Inverse problems of damped wave equations with Robin boundary conditions: an application to blood perfusion
Inverse problems of damped wave equations with Robin boundary conditions: an application to blood perfusion
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DOI：
10.1088/1361-6420/acca42
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发表时间：
2023-04
期刊：
影响因子：
2.1
通讯作者：
Yanqin Fang;D. Lesnic;M. Alosaimi
中科院分区：
文献类型：
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作者：
Yanqin Fang;D. Lesnic;M. Alosaimi
了解生物组织的特性对于监测任何可能正在形成并对器官功能障碍产生重大影响的异常情况是至关重要的。因此，必须及早发现和治疗这些疾病，以挽救生命，改善一般健康状况。在热疗的框架内，例如热疗或冷冻消融，对组织温度和血液灌注率的了解是至关重要的。因此，在这一重大生物医学应用的推动下，本文首次利用Carleman估计的强大技术，从Cauchy边界数据出发，研究了生物热传导的热波双曲模型中与空间相关的(非均相)灌流系数的唯一性和稳定性。另外的新奇之处在于考虑了Robin边界条件，以及发展了一种数学分析，与双曲型偏微分方程系数识别问题的文献中通常报道的相比，在更短的时间间隔内得到了更强的稳定性估计。在数值上，将系数反问题转化为使用共轭梯度法(CGM)求解的非线性最小二乘问题。准确的和有噪声的数据都是反转的。为了实现稳定，根据差异原则停止了CGM。文中给出了一个物理算例的数值结果，并对其进行了讨论，结果表明了该方法的收敛、精度和稳定性。
Knowledge of the properties of biological tissues is essential in monitoring any abnormalities that may be forming and have a major impact on organs malfunctioning. Therefore, these disorders must be detected and treated early to save lives and improve the general health. Within the framework of thermal therapies, e.g. hyperthermia or cryoablation, the knowledge of the tissue temperature and of the blood perfusion rate are of utmost importance. Therefore, motivated by such a significant biomedical application, this paper investigates, for the first time, the uniqueness and stable reconstruction of the space-dependent (heterogeneous) perfusion coefficient in the thermal-wave hyperbolic model of bio-heat transfer from Cauchy boundary data using the powerful technique of Carleman estimates. Additional novelties consist in the consideration of Robin boundary conditions, as well as developing a mathematical analysis that leads to stronger stability estimates valid over a shorter time interval than usually reported in the literature of coefficient identification problems for hyperbolic partial differential equations. Numerically, the inverse coefficient problem is recast as a nonlinear least-squares minimization that is solved using the conjugate gradient method (CGM). Both exact and noisy data are inverted. To achieve stability, the CGM is stopped according to the discrepancy principle. Numerical results for a physical example are presented and discussed, showing the convergence, accuracy and stability of the inversion procedure.