577 lines
18 KiB
TeX
577 lines
18 KiB
TeX
|
\documentclass[11pt]{article}
|
||
|
|
|
||
|
|
\usepackage[margin=1in]{geometry}
|
||
|
|
\usepackage{amsmath, amssymb, amsfonts}
|
||
|
|
\usepackage{bm}
|
||
|
|
\usepackage{booktabs}
|
||
|
|
\usepackage{enumitem}
|
||
|
|
\usepackage{hyperref}
|
||
|
|
|
||
|
|
\title{DeepHealth Burden Indices: Dynamic Organ and Functional Burden}
|
||
|
|
\author{}
|
||
|
|
\date{}
|
||
|
|
|
||
|
|
\begin{document}
|
||
|
|
|
||
|
|
\maketitle
|
||
|
|
|
||
|
|
\begin{abstract}
|
||
|
|
DeepHealth produces a query-time hidden representation \(h(t)\) and
|
||
|
|
disease-specific future risk functions \(p_d(h,\Delta)\). These disease-level
|
||
|
|
outputs are clinically granular but difficult to interpret directly as a
|
||
|
|
patient-level health state. We therefore define Burden Indices (BI) that
|
||
|
|
aggregate historical and predicted disease burden into higher-level,
|
||
|
|
interpretable dimensions. The Organ Burden Index (OBI) maps diseases to
|
||
|
|
anatomical systems, while the Functional Burden Index (FBI) maps diseases to
|
||
|
|
function- and frailty-related burden domains, anchored by CIHI-HFRM-style
|
||
|
|
diagnosis weights when available. For formed burden, we distinguish an
|
||
|
|
observed-anchored version based on actual historical diagnoses and a
|
||
|
|
model-weighted version based on DeepHealth's historical risk trajectory. The
|
||
|
|
indices are burden measures, not direct health reserve measures, because the
|
||
|
|
current model is supervised by disease events rather than direct functional
|
||
|
|
outcomes such as ADL/IADL, gait speed, grip strength, cognition, or recovery
|
||
|
|
capacity.
|
||
|
|
\end{abstract}
|
||
|
|
|
||
|
|
\section{Motivation}
|
||
|
|
|
||
|
|
At query time \(t\), DeepHealth produces a hidden state \(h(t)\) and
|
||
|
|
disease-level risk predictions
|
||
|
|
\[
|
||
|
|
p_d(h(t), \tau), \qquad d = 1,\ldots,D,
|
||
|
|
\]
|
||
|
|
where \(p_d(h(t),\tau)\) is the predicted probability of disease \(d\) occurring
|
||
|
|
within future horizon \(\tau\). These outputs are useful for disease-specific
|
||
|
|
risk prediction, but they do not directly answer patient-level questions such as
|
||
|
|
where disease burden is concentrated or how much functional vulnerability is
|
||
|
|
implied by the disease profile.
|
||
|
|
|
||
|
|
We introduce Burden Indices to summarize disease-level predictions into
|
||
|
|
interpretable state representations:
|
||
|
|
\[
|
||
|
|
\text{disease-level risk}
|
||
|
|
\quad \longrightarrow \quad
|
||
|
|
\text{system-level burden}.
|
||
|
|
\]
|
||
|
|
The indices combine two components:
|
||
|
|
\begin{enumerate}[leftmargin=*]
|
||
|
|
\item formed burden: disease burden already accumulated by query time \(t\);
|
||
|
|
\item future expected burden: disease burden expected to newly form within
|
||
|
|
horizon \(\tau\).
|
||
|
|
\end{enumerate}
|
||
|
|
|
||
|
|
At the current stage, these quantities should be called burden indices rather
|
||
|
|
than health reserve or health state scores. The current model is trained and
|
||
|
|
validated primarily on ICD disease events. Its directly verifiable semantics are
|
||
|
|
disease occurrence and disease risk. Without direct functional labels or a
|
||
|
|
calibrated healthy reference state, quantities such as \(100-\text{burden}\)
|
||
|
|
cannot be rigorously interpreted as remaining health reserve.
|
||
|
|
|
||
|
|
\section{Two Burden Spaces}
|
||
|
|
|
||
|
|
We define two complementary burden spaces.
|
||
|
|
|
||
|
|
\subsection{Organ Burden Index}
|
||
|
|
|
||
|
|
The Organ Burden Index (OBI) maps disease burden to anatomical systems. It
|
||
|
|
answers:
|
||
|
|
\[
|
||
|
|
\text{Which organs or anatomical systems carry the largest pathological burden?}
|
||
|
|
\]
|
||
|
|
Typical dimensions may include heart/vascular, brain/neurological, kidney,
|
||
|
|
lung, liver/digestive, metabolic/endocrine, musculoskeletal, hematologic, and
|
||
|
|
malignancy-related systems. The mapping matrix is denoted
|
||
|
|
\[
|
||
|
|
A^{\mathrm{organ}} \in \mathbb{R}_{\ge 0}^{K_o \times D},
|
||
|
|
\]
|
||
|
|
where \(A^{\mathrm{organ}}_{k,d}\) is the contribution weight from disease
|
||
|
|
\(d\) to organ dimension \(k\).
|
||
|
|
|
||
|
|
\subsection{Functional Burden Index}
|
||
|
|
|
||
|
|
The Functional Burden Index (FBI) maps disease burden to function- and
|
||
|
|
frailty-related diagnostic burden domains. It answers:
|
||
|
|
\[
|
||
|
|
\text{How much functional vulnerability is implied by the disease burden?}
|
||
|
|
\]
|
||
|
|
Candidate dimensions include mobility burden, cognition burden, mood burden,
|
||
|
|
sensory burden, nutrition burden, infection or immune vulnerability burden,
|
||
|
|
functional dependence burden, and comorbidity burden.
|
||
|
|
|
||
|
|
When CIHI-HFRM or another validated hospital frailty risk measure code list is
|
||
|
|
available, it should be used as the primary anchor for FBI. The mapping matrix
|
||
|
|
is denoted
|
||
|
|
\[
|
||
|
|
A^{\mathrm{func}} \in \mathbb{R}_{\ge 0}^{K_f \times D},
|
||
|
|
\]
|
||
|
|
where \(A^{\mathrm{func}}_{k,d}\) is the contribution weight from disease \(d\)
|
||
|
|
to functional burden dimension \(k\).
|
||
|
|
|
||
|
|
OBI and FBI are not redundant. OBI describes where pathology is concentrated,
|
||
|
|
whereas FBI describes how disease burden may translate into functional
|
||
|
|
vulnerability. For example, stroke contributes primarily to brain/vascular
|
||
|
|
burden in OBI, but may contribute to mobility, cognition, sensory, and
|
||
|
|
functional dependence burden in FBI.
|
||
|
|
|
||
|
|
\section{Model Outputs and Disease Risk Function}
|
||
|
|
|
||
|
|
For each hidden state \(h\), DeepHealth defines a disease-specific future risk
|
||
|
|
function
|
||
|
|
\[
|
||
|
|
p_d(h,\Delta),
|
||
|
|
\]
|
||
|
|
where \(\Delta \ge 0\) is the time horizon. The risk function is produced by the
|
||
|
|
all-future model. Let
|
||
|
|
\[
|
||
|
|
\eta_d(h) = \operatorname{risk\_head}(h)_d,
|
||
|
|
\qquad
|
||
|
|
\lambda_d(h) = \operatorname{softplus}(\eta_d(h)).
|
||
|
|
\]
|
||
|
|
|
||
|
|
For the exponential all-future model,
|
||
|
|
\[
|
||
|
|
p_d(h,\Delta)
|
||
|
|
=
|
||
|
|
1-\exp[-\lambda_d(h)\Delta].
|
||
|
|
\]
|
||
|
|
|
||
|
|
For the Weibull all-future model, with
|
||
|
|
\[
|
||
|
|
\rho_d(h)=\operatorname{softplus}(\operatorname{rho\_head}(h)_d),
|
||
|
|
\]
|
||
|
|
the risk function is
|
||
|
|
\[
|
||
|
|
p_d(h,\Delta)
|
||
|
|
=
|
||
|
|
1-\exp[-\lambda_d(h)\Delta^{\rho_d(h)}].
|
||
|
|
\]
|
||
|
|
|
||
|
|
The burden formulation below only assumes access to \(p_d(h,\Delta)\); the
|
||
|
|
exponential and Weibull cases are specializations.
|
||
|
|
|
||
|
|
\section{Formed Burden}
|
||
|
|
|
||
|
|
For a patient queried at time \(t\), let the available historical readout times
|
||
|
|
be
|
||
|
|
\[
|
||
|
|
t_0 < t_1 < \cdots < t_n \le t.
|
||
|
|
\]
|
||
|
|
For notational convenience, define
|
||
|
|
\[
|
||
|
|
t_{n+1}=t.
|
||
|
|
\]
|
||
|
|
The historical trajectory is partitioned into adjacent, non-overlapping
|
||
|
|
intervals
|
||
|
|
\[
|
||
|
|
[t_i,t_{i+1}], \qquad i=0,\ldots,n.
|
||
|
|
\]
|
||
|
|
Let
|
||
|
|
\[
|
||
|
|
h_i = h(t_i), \qquad \Delta_i = t_{i+1}-t_i.
|
||
|
|
\]
|
||
|
|
|
||
|
|
The interval-level model-implied probability for disease \(d\) is
|
||
|
|
\[
|
||
|
|
q_{d,i}(t) = p_d(h_i,\Delta_i).
|
||
|
|
\]
|
||
|
|
|
||
|
|
\subsection{Model-Weighted Formed Burden}
|
||
|
|
|
||
|
|
The model-weighted formed burden uses DeepHealth's own historical risk
|
||
|
|
trajectory to quantify how strongly disease \(d\) is represented as formed
|
||
|
|
burden by time \(t\). It is defined by noisy-or accumulation over historical
|
||
|
|
intervals:
|
||
|
|
\[
|
||
|
|
z^{\mathrm{model}}_d(t)
|
||
|
|
=
|
||
|
|
1-\prod_{i=0}^{n}\left[1-q_{d,i}(t)\right].
|
||
|
|
\]
|
||
|
|
Equivalently,
|
||
|
|
\[
|
||
|
|
z^{\mathrm{model}}_d(t)
|
||
|
|
=
|
||
|
|
1-\prod_{i=0}^{n}
|
||
|
|
\left[
|
||
|
|
1-p_d\!\left(h(t_i),t_{i+1}-t_i\right)
|
||
|
|
\right],
|
||
|
|
\qquad t_{n+1}=t.
|
||
|
|
\]
|
||
|
|
|
||
|
|
This definition uses each segment of the historical trajectory exactly once and
|
||
|
|
therefore avoids repeatedly counting overlapping predictions from multiple
|
||
|
|
historical states to the same query time.
|
||
|
|
|
||
|
|
\subsection{Observed-Anchored Formed Burden}
|
||
|
|
|
||
|
|
The observed-anchored formed burden treats historical diagnoses as factual
|
||
|
|
evidence. Define the observed historical disease indicator
|
||
|
|
\[
|
||
|
|
o_d(t)
|
||
|
|
=
|
||
|
|
\mathbb{I}\left\{
|
||
|
|
\exists j:\; \mathrm{event}_j=d,\; \mathrm{time}_j\le t
|
||
|
|
\right\}.
|
||
|
|
\]
|
||
|
|
The observed-anchored version is
|
||
|
|
\[
|
||
|
|
z^{\mathrm{obs}}_d(t)=o_d(t).
|
||
|
|
\]
|
||
|
|
|
||
|
|
This version is closest to diagnosis-code burden measures such as HFRM: once a
|
||
|
|
disease code has appeared before query time \(t\), the corresponding disease
|
||
|
|
burden component is considered present. It is maximally auditable and aligned
|
||
|
|
with code-based burden definitions, but it does not distinguish severity,
|
||
|
|
recency, or residual impact among patients with the same historical diagnosis.
|
||
|
|
|
||
|
|
\subsection{Choice of Formed Burden}
|
||
|
|
|
||
|
|
The two definitions represent different semantics:
|
||
|
|
\[
|
||
|
|
z^{\mathrm{obs}}_d(t)
|
||
|
|
=
|
||
|
|
\text{observed diagnostic burden},
|
||
|
|
\]
|
||
|
|
\[
|
||
|
|
z^{\mathrm{model}}_d(t)
|
||
|
|
=
|
||
|
|
\text{model-weighted state burden}.
|
||
|
|
\]
|
||
|
|
The observed-anchored version should be used when the goal is to reproduce or
|
||
|
|
extend diagnosis-code burden measures. The model-weighted version should be used
|
||
|
|
when the goal is to let DeepHealth assign a continuous burden strength based on
|
||
|
|
the historical hidden-state trajectory. In the formulas below, \(z_d(t)\)
|
||
|
|
denotes either \(z^{\mathrm{obs}}_d(t)\) or \(z^{\mathrm{model}}_d(t)\), depending
|
||
|
|
on the selected BI variant.
|
||
|
|
|
||
|
|
\subsection{Observed-Anchored versus Model-Weighted Burden}
|
||
|
|
|
||
|
|
The observed-anchored and model-weighted definitions share the same purpose:
|
||
|
|
both quantify disease burden already formed by query time \(t\), before adding
|
||
|
|
future expected burden. They also use the same downstream BI equations; the only
|
||
|
|
difference is the definition of \(z_d(t)\).
|
||
|
|
|
||
|
|
Their key difference is the evidence treated as primary. The observed-anchored
|
||
|
|
version treats diagnosis occurrence as the primary unit of evidence:
|
||
|
|
\[
|
||
|
|
z^{\mathrm{obs}}_d(t)=1
|
||
|
|
\quad\text{once disease } d \text{ has been observed before } t.
|
||
|
|
\]
|
||
|
|
This is appropriate when the burden index is intended to remain close to
|
||
|
|
diagnosis-code measures such as HFRM. It is transparent and robust to model
|
||
|
|
miscalibration, but it treats all historical occurrences of the same disease as
|
||
|
|
equally formed burden.
|
||
|
|
|
||
|
|
The model-weighted version treats the DeepHealth risk trajectory as the primary
|
||
|
|
unit of evidence:
|
||
|
|
\[
|
||
|
|
z^{\mathrm{model}}_d(t)
|
||
|
|
=
|
||
|
|
1-\prod_i
|
||
|
|
\left[
|
||
|
|
1-p_d\!\left(h(t_i),t_{i+1}-t_i\right)
|
||
|
|
\right].
|
||
|
|
\]
|
||
|
|
This version can assign different burden strengths to the same observed disease
|
||
|
|
depending on timing, surrounding history, extra-info context, and the hidden
|
||
|
|
state trajectory. It may better reflect state-dependent burden intensity, but it
|
||
|
|
can also downweight a disease that was truly observed if the model assigns low
|
||
|
|
historical probability.
|
||
|
|
|
||
|
|
Thus the two variants answer related but non-identical questions:
|
||
|
|
\[
|
||
|
|
z^{\mathrm{obs}}_d(t):
|
||
|
|
\text{Has disease } d \text{ been recorded as part of the patient's history?}
|
||
|
|
\]
|
||
|
|
\[
|
||
|
|
z^{\mathrm{model}}_d(t):
|
||
|
|
\text{How strongly does the model-implied trajectory support burden from }
|
||
|
|
d \text{ by time } t?
|
||
|
|
\]
|
||
|
|
For this reason, both should be considered useful sensitivity variants. The
|
||
|
|
observed-anchored version is preferable for auditability and alignment with
|
||
|
|
existing code-based indices. The model-weighted version is preferable when the
|
||
|
|
goal is to use DeepHealth as a continuous state model and allow the learned
|
||
|
|
trajectory to modulate burden strength.
|
||
|
|
|
||
|
|
\subsection{Cumulative Intensity Form}
|
||
|
|
|
||
|
|
Define the interval cumulative intensity
|
||
|
|
\[
|
||
|
|
\ell_d(h_i,\Delta_i)
|
||
|
|
=
|
||
|
|
-\log\left[1-p_d(h_i,\Delta_i)\right],
|
||
|
|
\]
|
||
|
|
and the accumulated historical intensity
|
||
|
|
\[
|
||
|
|
\Lambda^{\mathrm{model}}_d(t)=\sum_{i=0}^{n}\ell_d(h_i,\Delta_i).
|
||
|
|
\]
|
||
|
|
Then
|
||
|
|
\[
|
||
|
|
z^{\mathrm{model}}_d(t)=1-\exp[-\Lambda^{\mathrm{model}}_d(t)].
|
||
|
|
\]
|
||
|
|
|
||
|
|
For the exponential model,
|
||
|
|
\[
|
||
|
|
\ell_d(h_i,\Delta_i)=\lambda_d(h_i)\Delta_i,
|
||
|
|
\]
|
||
|
|
so
|
||
|
|
\[
|
||
|
|
z^{\mathrm{model}}_d(t)
|
||
|
|
=
|
||
|
|
1-\exp\left[
|
||
|
|
-\sum_{i=0}^{n}
|
||
|
|
\lambda_d\!\left(h(t_i)\right)(t_{i+1}-t_i)
|
||
|
|
\right].
|
||
|
|
\]
|
||
|
|
|
||
|
|
For the Weibull model,
|
||
|
|
\[
|
||
|
|
\ell_d(h_i,\Delta_i)
|
||
|
|
=
|
||
|
|
\lambda_d(h_i)\Delta_i^{\rho_d(h_i)},
|
||
|
|
\]
|
||
|
|
so
|
||
|
|
\[
|
||
|
|
z^{\mathrm{model}}_d(t)
|
||
|
|
=
|
||
|
|
1-\exp\left[
|
||
|
|
-\sum_{i=0}^{n}
|
||
|
|
\lambda_d\!\left(h(t_i)\right)
|
||
|
|
(t_{i+1}-t_i)^{\rho_d(h(t_i))}
|
||
|
|
\right].
|
||
|
|
\]
|
||
|
|
|
||
|
|
\section{Future Expected Burden}
|
||
|
|
|
||
|
|
The selected formed burden \(z_d(t)\) represents disease burden already formed
|
||
|
|
by query time \(t\). It can be either the observed-anchored burden
|
||
|
|
\(z^{\mathrm{obs}}_d(t)\) or the model-weighted burden
|
||
|
|
\(z^{\mathrm{model}}_d(t)\). The current future risk from query time \(t\) to
|
||
|
|
horizon \(\tau\) is
|
||
|
|
\[
|
||
|
|
p_d(h(t),\tau).
|
||
|
|
\]
|
||
|
|
The future expected newly formed burden for disease \(d\) is defined as
|
||
|
|
\[
|
||
|
|
f_d(t,\tau)
|
||
|
|
=
|
||
|
|
\left[1-z_d(t)\right]p_d(h(t),\tau).
|
||
|
|
\]
|
||
|
|
This term counts only the portion of disease burden that has not already formed
|
||
|
|
by time \(t\). The total dynamic disease burden contribution is
|
||
|
|
\[
|
||
|
|
b_d(t,\tau)
|
||
|
|
=
|
||
|
|
z_d(t)+f_d(t,\tau).
|
||
|
|
\]
|
||
|
|
Equivalently,
|
||
|
|
\[
|
||
|
|
b_d(t,\tau)
|
||
|
|
=
|
||
|
|
1-
|
||
|
|
\left[1-z_d(t)\right]
|
||
|
|
\left[1-p_d(h(t),\tau)\right].
|
||
|
|
\]
|
||
|
|
Thus \(b_d(t,\tau)\) can be interpreted as the probability that disease burden
|
||
|
|
for \(d\) has formed by time \(t\) or will newly form within the future horizon
|
||
|
|
\(\tau\).
|
||
|
|
|
||
|
|
\section{Burden Index Definition}
|
||
|
|
|
||
|
|
Let \(A \in \mathbb{R}_{\ge 0}^{K \times D}\) be a disease-to-burden mapping
|
||
|
|
matrix. The historical, future, and total burden indices for dimension \(k\)
|
||
|
|
are
|
||
|
|
\[
|
||
|
|
\operatorname{BI}^{\mathrm{hist}}_k(t)
|
||
|
|
=
|
||
|
|
\sum_{d=1}^{D} A_{k,d} z_d(t),
|
||
|
|
\]
|
||
|
|
\[
|
||
|
|
\operatorname{BI}^{\mathrm{future}}_k(t,\tau)
|
||
|
|
=
|
||
|
|
\sum_{d=1}^{D}
|
||
|
|
A_{k,d}
|
||
|
|
\left[1-z_d(t)\right]p_d(h(t),\tau),
|
||
|
|
\]
|
||
|
|
and
|
||
|
|
\[
|
||
|
|
\operatorname{BI}^{\mathrm{total}}_k(t,\tau)
|
||
|
|
=
|
||
|
|
\operatorname{BI}^{\mathrm{hist}}_k(t)
|
||
|
|
+
|
||
|
|
\operatorname{BI}^{\mathrm{future}}_k(t,\tau).
|
||
|
|
\]
|
||
|
|
Equivalently,
|
||
|
|
\[
|
||
|
|
\operatorname{BI}^{\mathrm{total}}_k(t,\tau)
|
||
|
|
=
|
||
|
|
\sum_{d=1}^{D}
|
||
|
|
A_{k,d}
|
||
|
|
\left\{
|
||
|
|
1-
|
||
|
|
\left[1-z_d(t)\right]
|
||
|
|
\left[1-p_d(h(t),\tau)\right]
|
||
|
|
\right\}.
|
||
|
|
\]
|
||
|
|
|
||
|
|
For OBI, \(A=A^{\mathrm{organ}}\). For FBI, \(A=A^{\mathrm{func}}\).
|
||
|
|
|
||
|
|
\section{Constructing the Mapping Matrices}
|
||
|
|
|
||
|
|
\subsection{Organ Mapping Matrix}
|
||
|
|
|
||
|
|
The organ mapping matrix should be constructed from code taxonomy or validated
|
||
|
|
clinical grouping systems rather than manually assigned arbitrary weights. In
|
||
|
|
the current ICD-token setting, the first version can use ICD chapters or
|
||
|
|
predefined ICD ranges to construct a sparse disease-to-organ mask
|
||
|
|
\[
|
||
|
|
M^{\mathrm{organ}}_{k,d}\in\{0,1\}.
|
||
|
|
\]
|
||
|
|
Examples include:
|
||
|
|
\begin{center}
|
||
|
|
\begin{tabular}{ll}
|
||
|
|
\toprule
|
||
|
|
Dimension & Example ICD groups \\
|
||
|
|
\midrule
|
||
|
|
Heart/vascular & I00--I99, optionally split into cardiac and vascular groups \\
|
||
|
|
Brain/neurological & G00--G99, F00--F09, I60--I69 \\
|
||
|
|
Kidney/urogenital & N00--N39, especially N17--N19 \\
|
||
|
|
Lung/respiratory & J00--J99 \\
|
||
|
|
Metabolic/endocrine & E00--E90 \\
|
||
|
|
Liver/digestive & K00--K93, especially K70--K77 \\
|
||
|
|
Musculoskeletal & M00--M99 \\
|
||
|
|
\bottomrule
|
||
|
|
\end{tabular}
|
||
|
|
\end{center}
|
||
|
|
|
||
|
|
The simplest organ weights are
|
||
|
|
\[
|
||
|
|
A^{\mathrm{organ}}_{k,d}=M^{\mathrm{organ}}_{k,d}.
|
||
|
|
\]
|
||
|
|
If longitudinal organ endpoint labels are available, the weights can be learned
|
||
|
|
under the mask:
|
||
|
|
\[
|
||
|
|
A^{\mathrm{organ}}_{k,d}\ge 0,
|
||
|
|
\qquad
|
||
|
|
A^{\mathrm{organ}}_{k,d}=0
|
||
|
|
\quad\text{if}\quad
|
||
|
|
M^{\mathrm{organ}}_{k,d}=0.
|
||
|
|
\]
|
||
|
|
This keeps the projection clinically interpretable while allowing data-driven
|
||
|
|
calibration.
|
||
|
|
|
||
|
|
\subsection{Functional Mapping Matrix}
|
||
|
|
|
||
|
|
The functional mapping matrix should be anchored by a validated frailty-related
|
||
|
|
diagnosis code set whenever possible. CIHI-HFRM or a closely related Hospital
|
||
|
|
Frailty Risk Measure provides a suitable starting point because it defines
|
||
|
|
frailty burden from diagnosis codes and associated weights.
|
||
|
|
|
||
|
|
Let \(w^{\mathrm{HFRM}}_d\ge 0\) be the HFRM weight mapped to DeepHealth disease
|
||
|
|
token \(d\). If the HFRM code list is more granular than the DeepHealth ICD
|
||
|
|
token vocabulary, weights should be mapped by code prefix. For three-character
|
||
|
|
ICD tokens, a conservative default is
|
||
|
|
\[
|
||
|
|
w^{\mathrm{HFRM}}_d
|
||
|
|
=
|
||
|
|
\max_{c:\, c \text{ maps to token } d}
|
||
|
|
w^{\mathrm{HFRM}}_c.
|
||
|
|
\]
|
||
|
|
|
||
|
|
For total functional burden, the one-dimensional mapping is
|
||
|
|
\[
|
||
|
|
A^{\mathrm{func,total}}_{1,d}
|
||
|
|
=
|
||
|
|
w^{\mathrm{HFRM}}_d.
|
||
|
|
\]
|
||
|
|
For domain-specific functional burden, define a grouping mask
|
||
|
|
\[
|
||
|
|
G_{k,d}\in\{0,1\},
|
||
|
|
\]
|
||
|
|
where \(G_{k,d}=1\) means HFRM-associated disease token \(d\) belongs to
|
||
|
|
functional burden domain \(k\). Then
|
||
|
|
\[
|
||
|
|
A^{\mathrm{func}}_{k,d}
|
||
|
|
=
|
||
|
|
G_{k,d} w^{\mathrm{HFRM}}_d.
|
||
|
|
\]
|
||
|
|
|
||
|
|
Candidate functional domains include mobility, cognition, mood, sensory,
|
||
|
|
nutrition, infection or immune vulnerability, functional dependence, and
|
||
|
|
comorbidity burden. These domain labels should be treated as diagnostic-burden
|
||
|
|
proxies unless direct functional measurements are available for calibration.
|
||
|
|
|
||
|
|
\section{Normalization and Reporting}
|
||
|
|
|
||
|
|
The raw burden index is an additive weighted burden:
|
||
|
|
\[
|
||
|
|
\operatorname{BI}_k(t,\tau)
|
||
|
|
=
|
||
|
|
\sum_d A_{k,d} b_d(t,\tau).
|
||
|
|
\]
|
||
|
|
For interpretability, the system should report the decomposition
|
||
|
|
\[
|
||
|
|
\operatorname{BI}^{\mathrm{hist}}_k(t),
|
||
|
|
\qquad
|
||
|
|
\operatorname{BI}^{\mathrm{future}}_k(t,\tau),
|
||
|
|
\qquad
|
||
|
|
\operatorname{BI}^{\mathrm{total}}_k(t,\tau).
|
||
|
|
\]
|
||
|
|
|
||
|
|
Optionally, a normalized burden can be reported as
|
||
|
|
\[
|
||
|
|
\widetilde{\operatorname{BI}}_k(t,\tau)
|
||
|
|
=
|
||
|
|
\frac{
|
||
|
|
\sum_d A_{k,d} b_d(t,\tau)
|
||
|
|
}{
|
||
|
|
\sum_d A_{k,d} + \epsilon
|
||
|
|
},
|
||
|
|
\]
|
||
|
|
where \(\epsilon>0\) prevents division by zero. This normalized score lies on a
|
||
|
|
dimension-comparable scale when \(b_d(t,\tau)\in[0,1]\) and \(A_{k,d}\ge 0\).
|
||
|
|
|
||
|
|
For cohort-level interpretation, an additional percentile score can be computed
|
||
|
|
within age- and sex-specific reference strata:
|
||
|
|
\[
|
||
|
|
\operatorname{PercentileBI}_k(t,\tau)
|
||
|
|
=
|
||
|
|
\operatorname{rank}_{\mathrm{age,sex}}
|
||
|
|
\left(
|
||
|
|
\operatorname{BI}^{\mathrm{total}}_k(t,\tau)
|
||
|
|
\right).
|
||
|
|
\]
|
||
|
|
This percentile is a relative burden ranking, not a health reserve percentage.
|
||
|
|
|
||
|
|
\section{Validation}
|
||
|
|
|
||
|
|
OBI and FBI should be validated against different endpoints.
|
||
|
|
|
||
|
|
For OBI, validation endpoints should be organ-system-specific future events, for
|
||
|
|
example cardiac events for heart/vascular burden, stroke or dementia for
|
||
|
|
brain/neurological burden, CKD progression for kidney burden, and respiratory
|
||
|
|
events for lung burden.
|
||
|
|
|
||
|
|
For FBI, validation should use CIHI-HFRM-style frailty burden, frailty-related
|
||
|
|
diagnosis endpoints, hospitalization, mortality, care dependence proxies, or
|
||
|
|
direct functional outcomes if available. If direct functional labels such as
|
||
|
|
ADL/IADL, gait speed, grip strength, cognitive tests, or recovery measures are
|
||
|
|
not available, FBI should be reported as a diagnosis-risk-based functional
|
||
|
|
burden proxy rather than a direct functional reserve measure.
|
||
|
|
|
||
|
|
\section{Summary}
|
||
|
|
|
||
|
|
DeepHealth Burden Indices transform disease-level risk predictions into
|
||
|
|
interpretable burden representations. Formed burden can be defined either as
|
||
|
|
observed-anchored burden \(z^{\mathrm{obs}}_d(t)\), which follows factual
|
||
|
|
diagnosis history, or as model-weighted burden \(z^{\mathrm{model}}_d(t)\),
|
||
|
|
which accumulates DeepHealth's predicted interval risks along the hidden-state
|
||
|
|
trajectory. The future expected burden is the residual future risk among disease
|
||
|
|
burden not already formed. OBI uses anatomical disease groupings to summarize
|
||
|
|
where pathological burden is concentrated. FBI uses CIHI-HFRM-style
|
||
|
|
frailty-related diagnosis weights to summarize functional vulnerability burden.
|
||
|
|
Together, they provide two complementary views of disease burden while allowing
|
||
|
|
the formed-burden semantics to be chosen explicitly.
|
||
|
|
|
||
|
|
\end{document}
|