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Although the original conception of LATTICE did not prescribe any rigorous symmetry, and early clinical implementations relied on manual vertex placement tailored to tumor shape and anatomical constraints, more recent automated approaches have introduced regular patterns such as simple cubic or hexagonal arrangements. These rigid configurations, while convenient, may reduce the flexibility needed to accommodate irregular tumor geometries and nearby critical structures, potentially resulting in unintended hotspots or under\u2010treatment. Optimizing the placement of vertices in LATTICE is beneficial for precisely targeting high\u2010dose regions within the tumor while minimizing radiation exposure to adjacent healthy tissue, but there is still no optimization method available for solving the positions of fully flexible placed vertices. The great challenge in such optimization lies in handling the constraints on the relative positions between different vertices.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Purpose<\/jats:title>\n                    <jats:p>This work aims to develop a new treatment planning method for LATTICE with fully flexible placement of vertices and simultaneous optimization of the position of each lattice vertex and dose, to improve overall plan quality compared with conventional LATTICE planning methods relying on manual regular placements of lattice vertices.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Methods<\/jats:title>\n                    <jats:p>The proposed method simultaneously optimizes each lattice vertex position and other plan optimization variables (proton spot weights or photon fluences) during the dose optimization process. This is formulated as a new constrained optimization problem by adding each lattice vertex position to optimization variables with appropriate constraints to meet the requirements of the LATTICE vertices placement guideline on the 1) center\u2010to\u2010center distance between lattice vertices and 2) distance of lattice vertices to the target boundaries. The optimization problem is solved by the alternating direction method of multipliers and iterative convex relaxation methods.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Results<\/jats:title>\n                    <jats:p>\n                      Plans generated using our proposed method (NEW) were compared with conventional LATTICE plans for two representative patient cases presented in the main manuscript: one abdominal and one lung tumor. To maintain brevity, two additional patient cases are included in the Supporting Information to further demonstrate the performance and generalizability of the proposed method. For each case, we generated 100 LATTICE plans with varying vertex positions. From these, three plans\u2014termed BEST, MID, and WORST\u2014were selected based on the largest, median, and smallest total optimization objective value . All LATTICE plans optimized with the NEW method showed results comparable to, or better than, the BEST plans. For example, for photon LATTICE abdomen plans, the values of\n                      <jats:italic>F<\/jats:italic>\n                      were 1.92 (NEW), 2.79 (WORST), 2.27 (MID), and 1.96 (BEST), representing a 31.1% improvement from WORST to NEW; the PVDR values were 5.88 (NEW), 3.00 (WORST), 4.33 (MID), and 5.16 (BEST), representing 96.0% and 14.0% improvements relatively from WORST and BEST, respectively to NEW.\n                    <\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Conclusion<\/jats:title>\n                    <jats:p>A new LATTICE treatment planning approach is introduced, in which lattice positions are fully flexible and optimized simultaneously with dose distribution, leading to improved target PVDR and OAR sparing compared to conventional LATTICE methods with regularly spaced vertices.<\/jats:p>\n                  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