Rethinking and Formulating Integrative Methodologies for a Better Understanding of Quantitative Drug-Related Issues

随着数学在药物研究领域的作用越来越重要，这里提出的研究计划解决了高度热门的问题。它侧重于发展先进和新颖的数学方法，这些方法解释了患者药物摄入、体内药物处置及其作用产生的变异性和非线性，目的是更新最广泛使用的（药理学）指标，以前以确定性和静态的方式制定，并提高对这种复杂关系的整体机制理解。这种重新表述有不同的方面：1)整合可用的药代动力学（PK）和药效学（PD）指标，通常在药物评估中单独考虑，以获得更多的信息负载指标，然后2)将药物输入的可变性添加到这些新的综合PKPD指标中，3)通过考虑在实践中不易测试的实际临床情况，从有限的药物信息中追溯患者的药物史，4)首次制定，分析或近似封闭形式，具有吸收和/或消除混合性质的更复杂药代动力学模型的动力学。所有这些数学方面都是至关重要的，使药物过程更加高效和有效，增加了药物计量学新兴领域的严谨性和合理性。反过来，数学科学可以从新的药理学问题的出现中受益，这些问题在其定量性质上具有挑战性，并导致新的数学主题，例如当前提案中描述的那些，超越了为描述生物现象而建立生物数学模型的传统领域。沿着这条道路走下去，将有可能丰富当前的建模视野，并有可能开辟应用数学的新领域。事实上，这些主流药物中心问题属于公共利益范围，具有高度的社会经济影响。解决这些数学问题带来了概念和技术上的机遇和挑战，这将使未来几代数学家的作用受到重视

The research program proposed here addresses highly topical issues that became very important with the increasing role that mathematics play in the drug research area. It focuses on the development of advanced and novel mathematical approaches that account for the variability and nonlinearity emanating from the patient's drug intake, drug disposition in the body and its action on it, with the aim to update the most widely used (pharmaco) metrics, previously formulated in a deterministic and static way, and to improve the overall mechanistic understanding of this complex interlationship. This reformulation takes different aspects : 1) with the integration of available pharmacokinetic (PK) and pharmacodynamic (PD) metrics together, usually considered separately in the drug assessment, to obtain more information-loaded metrics, then 2) the addition of the variability of drug input into these new integrative PKPD metrics, 3) retrace the patient's drug history from limited drug information, by considering realistic clinical situations that cannot be easily tested in practice, 4) formulate, for the first time, analytically or in approximative closed forms, the kinetics of more complex pharmacokinetic models with absorption and/or elimination of mixed nature. All these mathematical aspects are crucial to make the drug process more efficient and effective, adding rigour and rationality to the emerging area of pharmacometrics. Mathematical sciences can benefit, in turn, from the emergence of new pharmacological problems that are challenging in their quantitative nature and lead to new mathematical topics, such as those described in the current proposal, beyond the traditional area of establishment of biomathematical models for the description of biological phenomena. Going through this path, it will be possible to enrich the current modeling vision and potentially open a new area in applied mathematics. In fact, these mainstream pharmacomentric problems lie within the scope of public interest and are of high socioeconomic impact. Solving these mathematical problems presents conceptual and technical opportunities and challenges that will value the role of mathematicians for generations to come.**