Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Probabilistic recovery of neuroendocrine pulsatile, secretory and kinetic structure: An alternating discrete and continuous scheme

Authors: Somesh Chattopadhyay, Daniel M. Keenan and Johannes D. Veldhuis
Journal: Quart. Appl. Math. 66 (2008), 401-421
MSC (2000): Primary 62F15; Secondary 62P10
DOI: https://doi.org/10.1090/S0033-569X-08-01024-4
Published electronically: March 18, 2008
MathSciNet review: 2445520
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The brain (hypothalamus) directs hormone secretion by the pituitary gland via burst-like (pulsatile) release of specific peptides at inferentially random times. These pulsatile signals supervise growth, reproduction, lactation, stress adaptations, water balance and immune responses. However, hypothalamic molecules are diluted $ >$ 3000-fold in systemic blood, leaving pituitary-hormone pulses as measurable surrogates. The latter (roughly) mirror hypothalamic peptide bursts on a 1:1 basis, albeit being observed in a noisy environment. As a window to the brain, one must accurately recover the pulse (onset) times, and thereby estimate hormone secretion and kinetic parameters ( $ \theta \in \overline{\Theta}$) without distortion. Based upon limited observed data, one would like to obtain probability statements about underlying pulsatility, secretion and kinetics. Moreover, to be applicable in today's clinical setting, it is important that any such procedure require minimal or no human input. We propose and justify the following method. First, the data (a pituitary hormone concentration time-profile) is ``selectively smoothed'' by a nonlinear diffusion equation, whose diffusion coefficient is inversely related to the degree of rapid increase. This procedure generates a collection of potential pulse time sets

$ (\mathbb{T})$. Then, via an algorithm which alternates between a Metropolis algorithm on $ \mathbb{T}$ and a time-homogeneous diffusion process on $ \overline{\Theta}$, a compact manifold with boundary, simulation from an appropriately formulated (posterior) probability measure is achieved. The method is applied to recover the underlying structure of brain-pituitary regulation in disease and aging.

References [Enhancements On Off] (What's this?)

  • 1. Luis Alvarez, Pierre-Louis Lions, and Jean-Michel Morel, Image selective smoothing and edge detection by nonlinear diffusion. II, SIAM J. Numer. Anal. 29 (1992), no. 3, 845–866. MR 1163360, https://doi.org/10.1137/0729052
  • 2. William M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, No. 63. MR 0426007
  • 3. Chattopadhyay, S. (2001). Simultaneous Hormone Pulse Time and Secretion/Elimination Estimation: An Alternating Metropolis and Diffusion Scheme. Ph.D. thesis, Department of Statistics, University of Virginia, Charlottesville, Virginia.
  • 4. Geman, S. and Hwang, C.-R. (1986). Diffusions for global optimization. SIAM J. Control and Optimization 24 1031-1043. MR 0854068 (87j:49064)
  • 5. Grenander, U. (1983). Tutorial in Pattern Theory. Division of Applied mathematics, Brown University, Providence, Rhode Island.
  • 6. Ulf Grenander and Michael I. Miller, Representations of knowledge in complex systems, J. Roy. Statist. Soc. Ser. B 56 (1994), no. 4, 549–603. With discussion and a reply by the authors. MR 1293234
  • 7. Seizô Itô, Fundamental solutions of parabolic differential equations and boundary value problems, Jap. J. Math. 27 (1957), 55–102. MR 0098240
  • 8. John, F. (1982). Partial Differential Equations. Fourth Edition. Springer-Verlag, New York. MR 0831655 (87g:35002)
  • 9. Keenan, D.M., Alexander, S.L., Irvine, C.H.G., Clarke, I.J., Canny, B.J., Scott, C.J., Tilbrook, A.J., Turner, A.I., Veldhuis, J.D. (2004). Reconstruction of in vivo time-evolving neuroendocrine dose-response properties unveils admixed deterministic and stochastic elements in interglandular signaling. Proc. Natl. Acad. Sciences 101 6740-6745.
  • 10. Daniel M. Keenan, Somesh Chattopadhyay, and Johannes D. Veldhuis, Composite model of time-varying appearance and disappearance of neurohormone pulse signals in blood, J. Theoret. Biol. 236 (2005), no. 3, 242–255. MR 2157306, https://doi.org/10.1016/j.jtbi.2005.03.008
  • 11. Keenan, D. M., Licinio, J. and Veldhuis, J. D. (2001). A feedback-controlled ensemble model of the stress-responsive hypothalamo-pituitary-adrenal axis. Proc. Natl. Acad. Sciences 98 7:4028-4033.
  • 12. Daniel M. Keenan and Paula A. Shorter, Simulation by diffusion on a manifold with boundary: applications to ultrasonic prenatal medical imaging, SIAM J. Appl. Math. 64 (2004), no. 3, 932–960. MR 2068448, https://doi.org/10.1137/S0036139902408904
  • 13. Daniel M. Keenan, Weimin Sun, and Johannes D. Veldhuis, A stochastic biomathematical model of the male reproductive hormone system, SIAM J. Appl. Math. 61 (2000), no. 3, 934–965. MR 1788025, https://doi.org/10.1137/S0036139998334718
  • 14. Keenan, D. and Veldhuis, J. D. (2001). Disruptions in the hypothalamic luteinizing-hormone pulsing mechanism in aging men. Amer. J. Physiol., 281: R1917-R1924.
  • 15. Keenan, D. M., Veldhuis, J. D. and Yang, R. (1998). Joint recovery of pulsatile and basal hormone secretion by stochastic nonlinear random-effects analysis. Amer. J. Physiol. 275, R1939-R1949.
  • 16. Mauger, D.T., Brown, M. B. and Kushler, R. H. (1995). A comparison of methods that characterize pulses in a time series. Statistics in Medicine 14 311-325.
  • 17. Perona, P. and Malik, J. (1987). Scale space and edge detection using anisotropic diffusion. Proc. IEEE Computer Society Workshop on Computer Vision.
  • 18. D. Revuz, Markov chains, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematical Library, Vol. 11. MR 0415773
  • 19. Keniti Sato and Tadashi Ueno, Multi-dimensional diffusion and the Markov process on the boundary, J. Math. Kyoto Univ. 4 (1964/1965), 529–605. MR 0198547, https://doi.org/10.1215/kjm/1250524605
  • 20. Veldhuis, J. D. (1995). Pulsatile hormone release as a window into the brain's control of the anterior pituitary gland in health and disease: implications and consequences for pulsatile luteinizing hormone secretion. In The Endocrinologist, Editor: L. Loriaux. Williams & Wilkins, Baltimore.
  • 21. Yang, R. (1997). Maximum Likelihood Estimation Asymptotics for Parameter-Dependent Mixed Effects Models with Applications to Hormone Data. Ph.D. thesis, Division of Statistics, Department of Mathematics, University of Virginia, Charlottesville, Virginia.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 62F15, 62P10

Retrieve articles in all journals with MSC (2000): 62F15, 62P10

Additional Information

Somesh Chattopadhyay
Affiliation: Department of Statistics, Florida State University, Tallahassee, FL 32306-4330
Email: somesh@stat.fsu.edu

Daniel M. Keenan
Affiliation: Department of Statistics, University of Virginia, Charlottesville VA 22904
Email: dmk7b@virginia.edu

Johannes D. Veldhuis
Affiliation: Division of Endocrinology and Metabolism, Department of Internal Medicine, Mayo Clinic and Mayo Graduate School of Medicine, Rochester, MN 55905
Email: veldhuis.johannes@mayo.edu

DOI: https://doi.org/10.1090/S0033-569X-08-01024-4
Keywords: Pulse detection, simulation by diffusion, hormonal secretion, estimation
Received by editor(s): January 27, 2006
Published electronically: March 18, 2008
Additional Notes: Support provided by NSF DMS-0107680 and NIH AG19164, AG19695, AG23133, AG29215, AG14759, DK60717, and M01 RR00585
Article copyright: © Copyright 2008 Brown University
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society