Estimation of Ferrokinetic Parameters by a Mathematical Model in Patients with Primary Acquired Sideroblastic Anaemia

British]ournaf of Haemlofogy, 1978, 39.409.

Estimation of Ferrokinetic Parameters by a Model in Patients with Primary Acquired

Anaemia

Ma thematical Sidero blastic

G. BAROX,* M. CAZZOLA,t s. MORANDI,* M. STEFANELLI~AND s. PERUCINIt

*CIinica Medica I , t ls t i tuto di Patologia Medica 1 and Slstituto di Elettronica, University of Pavia, Pavia, Italy

(Received 17 October 1977; acceptedfor publication 29 November 1977)

SUMMARY. Ferrokinetic parameters were estimated in eight patients with primary acquired sideroblastic anaemia (PASA) by means of a mathematical model of iron kinetics. Both plasma 59Fe clearance and red cell utilization curves were used to identify the model parameters, The close agreement between experimental and theoretical data proved that the model can describe iron kinetics in sideroblastic anaemia. Ineffective erythropoiesis was found to be the major factor in the production of the anaemia. Various degrees and patterns of peripheral haemolysis were observed. The storage iron was always markedly increased and a large non-erythroid iron turnover was calculated in the most severely affected patients. The results are compared with present knowledge of erythropoiesis in PASA.

Sideroblastic anaemias are a group of dyserythropoietic disorders in which the most distinctive morphologic feature is the presence of ringed sideroblasts in the bone marrow (Bessis & Jensen, 1965; Mollin, 1965). Their common pathogenetic mechanism is an impairment in the haem biosynthesis which results in the mitochondria1 iron overload of the sideroblasts (Cartwright & Deiss, 1975; Ponka & Neuwirt, 1975). These cells are functionally abnormal (Wickramasinghe et all 1968; Freedman et all 1976) and many of them are destroyed in the marrow. The ineffectiveness of erythropoiesis in sideroblastic anaemias has been demonstrated by studies of early labelled bilirubin and carbon monoxide production (James & Abbott, 1961; Barrett et a l , ~966; White et al, 1967; Horne et all 1975; Samson et al , 1976b) and by ferrokinetic studies (Dacie et al, 1959; Verloop et af, 1962; Bernard et al , 1963; Zittoun etal, 1965; Wickramasinghe et all 1968; Tanaka et a l , 1972; Wagner et all 1973).

Previous ferrokinetic studies in sideroblastic anaemias did not provide a quantitative assessment of effective and ineffective red cell production. Only recently, Ricketts et a1 (1975) developed a method which allows measurement of the degree and effectiveness of erythroid activity and the red cell lifespan. At the same time, we elaborated a very sensitive method for liquid scintillation counting of 59Fe in plasma and whole blood (Perugini et a l , 1974) and formulated a seven compartment model of iron kinetics for the interpretation of ferrokinetic

Correspondence: Dr G . Barosi, Clinical Medica I ‘A. Ferrata’, Policlinico S. Matteo, 27100 Pavia, Italy. 0o07-1048/78/07o0-04og$0z.00 0 1978 Blackwell Scientific Publications

409

410 G. Barosi et al

data (Barosi et a l , 1976). Both plasma 59Fe clearance and red cell utilization curves were used to estimate the model parameters by minimizing a suitable function of the difference between experimental data and model prediction. Surface counting over sacral marrow, liver and spleen were compared with model predictions in order to test the reliability of the obtained estimates. The model proved adequate in estimating effective and ineffective erythropoiesis, red cell lifespan and non-erythroid iron turnover both in normal subjects and anaemic patients (Barosi et a l , 1976).

This paper presents the results obtained within a group of patients with primary acquired sideroblastic anaemia (PASA) by our ferrokinetic method. The significance of the ferrokinetic parameters with respect to erythropoiesis is analysed.

METHODS Patients

consent. Clinical and haematological data are shown in Table I. The study was carried out on eight patients who had previously given their fully informed

TABLE I. Clinical and haematological data of the examined patients -

Serum bilirubin Weight H b MCV M C H C Retics total/unconJugated Serum iron/ TlBC

Patient Age Sex (kg) (gldl) (jl) (gldl) (1091 l . ) ( W O ~ I l . ) ( f l m 0 V . )

I . A.Bi. 2. M.Ga. 3 . A.Gi. 4. O h . 5. C.Me. 6. E.Ma. 7. E.Mo. 8. P.Zu.

80 F 59 8.1 67 F 52 8. I

5 3 M 74 8.7 so M 65 5.6

78 M 67 6.5 78 M 59 7.2 67 M 62 5.9

62 F 76 6.2

31.6 60.5 3 0 . 3 20.4 3 1 . 1 31.6 30.8 38.2 30.6 54.0 31.7 69.0 31.6 12.2

31 .0 29.0

22.3112.3

3 7.8124.3 21.6112.2 21.6115.9 24. 511 3.7

37.011 5.4 60.1149.8

9.81 6.8

The diagnosis of PASA was established on the basis of the following criteria. The patients were adults who insidiously developed a chronic refractory anaemia. All had normal or reduced reticulocyte counts and normal or increased plasma iron with increased transferrin saturation. Bone marrow examination demonstrated an erythroid hyperplasia, increased stainable iron and numerous ringed sideroblasts. No patient had previous exposure to drugs or chemicals known to produce sideroblastic anaemia. There were no other associated diseases and family history was negative.

Ferrokirietic Studies Ferrokinetic studies began between 10 and 12 a.m. 10-1s pCi of sterile [59Fe]ferric citrate

having a specific activity greater than 10 pCi/pg (Sorin, Saluggia, Italy) was added to 12-1 5 ml

Ferrokinetics in Sideroblastic Anaemia 41 1

of fresh autologous plasma. The mixture was incubated at 37°C for 30 min with intermittent agitation. The latent iron-binding capacity in the employed volume of plasma was at least 6 times greater than the amount of radioiron added.

An aliquot of labelled plasma (about 0.2 pCi/kg) was injected intravenously after saving a sample for the preparation of standards. Venous blood (about 3 ml) was taken into a heparinized syringe at approximately 10, 30, 45 and 60 min, 3, 6, 12 and 24 h after radioiron injection. Thereafter, blood samples were taken every day for 14 d. No patient had signs of intravascular haemolysis. Furthermore, plasma was separated with great care to avoid artifac- tual haemolysis.

Plasma and red cell radioactivity were determined by a method for liquid scintillation counting of 59Fe (Perugini et al, 1974). Counting samples were prepared using I ml of plasma and 0.2 ml of whole blood, respectively. The average background was 30 cpm and mean counting efficiency was 85% for plasma and 82% for whole blood samples. Each sample was counted until 20 ooo counts or for IOO min; a total count of at least 4000 was obtained on low activity plasma samples. Quenching correction was made by external standard ratio and the standard deviation of each activity was calculated.

The plasma activities were not corrected for variation in plasma iron concentration because the authors have observed that this correction does not modify significantly either the shape of the clearance curve or the estimates of the ferrokinetic parameters (unpublished observations).

The plasma volume (PV) was calculated from the extrapolated zero time activity and the amount of radioactivity injected. The whole blood volume (WBV) was calculated from PV and the body haematocrit according to Albert (1965).

The red cell utilization curve was computed according to Hosain et a1 (1967). Serum iron and iron binding capacity were determined by the method of Olson & Hamlin (1969). The haemoglobin concentration and haematocrit were measured using a Coulter S cell counter.

Surface measurements of radioactivity over precordium, liver, spleen and sacral bone marrow were made using a Meditronic RHA I I ratemeter with four narrow well collimated NaI (Tl) detectors. The probes were positioned over the organs according to Finch et al(1970). Determinations were continuously made during the first hour and then at precisely timed intervals during the following 14 d. The count rates obtained over each organ were first corrected for background and physical decay and then expressed as a percentage of their extrapolated values at zero time (normalized counting rates). Finally, the normalized fixed activities in the liver, spleen and sacral bone marrow at each time were calculated by subtracting from the normalized counting rates over these organs the normalized counting rate over precordium at the same time.

Combined Studies with 59Fe and 51Cr In patients M.Ga. and E.Ma. the ferrokinetic study was combined with the 51Cr red cell

survival determination. Autologous red blood cells were labelled with [51Cr]chromate according to method A of the ICSH Panel (1971). About 90 min after 59Fe injection, the

Cr-labelled red cells were injected intravenously. Blood samples were drawn thereafter according to the ferrokinetic protocol. The 59Fe and 51Cr activities in blood were simul- taneously measured using a liquid scintillation counting method (Cazzola et al , 1976). The red cell lifespan was computed according to Bergner (1962).

412 G. Barosi et a1

a,, *

THE MODEL

Physiological and Mathematical Model The model of iron kinetics considered in the present work has the same structure as the

physiological model proposed by Ricketts et al (1975). It consists of two main pathways, as shown by Fig I : the erythroid and the non-erythroid pathways. In the first pathway, iron enters the erythroblasts and either leaves the bone marrow in mature red cells or reaches the reticulo-endothelial system (RES) following the destruction of immature cells (ineffective erythropoiesis) .

a3,

2 3

- Non-Erythroid Tissues

A

a72 1:: pla;ma

Iron in circulating red blood cells (RBC) returns to the plasma as a result of haem catabolism in RES. In the model in Fig I , RES is represented as a distinct compartment indicated as the haemoglobin catabolic system (HCS). The non-erythroid pathway involves a storage compartment exchanging with the plasma through an intermediate one. This latter represents both the extravascular circulation and tissue iron which exchanges rapidly with plasma. The model presented here differs from the model of Ricketts etal(1975) in what follows. It is not assumed that iron leaving the marrow in mature red cells will not return to plasma during the 14 d of the ferrokinetic study. The dynamics of red blood cell production is described by a two compartment model with the aim of introducing a delay to account for erythroid development. This functional subdivision of the erythroid compartment constitutes a first step towards the formulation of a distributed model which seems able to provide a more adequate description of the process (Colli Franzone et al, 1976).

The dynamics of the HCS is described by a single compartment model under the assumption that reticuloendothelial cells in marrow and in spleen behave identically. A two compartment model of HCS would lead to the introduction of another fractional transfer coefficient from HCS to plasma which could be estimated only with rather poor accuracy. The subdivision into spleen and marrow HCS is considered only in expressing in vivo measurements in terms of the model variables (see Appendix).

1

H C S

7 a17

Ferrokinetics in Sideroblastic Anaemia 413

A mathematical analysis of the kinetic model shown in Fig I is reported in the Appendix. The model parameters were estimated by minimizing a suitable function of the difference between model prediction and in witro measurements. Details about the employed numerical procedure are reported elsewhere (Berzuini et al , 1977).

Clinically Useful Parameters Derivable from the Model From a clinical point of view, the behaviour of internal iron exchanges derived from the

stationary iron content of all compartments is of great interest. Thus, assuming that the same model is suited for describing both iron metabolism and radioiron kinetics and that the fractional rate coefficients a,, (see Fig I ) have the same values for labelled and unlabelled iron, the size X, of the ith compartment can be evaluated (see Appendix).

Then, the following ferrokinetic parameters useful for diagnostic purposes can be derived (Barosi et al , 1976):

Plasma iron turnover ( P I T ) = (a2* +aJl ) X I pmolld Marrow iron turnover ( M Z T ) = U ~ ~ X ~ pmol/d Red cell iron turnover ( R C Z T ) = U ~ ~ X 3 pmol/d Ineffective red cell iron turnover (IIT)=azl X 1 pmol/d

This quantity may be more conveniently expressed as a percentage of MIT:

ZZT%=[I - ( U ~ ~ X ~ / U ~ ~ X ~ ) ] I O O

D a i l y haemoglobin synthesis ( D H S ) g/d

considering that I g of haemoglobin contains 62 pmol of iron.

Mean red cell lifespan ( M R C L ) = (RCV x MCHC x 6 2 ) / ( ~ 4 3 X 3 ) d

where RCV is the red cell volume in dl and MCHC is the mean corpuscular haemoglobin concentration in g/dl.

Notz-erythroid tissue iron turnover ( N E I T ) =as X1 pmol/d

In order to compare data from different subjects, PIT, MIT, RCIT, IIT and NEIT are expressed as pmol/l. blood/d, and DHS is expressed as g/l. blood/d.

RESULTS

The estimates of model parameters obtained from the ferrokinetic data of the eight patients with PASA are reported in Table 11. The computed values of the clinically useful parameters previously defined are summarized in Table 111. Mean values and standard deviations of the estimates obtained in normal subjects (Barosi et al , 1976) are also included in the tables.

As shown in Table 111, the following disturbances in iron metabolism and erythropoiesis were observed in all the patients: a variable increase in plasma iron turnover, a high degree of ineffective erythropoiesis and an increase in iron stores. The MRCL could be calculated only in the six patients who had not been recently transfused: it was nearly normal in two patients and markedly reduced in four patients.

414 G. Barosi et a1

TABLE 11. Model parameter estimates

Fractional transfer coefficients (d-l)

Patients a21 a51 0 1 5 a17 a 3 2 a72 a43 673 a65 a56

Group A A.Bi. M.Ga. A.Gi. 0 . Iv . C.Me.

Group B E.Ma. E.Mo. P.Zu.

34.98 3.20 2.m 0.62 0.99 0.61 0 . 1 1 0.17 1.99 40.87 1.92 1.98 0.23 1 . 0 1 1 .38 0.26 1.43 1.99 18.93 1.60 2.84 2 .15 1.01 0.76 0.012 0.067 2.20

32.56 1 . 5 0 2.14 1.46 0.34 0.44 0.021 0.063 2.60 37.69 1.80 2.10 0.86 1 . 1 5 0.68 0.030 0 .15 2.50

7.66 3 .01 2.29 2.89 0 . 3 5 0.037 1 . 3 8 0.66 1.37 12.91 4.86 0.96 1.45 1.06 0.87 0.025 0.19 2.18

9.74 3.48 0.70 1.67 0.90 1.56 0.019 0 . 1 1 2.34

Normal subjects Mean 7.95 2.64 0.55 1.09 0.54 0.07 0.50 0.01 1 . 1 3 0 . 3 5 SD 1 . 3 1 0.83 0.32 0.25 0.14 0.16 0.12 0.014 0.25 0.43

O n the basis of the computed PIT and NEIT and of surface activities, the examined patients were divided into two distinct groups. The first one (group A) was characterized by a PIT value markedly greater than the normal mean value (from 5.5 to 8.7 times). Such values were chiefly due to a marked increase of MIT (from 6.7 to I I times the normal), while NEIT was less than twice the normal. The second group (group B) included patients showing PIT values lower than those calculated for the patients belonging to group A, but 3-4 times that of normal. In this group, NEIT was about 4 times that of normal.

In all patients, estimated marrow iron content ( X , +X3 +A2,) was consistently increased (from 2 to 17 times) with respect to the normal. The computed values of DHS were normal to slightly increased in the patients with a reduced MRCL and low in the patients with a normal MRCL. IIT% ranged from 39% to 95%; patients with higher values had milder degrees of peripheral haemolysis.

The computed spleen iron context (X7 -X7) was significant only in those patients having a shortened MRCL.

As indicated in Table 111 in all cases except one, the size of compartment 6 (storage) was greater than xo6 pmol, since a value of aS6 less than I O - ~ d-' was computed. Thus, compartment 6 behaved as a trap for iron. This result has not been obtained from ferrokinetic data of normal subjects or patients with aplastic anaemia (Barosi et a l , 1976).

The ferrokinetic patterns of the two groups of patients showed interesting differences. The plasma clearance curve in the patients of group A presented a rapid initial clearance rate and plasma activity ranged from 0 . 3 I '3'0 to I .06% of the initial level at 6 h (normal range, 2-10%).

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fourteenth day about 0.5% of the injected activity was still present in the plasma. A clear hump was not present in these patients during the first days. In Fig z(a), the experimental plasma 59Fe clearance curve of M.Ga. (full circles) and E.Ma. (open circles), belonging to group A and B respectively, are compared with the prediction provided by the model. Secondary elevations in plasma activity were present in patients in either group with marked shortening of MRCL.

FIG 2. (a) Plasma 59Fe clearance and (b) red cell utilization curves in the patients M.Ga. (0) of group A and E.Ma. (0) of group B. Dots represent experimental data and lines the model prediction. The shaded areas represent normal ranges.

In most patients the red cell utilization curve increased slowly during the study and reached a markedly lower value than the normal on the fourteenth day. Only in patient E.Ma. having an increased DHS and a consistent peripheral haemolysis did the curve show a rapid initial increase, followed by a slower one. The red cell utilization curves, both experimental and theoretical, shown in Fig 2(b), illustrate the two different observed patterns.

If it is assumed that there is a fixed activity in the marrow, liver and spleen proportional to the variable w , (i= I , 2,3) as described in the appendix, examination of the surface radioactivity curves reveals several interesting points. The pattern of the marrow activity curves, shown in Fig 3(a), is suggestive of ineffective erythropoiesis. In all the patients there was a rapid accumulation of radioiron in the sacral bone marrow. The maximum radioactivity reached the normal level in the patients of group A (e.g. patient M.Ga., full circles, Fig 3a), while it was lower in the patients of group B (e.g. patient E.Ma., open circles, Fig 3a). While in normal subjects the fixed activity is already negligible 7-8 d after the injection of radioiron, in these patients on the fourteenth day it ranged from 25% to 72% of the maximum measured value. In group A patients, the fixed activity in the liver increased slowly after the first day and reached a value nearly twice the level measured at 24 h. In group B patients, the liver fixed a consistent amount ofradioiron during the first hours, and after 14 d the uptake was about 150% of the 24

Ferrokineticr in Sideroblastic Anaemia 417

h level. The model proved to be adequate in predicting this behaviour, as shown by experimental and theoretical data reported in Fig 3(b).

Finally, it is also of some interest to examine the data concerning the activity fixed in the spleen and to compare them with model prediction. In four of the patients studied, there was a secondary rise of 59Fe activity fixed in the spleen. Two different patterns were observed. In two patients (A.Bi. and M.Ga.), the activity increased rapidly during the first z d, reached a maximum on the third or fourth day and thereafter decreased. In the two remaining patients (C.Me. and E.Ma.) the rise of fixed splenic acitivity was slower, and the maximum occurred later. Fig 3(c) reports the data of the patients M.Ga. and E.Ma. Such experimental data

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FIG 3 . Normalized fixed 59Fe activites (a) in the bone marrow (b) in the liver and (c) in the spleen of the patients M.Ga. (0) of group A and E.Ma. (0) of group B. Dots represent experimental data and lines the model prediction. The shaded areas represent normal ranges.

suggested a significant breakdown of young labelled red cells during the period of the ferrokinetic study.

In order to interpret such data by the model herein presented, it was necessary to consider the possibility that a labelled red cell may be taken up by HCS during the ferrokinetic test. The red cell death probability function was defined according to Bergner (1962) and Garby et a l ( 1 9 6 9 ) , and the amount of radioiron fixed in red cells processed by splenic HCS for haemoglobin iron reutilization could be computed as described in the Appendix. A satisfactory agreement between experimental and theoretical data was obtained, as shown in Fig 3(c), by giving the patterns presented in Fig 4 to the red cell death probability function. Such patterns were identified from the red cell survival data reported in Fig 4 (solid line) following the approach described by Berzuini et al (1977); then, for low values of red cell age, they were modified (broken line) in order to account for an early splenic radioiron deposition. This latter phenomenon, probably due to an early haemolysis and/or to the removal of iron-loaded mitochondria from siderocytes is not revealed by red cell survival data.

418 G. Barosi et a1

Finally, it is worth noting that MRCL computed in patient M.Ga. from ferrokinetic data (76 d) is shorter than that one derived from 51Cr data (92 d). This finding agrees with the observed early splenic radioiron uptake. The calculated value of MRCL, however, is greater than the 65 d derived from the modified red cell death probability function (broken line in Fig 4). This discrepancy may be explained by the drastic simplifications introduced for describing the process of young red cell loss.

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FIG 4. Identification of the red blood cell death probability function p(q) in the patients M.Ga. (left side) and E.Ma. (right side). The upper part shows the comparison between experimental 51Cr survival data (dots) and the model prediction (line) obtained by giving to p(q) the patterns reported in the lower part (solid line). The broken line indicates the modification of p(q) required to account for the early splenic 59Fe uptake revealed by surface counting.

DISCUSSION

The studies reported here demonstrate that PASA is characterized by the following disturbances in iron metabolism and erythropoiesis: an increase in marrow iron turnover, a very high degree of ineffective erythropoiesis, a variable degree of peripheral haemolysis and an increase in iron stores.

O n the basis of ferrokinetic parameters, the patients were divided in two groups, A and B, that agree, more or less, with type I1 and 111 of Singh et af (1970). In group A, the predominant feature was ineffective erythropoiesis. In spite of a very great increase in marrow iron turnover, haemoglobin synthesis was normal or reduced. In the three patients who had a near normal haemoglobin synthesis, a significant reduction in MRCL was present. This reduction in MRCL may represent an alternative mechanism of the anaemia; erythroid cells that escaped intramedullary haemolysis may generate erythrocytes that survive a shorter time in circulation.


In group B, ineffective erythropoiesis is associated with a large non-erythroid iron turnover. These patients had a great need for transfusions (about 25 units per year) and a long history of anaemia. Surface counting showed a rapid accumulation of radioiron in the liver, which indicated a deposition of iron in parenchymal cells (Finch et al, 1970). This is a typical feature of aplastic anaemia, but in our cases marrow iron turnover was increased and bone marrow biopsy demonstrated an erythroid hyperplasia. Singh et al(1970) affirmed that the ferrokinetic pattern of type 111 (group B in our study) was identical with that seen in aplastic anaemia and was consistent with the concept of total marrow failure. Our results do not agree with this affirmation; besides, aplastic anaemia has been very seldom reported as an evolution of sideroblastic anaemia (Brittin et a l , 1968). Our findings suggest that the progression ofPASA is characterized by an increasing parenchymal iron uptake; this is not caused by a cessation of erythroid activity, which is always greater than in normal subjects. The mechanism of the increased body iron accumulation may be part of the lesion of the basic disorder (Bottomley, 1977).

Let us now examine the values assumed by some ferrokinetic parameters and consider their significance with respect to red cell economy. In sideroblastic anaemia, it is likely that ineffective erythropoiesis is a consequence of the failure in haem byosynthesis (Cartwright & Deiss, 1975). Studies of early labelled bilirubin and carbon monoxide production in these disorders have shown degrees of ineffective erythropoiesis between 49 and 76% (White et a\, 1967; Horne et al, 1975; Samson et a l , 1976b). In our study, this value ranged from 39 to 95%. Although it is hazardous to compare data collected from different patients, the estimates of ineffective erythropoiesis reported in this paper seem somewhat greater than those obtained by the above studies. However, the estimates obtained using our method in normal subjects are in good agreement with the figures obtained by these other studies (White et al , 1967; Berk et al, 1976; Samson et a l , 1976a).

The explanation of the above discrepancy probably lies in the fact that ferrokinetic studies cannot distinguish between the return of radioiron to plasma from marrow cells which are destroyed (ineffective erythropoiesis) and the return of excess marrow non-haem iron. Both these events occur in sideroblastic anaemia. Many sideroblasts containing haemoglobin iron and mitochondria1 non-haem iron are destroyed in the marrow; in this case, both iron and cell economy is impaired. Moreover, some of the iron loaded mitochondria are probably removed from sideroblasts via secondary lysosomes (Cartwright & Deiss, 1975). This represents ineffective utilization of iron but not ineffective erythropoiesis. Our data, however, seem to suggest that the latter event is of minor importance.

A variable reduction in MRCL as measured by the 51Cr method was observed in PASA (Dacie et al, 1959; Verloop et al, 1962; Zittoun et al, 1965; White et al, 1967; Wagner et a l , 1973). The present estimates for mean cell lifespan confirm these previously reported findings. Since in PASA about I % of red cells are siderocytes containing iron loaded mitochondria (Cartwright & Deiss, 1975), it is possible that ferrokinetic studies overestimate the haemoglobin synthesis and underestimate MRCL. The identification of the pattern of the increased red cell breakdown was explored in this study. In patient M.Ga., the near normal MRCL and the early radioiron uptake in the spleen suggested an early destruction of a small fraction of the circulating red cells. This phenomenon was clearly demonstrated by Riedler & Straub (1972) in a patient with preleukaemic sideroblastic anaemia. However, the removal of iron-loaded

420 C. Barosi et a1

mitochondria from siderocytes without their destruction might play a role in the spleen uptake (Singh et a l , 1970). In patient E.Ma., a severe random red cell destruction was present and splenic sequestration was marked.

The present findings demonstrate that several pathogenetic mechanisms are present in PASA, though ineffective erythropoiesis is the major factor in the production of the anaemia. Undoubtedly, some ferrokinetic abnormalities may represent different phases during the evolution of a single disease. Some findings, however, seem to be not related to the natural course, but to reflect likely an heterogeneity of PASA. This aspect deserves further study

ACKNOWLEDGMENTS

This work has been supported by C.N.R. grants and HUSPI project.

APPENDIX

Nonsteady State Model: Experimental Stage Let us denote by x8( t) the amount of radioiron in the ith compartment at time t and by a, the

fractional transfer coefficient between thejth and ith compartment. The flow d ( t ; x 3 ) indicates the amount of radioiron fixed in the red cells taken up per unit time by the HCS.

Thus, according to the scheme shown in Fig I , the mathematical model of radioiron kinetics consists in the following system of integro-differential equations:

The initial conditions representing the impulsive injection of radioiron into plasma a t t = o are:

X ~ ( O ) = I O O ; xi(o)=o i = 2 , ...., 7

Moreover, since the flow of radioiron to RBC compartment is given by a 4 3 ~ 3 ( t ) , d ( t ; x 3 ) can be completely defined by introducing a red blood cell death probability function p(q).

This function is such that p ( q ) d t gives the probability that a red cell of age q will be taken up by HCS within the time interval dt. From p(q) it is possible to derive the loss function g(q) which provides the relative amount of red cells of age q entering HCS per unit time. According to Bergner (1962) and Garby et al (1969),g(q) may be written:

v B(r)=P(vl)exP{- j P ( O d 5 )

0

hence

Ferrokinetics in Sideroblastir Anaemia 42 * In the time of the ferrokinetic test, p(q) can reasonably be assumed null in normal states and

in those pathological states which do not significantly shorten the red cell lifespan, while thic assumption would jeopardize the validity of the model in analysing ferrokinetic data from patients with an overt extra-meduliary haemolysis.

Finally, let us relate the state variables to the experimental observations. The variables xt and x4 give the time course of the normalized activities measured in plasma and in red blood cells, respectively. In order to predict the time course of the amount of radioiron fixed in the organs, the following simplifying hypotheses are formulated:

(i) Liver acts as a storage area which receives iron from plasma; then, the amount w1 of radioiron accumulated in the liver in per cent of the injected quantity can be written:

(ii) Iron in the marrow is incorporated in red cell precursors and in marrow reticuloendothelial cells; then, the amount w2 of radioiron accumulated in the marrow in percent of the injected quantity can be written:

w2=x2+x3+27

where i7 denotes the amount of radioiron in marrow reticuloendothelial cells and is the solution of the following differential equation:

27 = ~72x2 + a73x3 -a, 7 2 7

(iii) All iron fixed in the spleen comes from the catabolism of red cell haemoglobin; then, the amount w3 of radioiron accumulated in the spleen in per cent of the injected quantity can be written:

1 w3 =x7 -x7

Now, except for a scaling factor, the variables w,, i = ~ , 2, 3 , give the time course of the normalized fixed activities measured over the liver, the sacral bone marrow and the spleen, respectively.

Steady State Model: Basal Stage The behaviour of internal iron exchanges a t the steady state is defined by the algebraic

system which is obtained’by setting the time derivatives xi, i=2, 3 , 5 , 6 , 7 , in ( I ) equal to zero. Since plasma iron X1 and RBC iron X4 are directly measurable during the ferrokinetic

study, the following relations provide an evaluation of the size of the remaining compartments:

422 G. Burosi et al

Whenever a red cell death probability function p(q) is known and assigned in the model, an estimate of RBC iron is given by the following relation:

X4=@43X3 m

Where q= J q g(q) dq is the mean cell lifespan derived from p(q) (Bergner, 1962). 0

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Estimation of Ferrokinetic Parameters by a Mathematical Model in Patients with Primary Acquired Sideroblastic Anaemia