ON THE FIRST FUNDAMENTAL FORMULA IN THE ALONPA HYPOTHESIS

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SUT Journal of Mathematics 〔Fonner］．y TRU Mathematics） Vb1㎝e 25， Ntmber 2〔1989），197−198

℃N．THE FIRST FUNDAMENIIAL「FOR随几A

IN THE ALONPA HYP（皿｛ESIS

Motosaburo MASUYAMA．．（Received September 29，1989；Revised Nbvenber 28，1989） ABSTRACT・ The first fundamental formula was introduced ㎝唄1「1cally ba・ed Cn th・intuiti・n th・t inqividual・va・i・bility in a fundaitiental biological process Xrt i e／ is quite Small and it is』「⑤「esented・by・n・… a㎞・t−・n・par・meter〔in・㎞・t AIDNPA），by ぬi・in we皿ean tw・，・r m・re highly・c・rre1・t・d・param・ter・［1，2］． In this paper， this formula is derived from a plausible mat』matica1頑・1・砲id坤li・・。ne c・nditiOn p。sed・n X（t l eノ．・ AUS ヱ980 subjeet elαssiプ諺cationε． Primary 92AO7； Secondary 92AOg Key w°rds・A㎞・st−・n・pa・ameter（ALONPA〕，・ne c・mpartm・nt m・d・1， proper t1皿e．’ ﹁ 1． Proposed Mヒ）de1 ．． Letち「tノ＝Xrt i ei）b・ava・i・bl・whi・h・haract・・i・es th・．・i−th血dividu。i・ at timeちaPd K。「カノーX帥。ノb・t｝品t・f the reference per・・n．’ Assu「ni”g that th・ differe・ce・グθ。 is suffi・i・ntly・ma11，泌・et

（1）毛＝ち≠°。r・グ・。ノ・・。一・胸1。・・

or

〔2） XゲE。・D。・i・E。−X。一・。・。・

Denoting the mean with respect to i by，ldot，°， we get

〔3） X・−E。・D。θ…

hence it fbllows that

〔4） Xi 一一 “i・’biX・・bi・一・i／・・釦d％−E。r・−biノ，

aSsuming that θ． ≠ 0． The last equation （4） 100ks like the first fmdamental formula in the AL（）NPA hypothesis， but there exists one difference． That is，（5〕 E。−xrtle。ノーθ。ax／∂θ1。 i・・in・9・neral， d・pend・・t・・n・t． F・r it・i・d印・nd・nce， E／e2 i・垣t。grab1。 i。 th。

197

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198

FIRST FORMUIA IN THE ALONPA HYPOTHESIS ・・ig地・rh・・d・fθ。孤d日お・he． f・11・・pg f°m（see．［3］！P47・）（6） XCtlθ）＝ θρ（t） ≠ 〈1（9）， wh。re p i，垣d・P・nd・nt・f・’ End q i・i・d印・nd・nt・f t・its s・ffi・i・n・y i・・b・i・u・・th飢・he c・rre・a・i・n・・effl・i・n・り典・n・、’・nd bi is exactly equal t・一ヱ・when E。＞0・ If the last condition is satisfied， the coordi皿ates of the fixed point cm rXつ聖一pl・ne are given by「E。・E。ノ・ 2．AS」呼1・Ex卿1e・〔in・C・叩ar血・nt・M・d・1 in Ph・rmacOkin・tics L・t・irt／b・the c・ncen・ragi・・6f・d・g9・i・6…d・whiCh．wa・a血i・i・te「ed once at カ＝ O to the i−th patient． Then we have ．〔7） dCi （t）／dt−一ち％rカノ・ら「・ノー°。〔i・d・p・㎡ent・°珂・ where瓦． denotes his rate constant． t ， S・tt血9’Xi （tノーZ・（㌃「tノ・肥・btain

（8） X汐コ。−kiちE。＝Z” C。・，

which has the fom〔6〕，and

（9） Xi・（・）一・i・biX・rtノ・bi−ki／た・and・i−・。「・一ゐ∂・

If w・u・e hi・p・叩er t血・d・fin・d by・i−baちth・n it f・11°W・that

（10） X汐ノーE。一た・・ジ

・・th・・Xi i・i・d・p・・d6・t・fら岨…i−Ti・i・dep・nd・nt・鉱That’is・if

their proper ti皿es are equal to a fixed value T alnong Persons， then their concentrations are the same among them．工n other words， the individual variability among X．rtノ’s vanishes， if their proper times are fixed to be T． The author ac㎞owledges the assistance rendered by Dr． H． Sato ln revlslng the manuscript． REE…REN（コ三S ．．・［1］ Masuyama， M．（1979〕：Hzanαn Bioohem2 cαZ JndividuαZ VntctbiZities an（i TheZr ． Tokyo． euasZ−oonstancy． The Research Institute of Life Insurance Welfare，［2］ Masuyama， M．〔1985〕：tlVze AZmost・−one Pctr・ameteT Hypothesis App Zied to Azano Zogy and Ph（zr7nαookinetios． The Research Institute of Life Insurance Welfare， Tokyo．［3］ ByTKoBcK磁， A．T．〔1979〕：Xapcllcmepucmatnu（：ZlcmeM c Pαcηρθ∂evzeHnaLMu flap（zMepm・ dnpαθOUHoe刀tZco6ue． HayKa． Depart皿ent of Applied Mathematics Science Univers ity of Tokyo 1−3， Kagurazaka， Shinj uku一㎞，． 1 ．・， Tokソ0 162． Japan．