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Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
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Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the...
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SPAC: a scalable and integrated enterprise platform for single-cell spatial analysis.

Fang Liu1, Rui He2, Thomas Sheeley3

  • 1Biomedical and Computational Science Directorate, Frederick National Laboratory for Cancer Research, Rockville, MD, USA.

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|January 29, 2026
PubMed
Summary
This summary is machine-generated.

SPAC is a new web-based platform that makes spatial single-cell analysis easier and faster for all researchers. It significantly speeds up data processing and enables detailed exploration of tissue microenvironments.

Keywords:
High-performance computingInteractive visualizationMultiplex imagingScalable analysisSingle-cell analysisSpatial omicsSpatial proteomicsTumor microenvironment

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Area of Science:

  • Single-cell spatial biology
  • Computational biology
  • Bioinformatics

Background:

  • Spatially resolved single-cell technologies offer transformative insights into tissue architecture and disease.
  • Analyzing high-dimensional spatial datasets is computationally intensive and lacks user-friendly tools for researchers.

Purpose of the Study:

  • To develop a scalable, web-based platform for efficient and reproducible spatial single-cell data analysis.
  • To bridge the gap between experimental and computational researchers by providing an accessible tool.

Main Methods:

  • SPAC utilizes a four-tier architecture: Python analysis engine, HPC/GPU integration, no-code browser interface, and Shiny for Python visualization.
  • The platform supports distinct user roles, enabling customization for data scientists and ease of use for bench scientists.
  • Features include built-in reproducibility and collaborative workflow support.

Main Results:

  • SPAC achieved a >20-fold speedup in unsupervised clustering on a large dataset (2.6 million cells), reducing time from ~3 hours to <10 minutes with GPU acceleration.
  • Enabled fine-grained spatial profiling of tumor microenvironment compartments.
  • Demonstrated platform scalability and performance on complex biological data.

Conclusions:

  • SPAC overcomes key challenges in spatial single-cell analysis by combining an intuitive interface with high-performance computing.
  • The platform streamlines complex analyses, fostering collaboration and accelerating the translation of spatial data into biological insights.