曹慶發(fā)，胡 彬，萬(wàn) 殊，王澤文

生物傳熱方程中灌注率函數(shù)的數(shù)值反演算法

曹慶發(fā)，胡 彬，萬(wàn) 殊，*王澤文

（東華理工大學(xué)理學(xué)院，江西，南昌 330013）

本文研究了一類生物傳熱方程的灌注率函數(shù)反演問(wèn)題?；诟郊拥姆蔷植織l件和有限差分的Crank-Nicolson方法，構(gòu)造了重建灌注率函數(shù)的迭代算法；經(jīng)進(jìn)一步簡(jiǎn)化后，得到了反演灌注率的一個(gè)顯格式。為克服計(jì)算的不穩(wěn)定性，引入移動(dòng)平均濾波方法對(duì)誤差數(shù)據(jù)進(jìn)行去噪，算例結(jié)果表明結(jié)合移動(dòng)平均濾波去噪的數(shù)值反演算法是可行的，能有效反演出灌注率函數(shù)。

生物傳熱方程，灌注率，反問(wèn)題，有限差分，移動(dòng)平均

0 引言

生物醫(yī)學(xué)傳熱研究不僅在理論上很重要，而且具有重要的實(shí)際應(yīng)用價(jià)值[1]。例如，生物傳熱方程已被應(yīng)用于模擬高熱、血栓形成和血管硬化等研究中[2]?？茖W(xué)家提出了若干不同生物組織內(nèi)的傳熱模型，例如最常用的是由Pennes提出的生物傳熱模型[3]，以及文獻(xiàn)[4-6]提出連續(xù)型生物傳熱模型。

由Pennes提出的生物傳熱方程[3]為

它反映了血液灌注率，故稱其為血液灌注率函數(shù)。

和附加的非局部條件

不同于文獻(xiàn)[11-12]，本文受文獻(xiàn)[16]中研究的啟發(fā)，提出一種有限差分的反演算法，該方法無(wú)需事先將反問(wèn)題轉(zhuǎn)化為源項(xiàng)反演。本文接下來(lái)安排如下：第二小節(jié)基于Crank-Nicolson格式給出兩種有限差分的數(shù)值反演算法；第三小節(jié)給出反問(wèn)題的數(shù)值算例。

2 反問(wèn)題的有限差分解法

2.1 反問(wèn)題的數(shù)值解法

利用有限差分的Crank-Nicolson方法，將方程（2）離散為

由邊界條件（3）-（5），有

對(duì)于附加條件（6），利用數(shù)值積分的復(fù)化梯形公式得

將（13）-（14）改寫成矩陣形式為：

Step 3. 計(jì)算

上述差分格式的矩陣形式為

2.2 噪聲處理的移動(dòng)平均濾波

3 數(shù)值算例

算例1 考慮生物傳熱反問(wèn)題：

其中精確解為

圖1 算例1的反演結(jié)果對(duì)比

算例2 考慮生物傳熱反問(wèn)題：

其中精確解為

圖2 算例2的反演結(jié)果對(duì)比

數(shù)值算例的結(jié)果圖1，圖2表明所給出的算法是可行的，且迭代算法的數(shù)值反演效果更佳，特別是在第二個(gè)數(shù)值算例中算法體現(xiàn)了較強(qiáng)的抗噪能力，這可能是移動(dòng)平均濾波對(duì)算例2的數(shù)據(jù)去噪效果更佳的緣故。該方法也可以推廣到求解高維生物傳熱方程的相關(guān)反問(wèn)題。

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NUMERICAL METHOD FOR RECOVERING PERFUSION COEFFICIENT IN A BIOLOGICAL HEAT TRANSFER EQUATION

CAO Qing-fa, HU Bin, WAN Shu,*WANG Ze-wen

（School of Science, East China University of Technology, Nanchang, Jiangxi 330013, China）

The inversion of the perfusion coefficient function of a class of bioheat transfer equations is studied in this paper. Based on the additional non-local conditions and the Crank-Nicolson method of finite difference, an iterative algorithm for reconstructing the perfusion coefficient function is constructed; after further simplification, an explicit scheme for retrieving perfusion coefficient is obtained. In order to overcome the instability of calculation, the moving average filtering method is introduced to denoise the error data. The results of calculation examples show that numerical inversion algorithms combined with the moving average filtering denoising are feasible and effective for retrieving perfusion coefficient function.

bioheat transfer equation; perfusion coefficient; inverse problem; finite difference, moving average

1674-8085（2022）02-0022-06

O29

A

10.3969/j.issn.1674-8085.2022.02.004

2021-08-01；

2021-09-18

國(guó)家自然科學(xué)基金項(xiàng)目(11961002，11761007)；江西省教育廳科技計(jì)劃項(xiàng)目(GJJ170444)；東華理工大學(xué)大學(xué)生科技創(chuàng)新基金項(xiàng)目

曹慶發(fā)(1996-)，男，江西贛州人，碩士生，主要從事一般反問(wèn)題的計(jì)算方法研究(E-mail:cqingfa58@163.com)；

*王澤文(1974-)，男，江西上饒人，教授，博士，主要從事一般反問(wèn)題的計(jì)算方法研究(E-mail:zwwang6@163.com).