Computer study of iodide transport in thyroid slices in vitro

COMPUTERS

Computer

AND

8,405-422 (1975)

BIOMEDICAL

RESEARCH

Study

of Iodide

Transport

in Thyroid

Slices

in Vitro*

F. R. L. CANTRAINE~ Institut de Recherche Interdisciplinaire en Biologic Humaine et Nucle’aire (Biotnathematical Group), Faculty of Medicine, Free University of Bnrssels, Belgium

Model

AND

A. M. JORTAY Department of Surgery, Institute Jules Bordet, Faculty of Medicine, Free University of Brussels, Belgium Received February 1,1974 We describe a model for iodide transport in thyroid slices incubated in vitro. This model is based on the anatomical structure of the slices and the experiments of Doniach and Logothetopoulos. Assuming that the flows are instantaneously adjusted to the pools, we obtain a compartmental model. Its stationary state is demonstrated and its compartmentalization is justified by the rapid diffusion of iodide, The sensitivity analysis determines the best period of time for the determination of the turnover rates. The simulation of the radioactivity curves in the “cell” and “lumen” compartments suggests an experimental test of the kinetic model by autoradiography. INTRODUCTION

The thyroid gland traps iodide from serum (3) and sets up a concentration gradient between glandular and circulating iodide (4). In uiuo, this gradient is expressed by the thyroid to serum radioiodide (T/S) ratio (5), and in vitro, by the thyroid to medium radioiodide (T/M) ratio (4-7). Iodide trapping is carried out by the thyroid cells encircling the follicular lumen. It is subsequently bound to thyroglobulin. However, the effect of the uptake process on iodine binding to thyroglobulin has not yet been determined (8). One method to study the iodide exchanges between thyroid and plasma is to block organification with antithyroid drugs such as propylthiouracil (PTU) or methimazole (MMI). Such methods have been used both in vivo and in vitro. Some of * Work realized under Contract of the Minis&e de la Politique Scientifique within the framework of the Association Euratom-University of Brussels-University of Pisa. This work is part of a Ph. D. thesis. t To whom inquiries should be addressed. 405 Copyright 0 1975 by Academic Press, Inc. All rights of reproduction Printed in Great Britain

in any form

reserved.

406

CANTRAINE AND JORl-AY

the more significant in vivo experiments have been performed by Wollman and Reed (4). The necessity to use an animal, for a single experimental point, has hindered the definitive kinetic demonstration of the existence of two iodide compartments I luminal and cellular. In vitro methods are better suited for the study of iodide transport. W~olH‘(9). Scranton (IO), Surks (I I) and our group (I2,13) have incubated sheepor dog thyroid lobes or slices.Numerous samplescan be obtained during the incubation procedure. Thyroid

slices

Medium

Total

medium

( 1 gland KRP

1

(27mt

I

: 30 ml Incubatorlshaking Ltic pump

KR P:

FIG.

Krebs Ringer Phosphate glucose 8 mM methimazolo 2 mM K I 1 pM 1 specitic activity

buffer

1.Apparatusfor continuousrecordingof the

300

100/mm

1 4 mt/min

’

)

I pH 7.L )

C/M

medium

1

radioactiwty.

Nevertheless the insufficient number of experimental points, the animal variability and the thyroid heterogeneity induce such variations in the T/M ratio that a precise mathematical analysis is impossible. Recently, Jortay et al. (22, 13) have developed a technique of continual measurement of iodide uptake in dog thyroid slicesincubated in vitro. This method appears to be sensitive and accurate (Fig. 1). The purpose of the present study is to demonstrate how to handle the data thus obtained by compartmental and sensitivity analysis. We propose mathematical solutions to the generally accepted physiologic models. In addition, we suggest new methods to check the validity of the proposed model. The present analysis can be applied to any three-compartment system. EXPERIMENTAL METHODOLOGY(13)

(Figure I )

Thyroids from dogs pretreated with thyroid extract for three days were cut in sliceslessthan 0.5 mm thick and preincubated at 37°C for 60 min in an iodide-free

IODIDE

407

TRANSPORT

Krebs Ringer phosphate (KRP) medium. 500 to 1000 mg of the preincubated slices were then incubated at 37°C in 30 ml KRP with methimazole (2 mM). The flask was continually agitated at 100 strokes/min. Stable (12’1, KI- 10m6 M) and radioactive (1311 or 1251)iodide (50 to 100 PC) were added to the medium prior to immersion of the slices. A fraction of the medium was continually circulated through a polyethylene coil placed in a counting well with the aid of a peristaltic pump. The radioactivity in the coil was recorded in the memories of a multichannel analyzer (Intertechnique SA40B) every 80 set for 8 hr (Fig. 2). KRP METH

ACTIVITY

IN THE

MEDIUM

1131

NaCl0~

100103 ; IH

FIG.

,L 2H

3H

LH

5H

2. Typical experiment showing evolution of the radioactivity

6H

7H

in the incubation medium.

MODELS

Physiologic Model (Figure 3)

The thyroid slice consists of numerous follicles separated by connective tissue. Iodide diffuses and is trapped by the cells bordering the thyroid follicles (14). From there it either enters the follicular lumen, or returns to the incubation medium. We choose the following three compartments system (6) (Fig. 3) : the first compartment is the medium, including the connective tissue; the second compartment includes the cells; the third compartment is constituted by the follicular lumens. Mathematical Model (Figure 3) I?, = -k,, Rx f k,, Rz 2, = k,, RI - (klz + kx) Zt, = k,, R2 - k,, R,.

R2

+ k,,

R3

(1)

408

CANTRALNE AND lORTA\r PHYSIOLOGICAL

MODEL

MATHEMATICAL

MODEL

FIG. 3. Physiologic and mathematic models. If the model ISvalid the flows between medium and lumen are nil. The R, represents the radioactivity in the i compartment; the ki,i the turnover Hereafter the system will be written under matricial form IAl = lIKiI*JRI.

rates. (2)

ESTIMATION OF THE PARAMETERSOF THE SYSTEM Initially, the compartments S2 and S, are empty of stable iodide. They have been discharged of their stable iodide during the preincubation. As the tracer and the stable iodide (the tracee) move simultaneously and identically. the specific activity of each compartment (Ai = RJS,) is constant and identical to the medium specific activity. Therefore 1311in each compartment is proportional to the 12’T mass. Since the volumetric flows and the volume of the compartments are fixed by the structure of the slice, the turnover rates are constant (15). In contrast to the classical theory (1618), in our model the masses and mass flows vary contemporarily and in parallel while the specific activity remains constant. Experimentally, we measure continually the residual activity of the medium. Since the three compartment system is closed, the radioactivity curve is described by two exponentials and one constant; we normalize to the injected dose R. X,(O) = R1(0)/Ro = I and we write: X,(t) = Ae-“’ + Be-l” + C. We estimate the parameters of the system by identifying the coefficients obtained by exponential fitting with the formal expression of amplitudes A, B, Cfor the initial conditions : X,(O) = 1, X,(O) = X,(O) = 0.

IODIDE

409

TRANSPORT

which gives C=l-A-B=k12k,,/afi

(3)

A=-a2+ ah +k32+b3)- k12 k23 a@- 4

(4)

The equations binding the parameters of the system to the exponents of the exponentials are obtained by considering the roots si of the matrix I/K - s/I -TrllRlj=

~~~si=a+B=kll+k~l+k23+k32

(5) (6)

01

When reporting Eqs. (5) and (3) in Eq. (4) it becomes kzI = Aa + BP.

(7)

When taking Eqs. (7) and (3) and substituting them in Eq. (6) it becomes

When taking Eqs. (3) and (8) and substituting them in (4), taking (7) into account, k

12

=a2A+B2Bek k 21

21,

From Eq. (3) we draw immediately k,, = (1 - A - B)aj?/k,,.

(10)

From Eq. (5) we then draw k 32

--

(11)

Table 1 presents a series of typical constants from some experiments. MULTIEXPONENTIAL

FITTING

OF EXPERIMENTAL

CURVES

Correction of Data for Systematic Errors The recorded data are corrected for the following systematic errors : background, dead time, and radioactive decay, according to the formulas proposed in (12). It should be noted that in our case the correction for radioactive decay is legitimate as the system studied is homogeneous (19). Methodr for Exponential Curve Fitting When the sampled data are equally spaced in time and without error, the exponential curves are easily determined by the method of Prony (20,21) or by the method

39.55 975.8

27.02

1073 81.16 971.3

40.93 1006

27/2/70 cr

14/3/70

cr 21,‘3/70 0

15/5!70 (T

4.804 0.210

0.862 4.814 0.098

11.06

2.191 0.065

9.207 4.621

75.24 922.9

681.7 179.8 951.5

154.2

72.70 391.7

24.92 1676

0.769 0.012

0.007 0.876 0.005

1.189

0.326 0.012

1.745 0.106

113.1 178.3

1.11 3.1

5.6 1.27 1.3

130.4

1.59

0.517 1.5

0.442 24.4

171.7 116.6

121.7

101.1 879.6

180.1 132.5

2.78 5.0

2.42 2.5

11.1

5.72

1.27 3.0

5.28 54.2

1.03 4.8

1.31 2.2

5.0

4.02

0.461 4.6

2.63 67.7

0.656 1.7

3.6 0.693 0.7

0.924

0.266 4.6

2.61 13.2

MODEL"

4.09

1.75

3.30

1.30

3.97

2’4

227

172

221

158

16 7

65.5

15.5

15.3

4.29

0.47s

0.624

0.729

0.290

0.263

0.578

0.493

0.549

0.377

1.700

a The amplitudes are expressed in 1000 counts. 80 set: x, /I, and /c; as hr ’ J = [l (A, -- !I) 1 [(VI h 1 -.-. J‘,,p ,I2 ,VLri, $1, iL,, = IlLlIT 1b er ddata points. tt numbrr ol’ajjustable parameters, 5 for S2. 3 fur S, I’hc szwdard deviations of amplitudes A, B. C arc expressed in counts,80 SK; the standard deviation\ ofr. /{as hr ‘. and X,, in pcrccnt .S,. /i,‘,. Xi1 correspond to the parameters of a single compartment th) roid model wesented for comparison nith the TV o compartment model.

5.379 1626

20/2/70 D

1

FITTEDPARAMETERSOFTHETHYROID

TABLE

I <

3 g

E >

?

n is

IODIDE

of Gardner (22). These methods in each experimental point. We which allows us to impose the method does not necessitate the of Newton-Raphson (23).

411

TRANSPORT

cannot be applied in practice because of the errors have used the iterative method of Gauss-Newton fitting by a sum of decreasing exponentials. This calculation of second derivatives as in the method

First Approximation of Parameters The method of Gauss-Newton requires a first approximation of the parameters to initiate the iterative procedure. These initial values are obtained by graphical peeling in semilogarithmic coordinates. This method leads to a progressively increasing error resulting from the accumulation of errors in the slower exponentials. Moreover, on semilogarithmic paper, this error is systematic as the linear distance between the experimental and the theoretical curves does not have the same weight all along the curve; it is thus necessary to improve the values estimated by the peeling method. Exponential Fitting We have to fit f

(tj,

c,,

&)

=

Co

+

t:

ecAktJ

c,

k

to the experimental data using the first approximations obtained by the peeling method. Say dyj, the difference between the theoretical and experimental values at iteration I of the function at pointj, AYjl

=

Yj

-f

(tj9

ckl,

(12)

&cl).

By expanding the fitting function to first order in a Taylor series and substituting the result into Eq. (12) (13)

This system of equations resolved for all the points is, in general, incompatible. In contrast, when minimizing the sum of the weighted square of the errors

with regard to the parameter increments, we obtain one system of equations where AC,, l+l and 4 ltl are solutions. When cancelling the partial derivatives with respect to any parameter asladn, ,+1 = 0

as/a&.,

,+1 = o

412

CANTRAINE

AND

JORTAI

we derive the system of equations jAlsidals

= IDi,

(15)

where

and

&denotes the components of the parameters (C,, 2,). At the iteration f, we calculate the vector of the parameter increments {AC,, l A&, l}. These values are added. to the parameters {ck, 1, i k, 1 } defining their new values. The iterations are achieved when ICsj - sj+l)/sj+l I < lo-” /AC,, l+I/Ckl and /A&, I+l/&( < IO-”

for each k.

or when the apriori fixed maximum number of iterations is achieved. All the curves were resolved as the sum of two exponentials and one constant (Table 1). Errors on the Parameters of the Exponential Curves

The unbiased estimates variance-covariance Ok

2=---...- ’ N,-n

B

of the parameters equal

Gkk’ = N,-n’

kk

Bkk”

N, being the number of experimental points, n the number of adjusted parameters and Bkk, the element kk’ of the inverse matrix (15) (24). Table 1 gives a series of typical constants and their standard deviation. The “confidence interval” is ak -

uak

t4/2

< Ak

<

ak f

auk

tpizy

where tp12 is the 1008,2 percentage point of the t-distribution of N, .- iz degrees of freedom. The ?,, are accurately determined, but the amplitudes have a larger confidence interval. Usually, the best accuracy is obtained for the constant, followed by the parameters of the slower exponential and finally the faster exponential. This is due to the number of points at our disposal for each exponential. Errors on the Parameters of the Model

Having the variance-covariance matrix on the Ck, & we determine approximately the error on the parameters of the model by using the following relationship (24). 2- --* Ok IJ

s

ND-n

ak

,

cc

j ak ij B

m 32,

acr, h’

(17)

IODIDETRANSPORT

413

where CQreplaces C, and I,. This general formulation allows the calculation of the error of a model’s parameters (kij) knowing the explicit relations between the kij and the (C,, &); the variances are reported in Table 1. According to Atkins (25), the relative errors on the kij are of the same order of magnitude as the ones obtained on the exponential parameters. SENSITIVITYOFTHEMODELTOITSPARAMETERS

To guide future experiments, let us consider the sensitivity of the model’s solution to variation of its parameters. We define the sensitivity as the derivatives of the model’s solutions with regard to the parameters:

When deriving the two members of the system (2) with respect to parameters kij it becomes : la&i3kijj = Ilak/akijll - /RI + I/K;/* laR/akijl.

The parameters are time-independent form

(18)

and thus we rewrite the system under the

Idtij/dt( = !IKli * 14ijj +

liW%jll */RI.

(19)

Our system being linear in the parameters k,j, ili3k/akijll include only terms equal to 0 and two terms equal to +I and -1, respectively. The sensitivity of the solutions to the parameters klz, for example, will be obtained by integration of the system (19) which is reduced to

dt:, - dt = -kx t:, + k,, t:, + Rz, dt:, = ka t:, - (k,, + kz) i”:, + k,, t:, - Rz, dt

Pa

d5:, Equations (20) with their initial conditions t!, = 0 (i = I, 2,3) constitute one of the cosystems of the prototype system (2) according to King (26). Figure 4 gives the sensitivities of the curve R, to the model’s four parameters. Sum of the Sensitivities of Solutions to One of the Parameters

The sum of the sensitivities of solutions to one of the parameters is zero. Indeed, our system being closed, the sum of the elements in each column of the matrix /I#lj and of its derivatives is zero. As a consequence, the sum of the sensitivities is constant upon time. As each term equals zero initially, the sum is zero at each moment.

414

CANTRAINEANDJORTAY

I\, - 100000

_---- 21

-00 L

i-M-----

u.dO

i .2a

2 .40

3 30

7----6a 4 30

H

FIG. 4. Sensitivities of the medium radioactivity curve to the parameters of the model (& = aR,/3kij) in the third (a) and fourth (b) experiment (Table 1).

Asymptotic Sensitivities Asymptotically, the sensitivities are solutions of the system liKli-15ijl

=-iitX/cYk~j/l*IRI.

Since the Ri(co) are nonnuli constants (closed system), the asymptotic sensitivities are constant. In the case of our model : rY:z= (kzdkd Gz Rz(aJ) + $$Y;z). 5VI12 = k121i SENSITIVITIESOFTHEMODELSOLUTIONSANDVARIANCESOFTHEPARAMETERS

We will now show that the sensitivities of a model to its parameters determine the precision one can expect in the determination of the model parameters.

415

IODIDETRANSPORT

The difference dy, between the experimental fi(ti, k) at the first order is given by

point yi and the theoretical point

Ay, = 2 (afipk,) 6k, + E. 1 The criterion to be minimized is then written 2

i(

S = 2 Wi Ay, - $ (afi/dkl) 6kl

)

*

The criterion gives a modified x2, taking for Wi the inverse of the experimental value at point i. At the nadir of the criterion, when the quadratic form S is stationary, the normal equations are written

The sum of the products of the sensitivities appears in the left member; in the right member, the sensitivities to the parameters of the fitted curve occur as a weighed function of the difference between the experimental and fitted curves. Therefore, the most sensitive solutions of the model determine best the parameters. The precision of the parameters kzl is almost the best (Table 1) in experiments 3 and 4 according to Eqs. (21) and to the sensitivity curves of Fig. 4. The sensitivity analysis is of interest because it determines apriori the informational content of each point of the expected curve. Indeed, by analogy with Eqs. (16) the variance-covariance matrix has its terms proportional to Olj

N

11 F Wi (li

(21)

(jii~-‘*

It should be noted that if one sometimes refers to the logarithmicsensitivity when comparing the sensitivities (26) one must consider the natural sensitivity when fitting. FAMILY OF 3 COMPARTMENT

SYSTEMS COMPATIBLE

WITH THE EXPERIMENTAL

DATA

Experimentally, the evolution of the radioactivity in the medium is recorded, After normalization to the injected dose R,, the radioactivity is written xl = 2 = Ae-“’ + Be-fit + Cc-Y’. 0

To find the general solutions of the system several constraints must be taken into account :

(1) We determine 6 independent parameters A, B, C, GI,/I, y by fitting. (2) Initially: X,(O) = 1 and X,(O) = X,(O) = 0.

416

CANTRAlNE

AND

JORTAY

(3) In our closed system, we have at each instant: The last exponential is a constant and thus y equals zero. The system possesses 12 degrees of freedom, nine amplitudes and three exponents. Having 10 conditions on the amplitudes and exponents, the system still possesses two degrees of freedom. The general solution will thus be of the type X, = Ae-” + Be-“’ + ( 1 - A - B) X2 = -(A - x) cent - (B-y)e-“‘+(A+B-x-y) X3 = -xe-‘“’ To generate the matrix relation (26)

(22)

- ye+’ + (x + 4’). of the transfer

coefficients

of the system. we use the

!iK;: = - ~A~~+~j.,jA~~-‘.

(23)

where A is the matrix of the amplitudes and A the diagonal matrix of the exponents

(a, P,0). Say P = Ay - Bx, IlAl(-l becomes: P PP The relation (23) leads to the following i

(Aa + BP) P ((x-A)a+j?(y-B))P -(BY + ax> P

P-y

P-js+B P

matrix of turnover rates

zA(P-y)+ B(P+x)fi (x-A)a(P-y)+(.rB)/?(P+x) -XX(P-I’)-&(P$x) zA(P+B--y)+ BP(P+x-A) r(x-A)(P+B-y)+(y-B)p(P+.r-A) -ctx(P+B-JX)-yfi(P+x--

.

(24)

A)

To get a compartmental matrix ;I/(,!, the diagonal elements must be negative; the others (kij with i #j) must be nonnegative. Figure 5 presents the lines liij in the (x, ~1) space. We eliminate the zones giving turnover rates incompatible with a compartmental system, and so we obtain the two hatched areas of this figure. The points in these zones correspond to the physically admissible systems. The existence of two zones reflects the multiplicity of the model’s solutions. corresponding to the permutation of indices 2 and 3 of the compartments. Inside each zone, the system presents simple degenerations (border points) or double degenerations (tops). This representation shows that some h-ij should not simultaneously vanish. In this manner, we obtain the radioactivity at the equilibrium for the different

IODIDE

TRANSPORT

417

systems compatible with the evolution of the observed medium radioactivity. In our system, the intersection point of lines k 13 = kS1 = 0 gives the analytical solutions of the system (22). At equilibrium, the amplitude (X + y) in compartment 3 must satisfy the conditions o
FIG. 5. Models compatible with the inequations obtained from the matrix (24). The lines limit the zone of points whose coordinates x and y fit the inequations derived from the matrix (24). The two hatched areas correspond to the points whose coordinates fit at the same time every inequation. The two peaks kjI = k13 = 0 and kll = k,, = 0 correspond to the model (Fig. 3) for the fifth experiment shown in Table 1. DISCUSSION

Discuss{onof the Hypotheses (I) Initial Conditions. We assumed that the cellular and luminal compartments are initially empty of iodide (23). In the course of preparation of the thyroid slices at 4°C most of the endogenous iodide is washed out into the medium by inhibition of the transport and by dilution. Moreover, the pretreatment of the animals with thyroid extracts diminishes the rate of proteolysis of endogenous organic iodine, and thus reduces the contamination with iodide resulting from intrathyroidal deiodination. Finally the preincubation in

418

CANTRAINE

AND

JORTA\r

an iodide free medium reduces the concentration of iodide in the slices and allows the release of colloid from the opened follicles achieved by mechanical agitation. (2) Stationary- State qf the System. We assumed that the turnover rates of the model are time independent. This hypothesis is based on the assumption that the system is linear and stationary as confirmed by two tracer injections. The kinetics of disappearance of lZ51and J311at a 5-hr interval are parallel on semilogarithmic coordinates (Fig. 6). CPi80sec ‘.

RADIOXTIWTY

L___-

..---..1

OF THE ‘NCUBATION MEDIUM

_~. 2

3

/.

H

FIG. 6. Illustrationof thestationarystateof the system. 13’1 and I23 radioactivitycurvesar a 5 hr intervalareremarkablysuperimposible after normalization.

(3) System compartmentalization. Compartmentalization of theisystem is justified Firstly iodide has a diffusion coefficient D in water of 2.10-’ cm’:‘sec (27). The

intracellular distribution thus becomes homogeneous within a few seconds. Moreover, the small size of the iodide anion does not impede its diffusion into the follicular lumen. Lehman and Pollard (28) have shown that iodide diffuses almost as rapidIy in a gel and in the cells as in water. Andros and Wollman (29) were unable to demonstrate by autoradiography an intracolloidal iodide gradient even 2 min after radioiodide injection. (43 Distribution of the Follicles Radii. The preparation of thyroid slices (500 pm thick) opens the follicles whose radii are greater than 250 pm. The larger peripheral follicles are cut during the slicing at random and are cleared from their colloid during the preincubation period. Histologically, the size of the follicles in the slices is similar. The importance of the follicular radius and of the diffusion is discussed elsewhere (30). Fitting of the Experimental Curves

Wollman (6) utilized a single compartmental thyroid model to determine the plasma iodide clearance by the thyroid. The existence of an iodide concentration gradient (I) between the lumen and the cellular compartment led him to propose a thyroid model with two compartments. The accuracy of our data allows us to check

IODIDE

TRANSPORT

419

the validity of such an approach. If some of our experiments are in agreement with a single compartment model, in such cases the fitted curve presents systematic differences with the experimental curve (short and long time). In support, we mention the turnover rates associated to such a model (Table 1). In this case kLl/ki2 = S,jS,. We can thus estimate the ratio of thyroid (S,) to medium iodide (S,) masses. The weighed sum of the squared differences obtained after fitting of two exponentials and one constant is strongly reduced in each case (Table 1). The fitted curve crosses on an average the experimental points. The square root of the mean relative quadratic difference is compatible with the expected fluctuation, taking into account the activity of the radioactive source. This two compartmental thyroid model agrees with the existence of two follicular iodide pools (cellular and luminal) suggested by Doniach (I). The radioactivity of these pools depends on the functional state of the gland. The gradient is more marked when iodide organification has been inhibited (2). Without supplementary physiological constraints this model already possesses two degrees of freedom, Eqs. (22). In order to keep a model coherent with the physiologic data we restrict ourselves to a thyroid model with two compartments (Fig. 3). Model Fitting We did not adjust directly the model to the data, because the multiexponential analysis allows a general treatment of all the models compatible with the observed residual radioactivity curve. If forthcoming verifications ensure a sufficient reliability of the model, the parameters will be fitted without multiexponential analysis. Here the turnover rates estimated by both methods are identical. The sensitivity analysis gives us two points of information: (1) the importance of the modifications of the solutions for a small increment of the parameters, e.g., the model is very sensitive to the variations of k,, which is related to the plasma iodide clearance by the thyroid studied by Wollman (6), and (2) the a priori precision which may be expected in the estimation of the model parameters. Discussion of the Model At the end of the experiment, the injection of antithyroid drugs into the medium (SCN-, ClO,-) allows an analysis by a perturbation method (31) their well-known effects (32). The experimental protocol does not give any information on the intrathyroidal iodide distribution. This distribution can be evaluated by autoradiography. Figure 7 gives the predicted total radioactivity curves in the two compartments of the thyroid. The lumen radioactivity equals the cell radioactivity at 2 hr, and is higher afterwards. This prediction corroborates the observations of Andros (29). If the ratio

420

CANTRAINEANDJORTAY

of the cell to lumen radioactivities differs from the ratio predicted by the model, we have to consider among the possible models those having a given radioactivity ratio. This is the ratio for which an asymptotic expression is written Eq. (22). gives the straight line y = [(I - C)/(l + pj] - x along which lies the systems presenting the correct radioactivity gradient. If this line passes through the point k13 = k,, = 0 of Fig. 5, we will obtain a confirmation of our model; otherwise, we will know which parameters have to be introduced.

MEDIUM

.-

--

LUMEN CELL

--

TIME 0

rxprrimenlal

computed

I HOURS 1

2 3 0 5 FIG. 7. Fitting of the medium radioactivity curve. The turnover rates are indicated. The radioactivities of the cellular and fuminal compartments, predicted by the model, were calculated. I

CONCLUSIONS

The proposed experimental investigation system allows an automatic continuous and precise sampling of the medium radioactivity. By extension of the compartment theory, the studied system is treated as a compartmental system. The turnover rates of the system can be solved by exponential fitting. The hypotheses about the stationary state, the compartmentalization of the system, and the initial conditions are satisfied. The good quality of the collected data is preserved. The precision of the turnover rates is of the same order as the precision of the data. Under the condition where the differences in follicular radii do not induce important kinetic differences among the turnover rates of the different follicles, we

IODIDE

TRANSPORT

421

propose that the two kinetic pools identified are the cellular and luminal thyroid pools. The study of the systems compatible with the disappearance curve of the radioactivity in the medium suggests a simple method for evaluating the adopted model, i.e., determination of intrathyroidal iodide distribution by autoradiography. These results will either confirm the model or indicate which new parameters must be introduced. ACKNOWLEDGMENTS The authors thank Dr. C. Delcroix, Dr. J. E. Dumont and Dr. P. Rocmans for their stimulating and helpful discussions. We are grateful to Dr. P. Rocmans, S. Refetoff and E. Schell for their help in the improvement of the style and grammar of the manuscript. We also thank Mrs. M. P. Ben Moussa and D. Legrand for their careful preparation of the manuscript. REFERENCES I. AND L~GOTHETOPOULOS, J. Endocrinol. 13,65 (1955).

I. DONUCH,

J. H. Radioautography

of inorganic iodide in the thyroid.

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