Biophysical, Biochemical and Mathematical Doctrine of the Median: The Algorithms of the Unified Bio‐Dynamics Theory for Basic Biomedical R&D and Regulatory Guidance
Authors: Chou Ting-Chao 
Published: April 2020

System analysis via pattern‐combinatory analysis and mathematical induction‐deduction on the mass‐action law (MAL), yields the Median‐Effect Equation (MEE), fa/fu = (D/Dm)m where the Ratio of fraction affected (fa)/fraction unaffected (fu) equals to the Ratio of mass dose (D)/median mass dose (Dm); m is the dynamic order of action. MEE describes the general “Dose and Effect” or “Mass and Action” mathematical relationship and algorism, and indicates that Dose and Effect are interchangeable. This leads to the Doctrine of the Median (DoM), where Dm signifies potency, and m signifies the shape (of the Dose‐Effect Curve, DEC). It further reveals that Dm is the universal reference‐point and the common dynamic‐link at different dynamic orders. In addition, Dm is the Harmonic Mean of kinetic constants, Kii and Kis; and Ki can be calculated from Dm (IC50), and Ki can never greater than IC50. Further, MEE of DoM is the Unified Pharmacodynamics (PD), Biodynamics (BD) and Bioinformatics (BI) Theory [MAL‐PD/BD/BI], encompassing Michaelis‐Menten Eq. of enzyme substrate half‐saturation, Henderson‐Hasselbalch Eq. of pH half‐ionization, Hill Eq. of ligand higher‐order half‐occupancy, and Scatchard Eq. of receptor half‐bound and half‐free.

MAL–MEE reveals two major paradigms: (i) All DECs can be linearized by the Median‐Effect Plot [MEP], x= Log (D)
vs Log (fa/fu), where fa+fu = 1, and when fa=fu, it is “Dm”. By computer simulation (e.g., CalcuSyn or CompuSyn), MEP yields PD‐parameter values, m as slope, and the x‐intercept as [antilog of Dm] (and thus, for the Dm value), instantly. Therefore, DEC is instantly simulated. (ii) Since MEP linearizes all DEC, therefore, based on MEE, we need a minimum of only two data points to simulate the entire dose effect curve instantly. This “Two‐Data Point Theory” (TDPT) should have great impact for animal studies and clinical trial protocol design, since in vivo cannot use too many doses, to avoid severe toxicity or very week effect to measure. TDPT simplicity also raises the issue that single‐dose clinical trials are not the PD Studies. Extension of MEE leads to Combination Index Equation (CIE) and diagnostic Fa‐CI plot, where, CI <1, =1 and >1, indicates/quantitates Synergism, Additive Effect, and Antagonism, respectively, at different dose or effect levels, by computer simulation for 2 to n drug combinations. It also leads to Dose‐Reduction Index Equation (DRIE) and diagnostic Fa‐DRI plot, to calculate how much dose reductions for each drugs as the results of synergy combinations. Also leads to computerized auto‐construction of “Isobologram”, “Dose‐Normalized Isobologram ” and “Polygonogram” for drug synergy graphic analysis.

The MAL‐PD/BD/BI theory, based on MEE, CIE, and its computer‐software, has been applied for single drug or drug combinations, in vitro, in vivo, in animals study and in clinical trial protocol design. Two most relevant references are: 1. Chou TC. Pharmacology. Rev. 58: 621–681, 2006 (Cited 3,119 times in 914 journals) and 2. Chou TC and Talley P. Adv. Enz. Regal. 22: 27–55, 1984 (Cited 6,275 times in 1,249 journals, as of 10.3.2019).

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The Unified PD/BD Theory of The Mass‐Action Law

Figure 1

Theoretical Basis, Experimental Design, and Computerized Simulation of Synergism and Antagonism in Drug Combination Studies

Figure 2