The freezing step plays a key role in the overall economy of the vacuum freeze-drying of pharmaceuticals, since the nucleation and crystal growth kinetics determine the number and size distribution of the crystals formed. In this work, a new mathematical model of the freezing step of a (bio)pharmaceutical solution is developed and validated. Both nucleation and crystal growth kinetics are modeled and included in a one-dimensional population balance (1D-PBM) that describes, given the product temperature measurement, the evolution of the pore size distribution during freezing. The developed model is coupled with the real-time measurements obtained from an infrared video camera. The ending time of the primary drying stage, and the maximum temperature inside the material, simulated through a simplified model of the process and the pore distribution forecast, resulted in good agreement with experimental values. The resulting Process Analytical Technology (PAT) has the potential to boost the development and optimization of a freeze-drying cycle and the implementation of a physically grounded Quality-by-Design approach in the manufacturing of pharmaceuticals. A more general mathematical model, including the aforementioned population balance, of a vial filled with a solution of sucrose was also developed and used to further validate the approach.

中文翻译：

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一种新的数学模型，用于监控冷冻过程中药物溶液中冰晶尺寸分布的时间演变。

冷冻步骤在药物真空冷冻干燥的总体经济中起着关键作用，因为成核和晶体生长动力学决定了所形成晶体的数量和大小分布。在这项工作中，开发并验证了（生物）药物溶液冷冻步骤的新数学模型。成核和晶体生长动力学均已建模，并包含在一维总体平衡（1D-PBM）中，该平衡描述了给定的产品温度测量结果，表明了冷冻过程中孔径分布的演变。所开发的模型与从红外摄像机获得的实时测量相结合。初步干燥阶段的结束时间以及物料内部的最高温度，通过简化的过程模型和孔分布预测进行模拟，结果与实验值吻合良好。由此产生的过程分析技术（PAT）有潜力促进冷冻干燥周期的开发和优化，以及在药品制造中实施基于物理的按设计质量方法。还开发了一个更通用的数学模型，包括上述填充蔗糖溶液的小瓶的种群平衡，并将其用于进一步验证该方法。由此产生的过程分析技术（PAT）有潜力促进冷冻干燥周期的开发和优化，以及在药品制造中实施基于物理的按设计质量方法。还开发了一个更通用的数学模型，包括上述填充蔗糖溶液的小瓶的种群平衡，并将其用于进一步验证该方法。由此产生的过程分析技术（PAT）有潜力促进冷冻干燥周期的开发和优化，以及在药品制造中实施基于物理的按设计质量方法。还开发了一个更通用的数学模型，包括上述填充蔗糖溶液的小瓶的种群平衡，并将其用于进一步验证该方法。