American Institute of Mathematical Sciences

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2011, 8(3): 723-732. doi: 10.3934/mbe.2011.8.723

Optimal nutritional intake for fetal growth

Chanakarn Kiataramkul 1, , Graeme Wake 2, , Alona Ben-Tal 3, and  Yongwimon Lenbury 4,

1. 

Department of Mathematics, Mahidol University, Bangkok 10400, Thailand & National Research Centre for Growth and Development, Auckland, New Zealand
2. 

National Research Centre for Growth and Development & Institute of Information and Mathematical Sciences, Massey University, Private Bag 102904, Albany, Auckland
3. 

Institute of Information and Mathematical Sciences, Massey University, Private Bag 102904, Albany, Auckland, New Zealand
4. 

Department of Mathematics, Mahidol University, Bangkok 10400, Thailand & Center of Excellence in Mathematics,, PERDO Commission on Higher Education, Si Ayudhya Rd., Bangkok 10400, New Zealand

Received  August 2010 Revised  November 2010 Published  June 2011

The regular nutritional intake of an expectant mother clearly affects the weight development of the fetus. Assuming the growth of the fetus follows a deterministic growth law, like a logistic equation, albeit dependent on the nutritional intake, the ideal solution is usually determined by the birth-weight being pre-assigned, for example, as a percentage of the mother's average weight. This problem can then be specified as an optimal control problem with the daily intake as the control, which appears in a Michaelis-Menten relationship, for which there are well-developed procedures to follow. The best solution is determined by requiring minimum total intake under which the preassigned birth weight is reached. The algorithm has been generalized to the case where the fetal weight depends in a detailed way on the cumulative intake, suitably discounted according to the history. The optimality system is derived and then solved numerically using an iterative method for the specific values of parameter. The procedure is generic and can be adapted to any growth law and any parameterisation obtained by the detailed physiology.
Keywords: controlled nutrition, dynamical system., Logistic.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Chanakarn Kiataramkul, Graeme Wake, Alona Ben-Tal, Yongwimon Lenbury. Optimal nutritional intake for fetal growth. Mathematical Biosciences & Engineering, 2011, 8 (3) : 723-732. doi: 10.3934/mbe.2011.8.723
References:
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C. W. Clark, "Bioeconomics: The Optimal Management of Renewable Resources," Wiley, New York, 1976.

[2]

H. R. Joshi, Optimal control of an HIV immunology model, Optim. Control Appl. Methods, 23 (2002), 199-213. doi: 10.1002/oca.710.

[3]

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems-Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473.

[4]

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D. Kirschner, S. Lenhart and S. Serbis, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792. doi: 10.1007/s002850050076.

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L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes" (ed. L. W. Neustadt), Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.

[8]

D. A. Redmer, J. M. Wallace and L. P. Reynolds, Effect of nutrient intake during pregnancy on fetal and placental growth and vascular development, Domestic Animal Endocrinology, 27 (2004), 199-217. doi: 10.1016/j.domaniend.2004.06.006.

[9]

G. Wu, F. W. Bazer, T. A. Cudd, C. J. Meininger and T. E. Spencer, Maternal nutrition and fetal development, Journal of Nutrition, 134 (2004), 2169-2172.

[10]

G. Wu, F. W. Bazer, J. M. Wallace and T. E. Spencer, Board-invited review: Intraurine growth retardation: Implications for the animal sciences, Journal of Animal Science, 84 (2006), 2316-2337. doi: 10.2527/jas.2006-156.

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G. Zaman, Y. H. Kang and I. H. Jung, Optimal vaccination and treatment in the SIR epidemic model, Proc. KSIAM, 3 (2007), 31-33.

show all references

References:
[1]

C. W. Clark, "Bioeconomics: The Optimal Management of Renewable Resources," Wiley, New York, 1976.

[2]

H. R. Joshi, Optimal control of an HIV immunology model, Optim. Control Appl. Methods, 23 (2002), 199-213. doi: 10.1002/oca.710.

[3]

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems-Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473.

[4]

K. L. Gatford, J. A. Owens, S. Li, T. J. M. Moss, J. P. Newnham, J. R. G. Challis and D. M. Sloboda, Repeated betamethasone treatment of pregnant sheep programspersistent reductions in circulating IGF-I and IGF-binding proteins in progeny, Am. J. Physiol. Endocrinol. Metab., 295 (2008), 170-178. doi: 10.1152/ajpendo.00047.2008.

[5]

D. Kirschner, S. Lenhart and S. Serbis, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792. doi: 10.1007/s002850050076.

[6]

S. Lenhart and J. T. Workman, Optimal control applied to biological models, in "Chapman & Hall/CRC Mathematical and Computational Biology Series," Chapman & Hall/CRC, Boca Raton, FL, 2007.

[7]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes" (ed. L. W. Neustadt), Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.

[8]

D. A. Redmer, J. M. Wallace and L. P. Reynolds, Effect of nutrient intake during pregnancy on fetal and placental growth and vascular development, Domestic Animal Endocrinology, 27 (2004), 199-217. doi: 10.1016/j.domaniend.2004.06.006.

[9]

G. Wu, F. W. Bazer, T. A. Cudd, C. J. Meininger and T. E. Spencer, Maternal nutrition and fetal development, Journal of Nutrition, 134 (2004), 2169-2172.

[10]

G. Wu, F. W. Bazer, J. M. Wallace and T. E. Spencer, Board-invited review: Intraurine growth retardation: Implications for the animal sciences, Journal of Animal Science, 84 (2006), 2316-2337. doi: 10.2527/jas.2006-156.

[11]

G. Zaman, Y. H. Kang and I. H. Jung, Optimal vaccination and treatment in the SIR epidemic model, Proc. KSIAM, 3 (2007), 31-33.

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