Experimental Gerontology, Vol. 25, pp. 483495, 1990
05315565/90 $3.00 + .00 Copyright © 1990 Pergamon Press plc
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A MATHEMATICAL MODEL OF ERYTHROPOIESIS SUGGESTS AN ALTERED PLASMA VOLUME CONTROL AS CAUSE FOR ANEMIA IN AGED MICE
M. LOEFFLER 1 and K. PANTEL2 tUniversitaetsklinik Koeln, Medizinische Klinik I, LFIEDV, JosefStelzmannStrasse 9, D5000 Koeln 41, FRG and 2Institut fur lmmunologie der LMU MiJnchen, Goethestr. 31, 8000 Miinchen 2
Abstract   In order to evaluate whether the anemia observed in aged C57B1 and B6D2F1 mice reflects a defect in the control mechanisms regulating erythropoiesis a mathematical model of erythropoietic control is employed, validated previously. In the model it is hypothesized that the most important mechanism for compensating an actual demand of red blood cells is an increase in the mitotic amplification (number of mitoses) of erythroid progenitors (CFUE, erythroblasts). The same sigmoidal doseresponserelationship between mitotic amplification and hematocrit (Hct) is proposed for young and aged mice. It is mediated by erythropoietin (EPO). Using this relationship one can demonstrate that the expansion of the plasma volume (PV) observed in aged mice is appropriately compensated by an increase in the mitotic amplification of CFUE and erythroblasts. This implies that aged mice operate in a stimulated state of erythropoietic amplification which is closer to the maximum of the doseresponserelationship than the steady state in young mice. This explains the finding of a reduced proliferative reserve in aged mice following further erythropoietic stimulation. An additional analysis regarding the response of aged mice to bleeding anemia is consistent with the view that young and aged mice share the same doseresponserelationship but start from different steady states. These findings suggest that the control mechanisms regulating erythropoiesis in young and aged mice are similar and that the anemia is due to alterations in the PV control.
Key Words: aging, mathematical model, erythropoiesis, anemia
INTRODUCTION DEVELOPMENTOF mild anemia with age is a wellknown fact for mice of different strains (e.g., Strong and Francis, 1940, in CBA; Ewing and Tauber, 1964, and Finch and Foster, 1973, in C57B1/6; Silini and Andreozzi, 1974, in C3H × C57Bl; Tyan, 1982, in B6D2F1). Recently, the results in mice have been reviewed by Boggs and Patrene (1985a; 1985b). In additional experiments, they foun I an increase of plasma volume (PV), blood volume (BV), and total red cell volume (RCV) in ~ gmonthold B6D2F1 (C57B1 × DBA) mice. The changes of PV, BV,
Correspondence to: M. Loeffler.
(Received 31 May 1989; Accepted 20 February 1990)
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M. LOEFFLER and K. PANTEL
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Fit. 1. Plasma volume (PV), blood volume (BV), and red cell volume (RCV) data recalculated and replotted from Boggs and Patrene (1985). a). The volumes per body weight as percentage of the young mice PV (©), BV (rT), RCV (A) (100% represents: PV: 0.032; RCV: 0.032; BV: 0.064 ml/g body weight), b). The absolute volumes of PV (0), BV (i), V (A) expressed as percentage of the young mice. (100% represents: PV: 0.75 ml; RCV: 0.76 ml; BV: 1.51 ml). The PV values were calculated from BV and HCT by using the formula PV = BV*(100HCT).
and RCV with age are given in Figure 1. One can express these parameters either in absolute units (e.g., ml) or in % of body weight. Both representations are given. They differ somewhat because mice aged 1328 months tend to have a 2030% larger body weight compared to 3 month old animals. However, irrespective of the type of calculation, clear increases in PV and RCV can be noted, with the PV increasing more than the RCV. From these data Boggs and Patrene conjectured that a defect in PV control leads to a dilution anemia, which is supported by the fact that the mean corpuscular volume and red cell life span are identical in young and aged mice. This conclusion contrasts with findings in C57B1/6 mice of a reduced proliferative capacity of marrow erythroblasts which was interpreted as a sign of hemopoietic senescence (Udupa and Lipschitz, 1984b). In addition, the pattern of erythropoietic recovery following bleeding seems to provide evidence that the control mechanisms regulating the production of red blood cells in 25 to 28monthold B6D2F1 and C57B1 mice are somewhat impaired (Boggs and Patrene 1985a; Harrison, 1975b). It is the objective of the present study to demonstrate that these data are not necessarily in conflict but can be consistently explained within the framework of a recently proposed quantitative concept of erythropoietic control. The basic assumptions of the concept have previously been formalized as a mathematical model of murine erythropoiesis (Wichmann, 1983; Loeffler et al., 1989).
THE M O D E L Model description The mathematical model of erythropoiesis is used as described previously (Loeffler et al., 1989). Briefly, the model consists of a series of compartments representing different maturation steps in the erythropoietic lineage (Fig. 2). Assuming a constant input of cells from early erythroid progenitors (BFUE) the compartments for late erythropoietic progenitors (CFUE),
ANEMIAINAGEDMICE
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BFUE
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F1c. 2. Block diagram of a mathematical model of normal erythropoiesis. Thick arrows represent the cell flux between cell compartment or production and destruction of EPO. Small arrows indicate regulatory influences. The model has been described elsewhere in detail (Loeffler e t al., 1989).
proliferating and nonproliferating erythroblasts (including marrow reticulocytes), blood reticulocytes, and erythrocytes are considered. The model describes the total erythroid cell production in the marrow and spleen. The mathematical description of the model compartments uses ordinary differential equations. A brief summary of the formulae used is given in the Appendix. The primary parameter of erythropoietic control is assumed to be the doserelationshipresponse between the hematocrit (Hct) (or the partial tissue oxygen pressure pt02) and the mitotic amplification factor f in the cell compartments CFUE (fCFUE) and proliferating erythroblasts (fPEP). The cumulative doseresponserelationship (f = fCFUE* fPEP) is displayed in Figure 3. If the Hct is reduced below its normal value (of 0.5) additional mitoses take place which increase the amplification up to a certain maximum level that cannot be exceeded. The maximum is assumed to be sixfold higher than normal. This implies that per BFUE cell in the course of erythroid amplification and maturation six times more erythrocytes are produced than under normal circumstances. In the normal steady state it is assumed that one BFUE gives rise to 2048 erythrocytes (amplification of 2048fold). With this in vivo doseresponsecurve a comprehensive quantitative understanding of many experimental situations such as bleeding,
486
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Hct Flo. 3. Dose response relationship between the total erythropoietic amplification factorfand Hct (or partial tissue oxygen pressure pt02) as derived from a previous analysis of erythropoietic control in mice (Loeffler et al. , 1989). The normal (young mouse) Hct value is assumed to be 0.50. A decrease in Hct leads to an increase in the amplification factor (full line) leveling off at a sixfold higher value. The amplification factor f and the number of mitoses m are related by the formulaf = 2'". hemolytic and dilution anemia, hypoxia, injection of EPO, hypertransfusion, hyperoxia, posthypoxia, and dehydration could be obtained (Wichmann et al., 1989; W u l f f et al., 1989). It should be noted, that the model parameters used in this study are identical with those established in the quoted references and that they are not fitted to the particular mouse strains considered in the following. MODEL SIMULATIONS Steady state
The model is used to simulate the erythropoietic compensation of a permanent expansion of the plasma volume. Technically, the simulations start at the normal steady state values (i.e., Hct = 0.50) and then a change o f the PV to a certain fixed value (e.g., 140%) is introduced. New steady states develop according to the model equations. These steady state values of Hct, RCV, etc. can be plotted against the PV used as input. This leads to nomograms predicting the steady state values for Hct or R C V for any given PV. This exercise is performed for two different model scenarios:
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1. Intact erythropoietic feedback with amplification regulated according to the doseresponsecurve in Figure 3; 2. Absence of erythropoietic regulation with a constant erythropoietic amplification independent of Hct.
Proliferative reserve The authors define proliferative reserve (PR) of mitotic amplification as the factor by which amplification can be increased from a given actual value to the maximum value. Given the maximum (fmax), normal (fno~), and actual operating (fact) values of the doseresponsecurve one can calculate the normal PR as PRnorm = (fmaxfno~m)/fnormand the actual PR a s P R a c t = (fmaxfact)/fact. Consequently, the PR for a given situation compared to normal is given by the simple ratio: PR
=
PRact/PRnorm,
100%.
Recovery following bleeding The assumptions for simulating recovery from bleeding anemia have been described previously (Wichmann et al., 1989). Briefly, Hct is reduced to the experimental nadir observed initially after bleeding. The model equations then predict the temporal pattern of regeneration in all compartments. Here three different regulatory scenarios are compared: 1. Intact erythropoietic feedback with a normal PV; 2. Intact erythropoietic feedback with a 40% larger PV; 3. Absence of erythropoietic regulation (constant amplification independent of Hct) with a 40% larger PV. RESULTS
Steady state analysis The previously established mathematical model of erythropoietic control can be used to predict the steady state Hct that establishes during a permanent expansion of the plasma volume (PV). If erythropoietic feedback is active, the steady state Hct maintained is considerably higher than in an uncompensated case. Using the model the difference can be quantified. This is illustrated in Figure 4a. The full curves represent the model steady state Hct values that establish for a certain fixed increase of PV provided the erythropoietic regulation is intact. For example, for a PV increase to 160% a reduction of the Hct to almost 47% can be expected. In a second scenario the steady state Hct values are calculated that one would expect under the extreme assumption that no regulation of erythropoiesis takes place. In this case, the steady state Hct values are much lower and for a PV increase to 160% a reduction of the Hct to 39% can be expected (lower curve in Fig. 4a). The data obtained by Boggs and Patrene (1985a; 1985b) can be compared to these model predictions. It is apparent that the full symbols (derived from absolute values; see Fig. 1 are close to the model curves for compensated anemia, while the
488
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FIG.4. a) Nomogram relating hematocrit (Hct) and plasma volume (PV) in steady state. Data deduced from Boggs and Patrene (1985a; 1985b) are plotted in the diagram: PV calculated from ratios of volume per g body weight (©) or as absolute volumes (ml) (O). In both cases, the data are expressed as percentage of the values obtained in young mice (see Fig. 1). The error bars represent the combined standard error for PV and Hct. The full upper curve represents a model calculation of compensated dilution anemia. The lower dashed curve represents the corresponding model curve if no erythropoietic compensation takes place (see constant dashed curve in Fig. 3). b) Nomogram relating red cell volume (RCV: A, &) and plasma volume in steady state.
open symbols (derived from values expressed as % body weight (see Fig. 1) indicate a slightly lower degree of erythropoietic compensation. A similar result is obtained if one plots the corresponding nomogram for PV versus RCV (Figure 4b). The full curve is the model prediction for the steady state in case of active erythropoietic control. Plotting the corresponding data into the same diagram one finds that the full symbols are again very close to the model curve for compensated anemia while the open symbols suggest a slightly impaired compensation.
Proliferative reserve (PR) The doseresponse curve given in Figure 3 for the amplification f a c t o r f has implications for the proliferative reserve. Young mice are assumed to have a steady state Hct of 0.50. From this level erythroid amplification can be increased sixfold until the maximum of the doseresponse curve is reached. Since aged mice exhibit a lower Hct than young mice their corresponding actual amplification factor is already increased under steady state conditions. Thus, their amplification is closer to the m a x i m u m than for young mice. In other terms, the proliferative reserve (PR) of aged mice is reduced compared with young mice. Table 1 gives the model predictions for the actual increase of the amplification factor at a given Hct and the respective proliferative reserve remaining. Lowering Hct gives higher amplification factors and lower PR; e.g., a Hct of 0.46 can be maintained by a 1.45fold increase in amplification. But this goes parallel with a reduction of the PR to 63% of normal. Recently, Udupa and Lipschitz (1984b) examined the proliferative reserve. The data revealed a moderate decrease in PR to 68% of normal for a Hct of 0.47 which is similar to the values predicted from our model calculation (see Table 1).
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TABLE 1. AMPLIFICATIONFACTORAND PROLIFERATIVERESERVE
Hct Model
Data
0.6 (normal) 0.48 0.46 0.44 0.42 0.40 0.30 <0.25 0.47 _ 0.02 ¢
Actual amplification Factor fa (Times normal)
Proliferative Reserve b (Percent of normal)
1.00 1.21 1.45 1.75 2.10 2.50 5.28 6.00
100 79 63 49 37 28 3 0
not accessible
68 + 8% a
aValues are obtained by calculating formula (7)(12) with f = f c e v  E*feEB from Loeffler et al. (1989). bRelative Proliferative Reserve = PR = PRacr/PRnorm*100%. CValues from Udupa & Lipschitz (1984a) Figure 1. dValues from Udupa & Lipschitz (1984b) Table 1.
Recovery from bleeding It was argued by Boggs and Patrene (1985a; 1985b), that a slower recovery from bleeding could indicate a defective erythropoietic regulation in old mice. Their Hct data are redrawn for 3month and 28monthold animals in Figure 5a. In Figure 5b corresponding model simulations are shown. Three model scenarios are shown, all starting with a Hct of 0.22 at day 0 (taken from the data): 1. The upper line indicates a recovery assuming that an erythropoietic feedback is active with the dose response curve given in Fig. 3. Here, a PV expansion is not assumed. This simulation is comparable to the data from young animals (Fig. 5a, upper curve). 2. In the middle curve, the same erythropoietic feedback is assumed as before. In addition, a permanent plasma expansion by 40% is taken into account. This model curve does not differ from the previous curve for the first two days. Subsequently, the recovery velocity slightly declines. Finally, a different steady state plateau is achieved. The shapes of both curves are similar. Indeed, data for young and old mice also exhibit a similar pattern but differ in plateau levels by the same amount as the model curves (compare Fig. 5a and 5b). Data by Harrison (1975a, 1975b) (not shown) also indicate an identical recovery for the first two days and completed recovery at different Hct levels after day 8. 3. The lower curve in Figure 5b indicates the recovery predicted after bleeding if erythropoietic regulation is assumed to be completely absent. In this case, red cell production continues at its normal rate resulting in a slow increase in Hct. It is evident that the data for old and young mice clearly contradict this simulation. Thus, the recovery after bleeding anemia does not provide evidence for a relevant defect of the erythropoietic control system.
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F~G.5. Recoveryfrom bleeding anemia: a) Data redrawn from Boggs and Patrene (1985a; 1985b) for 3monthold (+) and 28monthold mice (×) after severe bleeding anemia, b) Model calculations: The upper full line indicates the model curve for young mice assuming an intact erythropoietic feedback and no plasma expansion. The intermediate model curve is obtained by assuming an intact erythropoietic feedback and a 40% plasma volume expansion. The lower curve represents the model recovery for old mice (40% plasma volume expansion) after bleeding if no erythropoietic regulation takes place. This curve is not compatible with the data. DISCUSSION Based on a mathematical model of erythropoietic control validated previously (Loeffler et al., 1989) it can be concluded that the control mechanisms regulating erythropoiesis in young and aged mice are similar and that the anemia in aged mice is more likely due to alterations in the PV control. Such PV changes seem to drive the erythropoietic system into different operational steady states which are located on a dose response curve for mitotic amplification of CFUE and erythroblasts. Many observations on steady state and recovery behaviour in aged and young mice can apparently be explained by this concept of different steady states on a dose response curve which is identical for young and aged mice. This explanation is conceptually different from explanations of a defect erythropoietic control which would imply different dose response relationships for aged and young mice. This analysis shows that there is not enough evidence to conclude such a systematic regulatory difference. Steady state values for Hct clearly indicate the presence of capable erythropoietic compensation mechanisms. Although the PV increase is substantial the Hct is only slightly reduced, much less than what one would expect in an uncompensated circumstance. Previous reports using C57B1 mice showed a reduced proliferative reserve if stimulated by erythropoietin (Udupa and Lipschitz, 1984a; 1984b). This was interpreted as a defect in erythropoietic control. However, these findings can consistently be explained on the basis of an intact feedback regulation. Since Hct is decreased in aged mice, the corresponding mitotic amplification is already increased under steady state conditions. Consequently, the remaining reserve of mitotic amplification from this operational steady state to its maximum is reduced compared to that of young (nonanemic) mice. Indeed, the degree of the reduction predicted by the model as a consequence of the dilution anemia observed is similar to the reduction measured in mice by Udupa and Lipschitz (1984b).
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On the same basis, one can explain the response of aged mice to bleeding by an intact erythropoietic regulation. The different recovery pattern as compared to that of young mice can be understood quantitatively by the expanded PV. Likewise the model predicts that a response to continued hypoxia should result in a slightly slower increase of Hct in old mice compared to young reaching also different plateau levels. This has actually been found by Udupa and Lipschitz (1984a) although one has to be cautious due to the large variance in their data. Although the model curves approximated the data well the fit was not always perfect. The model curves after bleeding show a faster recovery (day 616) than is actually found in the data (see Fig. 4a, b). This discrepancy is most likely dueto the oversimplified model assumption that the total blood volume after bleeding and during the recovery is constant. This may not be entirely justified but the error should affect both the upper and the middle model curves in a similar way. Thus, their relation to one another will change if this assumption is corrected. Therefore, the differences observed should be comparable to the differences predicted by the model. It should be noted that the model was designed as an average mouse model to represent general patterns of erythropoietic regulation. This implies that the parameters are not adapted to a particular mouse strain such as B6D2F1 or C57B1 mice. This might also account for some differences between the model and the data. Thus, this analysis is not sensitive enough to exclude subtle differences in the doseresponse curves of young and old mice. It should also be noted that several studies could not show a significant difference in the steady state number of CFUE, erythroblast, or incorporation of 5°Fe between young and old B6D2F1 or C57B1 mice (Boggs and Patrene, 1985b; 1986; Udupa and Lipschitz, 1984a). However, this might not contradict the results presented here. The increase in the mitotic amplification predicted for the steady state conditions in old mice is only small. The corresponding increase in absolute numbers of CFUE and erythroblasts in the total animal might be too small (about 20% each) to be detectable, particularly as aged mice tend to be larger (2030% heavier) and correction procedures to estimate the total cell numbers from femur measurements are difficult. Recent results showed that spleens removed from 28monthold WCB6F1 and WBB6F1 donors and transplanted into constitutionally anemic SL/SL° recipients were less effective in correcting their anemia than spleens from 5 to 10montholddonors. This might indicate a decrease in the hematopoietic function of the spleen in aging mice (Harrison et al., 1988). Since the spleen has been shown to contribute to the erythropoietic recovery following bleeding anemia in mice (Loeffler and Wichmann, 1985), the modified recovery observed in 25 to 28monthold mice may result from a decrease in the erythropoesissupporting function of the spleen. Separate modeling of marrow and spleen conditions is underway to examine this point. Regarding hemopoietic stem cells several studies showed that the number of erythroid myeloid stem cells CFUS in the bone marrow remains constant with age (Chertkov and Gurevitch, 1981, in C57B1 x CBA/H mice) or even increases (Silini and Andreozzi, 1973, in C3H x C57B1 mice; Yuhan and Storer, 1967, in C57B1/6 mice; Ciggle et al., 1975, in SAS/4 mice; Tyan, 1982, C57B1). Only in the RFM/Un mouse strain Davic et al. (1971) found a decline in CFUS number after 12 months of age. Albright and Makinodan (1976) reported a reduced growth capacity of stem cells. However, the majority of the data suggests that the age related anemia is not caused by a change on the stem cell level. This conclusion is supported by the fact that bone marrow taken from 29 to 39monthold donor mice functioned as well as that from young mice (112 month) in establishing hematopoiesis in lethally irradiated recipients (Harrison, 1975a). In particular, injection of bone marrow from young mice into old
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M. LOEFFLER and K. PANTEL
mice could neither cure their anemia nor modify the response to bleeding (Harrison, 1975b). Recently Harrison e t a l . (1989) examined whether the primitive stem cells (PCS) change with age. These are more primitive than CFUS and give rise to lymphoid cells as well. It could be shown that PSC are about two times more frequent in old mice compared to young mice and that they can be effectively transplanted to give long lasting clonal growth in recipients. Thus, these studies do not provide conclusive evidence that the erythropoietic changes in old mice are caused by changes at the hemopoietic stem cell level. This justifies our model assumption to consider the input of cells into erythropoiesis as constant. On this basis, a separate modeling of CFUS and BFUE stages for young and old mice did not appear necessary. Finally, it should be stated that anemia in aged mice is not found in all mouse strains (e.g., not in C57BI × CBA mice, Chertkov and Gurevitch, 1981) nor in all individuals of an affected strain (Udupa and Lipschitz, 1984b). According to the authors' concept, these animals may not have the expansion of PV seen in others, indicating that their PVregulator may be more efficient than that in other strains or individuals. In this context, it should be noted that the reason for the expansion of PV in aged mice remains unclear. One possible mechanism might be a decrease in the renal or cardial function with increasing age, leading to an increase in water retention. Although the above model analysis suggests that young and aged mice have the same response, direct experimental proof would be desirable. How could it be obtained? At present, there is no reliable technique available to measure the in vivo doseresponse relationship directly. A possible approximation might be an experiment involving continuous longterm application of recombinant EPO at various dose levels. One could inject r EPO over two weeks to drive the erythropoietic amplification system into a new steady state (new operation point on the doseresponse curve). One should then try to quantify parameters of red cell production, e.g., by determining how many reticulocytes are produced per time unit during the steady state (using possibly techniques of cohort labelling). Accompanying measurements should quantify the total erythroblasts cell counts in marrow and spleen. This sort of experiment could be undertaken for various dose levels of r EPO achieving up to 100times normal serum levels. Plotting steady state EPOlevels versus steady state cell production for young and old mice separately should give very similar doseresponse curves. This procedure however requires a certain precision because one wants to demonstrate equivalence and not difference. Another indirect test of equivalence would be to bring young and old mice to similar hematocrits by infusion of plasma expander into young mice or by dehydration of old mice and to carry out similar erythropoietic manipulations like bleeding anemia, or short term application of r EPO. In conclusion, this analysis suggests that the anemia in aged B6D2F1 and C57B1 mice is caused by a yet unexplained plasma volume expansion and that the feedback control regulating the proliferative response of CFUE and erythroblasts in young and old mice is fairly similar providing no clear evidence for a defect of the erythropoietic control process. The authors thank W. Nijhof (Groningen) and H.E. Wichmann (Wuppertal) for valuable discussion and Stephan Gontardfor technical support. The research was supportedby grant LO 342/12of the Deutsche Forschungsgemeinschaft(FRG). K.P. was fellow of the DFG.
Acknowledgments 
REFERENCES ALBRIGHT,J.W., and MAKINODAN,T. Decline in the growth potential of spleencolonizingbone marrow stem cells of longlived aging mice. J. Exp. Med. 144, 12041231, 1976.
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BOGGS, D.R. and PATRENE, K.D. Hematopoiesis and aging III: Anemia and a blunted erythropoietic response to hemorrhage in aged mice. Am. J. Hemat. 19, 327338, 1985a. BOGGS, D.R. Hematopoiesis and aging IV: Mass distribution of erythroid marrow in aging mice. Exp. Hemat. 13, 10441047, 1985b. BOGGS, D.R. Hematopoiesis and aging V. A decline in hematocrit occurs in all aging female B6D2FI mice. Exp. Aging Res. 12, 131134, 1986. CHERTKOV, J.L. and GUREVITCH, O.A. Agerelated changes in hemopoietic microenvironment. Enhanced growth of hemopoietic stroma and weakened genetic resistance to hemopoietic cells in old mice. Exp. Geront. 16, 195198, 1981. DAVIS, M.L., UPTAN, A.C., and SATrERFIELD, L.C. Growth and senescence of the bone marrow stem cell pool in RFM/Un mice. Proc. Soc. Exp. Biol. Med. 137, 14521458, 1971. EWING, K.L. and TAUBER, O.E. Hematological changes in aging male C57B1/6 Ja x mice. J. Gerontol. 19, 165167, 1964. FINCH, C.E., FOSTER, J.R. Hematologic and serum electrolyte values of the C57Bl/6J male mouse in maturity and senescence. Laboratory. Animal Science 23, 339379, 1973. HARRISON, D.E. Normal function of transplanted bone marrow cell lines from aged mice. J. Geront. 30, 279285, 1975a. HARRISON, D.E. Defective erythropoietic responses of aged mice not improved by young marrow. J. Gerontol. 30, 286288, 1975b. HARRISON, D.E., ASTLE, C.M., and DELAITTRE, T. Effects of transplantation and age on immunohemopoietic cell growth in the splenic microenvironment. Exp. Hematol. 16, 213216, 1988. HARRISON, D.E., ASTLE, C.M., STONE, M. Numbers and functions of transplantable primitive immunohematopoietic stem ceils: Effects of Age. J. lmmunol. 142, 38333840, 1989. LOEFFLER, M. and WICHMANN, H.E. The role of the spleen in hemopoiesis. In: Mathematical model of stem cell proliferation. Stem cell regulation in hemopoiesis, Vol II., Wichmann, H.E. and Loeffler, M. (Editors), PP 163179. CRC Press, Boca Raton, 1985. LOEFFLER, M., PANTEL, K., WULFF, H., and WICHMANN, H.E. A mathematical model of erythropoiesis in mice and rats Part 1, Structure of the model, Cell and Tissue Kinet., 22, 1320, 1989. SILINI, G. and ANDREOZZI, U. Hematological changes in the aging mouse. Exp. Geront. 9, 99108, 1974. STRONG, L.C. and FRANCIS, L.D. Differences in hemoglobin values in the blood of breeder female mice: A comparison between cancer susceptible and cancer resistant strains. Am. J. Cancer 38, 399403, 1940. TYAN, M.L. Old mice: Marrow response to bleeding or endotoxin. Proc. Soc. Exp. Biol. Med. 169, 295300, 1982. UDUPA, K.B. and LIPSCHITZ, D.A. Erythropoiesis in the aged mouse. I. Response to stimulation in vivo. J. Lab. Clin. Med. 103, 574580, 1984a. UDUPA, K.B. and LIPSCHITZ, D.A. Erythropoiesis in the aged mouse. II. Response to stimulation in vitro. J. Lab. Clin. Med., 103, 581588, 1984b. WlCHMANN, H.E. Computer modeling of erythropoiesis. In: Current concepts in erythropoiesis, Dunn, C.D.R., (Editor), pp. 99141, Wiley & Sons, Chichester, 1983. WlCHMANN, H.E., LOEFFLER, M., PANTEL, K., and WULFF, H. A mathematical model of erythropoiesis in mice and ratsPart 2: Stimulated Erythropoiesis. Cell and Tissue Kinet., 22, 3149, 1989. WULFF, H., WlCHMANN, H., PANTEL, K., and LOEFFLER, M. A mathematical model of erythropoiesis in mice and ratsPart 3: Suppressed erythropoiesis. Cell and Tissue Kinet., 22, 51~51, 1989. YUHAN, J.M. and STOER, J.B. The effect of age on two modes of radiation death and on hematopoietic cell survival in the mouse. Rad. Res. 32, 596605, 1967.
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APPENDIX A brief summary of the essential model equations is given (for details and choice of parameters see Loeffler et al. 1989): The following notation is used for the model compartments: Y= f = "r =
number of cells in a compartment amplification factor (f= 2", m = number of mitoses) average transit time
Model equations of the compartments The compartment of the late erythropoietic progenitors (CFUE) is described by the differential equation: ~IrCFUE
=
fCFUE ' I

YCFU_E/TcFu_E.
(1)
I is the influx from the early erythropoietic progenitors (BFUE) which are not explicitly considered in the model. While I is constant, the amplification of this influx, fCFUE, depends on EPO. rCFUE is assumed as constant. The cells enter the second compartment of proliferative erythropoietic precursors (PEP): XlrpEP = fPEP " YCFUE/TCFUE  YPEP/TPEP"
(2)
Both the amplification factor fPEP and the transit time "ryeP depend on EPO. The erythroblasts loose their ability to proliferate and enter the compartment of the nonproliferative erythropoietic precursors (NPEP = erythroblasts and bone marrow reticulocytes) described by: ~IZNPEP = YpEI~TpE p  YNPEp/TNPEP,
(3)
with the transit time q'NPEP depending on EPO. The blood reticulocytes (RETI) are described by: XI~RETI = YNPEp/TNPEP  YRETI/TRETI .
(4)
The compartment of erythrocytes (ERY) is divided into two parallel parts. The first one, YERYA, obeys an almost age dependent destruction law (rectangular age structure) and represents the majority of cells produced at normal proliferation. The second part, YERYR, has an exponential age structure, suggesting a random destruction process. The distribution of the influx into the subcompartments YERYA and YERYR is governed by a function alpha (h(t)). Hence, ~rERYA
=
alpha(h(t)) • h(t)  alpha(h(t'rERYA))
" h(ITERYA),
(5)
with h(t) = YRETI(t)/'rRETI. YERVR = (1alpha(h(t)))" h(t)

YERYR/TERYR
(5')
with alpha(h(t)) = exp (  0.1 • (h(t)/hnorm)2). Finally, compartment EPO describes the feedback hormone erythropoietin. The model equation is: XlrEPO = PEPO  YEPO/TEPO'
(6)
with PEPO as the production rate depending on the oxygensupply and TEPO as the constant turnover time.
Feedback influences o f erythropoietin In the model it is assumed that EPO influences directly the amplification factorfcFu_ E andfpEp as well as
495
ANEMIA IN AGED MICE
the transit times TpEP and TNPEP. For these regulatory functions a sigmoid form is assumed: f(YEPo/YEPo n°rm) = A  B • exp (  C • (YEPo/YEPOn°rm)Bg),
(7a)
and T ( Y E P o / Y E P o n°rm) = A ' + B ' " e x p (  C '
• (YEPo,/YEPo norm) BR ).
(7b)
The constants A, B, C, and A ' , B ' , C' are positive and can be determined from measurements of the minimum, maximum, and normal values of the proliferation rates and transit times (Loeffler et al., 1989).
Feedback influences on erythropoietin production It is assumed that the production rate of erythropoietin PEPO depends exponentially on the tissue oxygen pressure PtO2: PEPo(ptO2/ptO2 n°rm) = D • exp (  E • PtO2/PtO2n°rm).
(8)
prO2 is determined by the tissue oxygen saturation (StO2) according to the Hill equation: p t 0 2 = p50 " ( S t O 2 / ( 1 0 0  S t O 2 ) ) l / n .
(9)
The parameters p50 and n characterize the oxyhemoglobin dissociation curve. StO 2 depends on the arterial saturation and the oxygen desaturation which is related to the hematocrit: StO 2 = SaO 2  F " (1/Hct 
1).
(10)
Readers interested to have a review of other models of erythropoiesis are referred to Wichmann (1983).