\documentclass[11pt]{article}

\usepackage[margin=1in]{geometry}
\usepackage{amsmath, amssymb}
\usepackage{booktabs}
\usepackage{enumitem}
\usepackage{hyperref}

\title{DeepHealth Disease Expression, Organ Involvement, and Frailty Risk Indices}
\author{}
\date{}

\begin{document}

\maketitle

\begin{abstract}
DeepHealth provides a query-time hidden state \(h(t)\) and disease-specific
risk functions \(p_d(h,\Delta)\). We use these outputs to define a continuous
disease expression rate \(z_d(t)\). This quantity should be interpreted as how
much disease \(d\) is model-implied to have formed or expressed by query time
\(t\), not as true physiological damage. Based on \(z_d(t)\), we define two
downstream indices: an organ involvement index, which summarizes whether an
organ-age-inspired clinical system is involved by any related disease process,
and a DeepHealth-HFRS frailty risk index, which is the original UK-HFRS weighted
sum with binary disease occurrence replaced by continuous disease expression.
\end{abstract}

\section{Disease Expression Rate}

For a patient queried at time \(t\), let the historical readout times be
\[
    t_0 < t_1 < \cdots < t_n \le t,\qquad t_{n+1}=t.
\]
For each interval \([t_i,t_{i+1}]\), DeepHealth produces a hidden state
\(h_i=h(t_i)\) and an interval risk
\[
    q_{d,i}(t)=p_d(h_i,t_{i+1}-t_i).
\]
The model-implied disease expression rate is defined by noisy-or accumulation:
\[
    z_d(t)
    =
    1-\prod_{i=0}^{n}\left[1-q_{d,i}(t)\right].
\]
Informally, \(z_d(t)\) is the degree to which disease \(d\) is expressed in the
patient by time \(t\). Unlike a raw diagnosis indicator, it is continuous and
can reflect heterogeneity within the same ICD label.

\section{Organ Involvement Index}

The organ index is not a frailty score, health reserve score, or organ age. It
is an organ involvement index. Let \(\mathcal{D}_k\) be the set of diseases
assigned to organ/system \(k\). Define disease expression intensity as
\[
    \Lambda_d(t)=-\log\left[1-z_d(t)\right].
\]
The equal-weight organ involvement index is
\[
    O_k(t)
    =
    1-\exp\left(
        -\sum_{d\in\mathcal{D}_k}\Lambda_d(t)
    \right).
\]
Equivalently,
\[
    O_k(t)
    =
    1-
    \prod_{d\in\mathcal{D}_k}
    \left[1-z_d(t)\right].
\]
Thus \(O_k(t)\in[0,1]\) is the probability-like degree to which organ/system
\(k\) is involved by at least one related disease process. In the current
version all diseases assigned to the same organ are equally weighted; this is a
first-stage structural definition. Future versions can introduce
organ-specific disease weights \(\alpha_{k,d}\):
\[
    O_k(t)
    =
    1-\exp\left(
        -\sum_{d\in\mathcal{D}_k}\alpha_{k,d}\Lambda_d(t)
    \right).
\]

\section{Organ List}

The organ/system categories are inspired by organ-age studies, especially
organ-specific plasma proteomic aging models, and are adapted to ICD disease
labels. The current list is:
\begin{center}
\begin{tabular}{ll}
\toprule
ID & Label \\
\midrule
brain\_neurologic & Brain and neurologic system \\
heart & Heart \\
artery\_vascular & Artery and vascular system \\
immune & Immune and infection-related system \\
intestine\_digestive & Intestine and digestive system \\
kidney & Kidney and urinary system \\
liver & Liver \\
lung & Lung and respiratory system \\
muscle\_musculoskeletal & Muscle and musculoskeletal system \\
pancreas\_endocrine & Pancreas and endocrine system \\
adipose\_metabolic & Adipose and metabolic system \\
female\_reproductive & Female reproductive system \\
male\_reproductive & Male reproductive system \\
neoplasm & Neoplasm \\
\bottomrule
\end{tabular}
\end{center}
The neoplasm category is retained as a disease-system category rather than
forced into a single anatomical organ. Sex-specific reproductive diseases are
separated into female and male reproductive systems.

\section{DeepHealth-HFRS Frailty Risk Index}

The original UK-HFRS is a weighted sum over binary disease occurrence:
\[
    \operatorname{HFRS}^{\mathrm{obs}}(t)
    =
    \sum_{d\in\mathcal{D}_{\mathrm{HFRS}}}
    w^{\mathrm{HFRS}}_d\,o_d(t),
    \qquad
    o_d(t)\in\{0,1\}.
\]
DeepHealth-HFRS keeps the published UK-HFRS weights and replaces the binary
disease state with the continuous DeepHealth disease expression rate:
\[
    \operatorname{HFRS}^{\mathrm{DH}}(t)
    =
    \sum_{d\in\mathcal{D}_{\mathrm{HFRS}}}
    w^{\mathrm{HFRS}}_d\,z_d(t),
    \qquad
    z_d(t)\in[0,1].
\]
This is a natural continuous extension of the original HFRS, so it can still be
called a frailty risk index. The semantic change is not the HFRS weight system;
the change is the disease state variable.

\section{Current Implementation}

The current code computes historical current-state indices only. No future
horizon is used. For each landmark age \(t\), it outputs:
\begin{itemize}[leftmargin=*]
    \item \(z_d(t)\) internally as model-implied disease expression;
    \item \(O_k(t)\) as equal-weight organ involvement;
    \item \(\operatorname{HFRS}^{\mathrm{DH}}(t)\) as DeepHealth-HFRS frailty
    risk.
\end{itemize}
The output table uses the columns
\[
    \texttt{index\_type},\quad
    \texttt{index\_id},\quad
    \texttt{index\_label},\quad
    \texttt{index\_value}.
\]

\end{document}
