Journal of Mathematical Biology

, Volume 25, Issue 2, pp 203–226 | Cite as

Inference for an age-dependent, multitype branching-process model of mast cells

  • Jerry Nedelman
  • Heather Downs
  • Pamela Pharr
Article
Revised:

Abstract

We consider an age-dependent, multitype model for the growth of mast cells in culture. After a colony of cells is established by an initiator type, the two possible types of cells are resting and proliferative. Using novel inferential procedures, we estimate the generation-time distribution and the offspring distribution of proliferative cells, and the waiting-time distribution of resting cells.

Key words

Cell kinetics Hemopoiesis Inference for stochastic processes 

List of Notations

Bi

cumulative distribution function for the time until branching of a cell of type i

bi

probability density function for the time until branching of a cell of type i

bi

b i (1−D i )

Di

cumulative distribution function for the time until death of a cell of type i

di

probability density function for the time until death of a cell of type i

\({{f_\Gamma (t;\gamma ,\eta ,\lambda ) = \lambda ^\eta (t - \gamma )^{\eta - 1} e^{ - \lambda (t - \gamma )} } \mathord{\left/ {\vphantom {{f_\Gamma (t;\gamma ,\eta ,\lambda ) = \lambda ^\eta (t - \gamma )^{\eta - 1} e^{ - \lambda (t - \gamma )} } {\Gamma {\text{(}}\eta {\text{)}}}}} \right. \kern-\nulldelimiterspace} {\Gamma {\text{(}}\eta {\text{)}}}}\)

probability density function of a gamma distribution

Gi

cumulative distribution function for the lifetime of a cell of type i

G1*2

Convolution of G1 and G2

¯Gi

1−G i

gi

probability density function for the lifetime of a cell of type i

Li

likelihood of a history of type i

m

average number of proliferative daughters produced by dividing cells

Mij(t)

the expected number of type-j cells in a colony at time t if that colony began at time 0 with one type-i cell

Mi+(t)

M i0 (t) + M i 1(t) + M i 2(t)

prs

probability that a dividing cell produces r proliferative and s resting daughters

ti

times defining colony histories. See IV.2.1

T0

time to division of an initiator cell

T1, T2

times from birth to division of the two daughters of an initiator cell

T(1), T(2)

order statistics of T1 and T2

γ

minimum value of a gamma distribution

λ

scale parameter of a gamma distribution or of an exponential distribution

μ

probability per unit time of death for proliferative and resting cells

πrs

expected value of p rs when there is heterogeneity

η

shape parameter of a gamma distribution

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References

  1. Bellman, R., Harris, T. E.: On the theory of age-dependent stochastic processes. PNAS 34, 601–604 (1948)Google Scholar
  2. Bellman, R., Harris, T. E.: On age-dependent binary branching processes. Ann. Math. 55, 280–295 (1952)Google Scholar
  3. Bennet, D. C.: Differentiation in mouse melanoma cells: Initial reversibility and an on-off stochastic model. Cell 34, 445–453 (1983)Google Scholar
  4. Bondurant, M., Koury, M., Krantz, S. B., Blevins, T., Duncan, D. T.: Isolation of erythropoietin-sensitive cells from Friend virus-infected marrow cultures: Characteristics of erythropoietin response. Blood 61, 751–758 (1983)Google Scholar
  5. Burns, F. J., Tannock, I. F.: On the existence of a G 0-phase in the cell cycle. Cell Tiss. Kinetics 3, 321–334 (1970)Google Scholar
  6. Ciampi, A., Kates, L. Buick, R., Kriukov, Y., Till, J. E.: Multi-type Galton-Watson process as a model for proliferating human tumour cell populations derived from stem cells: estimation of stem cell self-renewal probabilities in human ovarian carcinomas. Cell Tiss. Kinetics 19, 129–140 (1986)Google Scholar
  7. Cormack, D.: Time-lapse characterization of erythrocytic colony-forming cells in plasma cultures. Exp. Hematol. 4, 319–327 (1976)Google Scholar
  8. Crumps, K. S., Mode, C. J.: An age-dependent branching process model with correlations among sister cells. J. Appl. Probab. 6, 205–210 (1969)Google Scholar
  9. David, C. N., MacWilliams, H.: Regulation of the self renewal probability in Hydra stem cell colonies. PNAS USA 75, 886–890 (1978)Google Scholar
  10. Gill, P. E., Murray, W., Wright, M. H.: Practical optimization. London: Academic Press 1981Google Scholar
  11. Golde, D. W. (ed.): Methods in hematology. New York: Churchill Livingstone 1984Google Scholar
  12. Good, P. I.: A note on the generation age distribution of cells with a delayed exponential lifetime. Math. Biosci. 24, 21–24 (1975)Google Scholar
  13. Good, P. I., Smith, J. R.: Age distribution of human diploid fibroblasts, a stochastic model for in vitro aging. Biophys. J. 14, 811–823 (1974)Google Scholar
  14. Gusella, J., Geller, R., Clarke, B., Weeks, V., Housman, D.: Commitment to erythroid differentiation by Friend erythroleukemia cells: a stochastic analysis. Cell 9, 221–229 (1976)Google Scholar
  15. Harley, C. B., Goldstein, S.: Retesting the theory of cellular aging. Science 207, 191–193 (1980)Google Scholar
  16. Harris, T. E.: The theory of branching processes. Berlin, Heidelberg, New York: Springer 1963Google Scholar
  17. Hoel, D. G., Crump, K. S.: Estimating the generation-time distribution of an age-dependent branching process. Biometrics 30, 125–135 (1974)Google Scholar
  18. Jagers, P.: Stochastic models for cell kinetics. Bull. Math. Bio. 45, 507–519 (1983)Google Scholar
  19. Kirkwood, T. B. L., Holliday, R.: Commitment to senescence: a model for the finite and infinite growth of diploid and transformed human fibroblasts in culture. J. Theor. Biol. 53, 481–496 (1975)Google Scholar
  20. Korn, A. P., Henkelman, R. M., Ottensmeyer, F. P., Till, J. E.: Investigations of a stochastic model of hemopoiesis. Exp. Hematol. 1, 362 (1973)Google Scholar
  21. Lebowitz, J., Rubinow, S. I.: Grain count distributions in labeled cell populations. J. Theor. Biol. 73, 99–123 (1969)Google Scholar
  22. Linz, P.: Analytical and numerical methods for Volterra equations. Philadelphia: SIAM 1985Google Scholar
  23. Loeffler, M., Wichmann, H. E.: A comprehensive mathematical model of stem cell proliferation which reproduces most of the published experimental results. Cell Tiss. Kinetics 13, 543–561 (1980)Google Scholar
  24. Mackillop, W. J., Ciampi, A., Till, J. E., Buick, R. N.: A stem cell model of human tumor growth: Implications for tumor cell clonogenic assays. JNCI 70, 9–16 (1983)Google Scholar
  25. Mode, C. J.: Multitype branching processes: Theory and applications. New York: American Elsevier 1971Google Scholar
  26. Nedelman, J., Rubinow, S. I.: Investigation into the experimental kinetic support of the two-state model of the cell cycle. Cell Biophysics 2, 207–231 (1980)Google Scholar
  27. Paulus, J.-M., Prenant, M., Deschamps, J.-F., Henry-Amar, M.: Polyploid mekagaryocytes develop randomly from a multicompartmental system of committed progenitors. PNAS USA 79, 4410–4414 (1982)Google Scholar
  28. Pharr, P. N., Nedelman, J., Downs, H. P., Ogawa, M., Gross, A. J.: A stochastic model for mast cell proliferation in culture. J. Cell. Physiol. 125, 379–386 (1985)Google Scholar
  29. Pharr, P. N., Suda, T., Bergman, K. L., Avila, L. A., Ogawa, M.: Analysis of pure and mixed murine mast-cell colonies. J. Cell. Physiol. 120, 1–12 (1984)Google Scholar
  30. Potten, C. S. (ed.): Stern cells. New York: Churchill Livingstone 1983Google Scholar
  31. Prothero, J., Gallant, J. A.: A model of clonal attenuation. PNAS USA 78, 333–337 (1981)Google Scholar
  32. Rubinow, S. I., Lebowitz, J. L.: A mathematical model of neutrophil production and control in normal man. J. Math. Biol. 1, 187–225 (1975)Google Scholar
  33. Schofield, R. S., Lajtha, L. G.: Determination of the probability of self renewal in haemopoietic stem cells: a puzzle. Blood Cells 9, 467–473 (1983)Google Scholar
  34. Shall, S., Stein, W. D.: A mortalization theory for the control of cell proliferation and for the origin of immortal cell lines. J. Theor. Biol. 76, 219–231 (1979)Google Scholar
  35. Smith, J. A., Martin, L.: Do cells cycle? PNAS USA 70, 1263–1267 (1973)Google Scholar
  36. Smith, J. R., Pereira-Smith, O., Good, P. I.: Colony size distribution as a measure of age in cultured human cells. A brief note. Mechanics Ageing Develop. 6, 283–286 (1977)Google Scholar
  37. Smith, J. R., Pereira-Smith, O. M., Schneider, E. L.: Colony size distributions as a measure of in vivo and in vitro aging. PNAS USA 75, 1353–1356Google Scholar
  38. Sonoda, T., Kanayama, Y., Hara, H., Hayashi, C., Tadokoro, M., Yonezawa, T., Kitamura, Y.: Proliferation of peritoneal mast cells in the skin of W/W v mice that genetically lack mast cells. J. Exp. Med. 160, 138–151 (1984)Google Scholar
  39. Steinberg, M. M., Brownstein, B. L.: A clonal analysis of the differentiation of 3T3-L1 preadipose cells: Role of insulin. J. Cell. Physiol. 113, 359–364 (1982a).Google Scholar
  40. Steinberg, M. M., Brownstein, B. L.: Differentiation of cultured preadipose cells: A probability model. J. Cell. Physiol. Supp. 2, 37–50 (1982b)Google Scholar
  41. Suda, T., Suda, J., Ogawa, M.: Disparate differentiation in mouse hemopoietic colonies derived from paired progenitors. PNAS USA 81, 2520–2524 (1984)Google Scholar
  42. Tarella, C., Ferrero, D., Gallo, E., Pagliardi, G. L., Ruscetti, F. W.: Induction of differentiation of HL-60 cells by Dimethyl Sulfoxide: Evidence for a stochastic model not linked to the cell division cycle. Cancer Res. 42, 445–449, Erratum 42, 3259 (1982)Google Scholar
  43. Till, J. E., McCulloch, E. A., Siminovitch, L.: A stochastic model of stem cell proliferation, based on the growth of spleen colony-forming cells. PNAS USA 51, 29–36 (1964)Google Scholar
  44. Vogel, H., Niewisch, H., Matioli, G.: The self renewal probability of hemopoietic stem cells. J. Cell. Physiol. 72, 221–228 (1968)Google Scholar
  45. Vogel, H., Niewisch, H., Matioli, G.: Stochastic development of stem cells. J. Theor. Biol. 22, 249–270 (1969)Google Scholar
  46. Waugh, W. A. O'N.: An age-dependent birth and death process. Biometrika 42, 291–306 (1955)Google Scholar
  47. Wichmann, H. E., Loeffler, M.: Probability of self renewal: assumptions and limitations. Blood Cells 9, 475–483 (1983)Google Scholar
  48. Wu, A. M.: A method for measuring the generation time and length of DNA synthesizing phase of clonogenic cells in a heterogeneous population. Cell Tiss. Kinetics 14, 39–52 (1981)Google Scholar
  49. Wu, A. M.: Regulation of self-renewal of human T-lymphocyte colony-forming units (TL-CFU's). J. Cell. Physiol. 117, 101–108 (1983)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Jerry Nedelman
    • 1
  • Heather Downs
    • 2
  • Pamela Pharr
    • 2
  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA
  2. 2.VA Medical Center Research ServiceMedical University of South Carolina and VA Medical CenterCharlestonUSA

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