Computational Biomechanics of the Musculoskeletal

Computational Biomechanics of the Musculoskeletal System

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Edited by

Ming Zhang and Yubo Fan

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Contents Editors................................................................................................................................................ix Contributors.......................................................................................................................................xi MATLAB Statement........................................................................................................................ xv


Section I Foot and Ankle Joint Chapter 1 Foot Model for Investigating Foot Biomechanics and Footwear Design......................3 Ming Zhang, Jia Yu, Yan Cong, Yan Wang, and Jason Tak-Man Cheung Chapter 2 Female Foot Model for High-Heeled Shoe Design..................................................... 19 Jia Yu, Yubo Fan, and Ming Zhang Chapter 3 Foot and Ankle Model for Surgical Treatment........................................................... 37 Yan Wang and Ming Zhang Chapter 4 First Ray Model Comparing Normal and Hallux Valgus Feet................................... 49 Duo Wai-Chi Wong, Ming Zhang, and Aaron Kam-Lun Leung Chapter 5 Dynamic Foot Model for Impact Investigation........................................................... 61 Jia Yu, Duo Wai-Chi Wong, and Ming Zhang

Section II Knee Joint Chapter 6 Knee Joint Model for Anterior Cruciate Ligament Reconstruction............................ 75 Jie Yao, Ming Zhang, and Yubo Fan Chapter 7 Knee Joint Models for Kneeling Biomechanics.......................................................... 83 Yuxing Wang, Yubo Fan, and Ming Zhang Chapter 8 Knee Implant Model: A Sensitivity Study of Trabecular Stiffness on Periprosthetic Fracture�� 93 Duo Wai-Chi Wong and Ming Zhang

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Contents

Section III Hip and Pelvis Chapter 9 Femur Model for Predicting Strength and Fracture Risk......................................... 105 He Gong, Yubo Fan, and Ming Zhang Chapter 10 Hip Model for Osteonecrosis.................................................................................. 113


uo Wai-Chi Wong, Zhihui Pang, Jia Yu, Aaron Kam-Lun Leung, D and Ming Zhang Chapter 11 Pelvis Model for Reconstruction with Autografted Long Bones following Hindquarter Amputation�� 125 Wen-Xin Niu, Jiong Mei, Ting-Ting Tang, Yubo Fan, Ming Zhang, and Ming Ni

Section IV Lower Limb for Rehabilitation Chapter 12 Foot–Ankle–Knee Model for Foot Orthosis........................................................... 141 Xuan Liu, Yubo Fan, and Ming Zhang Chapter 13 Lower Residual Limb for Prosthetic Socket Design............................................... 153 Winson C.C. Lee and Ming Zhang Chapter 14 Residual Limb Model for Osteointegration............................................................. 163 Winson C.C. Lee

Section V Spine Chapter 15 Spine Model for Vibration Analysis........................................................................ 175 Lixin Guo, Ming Zhang, and Ee-Chon Teo Chapter 16 Cervical Spinal Fusion and Total Disc Replacement.............................................. 199 Zhongjun Mo, Lizhen Wang, Ming Zhang, and Yubo Fan Chapter 17 Spine Model for Disc Replacement......................................................................... 213 Qi Li, Lizhen Wang, Zhongjun Mo, and Yubo Fan Chapter 18 Spine Model for Applications in Aviation Protection............................................. 225 Cheng-fei Du, Lizhen Wang, Ya-wei Wang, and Yubo Fan

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Contents

Section VI Head and Hand Chapter 19 Head Model for Protection...................................................................................... 245 izhen Wang, Peng Xu, Xiaoyu Liu, Zhongjun Mo, Ming Zhang, L and Yubo Fan Chapter 20 Tooth Model in Orthodontics and Prosthodontics.................................................. 255 Chao Wang, Yi Zhang, Wei Yao, and Yubo Fan


Chapter 21 Eye Model and Its Application................................................................................ 271 Xiaoyu Liu, Lizhen Wang, Deyu Li, and Yubo Fan Chapter 22 Temporomandibular Joint Model for Asymptomatic and Dysfunctional Joints..... 283 Zhan Liu, Yuan-li Zhang, Ying-li Qian, and Yubo Fan Chapter 23 Fingertip Model for Blood Flow and Temperature................................................. 299 Ying He, Hongwei Shao, Yuanliang Tang, Irina Mizeva, and Hengdi Zhang

Section VII Bone Chapter 24 Micro-Finite Element Model for Bone Strength Prediction................................... 323 He Gong, Ming Zhang, and Ling Qin Chapter 25 Simulation of Osteoporotic Bone Remodeling....................................................... 331 He Gong, Yubo Fan, and Ming Zhang Index�� 343


Editors Ming Zhang is a professor of biomedical engineering and the director of the Research Center for Musculoskeletal Bioengineering, Interdisciplinary Division of Biomedical Engineering, The Hong Kong Polytechnic University. He obtained a BSc degree in automation control engineering and an MSc degree in mechanical engineering from Beijing Institute of Technology and a PhD degree in medical engineering and physics from King’s College, University of London. He is the secretary of the World Association for Chinese Biomedical Engineers (WACBE), council member of the World Council of Biomechanics, and a standing council member for the Chinese Society of Biomedical Engineering (CSBME) and the Chinese Rehabilitation Devices Association (CRDA). He has published about 200 journal papers and book chapters. His current research interests include computational biomechanics, bone biomechanics, foot biomechanics and footwear design, body support biomechanics, prosthetic and orthotic bioengineering, human motion, and body vibration analysis. Yubo Fan is the dean of the School of Biological Science and Medical Engineering at Beihang University and director of the Key Laboratory for Biomechanics and Mechanobiology of the Chinese Education Ministry. He is also currently the president of the Chinese Biomedical Engineering Society (2008–), the vice president of the Chinese Strategic Alliance of Medical Device Innovation (2010–), an American Institute for Medical and Biological Engineering (AIMBE) fellow (2014), and council member of the World Council of Biomechanics. Professor Fan specializes in biomechanics, with particular interest in computational biomechanics, the biomechanical design of implanting, musculoskeletal and dental mechanics, cardiovascular fluid mechanics, biomaterials, and rehabilitation engineering. He has successfully bid for research grants for over 30 million Chinese Yuan in the past 5 years. He has published over 100 international journal papers and has supervised more than 50 graduate students including MSc, PhD, and postdoctoral fellows in the area of biomechanical engineering.

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Contributors


Jason Tak-Man Cheung Li Ning Sports Science Research Center Beijing, China Yan Cong Interdisciplinary Division of Biomedical Engineering The Hong Kong Polytechnic University Hong Kong SAR, China Cheng-fei Du Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China Yubo Fan Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China He Gong School of Biological Science and Medical Engineering Beihang University Beijing, China Lixin Guo School of Mechanical Engineering and Automation Northeastern University Shenyang, China Ying He Department of Modern Mechanics University of Science and Technology Hefei, China

Winson C.C. Lee Interdisciplinary Division of Biomedical Engineering The Hong Kong Polytechnic University Hong Kong SAR, China Aaron Kam-Lun Leung Interdisciplinary Division of Biomedical Engineering The Hong Kong Polytechnic University Hong Kong SAR, China Deyu Li Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China Qi Li Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China Xiaoyu Liu Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China

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Xuan Liu Shenzhen Research Institute The Hong Kong Polytechnic University Shenzhen, China and


Interdisciplinary Division of Biodmedical Engineering The Hong Kong Polytechnic University Hong Kong SAR, China Zhan Liu Provincial Key Laboratory of Biomechanical Engineering Sichuan University Chengdu, Sichuan, China Jiong Mei Tongji Hospital Tongji University School of Medicine Shanghai, China Irina Mizeva Institute of Continuous Media Mechanics Academy Koroleva Perm, Russia Zhongjun Mo Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China and Interdisciplinary Division of Biomedical Engineering The Hong Kong Polytechnic University Hong Kong SAR, China Ming Ni Tongji Hospital Tongji University School of Medicine and Department of Orthopedics Pudong New Area People’s Hospital Shanghai, China

Contributors

Wen-Xin Niu Tongji Hospital Tongji University School of Medicine and Shanghai Key Laboratory of Orthopaedic Implants Shanghai, China and Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China and Interdisciplinary Division of Biomedical Engineering The Hong Kong Polytechnic University Hong Kong SAR, China Zhihui Pang First Affiliated Hospital of Guangzhou University of Chinese Medicine Guangzhou, China Ying-li Qian AVIC 611 Institute Chengdu, Sichuan, China Ling Qin Department of Orthopaedics and Traumatology Chinese University of Hong Kong Hong Kong SAR, China Hongwei Shao Department of Modern Mechanics University of Science and Technology Hefei, China Ting-Ting Tang Shanghai Key Laboratory of Orthopaedic Implants Shanghai, China

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Contributors

Yuanliang Tang Department of Modern Mechanics University of Science and Technology Hefei, China


Ee-Chon Teo School of Mechanical and Aerospace Engineering Nanyang Technological University Singapore Chao Wang Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China Lizhen Wang Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China Yan Wang Interdisciplinary Division of Biomedical Engineering The Hong Kong Polytechnic University Hong Kong SAR, China Ya-wei Wang Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China

Yuxing Wang National Key Lab of Virtual Reality Technology Beihang University and Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beijing, China Duo Wai-Chi Wong Interdisciplinary Division of Biomedical Engineering The Hong Kong Polytechnic University Hong Kong SAR, China Peng Xu Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China Jie Yao Key Laboratory for Biomechanics and Mechanobiology of the Ministry of Education School of Biological Science and Medical Engineering Beihang University Beijing, China Wei Yao Department of Biomedical Engineering University of Strathclyde Glasgow, Scotland, United Kingdom Jia Yu Interdisciplinary Division of Biomedical Engineering The Hong Kong Polytechnic University Hong Kong SAR, China

xiv

Hengdi Zhang Institute of Continuous Media Mechanics Academy Koroleva Perm, Russia


Ming Zhang Interdisciplinary Division of Biomedical Engineering The Hong Kong Polytechnic University Shenzhen Research Institute Hong Kong SAR, China

Contributors

Yi Zhang Chongqing Research Center for Oral Diseases and Biomedical Science College of Stomatology Chongqing Medical University Chongqing, China Yuan-li Zhang Provincial Key Laboratory of Biomechanical Engineering Sichuan University Chengdu, Sichuan, China

MATLAB Statement


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23

Fingertip Model for Blood Flow and Temperature Ying He, Hongwei Shao, Yuanliang Tang, Irina Mizeva, and Hengdi Zhang

Contents 23.1 Introduction.......................................................................................................................... 299 23.2 Two-Dimensional Thermal Model for Assessing Cold-Stressed Effects on the Human Finger......................................................................................................................302 23.3 Thermal Regulation Modeling............................................................................................. 303 23.4 Three-Dimensional Analysis of Blood Flow and Temperature Distribution.......................307 23.4.1 Image-Based Modeling of a Human Hand............................................................307 23.4.2 Mesh Generation....................................................................................................307 23.4.3 Modeling Blood Perfusion and Heat Transfer in Biological Tissues.....................309 23.4.3.1 Blood Perfusion Modeling Based on Darcy’s Equation.......................309 23.4.3.2 Heat Transfer Modeling Based on Pennes Equation............................. 310 23.5 Finite Element Analysis of Blood Flow and Temperature during Digital Thermal Monitoring........................................................................................................................... 312 23.6 Prediction of Temperature Oscillations after the Contralateral Cooling Test...................... 314 23.7 Concluding Remarks............................................................................................................ 317 Acknowledgments........................................................................................................................... 318 References....................................................................................................................................... 318

23.1 Introduction It has been shown that chronic illnesses such as systemic sclerosis and diabetes mellitus are related to some disorders at the capillary level, such as increased vessel permeability, the presence of avascular areas, enlarged loops, poor circulation, and increased tortuosity (Daly and Leahy 2013). There has been an increasing interest in using microcirculation as a marker for cardiovascular health and metabolic functions, as it may be related to the development of instruments for detecting a variety of pathological processes in the circulatory system. Since the skin is readily accessible, it provides an appropriate site to assess peripheral microvascular reactivity. Vascular reactivity is a primary feature of the circulatory system that enables the vasculature to respond to physiological and physical stimuli that require adjustments in blood flow, vessel tone, and vessel diameter. For more than two decades, methods that focus on the noninvasive exploration of cutaneous microcirculation have been mainly based on optical microscopy and laser Doppler techniques (Roustit and Cracowski 2012). In recent years, laser Doppler flowmetry (LDF), in combination with wavelet analysis of blood flow oscillations, has been increasingly used to detect alterations in vessel tone. Blood flow modulations in microvessels form five non-overlapping frequency bands within the wave range 0.0095–3 Hz (Stefanovska et al. 1999) and the lowest frequency range (0.0095–0.021 Hz) is related to the functional activity of the microvessel endothelium. Fedorovich (2012) investigated the correlation

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between metabolic and microhemodynamic processes in the skin and showed that improved oxygen uptake and glucose disposal by tissues is accompanied by a significant increase in endothelial rhythm amplitude. Bernjak et al. (2008) showed that congestive heart failure exhibits abnormally attenuated blood flow oscillations in the frequency interval 0.005–0.021 Hz, and that treatment with β 1-blockers (Bisoprolol) can move the spectral amplitude 0.005–0.0095 Hz to that of the healthy control subjects. Laser speckle contrast imaging is a recently developed optical noncontact technique that allows for the continuous assessment of skin perfusion over wide areas. In comparison to laser Doppler flowmetry, laser speckle contrast imaging can provide real-time monitoring of the microcirculation with high resolution (Basak et al. 2012). An extensive overview of these optical techniques may be found in a review of blood flow imaging by Daly and Leahy (2013). A major drawback of these methods is that the instruments they employ are complicated and may not be suitable for daily use. The fingertip temperature response to a thermal stimulus or to pressure loading depends on the amount of blood perfusion, which implies that monitoring alterations in fingertip temperature could be employed to study microvascular reactivity. Based on a bioheat transfer equation, Haga et al. (2012) employed an inverse analysis method for examining blood perfusion. The instrument that was developed from this method can be used to estimate blood perfusion according to the observed fingertip temperature response under a certain thermal stimulus and shows good measurement repeatability and sensitivity. Yue et al. (2008) presented a three-point method to measure blood perfusion, whereby a heater is wrapped around a cylindrical section of living tissue, and three-point skin temperatures are measured. The characteristic points are located at the center of the heater, 1 cm away from the edge of the heater, and 2 cm away from the heater, respectively. By constructing an objective function between the measured and calculated temperatures and minimizing the function, the optimal value for blood perfusion and thermophysical properties can be obtained. Apart from methods that are needed to load heat sources in living tissues, Nagata et al. (2009) established a passive method to evaluate blood perfusion only by using thermal information for the human body, whereby blood perfusion can be expressed in terms of the rate of temperature change at the contact sensor point and the initial skin temperature. Furthermore, a novel thermal method has been presented that uses the fingertip temperature, forearm temperature minus rectal temperature, and their changes across time to predict finger blood flow (Carrillo et al. 2011). Recently, a new thermal peripheral blood flowmeter has been developed that is integrated with a force sensor for force-compensated blood flow measurement (Sim et al. 2012). The important feature of this device is that, apart from the conventional metal resistance of the temperature detector, there is a membrane fabricated by surface and bulk micromachining techniques that is embedded with a piezoresistive force sensor. The compensated blood flow can be determined by detecting the rate of temperature change and the contact force. Due to simple structures and low costs, many thermal methods for peripheral blood flow measurement have been developed. Not only can the peripheral blood flow rate be determined from the skin temperature, but vasodilated function may also be detected. Fingertip digital thermal monitoring (DTM) during cuff-occlusive reactive hyperemia (RH) is a new, noninvasive method of vascular function assessment that is based on the premise that changes in fingertip temperature during and after an ischemic stimulus reflect changes in blood flow and endothelial function. DTM technology usually requires 2 min of cuff inflation to cause brachial artery occlusion and 5 min of deflation later to bring about blood reperfusion. During the occlusion stage, a vasodilatory response occurs in the peripheral arteries and capillaries due to the absence of blood flow. After the brachial occlusion is finished, blood rushes into the forearm and hand, and thus causes a transient temperature rebound in the fingertip. Ley et al. (2008, 2009) presented two mathematical models of heat transfer based on Pennes equation to estimate the influence of different factors on the dynamic temperature response in the fingertip during vascular occlusion and reperfusion. The models are based on different dimensions and anatomical details. One is a lumped parameter model that neglects tissue composition and the other


Fingertip Model for Blood Flow and Temperature

301

is a two-dimensional thermal model of a simplified finger. A comparison of the models indicates that the lumped parameter model is effective as long as the proper variation between the initial condition and environmental parameters that affect the response is adopted. For both models, the blood perfusion rate was considered to be an input and distributed uniform in any places of the finger. In subsequent work, Ley et al. (2011) measured the fingertip temperature, heat flux, and blood perfusion of 12 healthy volunteers. According to their heat transfer analysis, the magnitude of the laser Doppler signal was correlated with the local tissue temperature in an exponential manner. Moreover, they found that the initial and minimum temperatures had a significant effect on the thermal response during post-occlusion. Akhtar et al. (2010) proposed a three-dimensional thermal model of a simplified finger to estimate the sensitivity of DTM parameters currently in clinical use to assess RH. According to their simulation, temperature rebound was shown to have the best correlation with the level of RH with good sensitivity for the range of flow rates that were studied. Their results indicated that temperature rebound and the equilibrium initial temperature are necessary to identify the amount of RH and to establish criteria for predicting the state of a patient’s cardiovascular health. McQuilkin et al. (2009) developed a signal-processing model to establish the basis for the relationship between finger temperature and blood flow reactivity following a brachial artery occlusion and reperfusion procedure. They expressed the fingertip temperature as the convolution of the flowdomain signal and the exponential impulse response. The normalized flow-domain signals that were computed from Doppler and DTM sensors showed favorable agreement. Their work demonstrates that vascular reactivity indices measured by DTM can be evaluated accurately. Many researchers have also attempted to determine the correlation between skin temperature and blood flow oscillations. Liu et al. (2011) employed functional infrared imaging to identify lowfrequency temperature fluctuations on the surface of the skin. They found that large veins have stronger contractility in the range of 0.005–0.006 Hz, compared to microvasculature and skin areas without infrared(IR)-detectable vessels. Sagaidachnyi et al. (2012) calculated the rate of blood flow from measured skin temperature signals with the use of Pennes bioheat equation. By adding a time delay of 10–20 s caused by temperature waves, the calculated blood flow waves corresponded with measured blood flow signals by using photoplethysmography. Compared to detecting blood flow oscillations under normal thermal conditions, investigation of blood flow oscillations after a cooling test may reveal endothelial functions more accurately. Coldinduced vasodilation is an acyclic oscillation of blood flow that occurs upon exposure to the cold and commonly occurs in the extremities. The initial response to cold exposure is sympathetically mediated peripheral vasoconstriction, which results in reduced local tissue temperature. With continued cold exposure, this vasoconstriction may be interrupted, which results in periods of vasodilation. The vasodilation shows a characteristic cycle of increasing and decreasing blood flow. Smirnova et al. (2013) employed wavelet analysis to investigate skin temperature oscillations during a contralateral cooling test on type 2 diabetic patients. Fifty-five healthy subjects and 35 type 2 diabetic patients with retinopathy and nephropathy participated in the study. Their skin temperatures were measured on the palm surface of their index fingers of the right hand while their left hands were immersed in cold water at 5°C for 3 min. The results of a wavelet analysis show that during the cold stimulus test the amplitudes of the skin temperature oscillations on the index finger of the contralateral hand significantly decreased and increased after unloading the cold stimulus for the healthy subjects. However, for the type 2 diabetic patients, after the decrease in temperature fluctuation during the cold stimulus, there was no further reliable increase in the amplitudes of the temperature fluctuations. These results demonstrate that cold-induced vasodilation for type 2 diabetic patients is much weaker than that for healthy subjects. So far, we have seen that there are significant benefits in obtaining information on blood flow by employing thermal methods. In particular, temperature variation after brachial artery occlusion and a contralateral cooling test is closely related to significant variation in microvascular resistance. However, the mechanisms with regard to how vasomotion and peripheral temperature interact with


302


each other remain unclear. Quantitatively predicting the relationship between vascular reactivity, vessel diameter, and blood flow may give insights into the assessment of endothelial functions. In this review, the methods for modeling temperature and blood flow at the fingertips during cuffocclusive RH and a contralateral cooling test are discussed. The discussion centers on the developments of the authors’ modeling work in this aspect. This review is arranged into five parts. The first part deals with a two-dimensional model for assessing fingertip temperature variation during cold-water stimulation. Next, approaches for the simulation of peripheral thermoregulation are discussed. The third part focuses on the development of an image-based human model, a Darcy’s model, and a heat transfer model of living tissue. Finally, the applications of this model for assessing vascular reactivity after brachial artery occlusion and a contralateral cooling test are discussed.

23.2 Two-Dimensional Thermal Model for Assessing Cold-Stressed Effects on the Human Finger Figure 23.1 shows the anatomical structure of a finger. It can be seen that the arteries, veins, and nerves pass through the tissue and are located very close to each other. According to the anatomical structure of the finger cross-section, the finger is modeled as an elliptical cross-sectional area and consists of four parts—bone, tendon, dermis, and epidermis—as shown in Figure 23.2a. Figure 23.2b depicts a grid generation of this computation (He et al. 2001).

Figure 23.1 Anatomical structure of a human finger. (From Kahle W et al., Tachen atlas Der Anatomie, Bunkoudo, 1990.) Epidermis

Bone Tenton

(a)

Artery

(b)

Figure 23.2 (See color insert.) (a) Different materials in the finger and (b) a mesh network of the cross section of the finger.

303

Fingertip Model for Blood Flow and Temperature Time t, s

Time t, s 39 36

240

120

360

480

35

Temperature of water Tw = 10°C Temperature of air Tair = 22°C

480

360

33 32

30

Temperature T, °C

Temperature T, °C


240

34

33

27 24 21 18

400

800

31 30

Tw = 10°C Tair = 22°C

29 28 27 26

Experimental result 4ωdermis ωdermis

25

4ωdermis ωdermis

15 12 0

120

24 23 1200

22

480

600

720

Time t, s

Time t, s

(a)

(b)

840

Figure 23.3 (a) Predicted average skin temperature under cold water stimulation. (b) Comparison between predicted average skin temperature and experimental data. (From Clark RP and Edholm OG, Man and his thermal environment, Edward Arnold (Publishers) Ltd., London, 1985.)

In the simulation, the following computational conditions were set up. The environmental t emperature was 22°C and the wind velocity was 0.1 m/s. After immersion in 10°C cold water for 1 min, the finger was then exposed to indoor air at 22°C. The arterial temperature was considered to be constant at 37°C. The venous temperature was set to be equal to the tissue temperature. The predicted average skin temperature is plotted in Figure 23.3a. It can be seen that the skin temperature became the same as that at the resting stage within 5 min, when the blood perfusion of the dermis was four times as large as that at the resting stage. If the blood perfusion in the recovery stage remained the same as that of the resting perfusion, the skin temperature could not recover even up to 10 min later. Figure 23.3b clearly shows that the predicted temperature with larger blood perfusion after cold-water stimulation was in a good agreement with the experimental values. We can also see that there was a fluctuation in the measured skin temperature that was not revealed in the prediction, and that the fluctuation was not different from the heart rate. Hence, the fluctuation may have been related to regulation of the autonomic nervous system (ANS).

23.3 Thermal Regulation Modeling In order to simulate the temperature oscillation, we coupled a model of regulation of the ANS (Zhang et al. 2010). We developed an ANS model on the basis of earlier studies by Liang (2007) and Xu et al. (2008). The peripheral thermoregulation pathway comprises sensory nerves, receptors, the central nervous system (CNS), efferent nerves, and effectors. Receptors are located in the hypothalamus, efferent nerves are sympathetic vasoconstrictors, and effectors adhere to arteriole smooth muscle. A flow chart for signal transport is shown in Figure 23.4. The thermoregulation signal is considered to start from the current through the opening of ion channels in the sensory nerves, and the intensity of the current is related to the environmental temperature and the subject’s comfort threshold. The receptors receive the frequency of impulses from

304



Environmental temperature

Fingertip temperature

Sensory nerves

Blood perfusion in fingertips

Thermal receptors in hypothalamus

Arteriole smooth muscle

Central nervous system (CNS)

Sympathetic vasoconstrictors

Figure 23.4 Flow chart of the thermoregulation pathway. (From Zhang HD et al., Computers in Biology and Medicine, 40, 2010. With permission.)

external stimulation, and this frequency is a function of the current. Therefore, the frequency in the afferent fibers can be expressed as a function of temperature as follows:

faf =

faf ,min + faf ,max exp[(T ftip,t − Tcri ) / K ] 1 + exp[(T ftip,t − Tcri ) / K ]

(23.1)

Here, faf is the frequency of spikes in the afferent fibers, and faf,min and faf,max are the minimum and maximum values, respectively, of the frequency. K is the parameter that controls the slope between temperature Tftip,t and afferent frequency faf. If K is equal to 3, faf will reach the maximum value (15 Hz) when the fingertip temperature is 43°C. The threshold value of the tissue is 43°C (Xu et al. 2008). Thus, it is reasonable to choose K as being 3. Tcri was considered to be 30°C. After the thermal receptor receives the frequency in the hypothalamus, it delivers these signals to the CNS. The CNS is instrumental in gathering input information from the receptors and sending the lowest signal to the effectors. The expression for this process is as follows:

fef , in = min( fef ,in ,max , Taf ) (23.2)

Here, fef,in,max is the maximum constant frequency of discharge from the sympathetic nerves. The control signal from the CNS is subsequently transferred to the effectors. For thermoregulation during cold-water stimulation, a cutaneous vasoconstrictor nerve may be activated as the effector to allow the arteriole smooth muscle to constrict, thereby causing changes in blood perfusion. This variation is expressed as follows:

σ (t ) = G A log[ fef , in (t − DA ) − fefin ,min + 1] (23.3)

Here, fef , in ,min is the threshold of the sympathetic nerve, GA is the gain factor, and DA is the latency time of the nerves. Therefore, blood perfusion after cold-water stimulation can be expressed as follows:

d∆ω (t ) 1 = [ −∆ω (t ) + σ (t ) ] dt τA

305


ω (t ) = ω 0 + ∆ω (t ) (23.4)

Here, ω(t) is the blood perfusion at time t, ω 0 is the initial blood perfusion before stimulation, and Δω(t) is the variation in blood perfusion at time t. The equation for skin temperature variation in the fingertip is adopted from Ley et al. (2008) as follows:


ρVC p

dT = hair A(Tair − T ftip,t ) + ρbC pbω (t )(TA − T ftip,t ) (23.5) dt

Here, ρ is the tissue density of the fingertip, Cp is the specific heat of tissue, ρ b is the density of blood, Cpb is the specific heat of blood, TA is the temperature of arterial blood in the fingertip (assumed to be 34°C), Tair is the environmental temperature, Tftip,t is the fingertip temperature at time t, and hair is the heat transfer coefficient. V is defined as being the volume of the fingertip, which is assumed to be a hemisphere, and is expressed as follows: V = π D 3 / 12 (23.6)

Here, D is the average diameter of the fingertip. A is the skin area of the fingertip and is defined as follows: A = π D 2 / 2 (23.7)

Equations 23.1 to 23.5 were solved numerically by using the fourth-order Runge-Kutta method. Some important parameters related to the ANS model are listed in Table 23.1. The thermophysical properties that are used in Equation 23.5 are listed in Table 23.2. Figures 23.5a and b show simulated recovery temperatures with different nervous control signals after 1 min of immersion

Table 23.1 Some Important Parameters Used in the Thermal Regulation Model Minimum frequency Maximum frequency Sensitivity of receptor Time constant for vessels

1 Hz 15 Hz 3 6s 3s

fef,min, faf,min fef,max, faf,max K τA DA

Latency time of nerves

Table 23.2 Values of Thermophysical Parameters Parameters

Bone

Muscle

Blood

Skin

Density ρ (kg/m3) Heat capacity Cp(J/kg K) Heat conductivity (W/mK) Heat transfer coefficient at skin level (W/m2K) Initial blood perfusion (m3/s)

1418

1270

1100

1200

2049

3768

3300

3391

2.21

0.35

0.50

0.37 6

2.3 × 10–10

2.3 × 10–10

306

Computational Biomechanics of the Musculoskeletal System 40

Thumb Index finger Middle finger Ring finger Little finger

38 36 34


Temperature

32 30 28 26 24 22 20

0

100

200

300

400 Time/s (a)

500

600

40

800

Thumb Index finger Middle finger Ring finger Little finger

38 36 34 Temperature

700

32 30 28 26 24 22 20

0

100

200

300

400 Time/s (b)

500

600

700

800

Figure 23.5 Temperature recovery profiles with (a) periodic and (b) constant nervous control signals. (From Zhang HD et al., Computers in Biology and Medicine, 40, 2010. With permission.)

in 10°C cold water. When the nervous control signal was periodic and the efferent frequency was expressed as Equation 23.8, the temperature recovery profile was as shown in Figure 23.5a.

 1  fef = 1 + sin(π t ) faf , pre (23.8)  2 


307

Here, faf,pre is the afferent frequency in the previous time step. We can see that simulated variation profiles follow the same trend as those obtained from the measurements. When the nervous control signal was constant, the temperature recovery profiles were smooth and oscillation-free, similar to those plotted in Figure 23.5b. From these results, we can speculate that temperature oscillation in the measurements may have been due to periodic variation of the nervous control signal.


23.4 Three-Dimensional Analysis of Blood Flow and Temperature Distribution 23.4.1 Image-Based Modeling of a Human Hand Original magnetic resonance (MR) images were acquired from a volunteer’s hand. All of the procedures were handled according to an ethical protocol that was approved by RIKEN in Japan. A hand-fitted supporter made of sponge was designed in order to fix the volunteer’s hand before taking the images. A 1.5-T scanner (Excelart, Toshiba Medical Systems) was used with different sequences for taking images of blood flow and different tissues. The resolution of the MR images was 320 × 320 and the distance between two adjacent slices was set to be 1 mm. During image processing, a MATLAB program was developed to extract information on the edges and bones from the original sequential MR images semi-automatically. First, several image processing operators were applied in turn, such as blurring, sharpening in order to reduce noise, and enhancing the contrast. Usually, the gradient of the grayscale near the edge is largest so that the object can be extracted from the background. Due to the limited image quality of the bone regions, reparation was implemented manually to ensure the integrity of the bone regions. The edges and the bone were set in different pixel values separately, such as 255 for the edges and 100 for the bone. Finally, arteries and veins were identified for every slice with specific pixel values (200 for arteries and 180 for veins) according to their anatomical structures.

23.4.2 Mesh Generation Based on the pre-processed cross-section images of the hand, the transfinite interpolation (TFI) method was applied to generate a mesh for every cross-section. Subsequently, the mesh for every cross-section was connected successively by stacking them in turn. The TFI method is a kind of algebraic mapping method based on mesh generation. The concept of this method is that the physical coordinates of the nodes, which are treated as a function of the computational coordinates, are interpolated based on their values along the edges of the computational domain (Farrashkhalvat and Miles 2003). The one-dimensional interpolations are expressed as follows:

Pξ (ξ ,η ) = (1 − ξ ) r ( 0,η ) + ξ r (1,η ) (23.9)

Pη (ξ ,η ) = (1 − η ) r (ξ ,0 ) + ηr (ξ ,1) (23.10)

where ξ , η are the computational coordinates, r (ξ ,η ) are the physical coordinates on the edges of the computational domain, and Pξ , Pη are the transformations that map the nodes in the computational coordinates to the nodes in the physical coordinates. A two-dimensional interpolation can be constructed as a linear combination of two one-dimensional interpolations and their product as follows:

Pξ ⊕ Pη = Pξ + Pη − Pξ Pη (23.11)

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This formula is the basis of the TFI method in two dimensions. Based on this formula, a mesh is generated by taking discrete values ξi , η j of ξ and η with


0 ≤ ξi =

i −1 j −1 ≤ 1 and 0 ≤ η j = ≤1 iM − 1 jM − 1

i = 1,2,…, iM , j = 1,2,…, jM (23.12)

where iM and jM are the maximum numbers of nodes in the two directions. The TFI method, which is the most common approach to algebraic mesh generation, can produce excellent meshes effectively when other methods are difficult to apply, and it is convenient for adjusting the locations of the mesh nodes. In the present work, the TFI method was automatically applied to the processed image for quadrilateral mesh generation. Figure 23.6 shows the mesh generation for different layers. It can be seen that the numbers of domains are different for different layers. Since the mesh on each slice was generated, hexahedral mesh generation could be executed by connecting the corresponding quadrilateral elements in two adjacent slices. However, due to the existence of branches in the human hand, a special rule should be made to connect two elements in these areas to ensure the correct geometric shape. For example, there may be one region in the current slice, but three regions in the next slice. Therefore, a rule should be prescribed to determine which two elements will be connected. In this work, the distance between two corresponding regions that should be connected is the minimum. Therefore, by calculating the distances between the corresponding subregions in the adjacent

Figure 23.6 Mesh network in different layers of MR images



309

Figure 23.7 Mesh network of the real geometrical human hand. (From Shao HW et al., Computer Methods in Biomechanics and Biomedical Engineering, 2012. With permission.)

slices, the regions with the minimum distance were connected to generate the hexahedral mesh. We believe that the hexahedral mesh is beneficial for finite element method (FEM) to generate acceptable results, and it requires less solver time than the other kinds of mesh. Hence, this method for generating hexahedral mesh is more suitable for the finite element method solver and can be applied to the mesh generation of other tissues. A mesh network of the hand that was generated for further analysis is shown in Figure 23.7.

23.4.3 Modeling Blood Perfusion and Heat Transfer in Biological Tissues 23.4.3.1 Blood Perfusion Modeling Based on Darcy’s Equation Blood in the hand flows through arteries, capillaries, and veins in turn. In this work, blood flow was divided into two parts: blood flow in large vessels (diameters > 1 mm) and blood perfusion in microvessels. Blood flow in the large vessels was set as a time-dependent input in the numerical model. Blood perfusion in microvessels was considered to be a fluid phase in porous media. Pennes equation was numerically solved to describe the dynamic temperature distribution in the hand. Because of the huge number of microvessels, modeling all of them would have been difficult and unnecessary. An effective method to deal with this problem is to simplify the tissue with microvessels as porous media. The earliest model for fluid transport in porous media is considered to be Darcy’s law, which can be expressed as

µ ∇P = − VDarcy (23.13) k

where µ is the viscosity of blood, k is the permeability of the porous media, and VDarcy is the Darcy’s velocity. In the present study, blood perfusion in the tissue through the microvessels was considered to occur through seepage in the porous media. For different parts of the human hand, the permeability k varies with the density of the microvessels. In the analysis that is presented in Section 23.5, the

310


value of k for the fingertips, whereby abundant microvessels exist, is set to be 5.0 × 10 −13 m 2, while it is set to be 1.0 × 10 −13 m 2 for the other parts of the hand (Nield and Bejan 1998). Considering the continuity equation and the momentum equation, an equation for pressure in porous media can be obtained that is expressed as

µ µ ∇ 2 P = ∇ ⋅  − VDarcy  = − ∇ ⋅ VDarcy = 0  k  k

(23.14)


The dimensionless form of Equations 23.13 and 23.14 can be written as

∇ 2 P* = 0 (23.15)

* VDarcy = − Da ⋅ Re ⋅ ∇P* (23.16)

where Da is the Darcy number such that Da = k / D2 and Re is the Reynolds number such that Re = ρUD / µ . 23.4.3.2 Heat Transfer Modeling Based on Pennes Equation In order to obtain the temperature of living tissue, energy equations for the blood and tissue phase are needed, and this may cause the simulation to become complex. As an alternative, Pennes equation is used to investigate heat transfer in living tissues, which is expressed as

ρc

∂T = λ∇ 2T + Qm + ω b ρbcb (Tb − T ) (23.17) ∂t

where r indicates the tissue density, c is the specific heat of the tissue, rb and cb represent the density and specific heat of blood, respectively, Tb is the temperature of blood that perfuses the tissue, which is assumed to be constant, l is the thermal conductivity of the tissue, Qm is the heat production per unit volume, and w b indicates the blood perfusion rate. If the relationship between Darcy’s velocity and the blood perfusion rate is known, the local blood perfusion can be obtained. Thus, coupling Darcy’s equation and Pennes equation is an elegant method to describe the non-uniformity of blood perfusion. It is assumed that the diameter and length of the microvessels are the same in different parts of the tissue, thus, the ratio of the microvessel blood flow and microvessel volume can be written as

Qb / Vb = vb / L (23.18)

where vb is the blood velocity and L is the length of the microvessel. Since the microvascular volume and Darcy’s velocity are written as Vb = ϕVt and VDarcy = ϕ vb , where j is the porosity of the tissue, the blood perfusion can be expressed as

ω b = Qb / Vt = ϕ Qb / Vb = ϕ vb / L = VDarcy / L (23.19)

Substituting Darcy’s velocity for the blood perfusion rate, the dimensionless form of the energy equation can be expressed as

∂T * 1 2 * 1 * * = ∇T + Qm + βγ VDarcy (Tb* − T * ) (23.20) * ∂t Pe Pe

where Pe is the Peclet number such that Pe = UD / α , b is the ratio of the heat capacity of blood and tissue, and g is the ratio of the diameter to the length of the microvessel.

311


On the surface of the skin, convection, radiation, and evaporation are the main modes of heat transport between the tissue and the surrounding environment. The boundary condition on the surface of the skin is thus expressed as − λs


∂T = hc ( T − Tambi ) + hra ( T − Tambi ) + Esk (23.21) ∂n

where Tambi is the ambient temperature, λs is the thermal conductivity of skin, hc is the convective heat transfer coefficient, hra is the radiative heat transfer coefficient, and Esk is the heat loss due to evaporation. The heat loss due to evaporation can be written as Esk = he ( Pskin − Pambi ) (23.22)

where Pskin is the vapor pressure at the surface of the skin (kPa) and Pambi is the vapor pressure of air (kPa). The evaporation coefficient he is related to the air velocity and can be expressed as he = 124 Vair W / m 2 kPa (23.23)

The dimensionless form of the boundary condition equation can be written as −

∂T * * = Bi ⋅ (T * -Tambi ) + Esk* (23.24) ∂ n*

hc D . λs When the hand is in moving air whose velocity is in the range of 0