% Appendices
\appendix

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\section{Additional Dataset Information}
\label{app:dataset}
\textbf{ADNI}: Data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD). Information about the dMRI sequence used in the present study is provided below. For further details, please visit \href{https://adni.loni.usc.edu/}{https://adni.loni.usc.edu/}.

\textbf{NACC}: The NACC cohort began in 1999 and is comprised of dozens of Alzheimer’s Disease Research Centers that collect multimodal AD data$^a$. The overall intention of the NACC cohort is to collate a large database of standardized clinical/neuropathological data$^{b,c,d,e}$.  $^a$Beekly DL, Ramos EM, van Belle G, et al. The national Alzheimer's coordinating center (NACC) database: an Alzheimer disease database. Alzheimer Disease \& Associated Disorders. 2004;18(4):270-277. $^b$Beekly DL, Ramos EM, Lee WW, et al. The National Alzheimer's Coordinating Center (NACC) database: the uniform data set. Alzheimer Disease \& Associated Disorders. 2007;21(3):249-258.48. $^c$Besser LM, Kukull WA, Teylan MA, et al. The revised National Alzheimer’s Coordinating Center’s Neuropathology Form—available data and new analyses. Journal of Neuropathology \& Experimental Neurology. 2018;77(8):717-726.49. $^d$Weintraub S, Besser L, Dodge HH, et al. Version 3 of the Alzheimer Disease Centers’ neuropsychological test battery in the Uniform Data Set (UDS). Alzheimer disease and associated disorders. 2018;32(1):10.50. $^e$Weintraub S, Salmon D, Mercaldo N, et al. The Alzheimer’s disease centers’ uniform data set (UDS): The neuropsychological test battery. Alzheimer disease and associated disorders. 2009;23(2):91.

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\clearpage
\section{Visualization of Generated Samples}
\label{app:image}
\begin{figure}[h]
    \centering
    \begin{tabular}{ccccc}
        \textbf{Age: 64.0} & \textbf{Age: 53.8} & \textbf{Age: 70.9} & \textbf{Age: 76.9}
        \\
        \textbf{Sex: Male} & \textbf{Sex: Female} & \textbf{Sex: Female} & \textbf{Sex: Male}
        \\
        \includegraphics[width=0.16\textwidth]{f/c01.png}
        &
        \includegraphics[width=0.16\textwidth]{f/c02.png}
        &
        \includegraphics[width=0.16\textwidth]{f/c03.png}
        &
        \includegraphics[width=0.16\textwidth]{f/c04.png}
        \\
        \textbf{Age: 80.9} & \textbf{Age: 77.7} & \textbf{Age: 73.2} & \textbf{Age: 84.6}
        \\
        \textbf{Sex: Male} & \textbf{Sex: Female} & \textbf{Sex: Female} & \textbf{Sex: Female}
        \\
        \includegraphics[width=0.16\textwidth]{f/c05.png}
        &
        \includegraphics[width=0.16\textwidth]{f/c06.png}
        &
        \includegraphics[width=0.16\textwidth]{f/c07.png}
        &
        \includegraphics[width=0.16\textwidth]{f/c08.png}
        \\
    \end{tabular}
    \caption{Unconditional generated samples by MetaVoxel. All images, age, and sex shown are decoded from the latent space before visualization.\label{f:image}}
\end{figure}

\begin{figure}[h]
    \centering
    \begin{tabular}{ccccc}
        \includegraphics[width=0.16\textwidth]{f/ddpm1.png}
        &
        \includegraphics[width=0.16\textwidth]{f/ddpm2.png}
        &
        \includegraphics[width=0.16\textwidth]{f/ddpm3.png}
        &
        \includegraphics[width=0.16\textwidth]{f/ddpm4.png}
        \\
        \includegraphics[width=0.16\textwidth]{f/ddpm5.png}
        &
        \includegraphics[width=0.16\textwidth]{f/ddpm6.png}
        &
        \includegraphics[width=0.16\textwidth]{f/ddpm7.png}
        &
        \includegraphics[width=0.16\textwidth]{f/ddpm8.png}
        \\
    \end{tabular}
    \caption{Unconditional generated samples by LDM (comparison method).\label{f:ldm_image}}
\end{figure}

\begin{figure}[h]
    \centering
    \begin{tabular}{cccccc}
        \textbf{Input (Real)} & \quad\quad & \multicolumn{4}{c}{\textbf{Inpainting with Left Half of the Image Fixed}}
        \\
        \includegraphics[width=0.16\textwidth]{f/case1_input.png}
        &
        &
        \includegraphics[width=0.16\textwidth]{f/case1_inpaint1.png}
        &
        \includegraphics[width=0.16\textwidth]{f/case1_inpaint2.png}
        &
        \includegraphics[width=0.16\textwidth]{f/case1_inpaint3.png}
        &
        \includegraphics[width=0.16\textwidth]{f/case1_inpaint4.png}
        \\
        \includegraphics[width=0.16\textwidth]{f/case2_input.png}
        &
        &
        \includegraphics[width=0.16\textwidth]{f/case2_inpaint1.png}
        &
        \includegraphics[width=0.16\textwidth]{f/case2_inpaint2.png}
        &
        \includegraphics[width=0.16\textwidth]{f/case2_inpaint3.png}
        &
        \includegraphics[width=0.16\textwidth]{f/case2_inpaint4.png}
        \\
        \includegraphics[width=0.16\textwidth]{f/case3_input.png}
        &
        &
        \includegraphics[width=0.16\textwidth]{f/case3_inpaint1.png}
        &
        \includegraphics[width=0.16\textwidth]{f/case3_inpaint2.png}
        &
        \includegraphics[width=0.16\textwidth]{f/case3_inpaint3.png}
        &
        \includegraphics[width=0.16\textwidth]{f/case3_inpaint4.png}
        \\
    \end{tabular}
    \caption{Image inpainting with MetaVoxel. Pixels in the left half of the inpainting results are repeatedly overwritten with their known values from the noised input during the denoising process, while the remaining pixels are freely generated. Each row shows a different example. The real images are shown on the left. MetaVoxel produces visually coherent completions that smoothly blend with the fixed portion of the image.\label{f:inpaint}}
\end{figure}
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\cbstart
\section{Dataset Demographics}
\label{app:data_demo}

\begin{figure}[h]
    \centering
    \includegraphics[width=0.80\textwidth]{f/data_distribution.png}
    \caption{Demographic characteristics of the study cohort. (Left) Histogram of subject ages with the mean indicated by the dashed red line. (Right) Distribution of biological sex   
  across all subjects.\label{f:data_demo}}
\end{figure}



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\section{Results on Conditional Image Generation}
We compared MetaVoxel to a conditional LDM trained with age and sex as conditioning inputs. These conditions are integrated into both models through a combination of input-channel concatenation and cross-attention. We report FID and MMD in the Figure~\ref{f:cldm}.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.9\textwidth]{f/fid_mmd_by_age_sex.png}
    \caption{FID and MMD results by different age-sex groups.\label{f:cldm}}
\end{figure}

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