{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T16:56:57Z","timestamp":1792083417393,"version":"4.4.0"},"reference-count":16,"publisher":"Wiley","issue":"10","license":[{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2023YFF0613503"],"award-info":[{"award-number":["2023YFF0613503"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11975313"],"award-info":[{"award-number":["11975313"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["81801799"],"award-info":[{"award-number":["81801799"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["aapm.onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Medical Physics"],"published-print":{"date-parts":[[2026,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:sec>\n                    <jats:title>Purpose<\/jats:title>\n                    <jats:p>\n                      In pencil beam dose calculation, the convolution is usually performed in the frequency domain via FFT. The dose is assembled into a sparse dose\u2010influence matrix (d\n                      <jats:italic>ij<\/jats:italic>\n                      ) for optimization. We propose to perform the convolution directly in the spatial domain with a deep\u2010learning operator (Conv2d), and investigate the feasibility of a d\n                      <jats:italic>ij<\/jats:italic>\n                      \u2010free optimization.\n                    <\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Methods<\/jats:title>\n                    <jats:p>\n                      A 6 MV photon kernel (1 Gaussian penumbra and 3 singular value decomposition (SVD) scatter components, 0.5\u00a0mm resolution) was adopted. The Conv2d operator was validated against FFT on head\u2010and\u2010neck (HN), prostate and liver cases, with 7, 5, and 4 beams and corresponding 7503, 2649, and 845 bixels. The cost of building d\n                      <jats:italic>ij<\/jats:italic>\n                      was investigated to identify the bottleneck. A one\u2010layer network optimization framework was implemented to investigate the feasibility of the d\n                      <jats:italic>ij<\/jats:italic>\n                      \u2010free optimization (workflow A). Workflow A recomputes the dose with Conv2D at each iteration, with no pre\u2010calculated d\n                      <jats:italic>ij<\/jats:italic>\n                      . The traditional d\n                      <jats:italic>ij<\/jats:italic>\n                      \u2010based workflow (B) was implemented for comparison.\n                    <\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Results<\/jats:title>\n                    <jats:p>\n                      Across all 16 beams Conv2d reproduced FFT at machine precision (maximum absolute dose difference 1.35 \u00d7 10\n                      <jats:sup>\u221213<\/jats:sup>\n                      ; Pearson correlation 1.0). The computational time of FFT versus Conv2d was 0.995s versus 0.502s (HN), 0.650s versus 0.199s (prostate), and 0.370s versus 0.082s (liver). D\n                      <jats:italic>ij<\/jats:italic>\n                      construction time was 132.9s, 76.4s, and 31.9s, with assembly accounting for \u223c89%. Workflow A performs no build but a full forward and backward convolution (2.166s, 0.832s, 0.368s) per\u2010iteration; workflow B performs the one\u2010time d\n                      <jats:italic>ij<\/jats:italic>\n                      build (130.2s, 76.0s, 31.5s), then a sparse matrix\u2010vector product (0.25s, 0.18s, 0.15s). The break\u2010even points were 69, 116, and 145 iterations.\n                    <\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Conclusions<\/jats:title>\n                    <jats:p>\n                      Pencil beam kernel convolution can be formulated as Conv2d spatial convolution with machine\u2010precision equivalence to FFT and competitive speed. The spatial\u2010domain formulation enables a differentiable, d\n                      <jats:italic>ij<\/jats:italic>\n                      \u2010free optimization workflow that recomputes dose without storing a dose\u2010influence matrix and that, for short optimization runs, is competitive with the traditional FFT\u2010built sparse d\n                      <jats:italic>ij<\/jats:italic>\n                      pipeline.\n                    <\/jats:p>\n                  <\/jats:sec>","DOI":"10.1002\/mp.70689","type":"journal-article","created":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T14:40:45Z","timestamp":1790865645000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Direct spatial convolution for pencil beam dose calculation and feasibility of dij\u2010free IMRT optimization on GPU"],"prefix":"10.1002","volume":"53","author":[{"given":"Wen\u2010Long","family":"Xia","sequence":"first","affiliation":[{"name":"National Cancer Center National Clinical Research Center for Cancer Cancer Hospital Chinese Academy of Medical Sciences and Peking Union Medical College  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong\u2010Kai","family":"Wang","sequence":"additional","affiliation":[{"name":"National Cancer Center National Clinical Research Center for Cancer Cancer Hospital Chinese Academy of Medical Sciences and Peking Union Medical College  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bin","family":"Liang","sequence":"additional","affiliation":[{"name":"National Cancer Center National Clinical Research Center for Cancer Cancer Hospital Chinese Academy of Medical Sciences and Peking Union Medical College  Beijing 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