Longevity Formulas- Immortality Knowledge Base

Longevity Formulas

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Model Name	Formula (Simplified)	Description
Integrated Damage Model (proposed)	`ba(t) = P(t) · (1/N_r) · ∫[f_resp(u) / f(m,e)] du`	Refines Escala’s theory by integrating respiration cycles over time, adjusted for species-specific maintenance (f) and life-phase acceleration (P). Can explain aging inflections and outliers like bats. Mechanistic and time-continuous.
Escala’s Respiration-Cycle Law	`t_life ≈ N_r / f_resp`	Proposes organisms undergo a nearly fixed number (~10⁸) of respiration cycles per lifetime. Supported across ~300 species from microbes to mammals. Unifies metabolism and lifespan.
Entropy Generation Model (Silva & Annamalai)	`∫σ(t) dt ≈ constant`	Connects aging to total entropy generated over lifetime. Predicts human lifespan from heat dissipation per kg. Conceptually links thermodynamics and aging.
Reliability Theory of Aging	`Vitality(t) = V₀ - k·t + noise`	Models aging as loss of systemic redundancy. Predicts rising mortality even when components have fixed failure rates. Supports late-life mortality plateaus.
Kleiber’s Law	`B ∝ M^3/4`	Basal metabolic rate scales as body mass to the 3/4 power. Foundational to metabolic and lifespan scaling laws across all taxa.
Lifespan–Mass Allometry	`t_life ∝ M^1/4`	Empirical observation that lifespan increases with body mass. Follows from Kleiber’s law and constant energy-per-mass assumptions.
Rate-of-Living Hypothesis (Rubner)	`E_life/M ≈ constant`	Suggests that organisms expend the same amount of energy per gram over a lifetime. Holds across mammals but breaks down across classes (e.g. birds).
Gompertz–Makeham Law	`μ(t) = α·e^β·t + λ`	Classical model of mortality where death risk grows exponentially with age and includes a constant background risk. Very good empirical fit but not mechanistic.
Gompertz Law	`μ(t) = R₀·e^b·t`	Simpler form of Gompertz–Makeham. Predicts exponential rise in mortality with age. Used widely in demography and aging studies.
Weibull Mortality Law	`μ(t) = k·λ·t^k-1`	Predicts mortality using a power law. Used in reliability theory and some species. Less common for biological aging.
DNA Methylation Clock (Horvath)	`DNAmAge = β₀ + Σ(β_i·CpG_i)`	Uses weighted DNA methylation levels at CpG sites to predict biological age. Highly accurate.
Phenotypic Age (Levine)	`PhenoAge = f(albumin, WBC, etc.)`	Combines standard blood biomarkers to predict mortality risk and biological age.

Model Name	Formula (Simplified)	Description
Integrated Damage Model (proposed)	`ba(t) = P(t) · (1/N_r) · ∫[f_resp(u) / f(m,e)] du`	Refines Escala’s theory by mathematically integrating respiration cycles over continuous time, adjusted for species-specific maintenance factors and life-phase acceleration. It provides a mechanistic framework to account for aging inflections, shifts, and outliers like bats that live exceptionally long relative to their size.
Escala’s Respiration-Cycle Law	`t_life ≈ N_r / f_resp`	Proposes that organisms undergo a nearly fixed number (~10⁸ to 10⁹) of respiration cycles per lifetime across a vast spectrum of species from microbes to mammals. This bridges metabolism, physiological pacing, and maximum biological lifespan.
Bejan’s Constructal Law Model	`Flow access optimization → M^3/4 & M^1/4`	Derived from thermodynamic first-principles, it explains that flow systems (like vascular and respiratory networks) must evolve configuration to minimize flow resistance. This successfully derives why metabolic rates scale to the 3/4 power and frequencies/lifespans to the 1/4 power.
West, Brown, & Enquist (WBE) Model	`dm/dt = (B₀/E_c)m^3/4 - (B_c/E_c)m`	A foundational mathematical framework treating biological transport systems as space-filling, energy-minimizing fractal networks. It fundamentally derives Kleiber's Law and ontogenetic growth curves from cellular-level constraints and hydrodynamics.
Entropy Generation Model (Silva & Annamalai)	`∫σ(t) dt ≈ constant`	Connects organismal aging directly to the total thermodynamic entropy generated over a lifetime. It evaluates heat dissipation per kilogram to predict longevity, bridging classical thermodynamics with biophysical aging.
Reliability Theory of Aging	`Vitality(t) = V₀ - k·t + noise`	Models biological aging as the progressive loss of systemic redundancy and component failure in complex networks. It explains why mortality rates rise exponentially even when individual components maintain fixed failure rates, and accounts for late-life mortality plateaus.
Kleiber’s Law	`B ∝ M^3/4`	The empirical cornerstone of metabolic scaling, showing that basal metabolic rate scales non-linearly with body mass to the three-quarters power, demonstrating that larger organisms are more energetically efficient per gram of tissue.
Lifespan–Mass Allometry	`t_life ∝ M^1/4`	The empirical observation that maximum biological lifespan scales proportionally with body mass raised to the one-quarter power, directly matching the inverse scaling of metabolic and heart rates.
Rate-of-Living Hypothesis (Rubner)	`E_life/M ≈ constant`	Classic hypothesis suggesting that animals expend a roughly fixed amount of energy per gram of tissue over their entire lifetime. While valid across many mammals, it encounters limitations when applied across broader taxa like birds.
Gompertz–Makeham Law	`μ(t) = α·e^β·t + λ`	A classical demographic model describing human and animal mortality where death risk increases exponentially with age, combined with an added constant background risk factor representing environment-driven hazard.
Gompertz Law	`μ(t) = R₀·e^b·t`	A simpler demographic baseline omitting the external environmental term, predicting an exponential rise in baseline physiological mortality risk with age. It is a cornerstone of biogerontology and actuarial science.
Weibull Mortality Law	`μ(t) = k·λ·t^k-1`	Utilizes a power-law formulation to model failure rates and mortality across time. While heavily used in mechanical reliability engineering, it is also applied to certain biological aging studies.
DNA Methylation Clock (Horvath)	`DNAmAge = β₀ + Σ(β_i·CpG_i)`	A molecular epigenetic tool that uses weighted methylation levels at specific CpG sites across the genome to predict biological age and healthspan with high cross-tissue accuracy.
Phenotypic Age (Levine)	`PhenoAge = f(albumin, WBC, etc.)`	An aging biomarker model combining standard clinical blood chemistry panels (such as albumin, white blood cell counts, and metabolic markers) to quantify biological aging and mortality risk accurately.

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