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Mathematical Modeling of Tracer Kinetics for Medical Applications

机译：示踪剂动力学的医学应用数学建模

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A tracer kinetic model is a mathematical description of the transport processes of the tracer within the tissue. Various methods have been proposed for the quantification of kinetic parameters including perfusion. These methods can be roughly classified into model-dependent and model-independent methods.The model-dependent method usually employs compartment models and generally requires specific knowledge about the vascular and compartmental architecture. The kinetic parameters are generally estimated using the nonlinear least-squares (NLSQ) method. Although the NLSQ method is a powerful tool for estimating kinetic parameters, it is not efficient in terms of computation time. Furthermore, poor initial guesses result in long computation times or occasionally failure to converge on a good fit. As an alternative approach, the linear least-squares (LLSQ) method has been introduced, which is computationally efficient and does not need initial guesses of the parameters. The graphical method has also been proposed.The model-independent method yields kinetic parameters for a tissue region described as a "black box" and thus does not provide insight into its internal structure. For quantification of kinetic parameters using the model-independent method, deconvolution between the measured tissue residue curve and the vascular input function is needed to obtain the tissue response function. Deconvolution, however, is an ill-posed problem especially when there exists statistical noise. Thus, it is necessary to convert an ill-posed problem to a well-posed one using the so-called "regularization". Of the regularization methods, the truncated singular value decomposition (TSVD) is a popular method for obtaining regularized solutions. In this chapter, the use of TSVD in the model-independent method is introduced. The application of spectral analysis is also introduced. This method provides a spectrum of the kinetic components which are involved in the regional uptake and partitioning of tracer from the blood to the tissue and allows the tissue response function to be derived with minimal modeling assumptions.The kinetic models employed in compartment analysis can be classified into lumped parameter and distributed parameter models. In the lumped parameter model, the compartments are assumed to be well-mixed, that is, tracer movement is sufficiently fast and distributes evenly throughout the compartments, such that tracer concentration is only a function of time, but not space. On the other hand, a distributed parameter model attempts to account for tracer concentration gradient within the compartments, such that concentration can be a function of both time and space. One of the simple distributed parameter models is the tissue homogeneity model. In this chapter, adiabatic approximation to the tissue homogeneity model is also described.Finally, an empirical mathematical model is introduced. This model has been developed to describe contrast uptake and washout behavior without use of vascular input functions. This approach does not require making assumptions about underlying physiology or anatomy. The primary disadvantage of this approach, however, is that the parameters obtained by this approach do not correspond directly to identifiable physiological or anatomic features.This chapter will be helpful for gaining a deeper understanding of mathematical modeling of tracer kinetics for medical applications.
机译：示踪剂动力学模型是组织中示踪剂传输过程的数学描述。已经提出了用于定量包括灌注的动力学参数的各种方法。这些方法大致可分为模型相关方法和模型独立方法。模型相关方法通常采用隔室模型，通常需要有关血管和隔室结构的特定知识。通常使用非线性最小二乘（NLSQ）方法估算动力学参数。尽管NLSQ方法是估算动力学参数的强大工具，但在计算时间方面效率不高。此外，较差的初始猜测会导致较长的计算时间，或有时无法收敛于良好的拟合度。作为一种替代方法，已引入线性最小二乘（LLSQ）方法，该方法计算效率高，并且不需要参数的初始猜测。还提出了图形方法。模型无关方法产生描述为“黑匣子”的组织区域的动力学参数，因此没有提供对其内部结构的了解。为了使用独立于模型的方法对动力学参数进行量化，需要在测量的组织残留曲线和血管输入函数之间进行反卷积以获得组织响应函数。然而，反卷积是一个不适当地的问题，特别是当存在统计噪声时。因此，有必要使用所谓的“正则化”将不适定的问题转换为适度的问题。在正则化方法中，截断奇异值分解（TSVD）是一种获得正则化解的流行方法。本章介绍了TSVD在模型无关方法中的使用。还介绍了光谱分析的应用。该方法提供了一系列示踪剂的动力学成分，这些示踪剂参与了示踪剂从血液到组织的区域吸收和分配，并允许在最少的建模假设下得出组织响应函数。可以对分类分析中使用的动力学模型进行分类分为集总参数模型和分布式参数模型。在集总参数模型中，假定隔室混合良好，即示踪剂的运动足够快，并且在整个隔室中均匀分布，因此示踪剂的浓度仅是时间的函数，而不是空间的函数。另一方面，分布式参数模型试图解释隔室内的示踪剂浓度梯度，以使浓度可以是时间和空间的函数。一种简单的分布式参数模型是组织均匀性模型。本章还描述了组织均匀性模型的绝热近似。最后，介绍了经验数学模型。已经开发出该模型来描述对比剂摄取和洗脱行为，而无需使用血管输入功能。这种方法不需要对基础的生理或解剖结构做出假设。然而，这种方法的主要缺点是，这种方法获得的参数并不直接对应于可识别的生理或解剖特征。本章将有助于加深对医学示踪动力学的数学建模的了解。

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