Data-driven simulation of network-based tau spreading tailored to individual Alzheimer's patients

Abstract

Tau tangles in the brain cortex spread along the brain network in distinct patterns among Alzheimer's patients. We aim to simulate their network-based spreading within the cortex, tailored to each individual along the Alzheimer's continuum, without assuming any assumptions about the network architecture. A group-level intrinsic spreading network was constructed to model the pathways for the proximal and distal spreading of tau tangles by optimizing the biophysical model based on a discovery dataset of longitudinal tau positron emission tomography images for 78 amyloid-positive individuals. Group-level spreading parameters were also obtained and subsequently adjusted to produce individuated tau trajectories. By simulating these individuated tau spreading models for every individual in the discovery dataset, we successfully captured proximal and distal tau spreading, allowing reliable inferences about the underlying mechanism of tau spreading. Simulating the models also allowed highly accurate prediction of future tau topography for both discovery and independent validation datasets.

Appendices

Appendix

Appendix A. Derivation of the objective function for group-level optimization

We first note that the loss function $\ell \left( {{{\varvec{\uptheta}}};\left\{ {\left( {t_{i,j} ,{\tilde{\mathbf{x}}}_{i,j} } \right)} \right\}_{j = 0}^{{m_{i} - 1}} ,H_{1} } \right)$ in the group-level optimization objective (3) quantifies the discrepancy as the difference between the average rate of change in observed tau topographic maps over $\left[ {t_{i,0} ,t_{{i,m_{i} - 1}} } \right)$ and the expected instantaneous rates of changes evaluated by the tau spreading model. The average rate of change, denoted by ${\tilde{\mathbf{y}}}_{i}$, can be represented as follows:

${\tilde{\mathbf{y}}}_{i} = \frac{{{\tilde{\mathbf{x}}}_{{i,m_{i} - 1}} - {\tilde{\mathbf{x}}}_{i,0} }}{{t_{{i,m_{i} - 1}} - t_{i,0} }} = \sum\limits_{j = 0}^{{m_{i} - 2}} {w_{i,j} \frac{{{\tilde{\mathbf{x}}}_{i,j + 1} - {\tilde{\mathbf{x}}}_{i,j} }}{{t_{i,j + 1} - t_{i,j} }}}$,

where $w_{i,j} = \frac{{t_{i,j + 1} - t_{i,j} }}{{t_{{i,m_{i} - 1}} - t_{i,0} }}$ denotes the weight assigned to each sub-interval $\left[ {t_{i,j} ,t_{i,j + 1} } \right)$. Let ${\dot{\mathbf{x}}}\left( {t;{{\varvec{\uptheta}}}} \right)$ denote the rate of instantaneous rates of changes evaluated by the tau spreading model ${\mathbf{x}}\left( {t;{{\varvec{\uptheta}}}} \right)$. By the mean-value theorem for vector-valued function [80], for a function differentiable ${\mathbf{x}}\left( t \right) = \left[ {\begin{array}{*{20}c} {x_{1} \left( t \right)} & {x_{2} \left( t \right)} & \ldots & {x_{k} \left( t \right)} \\ \end{array} } \right]^{{\text{T}}}$ over a bounded open interval, there exist constants $t_{1} ,t_{2} ,...,t_{k}$ in the interval such that

$$\frac{{{\mathbf{x}}\left( b \right) - {\mathbf{x}}\left( a \right)}}{b - a} = \left[ {\begin{array}{*{20}c} {\dot{x}_{1} \left( {t_{1} } \right)} & {\dot{x}_{2} \left( {t_{2} } \right)} & \cdots & {\dot{x}_{k} \left( {t_{k} } \right)} \\ \end{array} } \right]^{{\text{T}}} ,$$

with real numbers $a$ and $b$ in the interval. The exact values of the constants cannot be determined, however, so we tried to get the expectation of ${\dot{\mathbf{x}}}\left( {t;{{\varvec{\uptheta}}}} \right)$, rather than the raw value of ${\dot{\mathbf{x}}}\left( {t;{{\varvec{\uptheta}}}} \right)$, close to ${\tilde{\mathbf{y}}}_{i}$.

The expected value of ${\dot{\mathbf{x}}}\left( {t;{{\varvec{\uptheta}}}} \right)$ over the interval $\left[ {t_{i,0} ,t_{{i,m_{i} - 1}} } \right)$, denoted by ${\hat{\mathbf{y}}}_{i}$, can be represented as follows:

$${\hat{\mathbf{y}}}_{i} \left( {{\varvec{\uptheta}}} \right) = {\mathbb{E}}_{{T_{i} \sim p\left( {T_{i} } \right)}} \left[ {{\dot{\mathbf{x}}}\left( {T_{i} ;{{\varvec{\uptheta}}}} \right)} \right],$$

where $T_{i}$ denotes a random variable that can have a value between $\left[ {t_{i,0} ,t_{{i,m_{i} - 1}} } \right)$ with the probability density function $p\left( {T_{i} } \right)$. Since there is no information about the time constants except that they are in $\left[ {t_{i,j} ,t_{i,j + 1} } \right)$, we approximate $p\left( {T_{i} } \right)$ as a piecewise uniform distribution as follows:

$$p\left( {T_{i} } \right) = \left\{ \begin{gathered} w_{i,0} \cdot {\text{Uni}}\left( {t;t_{i,0} ,t_{i,1} } \right)\;\;\;\;\;{\text{if}}\;t \in \left( {t_{i,0} ,t_{i,1} } \right] \hfill \\ w_{i,1} \cdot {\text{Uni}}\left( {t;t_{i,0} ,t_{i,1} } \right)\;\;\;\;\;{\text{if}}\;t \in \left( {t_{i,1} ,t_{i,2} } \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \hfill \\ w_{{i,m_{i} - 2}} \cdot {\text{Uni}}\left( {t;t_{i,0} ,t_{i,1} } \right)\;\;{\text{if}}\;t \in \left( {t_{{i,m_{i} - 2}} ,t_{{i,m_{i} - 1}} } \right] \hfill \\ \end{gathered} \right.,$$

according to the principle of maximum entropy [81]. Herein, ${\text{Uni}}\left( {t;t_{i,j} ,t_{i,j + 1} } \right)$ denotes a continuous uniform distribution with the time interval. The expected value still cannot be evaluated due to intractable integration; we divided each interval $\left( {t_{i,j} ,t_{i,j + 1} } \right)$ into $H_{1}$ sub-intervals and approximated the continuous uniform distribution for each interval to a discrete uniform distribution ${\text{Uni}}\left\{ {t;t_{i,j} ,t_{i,j + 1} } \right\}$ whose support contains midpoints of the $H_{1}$ sub-intervals, according to the best discrete approximation having equally weighted support points [41]. The expected value is then computed as follows:

$$\begin{gathered} {\mathbb{E}}_{{T_{i} \sim p\left( {T_{i} } \right)}} \left[ {{\dot{\mathbf{x}}}\left( {T_{i} ;{{\varvec{\uptheta}}}} \right)} \right] \approx \frac{1}{{H_{1} }}\sum\limits_{j = 0}^{{m_{i} - 2}} {\sum\limits_{k = 1}^{{H_{1} }} {w_{i,j} \cdot {\dot{\mathbf{x}}}\left( {\left\langle {\tau_{i,j}^{{\left( {2H_{1} } \right)}} } \right\rangle_{2k - 1} ;{{\varvec{\uptheta}}}} \right)} } \\ = \sum\limits_{j = 0}^{{m_{i} - 2}} {w_{i,j} \cdot {\text{Avg}}_{k} \left( {{\dot{\mathbf{x}}}\left( {\left\langle {\tau_{i,j}^{{\left( {2H_{1} } \right)}} } \right\rangle_{2k - 1} ;{{\varvec{\uptheta}}}} \right)} \right)} . \\ \end{gathered}$$

It is worth noting that for taking midpoints of the $H_{1}$ sub-intervals, we divided each interval $\left( {t_{i,j} ,t_{i,j + 1} } \right)$ into $2H_{1}$ sub-intervals and took all odd-numbered points. Each value of ${\dot{\mathbf{x}}}\left( {\left\langle {\tau_{i,j}^{{\left( {2H_{1} } \right)}} } \right\rangle_{2k - 1} ;{{\varvec{\uptheta}}}} \right)$ was computed using the numerical simulation with the step size $h_{1} = {{\left( {t_{i,j + 1} - t_{i,j} } \right)} \mathord{\left/ {\vphantom {{\left( {t_{i,j + 1} - t_{i,j} } \right)} {2H_{1} }}} \right. \kern-0pt} {2H_{1} }}$. Finally, the loss function for the group-level optimization problem was defined as the squared $L_{2}$ norm of the average difference between ${\tilde{\mathbf{y}}}_{i}$ and ${\hat{\mathbf{y}}}_{i} \left( {{\varvec{\uptheta}}} \right)$, or equivalently formulated as follows:

$$\begin{gathered} \ell \left( {{{\varvec{\uptheta}}};\left\{ {\left( {t_{i,j} ,{\tilde{\mathbf{x}}}_{i,j} } \right)} \right\}_{j = 0}^{{m_{i} - 1}} ,H_{1} } \right) = \sum\limits_{j = 0}^{{m_{i} - 2}} {\left\| {{\tilde{\mathbf{y}}}_{i} - {\hat{\mathbf{y}}}_{i} \left( {{\varvec{\uptheta}}} \right)} \right\|_{2}^{2} } \\ \approx \left\| {{\text{Avg}}_{j} \left( {w_{i,j} \left\{ {\frac{{{\tilde{\mathbf{x}}}_{i,j + 1} - {\tilde{\mathbf{x}}}_{i,j} }}{{t_{i,j + 1} - t_{i,j} }} - {\text{Avg}}_{k} \left( {{\dot{\mathbf{x}}}\left( {\left\langle {\tau_{i,j}^{{\left( {2H_{1} } \right)}} } \right\rangle_{2k - 1} ;{{\varvec{\uptheta}}}} \right)} \right)} \right\}} \right)} \right\|_{2}^{2} . \\ \end{gathered}$$

Appendix B. Snapshot analysis

The snapshot analysis of tau tangle spreading encompassed two approaches: evaluating the relative importance of factors in the system and constructing vector fields to illustrate the proximal spreading patterns.

The relative influence of local reaction and diffusion is calculated for each node by dividing the absolute value of each term in Eq. (2) by their sum. Similarly, the relative influence of proximal and distal spreading on the diffusion of tau tangles was determined based on the decomposition ${\mathbf{L}} \cdot {\mathbf{x}}\left( t \right) = {\mathbf{L}}_{{{\text{prox}}}} \cdot {\mathbf{x}}\left( t \right) + {\mathbf{L}}_{{{\text{dist}}}} \cdot {\mathbf{x}}\left( t \right)$, where ${\mathbf{L}}_{{{\text{prox}}}}$ and ${\mathbf{L}}_{{{\text{dist}}}}$ represent weighted graph Laplacians of ${\mathbf{C}}_{{{\text{prox}}}}$ and ${\mathbf{C}}_{{{\text{dist}}}}$, respectively. The sign of the relative influence indicates whether tau SUVRs increase or decrease due to a specific term at that particular time.

The individualized vector field was constructed by assigning a gradient vector to each side of triangles within the pial surface mesh. For a triangle defined by vertices $v_{1}$, $v_{2}$, $v_{3}$ and sides $v_{1} v_{2}$, $v_{2} v_{3}$, $v_{3} v_{1}$, a gradient vector $\kappa \cdot \sqrt {c_{{v_{1} v_{2} }} } \cdot \left( {\mathbf{x}_{{v_{1} }} \left( t \right) - \mathbf{x}_{{v_{2} }} \left( t \right)} \right) \cdot \left( {{\mathbf{v}}_{1} - {\mathbf{v}}_{2} } \right)$, where ${\mathbf{v}}_{1}$ and ${\mathbf{v}}_{2}$ represent the corresponding Cartesian coordinate points, was assigned to the side $v_{1} v_{2}$. Similar gradient vectors were assigned to the sides $v_{2} v_{3}$ and $v_{3} v_{1}$. The representative vector of the triangle was determined as the Euclidean vector originating from its incenter and calculated as the average of the three gradient vectors of its sides.