Document ryL3oJKn1n5GX6x7QnVL38jE

The Growth Fraction of Human Myeloma Cells By B. Drewinko, R. Alexanian, H. Boyer, B. Barlogie, and S. I. Rubinow Greater reductions of tumor load in patients with multiple myeloma may result from therapeutic strategies that are based on a better knowledge of growth kinetics. We have previously shown that the labeling index of myeloma cells remains unchanged when tumor mass is reduced and that the cells of relapsing patients have different biologic prop- erties than the cells present before melphalan-prednisone therapy. This study investigated the growth fraction (GF) of myeloma cells at various disease stages using continu- ous i.v. infusions of tritiated thymidine. We studied 17 patients on 22 occasions (4 untreated, 2 unresponsive. 6 in remission, and 10 in relapse). All untreated and unrespon- sive patients and 5 of 6 patients in remission had a GF of less than 4%. GF was defined in these studies as the maximum percentage of labeled plasma cells exposed O NE APPROACH towards improving the treat- ment of multiple myeloma (MM) consists of designing therapeutic strategies based on a growth kinetics rationale.' Tumors are composed of two kinetically-distinct cell classes, proliferating and nonproliferating (quiescent) cells whose relative proportions change as tumor growth progresses. Because proliferating cells are more sensitive to most chemotherapeutic agents than quiescent cells, changes in their respective proportions will influence the design of treatment regimens, especially those that include cell cycle dependent drugs.6 The evolution of untreated MM is associated with a continuously decreasing proportion of proliferating cells so that the fraction of nonproliferating plasma cells is largest at the time of diagnosis.7'3 After successful therapy with melphalan and prednisone, the usual clinical course consists of an initial decrease in tumor load followed by a short duration of stabilized tumor mass, and then a final relapse with an increasing population of myeloma cells that are usually resis- tant to most antitumor drugs.' We have previously proposed that all untreated MM patients possess two neoplastic populations: a sensitive and a resistant frac- tion.9"#{176}"2With continuing therapy, the proportion of sensitive cells decreases and the tumor mass consists primarily of resistant cells. If the growth kinetics (i.e., proportion of proliferating cells) of sensitive and resis- tant cells differ, treatment regimens for relapsing patients must be modified in accordance with the properties of the dominant cell population. The present study was designed to evaluate the growth fraction (GF) and other growth kinetics parameters in patients with MM using prolonged intravenous infusions of tritiated thymidine (3H-TdR). Results in I 7 patients indicated that a markedly elevated GF occurred only during relapse continuously to tritiated thymidine. Relapsing patients. with the most rapid tumor doubling times, had GF ranging from 14% to 83%. The plasma cell transit time through the proliferative compartment for all of the relapsing patients ranged from 6.6 to 1 1 .9 days and the calculated intrinsic cell loss ranged from 50% to 86%. These findings support our model for the growth kinetics of multiple myeloma that assumes that the entire tumor mass issues from a small proportion of proliferating cells and that the growth kinet- ics of myeloma cells in relapsing patients differ from those in untreated and unresponsive patients. Therapeutic trials with cycle-active agents need further investigation in selected relapsing patients who are likely to have a high growth fractio.'. suggesting that further trials with cycle active agents should be conducted only at this phase of disease. MATERIALS AND METHODS Twenty-two studies were conducted in 17 patients with MM. Sixty-five percent were male and the median age was 54. All but 2 patients had an IgG myeloma protein peak to facilitate calculation of tumor mass doubling times as previously described.'0'2 Three patients were studied more than once. One patient (AU.) was evaluated once during remission and again during relapse; OR. was evaluated during two successive relapses, the first after melphalan- prednisone and the second after a response to adriamycin. W.S. was evaluated before initiating treatment, during remission and at two different stages of relapse. In another patient (F.V.), studies were conducted on bone marrow and pleural fluid plasma cells simulta- neously. All patients received detailed information concerning the investi- gational nature of the procedure and were required to sign an informed consent approved by the Human Surveillance Experimen- tation Committee of the Institution. Patients were given a single intravenous injection of 10 mCi of 3H-TdR (sp. act. 40-50 Ci/ mmole) followed by a continuous daily iv. infusion of 10 mCi of 3H-TdR/day in 5% dextrose (1000 mI/day) dispensed by an IVAC 500 infusion pump (IVAC Corp., San Diego, Calif.) for 8-10 days. Bone marrow aspirates were obtained I hr after the initial injection and at about 2-day intervals thereafter. Marrow aspirates were placed in a centrifuge tube containing 5 ml of McCoy's 5A medium with 660 U of heparin/mI and mixed by repeated pipette aspira- From the Department of Laboratory Medicine. the Department of Medicine. and the Department of Developmental Therapeutics. The University of Texas System Cancer Center. M. D. Anderson Hospital and Tumor Institute. Houston. Tex. and the Graduate School ofMedicalSciences. Cornell University, New York, N.Y. Supported in part by NIH Grants CA 05831 and CA /4528. Submitted August 14. /980; accepted October 7. 1980. Address reprint requests to Benjamin Drewinko, M.D., Ph.D.. Chief Section of Hematology. Department of Laboratory Medi- cine, M. D. Anderson Hospital and Tumor Institute. 6723 Beriner, Houston, Tex. 77030. 1981 by Grune & Stratton, Inc. 0006-4971/8//5702--0020$0l.00/0 Blood. Vol. 57. No. 2 (February), 1981 333 334 DREWINKO ET AL. tions. Smears were prepared by centrifuging cells directly onto slides by means of a cytocentrifuge and processed for autoradiogra- phy by the liquid emulsion technique with the use of Ilford K5 emulsion (Polysciences, Inc., Warrington, Penn.). After an exposure interval of 4-6 mo, the smears were developed, fixed, and dyed with May-Grunwald stain. Differential counts were made and 200-1000 plasma cells were examined. Cells were considered labeled when they exhibited at least 5 grains overlying the nucleus. Increments in labeling index were plotted as a function of time elapsed from the initial dose of 3H-TdR. In five relapsing patients with large GF, approximate estimates of cell cycle transit times were calculated from the continuous labeling curve using a heuristic growth kinetics assumptions. (1 ) The entire myeloma model based on the following cell population is comprised essentially of two distinct compartments, proliferating and nonpro- liferating cells. (2) The flux from the nonproliferating to the proliferating compartment is negligible over the duration of the experimental interval. (3) Following mitosis, half of the daughter cells in the proliferative compartment enter in a maturation phase, while the remaining daughter cells re-enter the G, phase, leading to subsequent division. (4) Cells in the maturation phase are lost at the rate at which they enter, so that the proliferative compartment during the experimental interval is in a steady state. (5) Variability of the durations of the phases of the proliferative cell cycle is considered to be negligible. The assumed theoretical scheme of proliferation is illustrated in Fig. I . Under these circumstances, it is a simple matter' to express mathematically the theoretical consequences of a continuous label- ing regimen (See Appendix). The schematic form of the resulting labeling index curve is shown in Fig. 2. The following features of the curve should be noted: (A) The "plateau" or asymptotic value of the labeling index, LI,. is an upper boundary of the GF which, strictly speaking counts only cells in G,, S and G2 phases. In what follows we interpret the LI,. as the GF, even though we recognize the approximate nature of the identification. (B) The initial labeling index, Ll(o) is given as LL TI/T where 1, is the duration ofS phase and T is the transit time through the proliferative compartment: T = TG + T, + T02 + TM (C) The transit time I may be estimated from the slope of the rapidly rising portion of the LI curve, which has the theoretical value 2 LI_IT. Knowing T, we can determine 1, from the initial labeling index, Ll(o). In addition to the above indicated paramenters, the growth rate of a tumor is influenced by the rate of cell attrition from the malignant population.'5 Classical methods to measure this rate of attrition are based on models that assume a homogeneous population and could, therefore, not be strictly applied to our myeloma cell population. Yet, in the interest of providing an approximate estimate of such values for MM tumors, we decided to treat the malignant plasma cells as a homogeneous population within the constrains of this evaluation. Fig. 1 . Diagram demonstrating the assumed phases of the proliferative compartment (G1 ; S; G3 and a maturation phase). The mitotic phase is assumed to be part of G2. The mean durations of th.se phases are: T,, ; T,; T,; and TM. respectively. Lb LI Li coTs T Il!, 1G2 TG2+TM t TG2+TGI Fig. 2. Schematic representation of the labeling index (LI) as a function of the time (t). based on equation 8 (See Appendix) with TM < T, . The slope on the first and third segments of the curves is Lloo/T,. while for the second segment. it is 2 Lloo/T. Based on generation time values calculated for the discrete experimental interval, further cytodynamics were estimated under the assumption of an exponential tangent to a Gompertzian func- tion. The potential doubling time (TDT) was estimated by TDT - T/GF; and the observed tumor doubling time (T) was computed from the curve depicting sequential increments in M-protein production rate.9"#{176C} ell loss was estimated as described by Steel,'3 cell lose factor - I - TDp/T, and considered independent of treatment since the doubling times were calculated from curves fitted to the changing M-protein production rate during relapse. RESULTS Two types of continuous labeling curves were obtained in patients receiving intravenous infusions of 3H-TdR (Fig. 3). very low proportion 8 days of infusion One untreated patient shows the of labeled cells ( I %) attained after in contrast to the very high level (47%) observed for a patient in relapse. The labeling index curves for patients in relapse were interpreted in accordance with the theoretical model described above. Variations in the G F of the myeloma tumor mass as a, (-3 0 E a 0. aa,, .0 a -I Ca, U a, 0. a 2 50 Ig ` G Status ` Untreated Ig G 4 6 8 Days 00 50 200 Hours Fig. 3. Increments in labeling index during prolonged infu- sions of tritiated thymidine. The figure contrasts the level of 47% labeled cells for a relapsing patient to the nearly constant value in an untreated patient. KINETICS OF MYELOMA CELLS 500 MR. MT. A.W. : oo 0 E50 6- S.F. o SF NN 0 10 20 30 10 20 300 Months . 10 20 30 40 Fig. 4. Growth fraction for patients in remission. Changes in myeloma protein production rate. as an index of tumor mass are correlated with durations of chemotherapy. a function of stage of disease are shown in the next series of representative graphs. The GF was low (<1%) for untreated patients and during the initial period of tumor mass reductions (Fig. 4). When rapid tumor relapse developed, GF values were very high (Fig. 5). During the last phase of relapse when the recurrent tumor mass exceeded the pretreatment level and growth rate slowed down, the GF was very low (Fig. 6). Figure 7 shows results for patients in whom at least two evaluations of GF were performed. The upper panel presents results for a patient evaluated during remission (GF = 8%) and then at relapse (GF = 1 8%); the mid-panel shows a GF of 20% during relapse that declined to I % during a second remission. The lower panel summarizes the data of4 separate GF evaluations. The GF was less than 1% before treat- ment and remained low (4%) during remission. As relapse began, the GF was 5.6% and increased to 14% as a higher tumor burden developed. Table I summarizes results from 22 studies 100 C.w. 335 G.F. <1% E0C,, I- C 5, 10 20 Months Fig. 6. Growth fraction determined entering a period of slow tumor growth. for a relapsing patient obtained in 17 patients. Although a low GF always corresponded to a very low initial LI, the reverse was not always true and a high GF may be present despite the low 1-hr LI. The correlation of GF values with stage of disease demonstrated that in all 4 untreated patients, in both unresponsive patients and in 5 of 6 patients in remis- sion the GF was less than 4% (usually less than 1%). In 7 or 10 patients in relapse, the GF exceeded 14%. All 7 patients were in the phase of rapid relapse. Only during very early and late phases of relapse was a low GF found. Table 2 presents values for growth kinetics parame- ters calculated for 4 patients with a GF exceeding 20%. Data obtained for 3 other relapsing patients were AH. CC,) ,, 0 E I- C 5, `3 5, Months Fig. 5. Growth fraction for patients in rapid relapse. Months Fig. 7. Individual patients with multiple growth fraction determinations. 336 DREWINKO ET AL. Table 1 . Growth Fractions of Patients With Multiple Myeloma Determined After Intravenous Infusions of Tritiated Thymidine Disease Stage Untreated Unresponsive Remission Early Late Relapse Rapid relapse Patient w.S. EM. CR. H.S. D.C. R.J. A.W. 0G. MR. MT. w.S. A.H. w.s. OR. C.w. OR. W.T. RN. F.V. A.H. CM. w.S. Labeling Index at 1 Hr 0.2 0.2 0.2 2.4 0.4 0.6 0.2 0.2 0.2 0.2 0.2 0.8 1.6 0.2 0.2 5.7 6.6 13.8 23.4 0.4 4.4 2.4 Growth Fraction (%) <1 <1 <1 3.7 <1 2.6 <1 <1 <1 <1 4 8 5.6 1 1 20 47 38 83 18 21 14.3 inadequate for these calculations. In one patient (F.V.), simultaneous studies were conducted for the myeloma cells present in the bone marrow and in the pleural effusion induced by a rib lesion. The transit time through the proliferation compartment of bone marrow myeloma cells ranged between 6.6 and I 1.9 days and the calculated S phase time ranged from I .2 to 3.3 days. The potential doubling time for the tumor mass of each of these patients ranged from 0.3 to 1 . I mo, values markedly shorter than the actual doubling times ( 1-8 mo) measured from changes in myeloma protein production rates. These differences are attrib- uted to the marked degree of cell loss experienced by the tumor mass computed at higher than 50% for all patients. Growth kinetics parameters of the myeloma cells in Table 2. Growth Kinetics of Myeloma Cells in Relapsing Patients F.v. o.R. W.T. RN. Bone Marrow Pleural Fluid Growthfraction.Ll_(%) LI (%) at 1 hr LengthofSphase Transittime Potential doubling Observed doubling Cellloss(%) time timet 20 5.7 1.9 6.6 1.1 8 86 48 6.6 1.2 8.9 0.6 1 .2 50 38 13.8 2.9 7.9 0.7 2 65 83 23.4 3.3 11.9 0.5 1 .3 62 95 10.5 0.8 7.2 0.3 1.3 77 Days. tMonths. the pleural fluid of patient F.V. differed markedly from those defined for the bone marrow. Thus, the transit time (7.2 days) and S phase duration (0.8 days) were only 61 % and 24% of the values defined for the bone marrow cells. The presence of numerous mitotic figures in the pleural fluid permitted an approximation of the length of G2 phase (1 1 hr) at the level of 50% labeled mitoses. After 22 hr, all mitoses were labeled. DISCUSSION To evaluate the percentage of proliferating plasma cells, patients with MM in different stages of disease received a prolonged intravenous infusion of 3H-TdR. If all marrow plasma cells participated in the prolifer- ative process, the percentage of labeled cells would approach 100% asymptotically.'5 In practice, the LI attains a "plateau" phase within the constraints of the experimental period of observation, and at a level that is usually significantly less than 100%. We believe that this feature reflects a fundamental inhomogeneity in the constituency of the plasma cells, and has led to our advocacy9"0"2 of the concept that plasma cells in MM patients, in the first approximation, should be consid- ered to behave as two distinct neoplastic populations, namely, proliferative (drug sensitive) and nonprolifer- ative (drug resistant). A mathematical realization of this concept is presented herein and forms the basis of a quantitative kinetic evaluation of MM patients. While this model represents a very simplistic version of a biphasic proliferating-nonproliferating cell popu- lation, it is doubtful whether a more elaborate version is either deserved by the data, or could alter treatment strategy in any significant degree. In our studies, the nonproliferating fraction deter- mined in this manner contains true quiescent cells, some proliferating cells with very long intermitotic times, and some proliferating cells that have entered the maturation phase. Although this approach under- estimates the size of the proliferative compartment, it provides valid information for potential clinical appli- cations; thus, if a cell does not traverse the cycle during an 8-10 day interval (as shown by 3H-TdR uptake), it should be considered out of cycle for purposes of sensitivity to antitumor agents delivered either as a bolus or in courses of less than 8 days. Conversely, the GF calculated from the "plateau" of labeled cells will be an overestimate because labeled proliferating cells that enter the maturation compart- ment are no longer destined to proliferate and indistin- guishable from cells in the proliferating compartment. Should quiescent cells reenter the proliferating compartment during the experimental interval, or if mature cells fail to disintegrate or leave the marrow, KINETICS OF MYELOMA CELLS 337 achievement of a plateau of labeled cells will be delayed. While more elaborate models have been constructed to calculate kinetic parameters from LI curves follow- ing continuously labeling,'56 the sampling frequency required by such analyses precludes their routine application in clinical procedures. Furthermore, these models are based on the assumption that the cell population under investigation consists entirely of proliferating cells, which we have seen is not supported by the data. In previous studies using the halving time of the median grain count of pulse labeled myeloma cells of relapsing patients, the generation time was longer than the study period of 4 days.'2 We concluded that the generation time of resistant myeloma cells was proba- bly much longer than the 2-4 days defined for sensi- tive plasma cells of untreated patients by Killman et al.,'6 Pileri et al.'8 and Riccardi et al.'9 This conclusion was confirmed by the present studies where the calcu- lated transit time ranged from 6.6 to 1 1 .9 days in 4 relapsing patients. Patient F.V. had different growth kinetics values for the myeloma cells growing in the bone marrow and the pleural fluid. The LI of the pleural fluid myeloma cells was less than half of the bone marrow cells, probably because a large fraction of these cells (included in the denominator of LI equations) consisted of shed nonviable elements. In contrast, the generation time and length of S phase were much shorter for the pleural fluid cells. These results were similar to those previously observed for the pleural effusion (79 hr) and bone marrow cells (> 4 days) of another patient with multiple myeloma in whom results were defined from the halving time of the median grain count.'2 Thus, secondary lesions appear to have shorter generation times than bone marrow plasma cells, perhaps because of reduced S phase transit time. This rapid proliferation rate could account for the increased sensitivity and more rapid reduction of tumor cells in secondary lesions than in the primary sites. The potential tumor doubling time calculated for relasing patients was about I mo, a value considerably shorter than the actual doubling times determined from increments in myeloma protein synthesis. These differences are attributed to the marked degrees of cell loss inherent in each tumor. Our data on relapsing patients Tarocco support the evidence presented by Pileri and for untreated patients with MM where a marked cell loss was considered responsible for the slow growth rate of the tumor mass)7 We do not know whether this cell loss issues from the proliferating or the quiescent pool; we also ignore whether it is a random process conditioned by as yet unexplained host-tumor cell immunologic relations or the conse- quence of normal maturation and senescence usually associated with cell renewal systems such as hemo- poietic tissues. What is most apparent is that the growth rate characteristics of the myeloma tumor mass are affected mostly by the GF and by the rate of inherent cell loss, and only partially by the generation time of the proliferating plasma cells. The magnitude of the Li did not correlate with the size of the GF. Although a low GF was always associated with a low LI, an initially low LI could correspond to large proportions of proliferating plasma cells suggesting that the size of the proliferating compartment varies greatly from patient to patient. These results were similar to those reported by Pileri et al.2#{176a}nd by Riccardi et al.'9 using a different method for determining the GF. These findings reem- phasize the inappropriateness differences in growth kinetics of the LI for comparing parameters among mdi- vidual patients.5"2'2' Our results also demonstrated that the GF of malig- nant plasma cells increases substantially only during the rapid period of relapse and it is only during this stage that the use of cell cycle sensitive drugs or treatment regimens based on a growth kinetics ration- ale (i.e., synchronization) are appropriate. Further- more, these results support our contention'#{176}"2 that the entire tumor mass of untreated and remission patients is maintained by a very small fraction of proliferating cells as also proposed by others.5'7 Hence, declining tumor loads may not be detected for long periods after treatment with effective agents that sterilize only the proliferating cells. To increase tumor cell kill beyond that presently obtained with conventional chemothera- py, agents must be developed that will kill nonprolifer- ating plasma cells or regimens must be designed that will recruit quiescent plasma cells into the proliferat- ing pool. These approaches might be enhanced by techniques for evaluating antitumor drugs on prima- ry22 or permanent23 cultures of myeloma cells. REFERENCES I . Alexanian R, Drewinko B: Multiple myeloma and related plasma cell syndromes, in Clark R, Howe C (eds): Cancer Patient Care. Chicago, Year Book Medical, 1976, p 199 2. Bergsagel D: Assessment of the response of mouse and human myeloma to chemotherapy and radiotherapy, in Drewinko B, Humphrey RM (eds): Growth Kinetics and Biochemical Regulation of Normal and Malignant Cells. Baltimore, Williams & Wilkins, 1977, p 705 3. Salmon 5, Dune BGM: Application of kinetics to chemother- apy for multiple myeloma, in Drewinko B, Humphrey RM (eds): 338 DREWINKO ET AL. Growth Kinetics and Biochemical Regulation of Normal and Malignant Cells. Baltimore, 4. Riccardi A, Martinotti Williams & Wilkins, 1977, p 865 A, Perugini S: Cytokinetic changes in 2 cases of plasma cell leukemia treated with a multipeptide devia- tion of m-[di(2-chloroethyl)amino]-L-Phenylalanine (peptichemio). Eur J Cancer 14: 1099, 1978 5. Pileri A, Conte PF: The biological and clinical features of human myeloma. Haematologica 62:202, 1977 6. Alberts DS, Dune BGM, Salmon SE: Treatment of multiple myeloma in remission with anticancer drugs having cell cycle specific characteristics. Cancer Treat Rep 6 1 :38 1, I 977 7. Mellestedt H, Killander D, Patterson D: Bone marrow kinetic studies on three patients with myelomatosis. Acta Med Scand 202:413, 1977 8. Salmon SE: Immunoglobulin synthesis and tumor kinetics of multiple myeloma. Semin Haematol 10:135, 1973 9. Hokanson J, Brown BW, Thompson J, Drewinko B, Alexanian R: Tumor growth patterns in multiple myeloma. Cancer 39:1077, I 977 I 0. Alexanian R, Hokanson JA, Drewinko B: Tumor kinetics in multiple myeloma, in Drewinko B, Humphrey RM (ed): Growth Kinetics and Biochemical Regulation of Normal and Malignant Cells. Baltimore, Williams & Wilkins, I 977, p 629 I I . Drewinko of chemotherapy B, Brown BW, Humphrey R, Alexanian R: Effect on the labeling index of myeloma cells. Cancer 34:526, 1974 1 2. Drewinko B, Alexanian R: Growth kinetics of plasma cell myeloma. J NatI Cancer Inst 58:1247, 1977 I 3. Salmon SE: Expansion of the growth fraction in multiple myeloma with alkylating agents. Blood 45:1 19, 1975 14. Rubinow SI, Lebovitz JL: A mathematical model of the acute myeloblastic leukemia state in man. Biophys J 16:897, 1976 I 5. Steel GG: Growth Kinetics of Tumors. Oxford, Clarendon Press, 1977, p56 16. Jansson B, Malahy MA, Drewinko B: Cell distribution over G, + S + G2 + M and over G0 and their use in the analysis of continuous labeling curves. Cell Tissue Kinet 12:675, 1979 17. Killman SA, Cronkite EP, Fliedner TM, Bond VP: Cell proliferation in multiple myeloma studied with tritiated thymidine i.n vivo. Lab Invest 11:845, 1962 18. Pileri A, Tarocco myeloma. Haematologica RP: In vivo kinetic 59:10, 1974 studies in human 19. Riccardi A, Martinotti A, Perugini S: Cytokinetic studies in two cases of plasma cell leukemia. Haematologica 62:58 1, 1977 20. Pileri A, Bernengo MG, Boccadoro M, Conte P. Marinone C, Masera P: Early recruitment in the human myeloma cell population after cytostatic treatment. Haematologica 61 : I 84, 1976 21. Baccarani M, Santucci A: Blood 46:650, 1975 (Letter to the Editor) 22. Hamburger A, Salmon SE: Primary bioassay of human myeloma stem cells. J Clin Invest 60:846, 1977 23. Burk KH, Drewinko B, Trujillo, JM, Ahearn, Mi: Establish- ment of a human plasma cell line in vitro. Cancer Res 38:2508, 1978 APPENDIX Assume that cells are traversing the phases of the proliferating compartment in a steady state manner (see Fig. I ). The mean transit times through these phases are designated T ; Ts; T02; and IN, respectively. With n equal to the steady state mitotic rate, the population of the phases of the proliferative compartment are: N0 - n TG,; N - n T5; NG, - n TG,; and NM ` TM. The total population N of the proliferative compartment is N = n T, where T is the total transit time. T=TG,+Ts+TG,+TM (I) Where N5 is the total number of labeled cell (Na, + N + N, + N); N is the total population of the prolifer- ative compartment; and N0 is the total population of the nonprolifer- ative compartment. Note that asymptotically, all of the cells in the proliferative compartment will be labeled so that N5 N and LI Li = N/(N + N0), which is essentially the growth fraction of the population. Hence the ratio N0/N can be expressed as I - LI N0/N LI (7) At time T - 0, continuous all cells in the proliferative labeled cells by an asterisk, labeling of all cells in S commences until compartment are labeled. Designating the labeled population is given as: 0 , 0<tsTG, N(t) - n t - T, , T62 < t T0, + TG, I T6 , T6, + T62 < t N(t) - nT5 , 0<t (2) (3) N,(t) - Io It nJ I TG, , 0 < t T62 , T0,<t , 0 < t T6, N(t) The labeling n t- I TM index, T, LI (t) , TG, < t TG, + TM , T0, + TM < t N5(t) is defined as LI (t) N + N0 (4) (5) (6) By substituting equations (I) through assuming that TM < T, , we obtain: (5), and (7) into (6), and LI' LI (t) - (T5+t Ts-TG,+ TM + T IT ,O<tsT0, 2t,TG,<tSTG,+TM + t , T6, < t TG, + T6, , T0, + TM < (8) The above equation is illustrated schematically in Fig. 2. Two features of the above equation should be noted. One is that TM and TG, must be interchanged if T, < TM. The other, is that if the concept of a maturation phase of the proliferative compartment is to be discarded, so that TM 0 in equation (8), then LI (t) becomes a ramp function represented by a single straight line of slope LI cx/T ainlternthaete inteirnvtaelrpretat0ion < t <of TeGx,perim+ entTa0l 2. Thceurvelsattersuchposassibiltihtyat shioswnan in Fig. 3, with "mending" of the curves due to cell-to-cell variability of phase duration.