For a Drug the Liver Extracts Almost Completely, Blood Flow Sets the Injection and Enzymes Set the Pill ── for a drug with extraction ratio 0.95, halving the liver's enzyme activity raises exposure to an injection only 1.050000-fold while exposure to a pill exactly doubles ── when liver blood flow falls to 70 per cent the roles reverse ── [Paper 591]

Abstract

It is said that when the liver weakens, drugs act too strongly. What this paper shows is that the same change in the liver acts differently on an injection and on a pill, and that for a drug the liver removes almost completely (high extraction ratio) the injection is governed by liver blood flow and the pill by the liver's enzyme activity. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──the well-stirred model of the liver, the extraction ratio, bioavailability, and the relation between exposure (the time integral of blood concentration) and clearance are all standard. Only the simplest model is used──a pill is taken to be completely absorbed; metabolism in the gut wall, elimination outside the liver, return of drug from bile to gut, and changes in protein binding are not included. A liver blood flow of 90 L/h is representative──the two drugs are set by extraction ratios of 0.95 and 0.05, and no drug is named. No dosing is advised──adjusting doses is for physicians and pharmacists; this paper looks only at the form of the equations. Relation to earlier papers: Paper 578 showed that what sets a dosing interval is the half-life and the width of the therapeutic window, and set aside in its scope note both absorption (taken as instant) and differences between people in liver function──this paper takes that seat and shows that a change in liver function moves exposure differently by injection and by pill. Paper 300 showed that sharing a root is decidable──exposure to the injection and exposure to the pill come from one formula and share a root; the limits of the extraction ratio separate them. What is added is setting out exposure ratios for two drugs and three changes, confirming by two routes, closed forms and equations of motion, that the pill's exposure does not depend on blood flow and exactly doubles when enzyme activity halves, and placing the separator on the extraction ratio. First, a drug of high extraction ratio barely reaches the body when swallowed──with a liver blood flow of 90 L/h and an extraction ratio of 0.95, only 0.050000 of a swallowed dose passes the liver, so a pill needs 20.000000 times the dose of an injection (Section 2). Second, and this is the core. When the liver's enzyme activity halves, exposure to the injection rises only 1.050000-fold, while exposure to the pill exactly doubles, 2.000000-fold──the same change in the liver acts almost twice as strongly by one route as by the other (Section 3). Third, when liver blood flow falls to 70 per cent, the roles reverse──exposure to the injection rises 1.407143-fold, and exposure to the pill stays at 1.000000, because the pill's exposure is the dose divided by the enzymes' capacity and contains no blood flow (Section 3). Fourth, for a drug of extraction ratio 0.05 the two routes agree──when enzyme activity halves, the injection rises 1.950000-fold and the pill 2.000000-fold, both nearly double (Section 3). Fifth, solving the equations of two compartments, body and liver, gives the same──eight exposures agree with the closed forms to a relative 5.4x10^-7 (Section 4). Sixth, the separator is whether the extraction ratio is near 1 or near 0──near 1, blood flow holds the injection and enzymes hold the pill; near 0, enzymes hold both (Section 5). It is said that when the liver weakens, drugs act too strongly. That depends on which function weakens, and by which route──a drug with extraction ratio 0.95 at a liver blood flow of 90 L/h reaches the body at only 0.050000 when swallowed, so a pill needs 20.000000 times the injected dose. When the liver's enzyme activity halves, injection exposure rises only 1.050000-fold while pill exposure exactly doubles, 2.000000-fold. When liver blood flow falls to 70 per cent the roles reverse: the injection rises 1.407143-fold and the pill does not move, because pill exposure is the dose divided by the enzymes' capacity and contains no blood flow. For a drug of extraction 0.05, halved enzymes raise the injection 1.950000-fold and the pill 2-fold: the two routes agree. Solving the equations of two compartments, body and liver, the eight exposures agreed with the closed forms to a relative 5.4x10^-7. The separator is whether the extraction ratio is near 1 or near 0──near 1, flow holds the injection and enzymes hold the pill; under Paper 300 the two come from one formula and share a root. Placed among the earlier papers──this takes the seat that Paper 578 left in its scope note, for differences in liver function and for absorption. To be honest──this is the simplest well-stirred model, with complete absorption and without metabolism in the gut wall, elimination outside the liver, return from bile to gut, or changes in protein binding. The liver blood flow is representative, no drug is named, and no dosing is advised. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: pharmacokinetics, hepatic extraction ratio, bioavailability, clearance, first-pass effect. ----- 肝臓の働きが落ちると、薬が効きすぎる、と言われる。本稿が示すのは、同じ肝臓の変化が静注と飲み薬とで別の向きに効き、肝臓が薬をほぼ取り切る(抽出率の高い)薬では、静注の効き目は肝血流で、飲み薬の効き目は肝臓の酵素の働きで決まることである。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──肝臓の中がよく混ざっているとする模型、抽出率、生物学的利用能、曝露(血中濃度の時間の積分)とクリアランスの関係は、すべて標準的である。最も単純な模型に限る──飲み薬は完全に吸収されるとし、消化管の壁での代謝、肝臓以外での消失、胆汁から腸へ戻る循環、血中の蛋白との結合の変化は入れない。肝血流 90 L/h は代表値である──二つの薬は抽出率 0.95 と 0.05 で置き、薬の名前は挙げない。服薬を指示しない──量の調整は医師と薬剤師が決めることで、本稿は式の形を見るだけである。既刊との関係:論文578 は、飲む間隔を決めるのが半減期と治療域の幅の二つであることを示し、射程注記で吸収を瞬時とし、肝臓の働きによる人の違いを扱わないとした──本稿はその席に座り、肝臓の働きが変わると、静注と飲み薬とで曝露が別の向きに動くことを示す。論文300 は同根か別根かが判定できると示した──静注の曝露と飲み薬の曝露は同じ一つの式から出る同根で、抽出率の極限が二つを分ける。加えたのは、二つの薬と三つの変化で曝露の比を並べたこと、飲み薬の曝露が肝血流に依らず、酵素が半分でちょうど 2 倍になることを、閉じた式と運動方程式の二つの道で確かめたこと、分離子を抽出率に置いたことである。 第一に、抽出率の高い薬は、飲むとほとんど体に届かない──肝血流 90 L/h で抽出率 0.95 の薬は、飲むと肝臓を一度通るあいだに体に届くのが 0.050000 で、静注と同じだけ効かせるには 20.000000 倍の量が要る(第2節)。 第二に、これが本稿の芯である。肝臓の酵素の働きが半分になると、静注の曝露は 1.050000 倍にしかならないのに、飲み薬の曝露はちょうど 2.000000 倍になる──同じ肝臓の変化が、投与の道で二倍近く違って効く(第3節)。 第三に、肝血流が 7 割に落ちると、向きが入れ替わる──静注の曝露は 1.407143 倍になり、飲み薬の曝露は 1.000000 倍のまま動かない。飲み薬の曝露は投与量を酵素の処理する力で割ったもので、肝血流を含まないからである(第3節)。 第四に、抽出率 0.05 の薬では、二つの道がそろう──酵素の働きが半分になると、静注は 1.950000 倍、飲み薬は 2.000000 倍で、どちらもほぼ倍になる(第3節)。 第五に、体と肝臓の二つの箱の運動方程式を数値で解いても同じである──八つの曝露が、閉じた式と相対差 5.4x10^-7 以内で一致する(第4節)。 第六に、分離子は、抽出率が 1 に近いか 0 に近いかである──1 に近い薬では、静注の曝露は肝血流が、飲み薬の曝露は酵素が握る。0 に近い薬では、どちらも酵素が握る(第5節)。 肝臓の働きが落ちると、薬が効きすぎる、と言われる。それは、どの働きがどの道で落ちたかによる──肝血流 90 L/h で抽出率 0.95 の薬は、飲むと体に届くのが 0.050000 で、静注の 20.000000 倍の量が要る。肝臓の酵素の働きが半分になると、静注の曝露は 1.050000 倍にしかならないのに、飲み薬の曝露はちょうど 2.000000 倍になる。肝血流が 7 割に落ちると向きが入れ替わり、静注は 1.407143 倍、飲み薬は 1 倍のまま動かない。飲み薬の曝露は投与量を酵素の処理する力で割ったもので、肝血流を含まないからである。抽出率 0.05 の薬では、酵素が半分で静注 1.950000 倍・飲み薬 2 倍と、二つの道がそろう。体と肝臓の二つの箱の運動方程式を数値で解いても、八つの曝露は閉じた式と相対差 5.4x10^-7 以内で一致した。分離子は、抽出率が 1 に近いか 0 に近いかである──1 に近い薬では、静注は血流が、飲み薬は酵素が握る。論文300 の基準で、二つは同じ式から出る同根である。既刊との位置──論文578 が射程注記で空けていた、肝臓の働きによる違いと吸収の席に座った。正直に言えば──肝臓の中がよく混ざるとする最も単純な模型で、吸収は完全とし、消化管の壁での代謝、肝臓以外での消失、胆汁から腸へ戻る循環、蛋白との結合の変化は入れていない。肝血流は代表値で、薬の名前は挙げず、服薬も指示しない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:薬物動態、肝抽出率、生物学的利用能、クリアランス、初回通過効果。