.17 Derived quantities 1 . 1 1 . 1 1 . 1 1.2 1 . 1 - 2.9 14 frozen - - C d C I CIICC change ( k , , , k , , ) and interstitial outflow (k,,) remain reliably determined. In fact, even the simplest model l b (see Fig. 3) is compatible with the tracer data and, according t o Fig. 5E, the parameters k,,+k,, and k z l ( Vl/ V,) are reliably determined. 42K A three-compartment system in series (model 3 in Fig. 3) was used in the analysis of the 42K data. This model differs from the open two-compart- ment model introduced by Conn & Robertson (S), and used by others (11, 15, 17, 29) in various re- spects. The plasma pool has not been considered as being infinite in size and has not been assumed to be in rapid exchange with the interstitial fluids, but has been described by an estimated “capillary plasma” forcing function. The governing function C: (the “capillary con- centration”) was estimated as a mean value of the concentrations in arterial plasma and cardiac lymph (coronary lymph -coronary venous plasma; cf. (1 1)). This approximation seems justified for all the ions studied in view of their rapid transcapillary exchange (cf. 30), but was only considered for potassium be- cause of the approximation for this ion has the greatest influence (see Figs 4 D , E and F). The parameters (klz + k O J , ( k d V,/ V d ) , k24 and k p 2 were estimated by fitting C,* to the lymph-tracer data, with the assumption that these data were representative of the interstitial fluids. The ratio V4/ V, does not influence the C,* values, but only the values of the intra-cellular compartment. The least- squares estimates are given in Table IV for all the Upsulu J M e d Sci 78 Parameters Dog 1 D o g 2 D o g 3 D o g 4 k , , x V l / V 2 (min-l) 0.16 0.045 0.03 0.11 k,, + k,, (min-l) 0.19 0.052 0.03 0.14 k,, (min-l) 1.38 0.194 0.22 0.17 k,, (min-’) 1.26 0.192 0.34 0.81 TLAG (min) 0 6 3 9 Derived quantities C&I 1.2 1.2 1.0 1 . 3 dogs and the corresponding simulated curve C,* is shown in Fig. 4 F for dog 2, compared with the experimental data. Model 3 was accepted as being compatible with the tracer data. The estimated parameter values are greatly dependent on the approximation of the “capillary concentration”, as tested by repeating the calculations, using the pri- mary plasma concentration values; the parameter values were then changed by factors ranging from 5 to 10. The steady-state equation for the intra-cellular compartment is Thus, in order to fulfil the constraint Cc N 30CI (cf. 13), the volume ratio V I / V c must be set equal t o -30 for dog 2. This is in conflict with common knowledge (V,-4VI, cf. 8). As was found in the analysis of the ,,Na data, model 3 was not adequate, because of the large uncertainty in the estimated parameter values kZ4 and k4,. The same conclusion applies t o model 3 for the description of the 42K data, as can be seen from Fig. 5 F , where the 95% confidence limits for the parameters k , , and kq2 are shown for dog 2. An attempt t o fit model 3 t o the data of dog 2 with the constraints C, - 3ocI and Vc-4VI did not produce an acceptable agreement with the experimental data. By adding a fourth compartment in series (model 5 in Fig. 3), it is possible to fulfil the concentration and volume constraints; the steady-state equations for compartments 4 and 5 give the relations (9) Electrolyte exchange kinetics in the dog’s heart 179 With the assumptions that C, = C, - 30 C, and V,+ Vs-4 V, it follows from the relations (9) and (10) that for example, for dog 2, where k , , - k 4 2 , k,, - 120k45, V , - 3 0 V 4 and v5-120V4-4v2. A spectrum of possible solutions thus presents itself, the limits of which are determined by the un- certainty of the parameter estimates of k,, and k , , (see Fig. 7F). The introduction of a special lymph compartment (model 4 in Fig. 3) does not present any new, or simpler solution of the present problems. The uncer- tainty concerning the exchange on the cellular level remains. DISCUSSION The studies presented here show that several com- partmental models for sodium and potassium ex- change (model 3 and model 5 ) could easily be con- formed so as to be compatible with the tracer data and it was also possible to make them consistent with independent relevant data from the literature. However, it was not possible to construct adequate models for the data, in the sense that the model parameters were reliably determined by the data. As the plots of the 95% confidence regions show (see Figs. 5 C, D, F), the uncertainty in the rate constants between the interstitial and the intracellular compart- ments is too large to permit any far-reaching conclu- sions. The tracer data only allowed for the estimation of transcapillary exchange for albumin and sulphate ions. The results so far are consistent concerning (1) the slower exchange of albumin between plasma and interstitial fluids, compared with the correspond- ing values for the sulphate ion (k=0.0011 +0.0002 (2 S.D.) min-l for I-albumin (dog 2) and k =0.0062 0.0002 min-’ for SO, (dog 2)), and ( 2 ) the influx rate of albumin and SO, to plasma (4-7 Yo and ca. 100 %, respectively, of the efflux rate from plasma). The coefficient of transfer of albumin across the capillary walls was here estimated to be 0.0011 0.0002 min-I (dog 2), i.e. about 0.1 Yo of the albumin in the plasma pool crossed the capillary walls each minute. This value is appreciably smaller than the values of 1 4 % previously reported from a similar study (2). The turnover time or the average time the albumin molecules spend in the interstitial space (Ilk,,,) was here estimated as 53 minutes for dog 2, a value which is also significantly smaller than the mean value of 257 minutes given by the same authors. The differences are due to the circumstance that these authors regarded the plasma pool as infinite, which was not the case in the present study (cf. page 173). The barrier t o potassium exchange has been claimed to be localized in the cell membrane by some authors (8, 1 1 , 15, 29), while other authors (23, 30) have presented some evidence for its localization in the capillary wall. As pointed out above, our 4 2 K data do not allow any detailed conclusions about the exchange between interstitial and intracellular spaces, but the estimates of the rate constants, taking into account their large confidence limits (see Table IV and Fig. 5F), still suggest that the barrier is situated at the capillary wall. However, it should be noted that it was necessary to introduce one more com- partment in series (see model 5 in Fig. 3 and the comments on page 178), in order to achieve compati- bility with both 4 2 K tracer data and data from the literature concerning the concentration and distribu- tion relations in interstitial and intracellular potas- sium spaces. Whether this auxiliary compartment should be interpreted as a second intracellular com- partment (cf. 24, 29) or as a reflection of slow mixing in an inhomogeneous interstitial compartment (cf. 20), cannot be answered by this study. It is thus also impossible to localize more precisely the potassium- exchange barrier. The analysis of 24Na data presented similar prob- lems to those discussed above for potassium (cf. also Sjostrand (25)). The data could not resoIve such com- plex models as discussed by Conn & Wood (9) for sodium exchange. In fact even the simplest two-com- partment model (model 1 b in Fig. 3) was compatible with our data. No adequate model could be found for the transfer kinetics of this substance. The uncertainty of the rate constants connecting interstitial and intra- cellular compartments was too large (see Fig. 5D) and could not be reduced to a meaningful level by introducing the further constraints presented by literature data on the concentrations and distribu- tion volumes of sodium (see Fig. 5 D). However, the 95 % confidence limits of these rate constants include the point estimates given for the same parameters by Conn & Wood (9) for a number of dogs. In summary, it can be concluded that 1 . experimental data of the type presented here Upsala J M e d Sci 78 180 G . Arturson et al. a n d previously used by other authors in the study of electrolyte kinetics in the dog’s heart, cannot be described adequately in terms of a few distinct compartments; and 2. accordingly the same data, d o not provide suf- ficient information for a reliable estimation of cellular exchange of sodium and potassium; 3. the transcapillary exchange of albumin and sulphate ions can be reliably estimated from the corresponding tracer data and compartmental mod- els, which are adequate for these substances. ACKNOWLEDGEMENTS T h e authors thank Prof. Lars Garby for valuable criticism, a n d C. Hallin, G. Montin, A. Persson, A. Wallenstll, B. Westerberg and B. d s t m a r k for skilful technical assistance. The investigation was supported financially by the Swedish Medical Research Council, grants B70-14X-2871-01A, B71- 14X-4252-01, and 0. and E. Ericsson’s Research Foundation. 14X-2871-02B, B72-14X-2871-03C, K73-17P-4141-01A, B74- REFERENCES 1. 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