Statistical Geometry and Topology of the Human Placenta
We present a method of characterising tree networks based on a structural triangulation of those networks. Each component triangle is assigned a generation number which reflects the distance of that component from the origin of the network. By interpreting the generation number as an energy level, we can associate a partition function with a tree network which, in terms of the usual statistical thermodynamic interpretation, enables the determination of the internal energy and entropy of the triangulation. These thermodynamic functions depend on a parameter analogous to an inverse temperature that assigns weights to different parts of the network based on the generation numbers of the triangular elements. The systematic variation of these weights permits an examination of the development of the network, from the initial stages at low temperatures, where lower generation numbers have the greatest weight, to the complete network at high temperature, where all generation numbers have similar weights. After working through several examples to illustrate our methodology, we analyze the arterial and venous vasculature of the chorionic plate of 13 human placentas. We attempt to examine the extent to which the entropy function is correlated to the infant birthweight with the sample set. A correlation is postulated as a key factor in determining lifelong health.
R.-K.S. was supported in part by Placental Analytics LLC. C.M.S. was partially supported by a NARSAD Young Investigator Award and by K23 MidCareer Development Award NIMH K23MH06785. This material is based upon work supported by the National Science Foundation under award DMS-1004694. This work was also supported by NO1-HD-5-3411 “NCS Formative Research Project 18–Placental Study” to Placental Analytics, LLC.
- 5.Cayley, A. On the theory of analytical forms called trees. Philos. Mag. 13 (1857), 172–176. Google Scholar
- 12.Euler L. Solutio problematis ad geometriam situs pertinentis. Comment. Acad. Sci. Imp. Petrop. 8 (1736), 128–140. [English translation: Newman, J. Leohnard Euler and the Königsberg bridges. Sci. Am. 189(1) (1953), 66–70] Google Scholar
- 15.Gill, J. S., Woods, M. P., Salafia, C. M. and Vvedensky, D. D. Probability distributions for measures of placental shape and morphology (unpublished). arXiv:1109.2057.
- 17.Kirchhoff, G. Über die Auflösung der Gleichungen, auf Welche man bei der Untersuchung der Linearen Verteilung Galvanischer Ströme Geführt Wird. Poggendorf Ann. Phys. Chem. 72 (1847), 497–508. [English translation: On the solution of the equations obtained from the investigation of the linear distribution of Galvanic currents. IRE Trans. Circuit Theory 5 (1958), 4–8.] Google Scholar
- 26.Schwartz, N., Coletta, J., Pessel, C., Feng, R., Timor-Tritsch, I. E., Parry, S. and Salafia, C. M. Novel 3-dimensional placental measurements in early pregnancy as predictors of adverse pregnancy outcomes. J. Ultrasound Med. 29 (2010), 1203–1212. Google Scholar
- 27.Schwartz, N., Mandel, D., Shlakhter, O., Coletta, J., Pessel, C., Timor-Tritsch I. E. and Salafia, C. M. Placental morphology and chorionic surface vasculature at term are highly correlated with 3-dimensional sonographic measures at 11–14 weeks. J. Ultrasound Med. 30 (2011), 1171–1178. Google Scholar
- 30.Uylings, H. B. M. Optimization of diameters and bifurcation angles in lung and vascular tree structures. Bull. Math. Biol. 399 (1977), 509–519. Google Scholar