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Research Papers

Mathematical Modeling of Mammary Ducts in Lactating Human Females

[+] Author and Article Information
S. Negin Mortazavi

Department of Mechanical Engineering,
University of Texas at Dallas,
Richardson, TX 75080
e-mail: negin@utdallas.edu

Foteini Hassiotou

Assistant Professor
School of Chemistry and Biochemistry,
University of Western Australia,
Crawley, Western Australia 6009, Australia
e-mail: foteini.hassiotou@uwa.edu.au

Donna Geddes

Associate Professor
School of Chemistry and Biochemistry,
University of Western Australia,
Crawley, Western Australia 6009, Australia
e-mail: donna.geddes@uwa.edu.au

Fatemeh Hassanipour

Assistant Professor
Department of Mechanical Engineering,
University of Texas at Dallas,
Richardson, TX 75080
e-mail: fatemeh@utdallas.edu

1Corresponding author.

Manuscript received April 16, 2014; final manuscript received October 17, 2014; published online June 3, 2015. Assoc. Editor: Alison Marsden.

J Biomech Eng 137(7), 071009 (Jul 01, 2015) (8 pages) Paper No: BIO-14-1167; doi: 10.1115/1.4028967 History: Received April 16, 2014; Revised October 17, 2014; Online June 03, 2015

This work studies a model for milk transport through lactating human breast ducts and describes mathematically the mass transfer from alveolar sacs through the mammary ducts to the nipple. In this model, both the phenomena of diffusion in the sacs and conventional flow in ducts have been considered. The ensuing analysis reveals that there is an optimal range of bifurcation numbers leading to the easiest milk flow based on the minimum flow resistance. This model formulates certain difficult-to-measure values like diameter of the alveolar sacs and the total length of the milk path as a function of easy-to-measure properties such as milk fluid properties and macroscopic measurements of the breast. Alveolar dimensions from breast tissues of six lactating women are measured and reported in this paper. The theoretically calculated alveoli diameters for optimum milk flow (as a function of bifurcation numbers) show excellent match with our biological data on alveolar dimensions. Also, the mathematical model indicates that for minimum milk flow resistance the glandular tissue must be within a short distance from the base of the nipple, an observation that matches well with the latest anatomical and physiological research.

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Figures

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Fig. 1

Anatomy of the lactating breast

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Fig. 2

Cross-sectional view of human lactating breast duct

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Fig. 3

Dichotomous branching of ducts and representative model

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Fig. 4

The architecture of each lobe, consisting of several lobules

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Fig. 5

Variation of diffusive and convective resistance versus bifurcation level (subject to average parameter values)

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Fig. 6

Optimum bifurcation level (subject to average parameter values)

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Fig. 7

A representation of the Alveolar sacs in the human lactating breast. The immunofluorescence micrograph on the left has been published in part in Hassiotou and Geddes [63]. In the immunohistochemistry micrograph on the right, the two black lines indicate how the two alveolar dimensions were measured.

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Fig. 8

Variation between and within tissues in alveolar diameter (T1–T6: lactating tissue 1–6). The number of alveoli analyzed per tissue is 30.

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