The Gompertz model, initially proposed for human mortality rates, has found various applications in growth analysis across the biotechnological field. This paper presents a comprehensive review of the Gompertz model's applications in the biotechnological field, examining its past, present, and future. The past of the Gompertz model was examined by tracing its origins to 1825, and then it underwent various modifications throughout the 20th century to increase its applicability in biotechnological fields. The Zwietering-modified version has proven to be a versatile tool for calculating the lag-time and maximum growth rate/quantity in microbial growth. In addition, the present applications of the Gompertz model to microbial growth kinetics and bioproduction (e.g., hydrogen, methane, caproate, butanol, and hexanol production) kinetics have been comprehensively summarized and discussed. We highlighted the importance of standardized citations and guidance on model selection. The Zwietering-modified Gompertz model and the Lay-modified Gompertz model are recommended for describing microbial growth kinetics and bioproduction kinetics, recognized for their widespread use and provision of valuable kinetics information. Finally, in response to the current Gompertz models' focus on internal mortality, the modified Makeham-Gompertz models that consider both internal/external mortality were introduced and validated for microbial growth and bioproduction kinetics with good fitting performance. This paper provides a perspective of the Gompertz model and offers valuable insights that facilitate the diverse applications of this model in microbial growth and bioproduction kinetics.

中文翻译：

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Gompertz 模型及其在微生物生长和生物生产动力学中的应用：过去、现在和未来

Gompertz 模型最初是针对人类死亡率提出的，现已在生物技术领域的生长分析中找到了多种应用。本文全面回顾了 Gompertz 模型在生物技术领域的应用，审视了它的过去、现在和未来。 Gompertz 模型的历史可以追溯到 1825 年，然后在整个 20 世纪经历了各种修改，以提高其在生物技术领域的适用性。 Zwietering 修改版本已被证明是计算微生物生长的滞后时间和最大增长率/数量的多功能工具。此外，还全面总结和讨论了Gompertz模型目前在微生物生长动力学和生物生产（例如氢气、甲烷、己酸、丁醇和己醇生产）动力学中的应用。我们强调了标准化引用和模型选择指南的重要性。建议使用 Zwietering 修改的 Gompertz 模型和 Lay 修改的 Gompertz 模型来描述微生物生长动力学和生物生产动力学，因其广泛使用和提供有价值的动力学信息而受到认可。最后，针对当前Gompertz模型对内部死亡率的关注，引入了考虑内部/外部死亡率的改良Makeham-Gompertz模型，并对其微生物生长和生物生产动力学进行了验证，具有良好的拟合性能。本文提供了 Gompertz 模型的视角，并提供了宝贵的见解，有助于该模型在微生物生长和生物生产动力学中的多样化应用。