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The Journal of the Royal Institute of Thailand Vol. 31 No. 1 Jan.-Mar. 2006 °“√ √â “ß·≈–«‘ ‡§√“–Àå µ— «·∫∫∑“ߧ≥‘ µ»“ µ√å ¢Õß°“√ √â “ß·≈– ≈“¬°√–¥Ÿ ° : º≈¢Õß°“√„Àâ ‡Õ ‚∑√‡®π∑¥·∑π 54 Abstract Construction and Analysis of a Mathematical Model of Bone Formation and Resorption: Effect of Estrogen Supplements Yongwimon Lenbury Fellow of The Academy of Science, The Royal Institute, Thailand Chontita Rattanakul Department of Mathematics, Faculty of Science, Mahidol University Bone, a major reservior of body calcium, is under the hormonal control of the parathyroid hormone (PTH). Estrogen plays an important role in bone physiology, and is extremely essential to maintain bone balance in adults. In order to provide a basis for understanding the underlying mechanisms of bone remodelling mediated by PTH as well as the action of estrogen therapy on bone turnover, we propose a mathematical model of the process. The nonlinear system model is then analyzed using singular perturbation arguments. Effect of estrogen administrations is investigated by extending on the core model, leading to clinically valuable discoveries. Key words : Bone resorption, estrogen supplements, mathematical model §«“¡µà “ß°— π¢Õߧ” çπ«— µ°√√¡é ·≈– çπ«°√√¡é §” çπ«— µ°√√¡é ‡ªì π»— æ∑å ∫— ≠≠— µ‘ ¡“®“°§”¿“…“Õ— ß°ƒ…«à “ çinnovationé æ®π“πÿ °√¡ ©∫— ∫√“™∫— ≥±‘ µ¬ ∂“π æ.». ÚıÙÚ „Àâ ∫∑π‘ ¬“¡‰«â ¥— ßπ’È π«— µ°√√¡ π. ‘Ë ß∑’Ë ∑”¢÷È π„À¡à À√◊ Õ·ª≈°®“°‡¥‘ ¡´÷Ë ßÕ“®®–‡ªì 𧫓¡§‘ ¥ «‘ ∏’ °“√ À√◊ Õ Õÿ ª°√≥å ‡ªì πµâ π (ª. π«µ + . °√⁄ ¡; Õ. innovation). à «π§” çπ«°√√¡é æ®π“πÿ °√¡ ©∫— ∫√“™∫— ≥±‘ µ¬ ∂“π ‰¥â „Àâ ∫∑π‘ ¬“¡‰«â «à “ çπ. °“√ °à Õ √â “ß.é ´÷Ë ß‡ªì π§”∑’Ë ¡’ §«“¡À¡“¬µ“¡∑’Ë „™â Õ¬Ÿà „π¿“…“∫“≈’ ¥— ßπ—È π §” çπ«°√√¡é ®÷ ߇ªì π§π≈– §”°— ∫ çπ«— µ°√√¡é ´÷Ë ß‡ªì π°“√ √â “ß»— æ∑å „™â ·∑π§”¿“…“µà “ߪ√–‡∑»