Physical models in the form of partial differential equations represent an important prior for many under-constrained problems. One example is tumor treatment planning, which heavily depends on accurate estimates of the spatial distribution of tumor cells in a patient's anatomy. Medical imaging scans can identify the bulk of the tumor, but they cannot reveal its full spatial distribution. Tumor cells at low concentrations remain undetectable, for example, in the most frequent type of primary brain tumors, glioblastoma. Deep-learning-based approaches fail to estimate the complete tumor cell distribution due to a lack of reliable training data. Most existing works therefore rely on physics-based simulations to match observed tumors, providing anatomically and physiologically plausible estimations. However, these approaches struggle with complex and unknown initial conditions and are limited by overly rigid physical models. In this work, we present a novel method that balances data-driven and physics-based cost functions. In particular, we propose a unique discretization scheme that quantifies the adherence of our learned spatiotemporal tumor and brain tissue distributions to their corresponding growth and elasticity equations. This quantification, serving as a regularization term rather than a hard constraint, enables greater flexibility and proficiency in assimilating patient data than existing models. We demonstrate improved coverage of tumor recurrence areas compared to existing techniques on real-world data from a cohort of patients. The method holds the potential to enhance clinical adoption of model-driven treatment planning for glioblastoma.

翻译：偏微分方程形式的物理模型为许多欠约束问题提供了重要的先验知识。肿瘤治疗规划便是其中一例，其高度依赖于对患者解剖结构中肿瘤细胞空间分布的准确估计。医学影像扫描能够识别肿瘤主体，但无法揭示其完整的空间分布。例如在最常见的原发性脑肿瘤——胶质母细胞瘤中，低浓度肿瘤细胞仍无法被检测到。基于深度学习的方法因缺乏可靠训练数据而难以估计完整的肿瘤细胞分布。因此现有研究多依赖基于物理的模拟来匹配观测到的肿瘤，从而提供解剖学和生理学上合理的估计。然而，这些方法在处理复杂且未知的初始条件时存在困难，并受限于过于僵化的物理模型。本研究提出一种平衡数据驱动与基于物理代价函数的新方法。特别地，我们设计了一种独特的离散化方案，用于量化学习到的时空肿瘤与脑组织分布对其对应生长方程和弹性方程的遵从程度。该量化作为正则化项而非硬性约束，使模型在融合患者数据时比现有方法具有更高的灵活性与效能。通过真实患者队列数据的验证，本方法在肿瘤复发区域覆盖范围上较现有技术展现出显著改进。该方法有望提升胶质母细胞瘤模型驱动治疗规划在临床中的采纳度。