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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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References

Lu, P., Sternlicht, M. D., and Werb, Z., 2006, “Comparative Mechanisms of Branching Morphogenesis in Diverse Systems,” J. Mammary Gland Biol. Neoplasia, 11(3–4), pp. 213–228. [CrossRef] [PubMed]
Riordan, J., and Wambach, K., 2010, Breastfeeding and Human Lactation, Jones & Bartlett Learning.
Love, S. M., and Barsky, S. H., 2004, “Anatomy of the Nipple and Breast Ducts Revisited,” Cancer, 101(9), pp. 1947–1957. [CrossRef] [PubMed]
Liu, Y., So, R., and Zhang, C., 2002, “Modeling the Bifurcating Flow in a Human Lung Airway,” J. Biomech., 35(4), pp. 465–473. [CrossRef] [PubMed]
Liu, Y., So, R., and Zhang, C., 2003, “Modeling the Bifurcating Flow in an Asymmetric Human Lung Airway,” J. Biomech., 36(7), pp. 951–959. [CrossRef] [PubMed]
Yang, X., Liu, Y., and Luo, H., 2006, “Respiratory Flow in Obstructed Airways,” J. Biomech., 39(15), pp. 2743–2751. [CrossRef] [PubMed]
Kulish, V. V., Lage, J. L., Hsia, C., and Johnson, Jr., R., 2002, “Three-Dimensional, Unsteady Simulation of Alveolar Respiration,” ASME J. Biomech. Eng., 124(5), pp. 609–616. [CrossRef]
Keyhani, K., Scherer, P., and Mozell, M., 1995, “Numerical Simulation of Airflow in the Human Nasal Cavity,” ASME J. Biomech. Eng., 117(4), pp. 429–441. [CrossRef]
De Backer, J., Vos, W., Devolder, A., Verhulst, S., Germonpré, P., Wuyts, F., Parizel, P. M., and De Backer, W., 2008, “Computational Fluid Dynamics can Detect Changes in Airway Resistance in Asthmatics After Acute Bronchodilation,” J. Biomech., 41(1), pp. 106–113. [CrossRef] [PubMed]
Mylavarapu, G., Murugappan, S., Mihaescu, M., Kalra, M., Khosla, S., and Gutmark, E., 2009, “Validation of Computational Fluid Dynamics Methodology Used for Human Upper Airway Flow Simulations,” J. Biomech., 42(10), pp. 1553–1559. [CrossRef] [PubMed]
Shome, B., Wang, L., Santare, M., Prasad, A., Szeri, A., and Roberts, D., 1998, “Modeling of Airflow in the Pharynx With Application to Sleep Apnea,” ASME J. Biomech. Eng., 120(3), pp. 416–422. [CrossRef]
Zhao, S., Xu, X., Hughes, A., Thom, S., Stanton, A., Ariff, B., and Long, Q., 2000, “Blood Flow and Vessel Mechanics in a Physiologically Realistic Model of a Human Carotid Arterial Bifurcation,” J. Biomech., 33(8), pp. 975–984. [CrossRef] [PubMed]
Rayz, V. L., Lawton, M. T., Martin, A. J., Young, W. L., and Saloner, D., 2008, “Numerical Simulation of Pre-and Postsurgical Flow in a Giant Basilar Aneurysm,” ASME J. Biomech. Eng., 130(2), p. 021004. [CrossRef]
Hughes, P., and How, T., 1996, “Effects of Geometry and Flow Division on Flow Structures in Models of the Distal End-to-Side Anastomosis,” J. Biomech., 29(7), pp. 855–872. [CrossRef] [PubMed]
Hofer, M., Rappitsch, G., Perktold, K., Trubel, W., and Schima, H., 1996, “Numerical Study of Wall Mechanics and Fluid Dynamics in End-to-Side Anastomoses and Correlation to Intimal Hyperplasia,” J. Biomech., 29(10), pp. 1297–1308. [CrossRef] [PubMed]
Lei, M., Giddens, D., Jones, S., Loth, F., and Bassiouny, H., 2001, “Pulsatile Flow in an End-to-Side Vascular Graft Model: Comparison of Computations With Experimental Data,” ASME J. Biomech. Eng., 123(1), pp. 80–87. [CrossRef]
Salzar, R. S., Thubrikar, M. J., and Eppink, R. T., 1995, “Pressure-Induced Mechanical Stress in the Carotid Artery Bifurcation: A Possible Correlation to Atherosclerosis,” J. Biomech., 28(11), pp. 1333–1340. [CrossRef] [PubMed]
Papaharilaou, Y., Ekaterinaris, J. A., Manousaki, E., and Katsamouris, A. N., 2007, “A Decoupled Fluid Structure Approach for Estimating Wall Stress in Abdominal Aortic Aneurysms,” J. Biomech., 40(2), pp. 367–377. [CrossRef] [PubMed]
Wang, D., Makaroun, M., Webster, M. W., and Vorp, D. A., 2001, “Mechanical Properties and Microstructure of Intraluminal Thrombus From Abdominal Aortic Aneurysm,” ASME J. Biomech. Eng., 123(6), pp. 536–539. [CrossRef]
Linninger, A. A., Tsakiris, C., Zhu, D. C., Xenos, M., Roycewicz, P., Danziger, Z., and Penn, R., 2005, “Pulsatile Cerebrospinal Fluid Dynamics in the Human Brain,” IEEE Trans. Biomed. Eng., 52(4), pp. 557–565. [CrossRef] [PubMed]
Moore, S., David, T., Chase, J., Arnold, J., and Fink, J., 2006, “3d Models of Blood Flow in the Cerebral Vasculature,” J. Biomech., 39(8), pp. 1454–1463. [CrossRef] [PubMed]
Bonfiglio, A., Leungchavaphongse, K., Repetto, R., and Siggers, J. H., 2010, “Mathematical Modeling of the Circulation in the Liver Lobule,” ASME J. Biomech. Eng., 132(11), p. 111011. [CrossRef]
Rani, H., Sheu, T. W., Chang, T., and Liang, P., 2006, “Numerical Investigation of Non-Newtonian Microcirculatory Blood Flow in Hepatic Lobule,” J. Biomech., 39(3), pp. 551–563. [CrossRef] [PubMed]
Debbaut, C., Vierendeels, J., Casteleyn, C., Cornillie, P., Van Loo, D., Simoens, P., Van Hoorebeke, L., Monbaliu, D., and Segers, P., 2012, “Perfusion Characteristics of the Human Hepatic Microcirculation Based on Three-Dimensional Reconstructions and Computational Fluid Dynamic Analysis,” ASME J. Biomech. Eng., 134(1), p. 011003. [CrossRef]
Cooper, A. P., 1840, On the Anatomy of the Breast, Longman.
Going, J. J., and Moffat, D. F., 2004, “Escaping From Flatland: Clinical and Biological Aspects of Human Mammary Duct Anatomy in Three Dimensions,” Pathology, 203(1), pp. 538–544. [CrossRef]
Ohtake, T., Kimijima, I., Fukushima, T., Yasuda, M., Sekikawa, K., Takenoshita, S., and Abe, R., 2001, “Computer-Assisted Complete Three-Dimensional Reconstruction of the Mammary Ductal/Lobular Systems,” Cancer, 91(12), pp. 2263–2272. [CrossRef] [PubMed]
Ramsay, D., Kent, J., Hartmann, R., and Hartmann, P., 2005, “Anatomy of the Lactating Human Breast Redefined With Ultrasound Imaging,” J. Anat., 206(6), pp. 525–534. [CrossRef] [PubMed]
Baum, K. G., McNamara, K., and Helguera, M., 2008, “Design of a Multiple Component Geometric Breast Phantom,” Proc. SPIE, 6913, p. 69134H. [CrossRef]
Geddes, D. T., 2007, “Inside the Lactating Breast: The Latest Anatomy Research,” J. Midwifery Womens Health, 52(6), pp. 556–563. [CrossRef] [PubMed]
Sohn, C., Blohmer, J.-U., and Hamper, U. M., 1999, Breast Ultrasound: A Systematic Approach to Technique and Image Interpretation, Thieme.
Moffat, D., and Going, J., 1996, “Three Dimensional Anatomy of Complete Duct Systems in Human Breast: Pathological and Developmental Implications,” J. Clin. Pathol., 49(1), pp. 48–52. [CrossRef] [PubMed]
Geddes, D. T., Kent, J. C., Mitoulas, L. R., and Hartmann, P. E., 2008, “Tongue Movement and Intra-Oral Vacuum in Breastfeeding Infants,” Early Hum. Dev., 84(7), pp. 471–477. [CrossRef] [PubMed]
Sakalidis, V. S., Kent, J. C., Garbin, C. P., Hepworth, A. R., Hartmann, P. E., and Geddes, D. T., 2013, “Longitudinal Changes in Suck-Swallow-Breathe, Oxygen Saturation, and Heart Rate Patterns in Term Breastfeeding Infants,” J. Hum. Lactation, 29(2), pp. 236–245. [CrossRef]
Mortazavi, S. N., Geddes, D., and Hassanipour, F., 2014, “Modeling of Milk Flow in Mammary Ducts in Lactating Human Female Breast,” Engineering in Medicine and Biology Society (EMBC), 2014 36th Annual International Conference of theIEEE, pp. 5675–5678. [CrossRef]
Waller, H., Aschaffenburg, R., and Grant, M. W., 1941, “The Viscosity, Protein Distribution, Andgold Number of the Antenatal and Postnatal Secretions of the Human Mammary Gland,” Biochem. J., 35(3), pp. 272–282. [PubMed]
White, F. M., and Corfield, I., 1991, Viscous Fluid Flow, Vol. 2. McGraw-Hill, New York.
Bates, J. H., 2009, Lung Mechanics: An Inverse Modeling Approach, Vol. 1. Cambridge University, Cambridge, Wiley, London.
Bejan, A., and Lorente, S., 2008, Design With Constructal Theory, Int. J. Engng Ed., 22(1), pp. 140–147, 2006. http://www.ijee.ie/articles/Vol22-1/IJEE1726.pdf
Bejan, A., 2005, “The Constructal Law of Organization in Nature: Tree-Shaped Flows and Body Size,” J. Exp. Biol., 208(9), pp. 1677–1686. [CrossRef] [PubMed]
Bejan, A., and Lorente, S., 2006, “Constructal Theory of Generation of Configuration in Nature and Engineering,” J. Appl. Phys., 100(4), p. 041301. [CrossRef]
Reis, A., Miguel, A., and Aydin, M., 2004, “Constructal Theory of Flow Architecture of the Lungs,” Med. Phys., 31(5), pp. 1135–1140. [CrossRef] [PubMed]
Bejan, A., and Lorent, S., 2000, Shape and Structure, From Engineering to Nature, Cambridge University, Cambridge, UK.
Ramsay, D. T., Mitoulas, L. R., Kent, J. C., Cregan, M. D., Doherty, D. A., Larsson, M., and Hartmann, P. E., 2006, “Milk Flow Rates can be Used to Identify and Investigate Milk Ejection in Women Expressing Breast Milk Using an Electric Breast Pump,” Breastfeed. Med., 1(1), pp. 14–23. [CrossRef] [PubMed]
Cregan, M. D., Mitoulas, L. R., and Hartmann, P. E., 2002, “Milk Prolactin, Feed Volume and Duration Between Feeds in Women Breastfeeding Their Full-Term Infants Over a 24 h Period,” Exp. Physiol., 87(2), pp. 207–214. [CrossRef] [PubMed]
Ramsay, D. T., Kent, J. C., Owens, R. A., and Hartmann, P. E., 2004, “Ultrasound Imaging of Milk Ejection in the Breast of Lactating Women,” Pediatrics, 113(2), pp. 361–367. [CrossRef] [PubMed]
Edgar, A., and Sebring, F., 2005, “Anatomy of a Working Breast,” New Begin., 22(2), pp. 44–50.
Neville, M. C., 1990, “The Physiological Basis of Milk Secretiona,” Ann. New York Acad. Sci., 586(1), pp. 1–11. [CrossRef]
Mills, A. F., 2001, Mass Transfer, Vol. 2. Prentice Hall, Upper Saddle River.
Liu, M., Nicholson, J. K., Parkinson, J. A., and Lindon, J. C., 1997, “Measurement of Biomolecular Diffusion Coefficients in Blood Plasma Using Two-Dimensional 1h-1h Diffusion-Edited Total-Correlation NMR Spectroscopy,” Anal. Chem., 69(8), pp. 1504–1509. [CrossRef] [PubMed]
Nakayama, A., Kuwahara, F., and Sano, Y., 2009, “Why do We Have a Bronchial Tree With 23 Levels of Bifurcation,” J. Heat Mass Transfer, 45(3), pp. 351–354. [CrossRef]
Comella, M. J., Cutnell, J. D., and Johnson, K. W., 1997, Physics, Wiley.
Neville, M. C., Keller, R., Seacat, J., Lutes, V., Neifert, M., Casey, C., Allen, J., and Archer, P., 1988, “Studies in Human Lactation: Milk Volumes in Lactating Women During the Onset of Lactation and Full Lactation,” Am. J. Clin. Nutr., 48(6), pp. 1375–1386. [PubMed]
Stacy, R. W., and Giles, F. M., 1959, “Computer Analysis of Arterial Properties,” Circul. Res., 7(6), pp. 1031–1038. [CrossRef]
Colley, J., and Creamer, B., 1958, “Sucking and Swallowing in Infants,” Br. Med. J., 2(5093), pp. 422–423. [CrossRef] [PubMed]
Prieto, C., Cárdenas, H., Salvatierra, A., Boza, C., Montes, C., and Croxatto, H., 1996, “Sucking Pressure and Its Relationship to Milk Transfer During Breastfeeding in Humans,” J. Reprod. Fertil., 108(1), pp. 69–74. [CrossRef] [PubMed]
Schrank, W., Al-Sayed, L. E., Beahm, P. H., and Thach, B. T., 1998, “Feeding Responses to Free-Flow Formula in Term and Preterm Infants,” J. Pediatr., 132(3), pp. 426–430. [CrossRef] [PubMed]
Medoff-Cooper, B., Bilker, W., and Kaplan, J. M., 2010, “Sucking Patterns and Behavioral State in 1-and 2-Day-Old Full-Term Infants,” J. Obstet. Gynecol. Neonat. Nurs., 39(5), pp. 519–524. [CrossRef]
Drewett, R., and Woolridge, M., 1979, “Sucking Patterns of Human Babies on the Breast,” Early Hum. Dev., 3(4), pp. 315–320. [CrossRef] [PubMed]
Mitoulas, L. R., Lai, C. T., Gurrin, L. C., Larsson, M., and Hartmann, P. E., 2002, “Effect of Vacuum Profile on Breast Milk Expression Using an Electric Breast Pump,” J. Hum. Lactation, 18(4), pp. 353–360. [CrossRef]
Mitoulas, L. R., Lai, C. T., Gurrin, L. C., Larsson, M., and Hartmann, P. E., 2002, “Efficacy of Breast Milk Expression Using an Electric Breast Pump,” J. Hum. Lactation, 18(4), pp. 344–352. [CrossRef]
Hassiotou, F., Hepworth, A. R., Beltran, A. S., Mathews, M. M., Stuebe, A. M., Hartmann, P. E., Filgueira, L., and Blancafort, P., 2013, “Expression of the Pluripotency Transcription Factor oct4 in the Normal and Aberrant Mammary Gland,” Front. Oncol., Vol. 3, Art. 79, pp. 1–15. [CrossRef]
Hassiotou, F., and Geddes, D., 2013, “Anatomy of the Human Mammary Gland: Current Status of Knowledge,” Clin. Anatomy, 26(1), pp. 29–48. [CrossRef]

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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