趙家維，曹秀玲，李 璇

(1. 天津大學(xué) 理學(xué)院，天津 300072；2. 天津財(cái)經(jīng)大學(xué) 珠江學(xué)院，基礎(chǔ)部 天津 301811)

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I型糖尿病患者血糖偏微分控制模型(下)

趙家維1，曹秀玲1，李 璇2

(1. 天津大學(xué) 理學(xué)院，天津 300072；2. 天津財(cái)經(jīng)大學(xué) 珠江學(xué)院，基礎(chǔ)部 天津 301811)

研究了I型糖尿病患者血糖控制系統(tǒng).在該系統(tǒng)中，針對(duì)I型糖尿病患者的生理特點(diǎn)，將熱傳導(dǎo)偏微分方程應(yīng)用到血糖控制系統(tǒng)中，從定向控制的角度研究控制輸入和控制目標(biāo)輸出的關(guān)系.運(yùn)用Matlab編寫程序進(jìn)行數(shù)值模擬，得出控制輸出目標(biāo)(人體正常血糖濃度)所對(duì)應(yīng)的精確控制輸入，更好地保證糖尿病患者的身體健康，以免出現(xiàn)低血糖的情況.

I型糖尿??；血糖控制；定向控制；數(shù)值模擬

(續(xù)接上期)

并且A = QUQ-1，其中

這時(shí)A = QUQ-1，并且U是對(duì)角矩陣，所以eAt=QeUtQ-1，其中

則

這里令

a(t)=

最后，將eAt代入式(11)得

2.3 耦合方程的解

前面，我們得到了u(x, t), i(t),Ge(t),G(t).下面將求解φ(t)與i(t),G(t)的關(guān)系.

2.3.1 φ(t)與i(t)的關(guān)系

已經(jīng)得到

u(x,t)=v(x,t)+w(x,t)=

(12)

和

(13)

(14)

將式(14)代入式(13)，得到

φ(s)ds

(15)

2.3.2 φ(t)與G(t)的關(guān)系

根據(jù)3.1和3.2，得到φ(t)與G(t)的關(guān)系：

e-(b1+b2)(t-s)φ(s)ds+

(16)

令t→∞，式(16)結(jié)果為：

(17)

這樣，可以說(shuō)

(18)

即G(t)依賴于式(16)中的主項(xiàng).

最后，得出φ(t)與G(t)的關(guān)系.

3 血糖-胰島素的數(shù)值模擬

本節(jié)對(duì)前面描述的模型進(jìn)行數(shù)值模擬.在參數(shù)的選擇上，沿用已有的研究結(jié)果.模型(1)所描述的胰島素類似物的皮下動(dòng)力學(xué)行為和[3]中的模型類似，所以參數(shù)M,B的選擇參照[3]；而模型(2)是KADIS@模型的一種特殊情況，因此參數(shù)選擇參照KADIS@ 模型.

各參數(shù)的取值如下：

M=9*10-5cm2min-1

B=1.3*10-2min-1ke=0.09min-1

Vd=12*103mLb0=8.0mg/kg

b1=0.018min

b2=0.094minb3=1.73mg·mU-1min-1

由于參數(shù)b0與糖尿病患者的體重有關(guān)，所以在模擬是我們?cè)O(shè)定患者的體重為60kg.當(dāng)空腹血大于等于 126mg/dL時(shí)，即可診斷為糖尿病，則假定Ge(0)= G(0)= 1.6mg/mL.其次，此模型是針對(duì)IDDM患者建立的，初始時(shí)刻體內(nèi)沒有胰島素，則i(0)= 0.為了方便，我們不考慮患者的飲食，即CHO(t)= 0.

該模型是從定向控制的方向建立的，即輸入不同的控制得出不同的結(jié)果，見圖4、5，左邊的曲線代表血液中胰島素濃度的變化，右邊的曲線代表血糖濃度的變化.我們選擇了注射10U和20U的胰島素進(jìn)行數(shù)值模擬.

圖4 φ(t) = 10 U時(shí)的胰島素和血糖變化

圖5 φ(t)= 20 U時(shí)的胰島素和血糖變化

首先，從圖4中發(fā)現(xiàn)，胰島素大概在10 min之后大量進(jìn)入血液，大概在35 min達(dá)到最大值，之后逐漸降低，證明了胰島素在皮下擴(kuò)散的過(guò)程以及胰島素在體內(nèi)和葡萄糖相互作用的過(guò)程.對(duì)于血糖濃度，大概在20 min時(shí)才開始下降，是因?yàn)槟Ｐ椭袥]有將患者機(jī)體正常運(yùn)轉(zhuǎn)所消耗的葡萄糖的量考慮在內(nèi)，即血糖沒有消耗，而內(nèi)源性糖平衡仍然正常地產(chǎn)生葡萄糖，所以出現(xiàn)血糖濃度持續(xù)升高的現(xiàn)象；血糖濃度大概在20 min之后逐漸下降，體現(xiàn)了胰島素和葡萄糖的相互作用，但是大概在50 min后血糖濃度變?yōu)榱素?fù)值，這并不符合實(shí)際情況，主要是因?yàn)樵谀Ｐ湍M時(shí)假設(shè)患者沒有任何飲食.從分析結(jié)果來(lái)看大致是符合實(shí)際情況的，對(duì)于患者機(jī)體正常運(yùn)轉(zhuǎn)所消耗的葡萄糖以及患者的飲食在此模型中是兩個(gè)難點(diǎn)，目前還沒有解決，希望以后有研究者解決.

其次，從兩個(gè)圖的對(duì)比來(lái)看，如果注射的胰島素的量較多，同一時(shí)間患者體內(nèi)胰島素的濃度相對(duì)高，而葡萄糖的濃度相對(duì)低.這是符合實(shí)際情況的，更多的胰島素能夠快速地降低葡萄糖的濃度.

最后，通過(guò)數(shù)值模擬的結(jié)果看出，本文提出的皮下胰島素動(dòng)力學(xué)模型是合理的，對(duì)于更加復(fù)雜的情況還待解決.

[1] NUCCI G, COBELLI C. Models of subcutaneous insulin kinetics. A critical review [J]. Computer methods and programs in biomedicine, 2000, 62(3): 249-257.

[2] MOSEKILDE E, JENSEN K S, BINDER C,etal. Modeling absorption kinetics of subcutaneous injected soluble insulin [J]. Journal of pharmacokinetics and biopharmaceutics, 1989, 17(1): 67-87.

[3] TRAJANOSKI Z, WACH P, KOTANKO P,etal. Pharmacokinetic model for the absorption of subcutaneously injected soluble insulin and monomeric insulin analogues [J]. Biomedizinische Technik Biomedical Engineering, 1993, 38(9): 224-231.

[4] BERGMAN R N, COBELLI C. Minimal modeling, partition analysis, and the estimation of insulin sensitivity [J]. Federation Proceedings, 1980, 39(1): 110-115.

[5] STURIS J, POLONSKY K S, MOSEKILDE E,etal. Computer model for mechanisms underlying ultradian oscillations of insulin and glucose [J]. American Journal of Physiology-Endocrinology and Metabolism, 1991, 260(5): E801-E809.

[6] PALUMBO P, PANUNZI S, GAETANO A D. Qualitative behavior of a family of delay-differential models of the glucose-insulin system [J]. Discrete and Continuous Dynamical Systems Series B, 2007, 7(2): 399.

[7] PALUMBO P, PEPE P, PANUNZI S,etal. Glucose control by subcutaneous insulin administration: a DDE modelling approach[C]//Milano: 18th IFAC World Congress, 2011. 1471-1476.

[8] SALZSIEDER E, VOGT L, KOHNERT K D,etal. Model-based decision support in diabetes care [J]. Computer methods and programs in biomedicine, 2011, 102(2): 206-218.

[9] CHEE F, SAVKIN A V, FERNANDO T L,etal. Optimal H∞insulin injection control for blood glucose regulation in diabetic patients [J]. Biomedical Engineering, IEEE Transactions on, 2005, 52(10): 1625-1631.

[10] MARCHETTI G, BAROLO M, JOVANOVIC L,etal. An improved PID switching control strategy for type 1 diabetes [J]. Biomedical Engineering, IEEE Transactions on, 2008, 55(3): 857-865.

[11] PAGUREK B, RIORDON J S, MAHMOUD S. Adaptive control of the human glucose-regulatory system [J]. Medical and biological engineering, 1972, 10(6): 752-761.

[12] RENARD E, COSTALAT G, CHEVASSUS H,etal. Closed loop insulin delivery using implanted insulin pumps and sensors in type 1 diabetic patients [J]. Diabetes Research and Clinical Practice, 2006, 74: S173-S177.

[13] ABU-RMILEH A, GARCIA-GABIN W. A gain-scheduling model predictive controller for blood glucose control in type 1 diabetes [J]. Biomedical Engineering, IEEE Transactions on, 2010, 57(10): 2478-2484.

[14] FABIETTI P G, CANONICO V, ORSINI-FEDERICI M,etal. Clinical validation of a new control-oriented model of insulin and glucose dynamics in subjects with type 1 diabetes [J]. Diabetes technology & therapeutics, 2007, 9(4): 327-338.

[15] ABU-RMILEH A, GARCIA-GABIN W, ZAMBRANO D. A robust sliding mode controller with internal model for closed-loop artificial pancreas [J]. Medical & biological engineering & computing, 2010, 48(12): 1191-1201.

Partial differential glycemic control model for type I diabetes(Ⅱ)

ZHAO Jia-wei1, CAO Xiu-ling1, LI Xuan2

(1. Department of Mathematics, Tianjin University, Tianjin 300072, China; 2. Department of Basic Course,Pearl River College, Tianjin University of Finance and Economics, Tianjin 301811, China)

In this paper, the glycemic control system of type I diabetics was discussed. In this system, the heat equation was applied to the glycemic control system based on the physiological character of type I diabetics, and the relationship between the control input and control objective was researched from the point of view of orientation control. Eventually, some numerical simulations were performed by Matlab program. From the results of numerical simulations, we got the accurate control input corresponding to the control objective guaranteeing the health of diabetics lest the appearance of hypoglycemia.

type I diabetes; glycemic control; orientation control; numerical simulation

2015-09-14.

國(guó)家自然科學(xué)基金(NSFC-61174080，61104130)

趙家維(1991-)，女，碩士，研究方向：控制論.

O174

A

1672-0946(2016)06-0682-04