fMRI can be used not only in humans but also in animals such as rats and monkeys. Among these, spared nerve injury (SNI) rats can produce persistent and stable symptoms such as hyperalgesia, allodynia, and spontaneous pain, which are commonly used in the study of neuropathic pain. Komaki et al. studied the restingstate fMRI changes in the SNI rat model and studied the node efficiency of some regions of interest and the functional connections between regions of interest by graph theory. They found that the centrality and node efficiency of the S1 region

**27**

*The Application of Functional Magnetic Resonance Imaging in Neuropathic Pain*

in the opposite side of the injured limb were significantly lower after injury, while the functional connection between the ACC and the posterolateral nucleus of the thalamus was significantly enhanced. This phenomenon may be related to the regulation of secondary nociceptor function by glial cells in the thalamus [24], and

mononuclear/macrophage and T lymphocytes may also be involved in [25]. The results of the above studies indicate that although many brain regions involved in pain perception overlap in their respective functions, their functional connections are regularly changing [26]. The lateral thalamus, S2, and insula may be involved in the perception of pain information, and excessive activation of the posterior parietal and prefrontal cortex (PFC) promotes cognitive attention to nociceptive stimuli. Different parts of the ACC are involved in the adaptation or emotional response to pain. Motion control areas (such as basal ganglia, SMA, cerebellum) are involved in the functional regulation of the pain-suppression system and the avoidance behavior of pain. The amygdala plays an important role in the processing of nociceptive information processing and participates in the regulation of medial prefrontal cortex and spinal cord excitability [27, 28]. Abnormal amygdala function is associated with the formation of neuropathic pain, and the destruction of the amygdala can reduce the incidence of neuropathic pain [29]. Through the study of these regular changes, we can deepen the understanding of pain information processing in the brain, and it also helps to define the neurobiological basis of

the formation of various typical symptoms of neuropathic pain.

The normal perception of pain and avoidance response is an important physiological protection mechanism. The chronic process of pain is the cause of neuropathic pain and the root of refractory disease. In the past, the understanding of the chronic process of neuropathic pain mostly stayed in the sensitization of nociceptors, the axonal buds of sensory neurons, etc. The understanding of advanced central structures and functional changes in the cerebral cortex was still limited. Nevertheless, fMRI is a powerful tool for studying the process of chronic pain. Both clinical observation and animal experiments have found that chronic process of neuropathic pain is associated with large-scale brain function changes and morpho-

Zhang et al. found that the connection strength of the brain's default network in patients with chronic neuropathic pain was significantly different from that of healthy controls, mainly as the weakening connections in characteristic areas of the default network itself (clamping back in the middle, back, inferior parietal lobule) and motor-related areas (superior parietal lobule, auxiliary sports area) [31]. Hubbard et al.'s task-state fMRI study of SNI rats found that when they stimulated the injured limbs in the injury (4 weeks after surgery), the activity of the contralateral somatosensory cortex (S1, S2), the posterolateral nucleus of the thalamus, and the dorsal striatum was enhanced compared with pre-injury and the control groups, whereas the activity of areas associated with painful emotional responses, such as contralateral insula, medial thalamus, and ipsilateral ACC, was inhibited. Restingstate fMRI study showed an enhanced functional connection between the nucleus accumbens (NAc) and the dorsal striatum. At the same time, molecular biology study found that the expression of dopamine 1A receptor and κ opioid receptor in NAc was downregulated. Moreover, the degree of functional compromise is proportional to the degree of downregulation of dopamine receptor gene expression. Inhibition of NAc's functional activity (injection of lidocaine) significantly reduced pain in SNI model animals [32]. At the late stage of injury (20 weeks after surgery), ACC, prefrontal, insula, basal ganglia, and S1 activity was significantly enhanced,

**3.2 Chronic process of neuropathic pain**

logical remodeling [30].

*DOI: http://dx.doi.org/10.5772/intechopen.89200*

**Figure 1.** *Schematic diagram of pain matrix [19].*

#### *The Application of Functional Magnetic Resonance Imaging in Neuropathic Pain DOI: http://dx.doi.org/10.5772/intechopen.89200*

in the opposite side of the injured limb were significantly lower after injury, while the functional connection between the ACC and the posterolateral nucleus of the thalamus was significantly enhanced. This phenomenon may be related to the regulation of secondary nociceptor function by glial cells in the thalamus [24], and mononuclear/macrophage and T lymphocytes may also be involved in [25].

The results of the above studies indicate that although many brain regions involved in pain perception overlap in their respective functions, their functional connections are regularly changing [26]. The lateral thalamus, S2, and insula may be involved in the perception of pain information, and excessive activation of the posterior parietal and prefrontal cortex (PFC) promotes cognitive attention to nociceptive stimuli. Different parts of the ACC are involved in the adaptation or emotional response to pain. Motion control areas (such as basal ganglia, SMA, cerebellum) are involved in the functional regulation of the pain-suppression system and the avoidance behavior of pain. The amygdala plays an important role in the processing of nociceptive information processing and participates in the regulation of medial prefrontal cortex and spinal cord excitability [27, 28]. Abnormal amygdala function is associated with the formation of neuropathic pain, and the destruction of the amygdala can reduce the incidence of neuropathic pain [29]. Through the study of these regular changes, we can deepen the understanding of pain information processing in the brain, and it also helps to define the neurobiological basis of the formation of various typical symptoms of neuropathic pain.

#### **3.2 Chronic process of neuropathic pain**

The normal perception of pain and avoidance response is an important physiological protection mechanism. The chronic process of pain is the cause of neuropathic pain and the root of refractory disease. In the past, the understanding of the chronic process of neuropathic pain mostly stayed in the sensitization of nociceptors, the axonal buds of sensory neurons, etc. The understanding of advanced central structures and functional changes in the cerebral cortex was still limited. Nevertheless, fMRI is a powerful tool for studying the process of chronic pain. Both clinical observation and animal experiments have found that chronic process of neuropathic pain is associated with large-scale brain function changes and morphological remodeling [30].

Zhang et al. found that the connection strength of the brain's default network in patients with chronic neuropathic pain was significantly different from that of healthy controls, mainly as the weakening connections in characteristic areas of the default network itself (clamping back in the middle, back, inferior parietal lobule) and motor-related areas (superior parietal lobule, auxiliary sports area) [31]. Hubbard et al.'s task-state fMRI study of SNI rats found that when they stimulated the injured limbs in the injury (4 weeks after surgery), the activity of the contralateral somatosensory cortex (S1, S2), the posterolateral nucleus of the thalamus, and the dorsal striatum was enhanced compared with pre-injury and the control groups, whereas the activity of areas associated with painful emotional responses, such as contralateral insula, medial thalamus, and ipsilateral ACC, was inhibited. Restingstate fMRI study showed an enhanced functional connection between the nucleus accumbens (NAc) and the dorsal striatum. At the same time, molecular biology study found that the expression of dopamine 1A receptor and κ opioid receptor in NAc was downregulated. Moreover, the degree of functional compromise is proportional to the degree of downregulation of dopamine receptor gene expression. Inhibition of NAc's functional activity (injection of lidocaine) significantly reduced pain in SNI model animals [32]. At the late stage of injury (20 weeks after surgery), ACC, prefrontal, insula, basal ganglia, and S1 activity was significantly enhanced,

*Medical Imaging - Principles and Applications*

treatment level of neuropathic pain [15].

nociceptive pain [23].

system. These spontaneous activities not only consume a lot of energy (60–80% of the total energy consumption of the brain) but also have an inherent spatial pattern called the resting brain functional network, which belongs to the study of functional integration between different brain regions of the brain [15, 16]. Studies have shown that many neurologically related diseases, including neuropathic pain, can have a characteristic impact on this resting brain functional network [17, 18], whereas the study of this brain functional network change is more conducive to the clarification of the disease mechanism and the improvement of the diagnosis and

**3. fMRI study of the central mechanism of neuropathic pain**

**3.1 Pain perception in the brain and the process of information transmission**

fMRI can be used not only in humans but also in animals such as rats and monkeys. Among these, spared nerve injury (SNI) rats can produce persistent and stable symptoms such as hyperalgesia, allodynia, and spontaneous pain, which are commonly used in the study of neuropathic pain. Komaki et al. studied the restingstate fMRI changes in the SNI rat model and studied the node efficiency of some regions of interest and the functional connections between regions of interest by graph theory. They found that the centrality and node efficiency of the S1 region

The brain's perception of pain has been one of the most interesting topics in the field of neuroimaging. Since the initial stage of fMRI technology, there have been a large number of related studies. With the in-depth study of the processing and transmission of pain information, the concept of "pain matrix" has gradually formed, which means that pain is achieved through the division of work between multiple regions of the brain, just like a network structure. The pain matrix mainly includes the thalamus, insula, primary somatosensory cortex (S1), secondary somatosensory cortex (S2), anterior cingulate cortex, periaqueductal gray (PAG), amygdala, etc. (**Figure 1**) [20–22]. Significant functional changes will occur in various brain regions within this pain matrix when suffering from acute

**26**

**Figure 1.**

*Schematic diagram of pain matrix [19].*

while activity in the medial thalamus and PAG areas was inhibited [33]. It can be seen that the formation of neuropathic pain is related to the inhibition of the function of the descending regulation system of pain [34]. ACC activity is inhibited in the early stages of nerve injury, and with the chronic process of pain, ACC activity is gradually enhanced, which may be related to the unpleasant emotional experience of neuropathic pain [29, 35, 36].

A recent resting-state fMRI study found that several major components of the limbic system, hippocampus, amygdala, striatum, and medial prefrontal cortex, are associated with neuropathic pain formation and maintenance [37]. Although SNI rats showed significant changes of activity degree in some brain regions in the early stage of pain, the functional connection between brain regions was not significantly different from that in rats receiving physiological pain stimulation. It can be considered as a normal response of pain matrix to noxious stimuli. However, with the extension of model establishment time, significant changes in the brain functional network occur, and the remodeled brain network has a specific topology. The vast majority of changes in long-term functional connections (97%) occurred within the edge system, and between the edge system and the nociceptive network, while there were no functional connection changes in the nociceptive network [38]. The limbic system neural network adjusts the reward and punishment, appetite, aversion, etc. to generate emotions and behaviors that can adapt to the pain state, so that the brain network gradually adapts to this pain state and the pain tends to be chronic.

#### **3.3 Hyperalgesia and allodynia**

The concept of hyperalgesia and allodynia is often confused, but in fact their formation mechanisms are different. Hyperalgesia refers to a phenomenon in which the pain threshold caused by tissue damage is reduced and the response to noxious stimulation is abnormally enhanced and prolonged. Maihofner et al. found that mechanical pain sensitivity led to abnormal activation of S1, S2, parietal association cortex (PA), insular, superior frontal cortex (SFC), and inferior frontal cortex (IFC) [39], whereas patients with hyperalgesia will have abnormal activation of S1, S2, PA, medial frontal cortex (MFC), ACC, and contralateral SFC and IFC [40]. Zambreanu et al. found that in resting state, patients with no spontaneous pain but hyperalgesia would have abnormal functional activities in multiple regions of the midbrain reticular formation in the brainstem region, namely, nucleus cuneiformis (NCF), rostral superior colliculi (SC), and PAG [41]. The resting-state fMRI of the SNI rat model showed a significant increase in the functional connection between the hippocampus and the striatum, and the intensity was inversely correlated with the mechanical pain threshold of SNI rats.

The clinical manifestation of allodynia is that non-noxious stimuli (such as light touch, mild rubbing, non-noxious cold stimuli) can cause pain, which is a manifestation of "misreading" of somatosensory information. Peripheral sensory nerves are classified into three types: Aβ, Aδ, and C fibers. Normally, the tactile signal is transmitted through the Aβ fiber to the mechanical stimulation zone of the spinal cord. However, in neuropathic pain, Aβ fibers may be abnormally linked to the pain transmission pathway, leading to symptoms such as allodynia [42, 43]. Task-state fMRI is the ideal tool for studying allodynia. Localization and functional connectivity of brain regions associated with allodynia can be achieved by comparing the states of no stimulation, stimulating pain hypersensitivity, and stimulating the same part of the contralateral body. Clinical studies have shown that somatic stimulation signals are amplified by the thalamic and thalamic-parietal circuits, causing excessive activation of the lateral pain sensory system and attention network (posterior parietal lobe). Unlike the response of the thalamic-parietal system,

**29**

*The Application of Functional Magnetic Resonance Imaging in Neuropathic Pain*

the role of ACC and medial prefrontal lobe in allodynia is more complicated. First, different parts of ACC respond differently to nociceptive stimuli. The central part of ACC was significantly activated during peripheral nerve injury-induced allodynia, whereas there was no significant change in the degree of activation in pain hypersensitivity caused by Wallenberg's lateral infarction. This may be due to the fact that the central part of ACC receives signals from the spinal thalamus bundle and Wallenberg's syndrome causes damage to the spinal thalamic bundle; nevertheless peripheral neuropathic pain does not. However, in most neuropathic pains, whether peripheral or central, the axons of ACC are shown to have a reduced degree of activation when allodynia is induced. The brain regions activated by different stimulating components are also different. Cold and mechanical stimulation can cause significant activation of the prefrontal cortex (PFC) and basal ganglia, and the degree of activation is related to the functional status of ACC [33], and insula

A similar phenomenon was also found in fMRI study of the SNI animal model by Komaki et al. According to the physiological characteristics of each nerve fiber, the Aβ fiber can be selectively excited by a current of 2000 Hz, 2.2 mA. When the hind paws of normal rats were administered with 2000 Hz, 2.2 mA DC stimulation, it caused only significant activation of the S1 region; however, when the same intensity of current was used to stimulate the pain-sensing hypersensitivity zone of SNI rats, it was found that not only S1, ACC, and thalamus were also significant

Spontaneous pain is generally difficult to study with fMRI. Because it is difficult

to obtain a comparison of pain and painlessness in the same patient under clinical conditions, this type of clinical study is still rare. A few studies include spinal cordectomy for cancer pain [46], persistent neuropathic pain before and after local anesthesia [47], and central pain before and after motor cortex stimulation [48]. A common finding of these studies is the reduction in local blood flow in the thalamus, that is, the decrease in thalamic activity, which is restored after pain relief. This depression of lateral thalamic function is found both in peripheral and central

In the study of trigeminal neuralgia and sphenopalatine neuralgia with typical spontaneous pain characteristics, it was found that the functional connection between the ipsilateral hypothalamus and the contralateral S1 and the ipsilateral wedge was weakened during the pain attack and remission. However, compared with the pain relief period, the functional connection between the hypothalamus and the S1, anterior wedge, and cerebellum is much less during

**3.5 Characteristic changes of brain function in different types of neuropathic**

In addition to the common changes in neuropathic pains, different types of neuropathic pain also have characteristic changes. By studying these changes, it is helpful to clarify its pathogenesis, and it is also possible to screen out specific neuro-

Neurovascular compression has long been considered to be the cause of primary trigeminal neuralgia. The 3D TOF MRA and 3D FIESTA sequences can clearly show the positional relationship between the nerves and blood vessels in the root enter zone (REZ) [52–54]. However, nerve compression is not the direct

imaging markers for the diagnosis and evaluation of diseases (**Figure 2**).

*DOI: http://dx.doi.org/10.5772/intechopen.89200*

signal significantly enhances during stimulation [44].

activated [45].

**3.4 Spontaneous pain**

neuropathic pain [49].

the onset of pain [50].

**pain**

*The Application of Functional Magnetic Resonance Imaging in Neuropathic Pain DOI: http://dx.doi.org/10.5772/intechopen.89200*

the role of ACC and medial prefrontal lobe in allodynia is more complicated. First, different parts of ACC respond differently to nociceptive stimuli. The central part of ACC was significantly activated during peripheral nerve injury-induced allodynia, whereas there was no significant change in the degree of activation in pain hypersensitivity caused by Wallenberg's lateral infarction. This may be due to the fact that the central part of ACC receives signals from the spinal thalamus bundle and Wallenberg's syndrome causes damage to the spinal thalamic bundle; nevertheless peripheral neuropathic pain does not. However, in most neuropathic pains, whether peripheral or central, the axons of ACC are shown to have a reduced degree of activation when allodynia is induced. The brain regions activated by different stimulating components are also different. Cold and mechanical stimulation can cause significant activation of the prefrontal cortex (PFC) and basal ganglia, and the degree of activation is related to the functional status of ACC [33], and insula signal significantly enhances during stimulation [44].

A similar phenomenon was also found in fMRI study of the SNI animal model by Komaki et al. According to the physiological characteristics of each nerve fiber, the Aβ fiber can be selectively excited by a current of 2000 Hz, 2.2 mA. When the hind paws of normal rats were administered with 2000 Hz, 2.2 mA DC stimulation, it caused only significant activation of the S1 region; however, when the same intensity of current was used to stimulate the pain-sensing hypersensitivity zone of SNI rats, it was found that not only S1, ACC, and thalamus were also significant activated [45].

#### **3.4 Spontaneous pain**

*Medical Imaging - Principles and Applications*

of neuropathic pain [29, 35, 36].

**3.3 Hyperalgesia and allodynia**

the mechanical pain threshold of SNI rats.

while activity in the medial thalamus and PAG areas was inhibited [33]. It can be seen that the formation of neuropathic pain is related to the inhibition of the function of the descending regulation system of pain [34]. ACC activity is inhibited in the early stages of nerve injury, and with the chronic process of pain, ACC activity is gradually enhanced, which may be related to the unpleasant emotional experience

A recent resting-state fMRI study found that several major components of the limbic system, hippocampus, amygdala, striatum, and medial prefrontal cortex, are associated with neuropathic pain formation and maintenance [37]. Although SNI rats showed significant changes of activity degree in some brain regions in the early stage of pain, the functional connection between brain regions was not significantly different from that in rats receiving physiological pain stimulation. It can be considered as a normal response of pain matrix to noxious stimuli. However, with the extension of model establishment time, significant changes in the brain functional network occur, and the remodeled brain network has a specific topology. The vast majority of changes in long-term functional connections (97%) occurred within the edge system, and between the edge system and the nociceptive network, while there were no functional connection changes in the nociceptive network [38]. The limbic system neural network adjusts the reward and punishment, appetite, aversion, etc. to generate emotions and behaviors that can adapt to the pain state, so that the brain

network gradually adapts to this pain state and the pain tends to be chronic.

The concept of hyperalgesia and allodynia is often confused, but in fact their formation mechanisms are different. Hyperalgesia refers to a phenomenon in which the pain threshold caused by tissue damage is reduced and the response to noxious stimulation is abnormally enhanced and prolonged. Maihofner et al. found that mechanical pain sensitivity led to abnormal activation of S1, S2, parietal association cortex (PA), insular, superior frontal cortex (SFC), and inferior frontal cortex (IFC) [39], whereas patients with hyperalgesia will have abnormal activation of S1, S2, PA, medial frontal cortex (MFC), ACC, and contralateral SFC and IFC [40]. Zambreanu et al. found that in resting state, patients with no spontaneous pain but hyperalgesia would have abnormal functional activities in multiple regions of the midbrain reticular formation in the brainstem region, namely, nucleus cuneiformis (NCF), rostral superior colliculi (SC), and PAG [41]. The resting-state fMRI of the SNI rat model showed a significant increase in the functional connection between the hippocampus and the striatum, and the intensity was inversely correlated with

The clinical manifestation of allodynia is that non-noxious stimuli (such as light touch, mild rubbing, non-noxious cold stimuli) can cause pain, which is a manifestation of "misreading" of somatosensory information. Peripheral sensory nerves are classified into three types: Aβ, Aδ, and C fibers. Normally, the tactile signal is transmitted through the Aβ fiber to the mechanical stimulation zone of the spinal cord. However, in neuropathic pain, Aβ fibers may be abnormally linked to the pain transmission pathway, leading to symptoms such as allodynia [42, 43]. Task-state fMRI is the ideal tool for studying allodynia. Localization and functional connectivity of brain regions associated with allodynia can be achieved by comparing the states of no stimulation, stimulating pain hypersensitivity, and stimulating the same part of the contralateral body. Clinical studies have shown that somatic stimulation signals are amplified by the thalamic and thalamic-parietal circuits, causing excessive activation of the lateral pain sensory system and attention network (posterior parietal lobe). Unlike the response of the thalamic-parietal system,

**28**

Spontaneous pain is generally difficult to study with fMRI. Because it is difficult to obtain a comparison of pain and painlessness in the same patient under clinical conditions, this type of clinical study is still rare. A few studies include spinal cordectomy for cancer pain [46], persistent neuropathic pain before and after local anesthesia [47], and central pain before and after motor cortex stimulation [48]. A common finding of these studies is the reduction in local blood flow in the thalamus, that is, the decrease in thalamic activity, which is restored after pain relief. This depression of lateral thalamic function is found both in peripheral and central neuropathic pain [49].

In the study of trigeminal neuralgia and sphenopalatine neuralgia with typical spontaneous pain characteristics, it was found that the functional connection between the ipsilateral hypothalamus and the contralateral S1 and the ipsilateral wedge was weakened during the pain attack and remission. However, compared with the pain relief period, the functional connection between the hypothalamus and the S1, anterior wedge, and cerebellum is much less during the onset of pain [50].

#### **3.5 Characteristic changes of brain function in different types of neuropathic pain**

In addition to the common changes in neuropathic pains, different types of neuropathic pain also have characteristic changes. By studying these changes, it is helpful to clarify its pathogenesis, and it is also possible to screen out specific neuroimaging markers for the diagnosis and evaluation of diseases (**Figure 2**).

Neurovascular compression has long been considered to be the cause of primary trigeminal neuralgia. The 3D TOF MRA and 3D FIESTA sequences can clearly show the positional relationship between the nerves and blood vessels in the root enter zone (REZ) [52–54]. However, nerve compression is not the direct

**Figure 2.** *Characteristic ROIs of trigeminal neuralgia [51].*

cause of trigeminal neuralgia. There are also many individuals in the population who have neurological compression but no clinical symptoms of trigeminal neuralgia [55]. Lin et al. found that there were no damage and functional changes in the white matter fibrin myelin or axons in patients with nerve compression but asymptomatic in their DTI study [56]. DeSouza found that the fractional anisotropy (FA) of REZ in patients with trigeminal neuralgia was 22% lower than that of the healthy side and 27% lower than that of healthy controls. Other white matter microstructural measurements of patients such as radial diffusivity (RD), axial diffusivity (AD), and mean diffusivity (MD) were higher than those of healthy controls, indicating demyelinating lesions without axonal injury may be an important factor in the pathogenesis of trigeminal neuralgia [57]. They then compared changes in white matter fiber connections in the REZ region before and after microvascular decompression. It is found that FA, MD, RD, and AD all recovered in the normal direction after treatment and the degree of recovery was proportional to the degree of pain relief [58]. It is suggested that nerve compression by blood vessels is only the inducement of trigeminal neuralgia, and the occurrence and maintenance of pain may be related to abnormal white matter fibers at REZ.

The task-state fMRI conducted by Moisset initially explored the effect of the trigger point of patients with trigeminal neuralgia on brain functional activity. They found that when the pain was onset, spinal trigeminal nucleus (SpV), thalamus, S1, S2, ACC, insular, premotor cortex, motor cortex, frontal nucleus, putamen, etc. were clearly activated in patients with trigger points. In patients without trigger points, the brain regions abovementioned were also activated except SpV, brainstem, and ACC. It can be considered that the structural or functional changes at these three regions may be related to the formation of the trigger point [59]. There are still a small number of studies on resting-state fMRI in patients with trigeminal neuralgia. Athinoula et al. found a weaker functional connection between amygdala and insula and S2 in a comparative study of migraine and trigeminal neuralgia [60].

**31**

*The Application of Functional Magnetic Resonance Imaging in Neuropathic Pain*

Wang et al.'s study showed a significant reduction in local consistency activity in the amygdala, hippocampus, and cerebellum in patients with trigeminal neuralgia [61]. The cause of residual limb pain or phantom pain after amputation has been controversial. Some people think that the sudden introduction of nerve afferents from the limbs will cause the expansion or displacement of the sensory cortex, which is called the incompatible remodeling of the cortex [62, 63]. This theory has been proposed for more than 20 years, but it is difficult to be verified before the rise of fMRI technology [64, 65]. Lotze et al. conducted a task-state fMRI study and found that cortical remodeling in patients with phantom limb pain is not limited to S1 but also includes some auxiliary sports areas [66]. For example, when the lips move, the activation degree of the representative region of the broken limb in M1 is also significantly enhanced, indicating that the lip represents region expands and displaces to the cortex of broken limb [67]. Moreover, the degree of displacement of the cortex to the representative area of the isolated limb is positively correlated with the degree of pain [68]. After mirror therapy, cortical displacement can be partially

Cauda et al. found that the functional connections between the ventral posterior

nucleus (VP) and the medial dorsal nucleus (MD) and the cerebral cortex were weakened in patients with diabetic peripheral neuropathy [70]. Cifre observed a weakened connection between resting thalamus and insula in patients with fibromyalgia [71]. In patients with postherpetic neuralgia, the reward circuit consisting of the striatum, prefrontal cortex, amygdala, and hippocampus and the circuits composed of striatum, thalamus, and insular leaves have very close functional connection [72]. With the advancement of fMRI data analysis methods and machine learning techniques, these seemingly cluttered functional connectivity features are highly likely to be used as neuroimaging markers for the diagnosis of neuropathic pain. In other studies of nervous system diseases, some scholars have successfully used fMRI technology to construct a resting brain functional network model for patients with Alzheimer's disease. And with the using of multimode variable analysis method, the sensitivity and specificity of early screening of Alzheimer's disease

**4. The value of fMRI in the clinical treatment of neuropathic pain**

In the past, the examination and evaluation of neuropathic pain relied mainly on medical history, symptoms, and signs and lacked tools for quantitative assessment. Even with the von Frey fiber test, the same stimulation site and strength often result in a lack of consistency [74]. The reason is that the activity of the nerve is disturbed by many factors. Inclusion of peripheral and central sensitization, genetics, cognition, and emotional response during testing will affect the signal transmission of noxious stimuli to the painful sensation area of the brain [75]. In addition, the subjective tendencies of participants and examiners may also cause serious bias. However, fMRI technology may provide us with qualitative and quantitative observations [76]. For example, PoCG's corresponding somatosensory representative area can help clinicians determine the pain site, and the degree of activation can also reflect the intensity of pain [51]. The local activity consistency of rostralanterior part of ACC in patients with postherpetic neuralgia was significantly correlated

fMRI can also be used to observe the effects of treatments on brain function or to assess the therapeutic effects of neuropathic pain. For example, after the application of opioids to achieve pain relief, an increase in ACC activation can be found. Thalamic electrical stimulation can significantly increase the activation of

*DOI: http://dx.doi.org/10.5772/intechopen.89200*

restored, and the pain intensity can be reduced [69].

in the general population is up to 90% [73].

with the anxiety and depression scores [77, 78].

#### *The Application of Functional Magnetic Resonance Imaging in Neuropathic Pain DOI: http://dx.doi.org/10.5772/intechopen.89200*

Wang et al.'s study showed a significant reduction in local consistency activity in the amygdala, hippocampus, and cerebellum in patients with trigeminal neuralgia [61].

The cause of residual limb pain or phantom pain after amputation has been controversial. Some people think that the sudden introduction of nerve afferents from the limbs will cause the expansion or displacement of the sensory cortex, which is called the incompatible remodeling of the cortex [62, 63]. This theory has been proposed for more than 20 years, but it is difficult to be verified before the rise of fMRI technology [64, 65]. Lotze et al. conducted a task-state fMRI study and found that cortical remodeling in patients with phantom limb pain is not limited to S1 but also includes some auxiliary sports areas [66]. For example, when the lips move, the activation degree of the representative region of the broken limb in M1 is also significantly enhanced, indicating that the lip represents region expands and displaces to the cortex of broken limb [67]. Moreover, the degree of displacement of the cortex to the representative area of the isolated limb is positively correlated with the degree of pain [68]. After mirror therapy, cortical displacement can be partially restored, and the pain intensity can be reduced [69].

Cauda et al. found that the functional connections between the ventral posterior nucleus (VP) and the medial dorsal nucleus (MD) and the cerebral cortex were weakened in patients with diabetic peripheral neuropathy [70]. Cifre observed a weakened connection between resting thalamus and insula in patients with fibromyalgia [71]. In patients with postherpetic neuralgia, the reward circuit consisting of the striatum, prefrontal cortex, amygdala, and hippocampus and the circuits composed of striatum, thalamus, and insular leaves have very close functional connection [72]. With the advancement of fMRI data analysis methods and machine learning techniques, these seemingly cluttered functional connectivity features are highly likely to be used as neuroimaging markers for the diagnosis of neuropathic pain. In other studies of nervous system diseases, some scholars have successfully used fMRI technology to construct a resting brain functional network model for patients with Alzheimer's disease. And with the using of multimode variable analysis method, the sensitivity and specificity of early screening of Alzheimer's disease in the general population is up to 90% [73].

## **4. The value of fMRI in the clinical treatment of neuropathic pain**

In the past, the examination and evaluation of neuropathic pain relied mainly on medical history, symptoms, and signs and lacked tools for quantitative assessment. Even with the von Frey fiber test, the same stimulation site and strength often result in a lack of consistency [74]. The reason is that the activity of the nerve is disturbed by many factors. Inclusion of peripheral and central sensitization, genetics, cognition, and emotional response during testing will affect the signal transmission of noxious stimuli to the painful sensation area of the brain [75]. In addition, the subjective tendencies of participants and examiners may also cause serious bias. However, fMRI technology may provide us with qualitative and quantitative observations [76]. For example, PoCG's corresponding somatosensory representative area can help clinicians determine the pain site, and the degree of activation can also reflect the intensity of pain [51]. The local activity consistency of rostralanterior part of ACC in patients with postherpetic neuralgia was significantly correlated with the anxiety and depression scores [77, 78].

fMRI can also be used to observe the effects of treatments on brain function or to assess the therapeutic effects of neuropathic pain. For example, after the application of opioids to achieve pain relief, an increase in ACC activation can be found. Thalamic electrical stimulation can significantly increase the activation of

*Medical Imaging - Principles and Applications*

*Characteristic ROIs of trigeminal neuralgia [51].*

cause of trigeminal neuralgia. There are also many individuals in the population who have neurological compression but no clinical symptoms of trigeminal neuralgia [55]. Lin et al. found that there were no damage and functional changes in the white matter fibrin myelin or axons in patients with nerve compression but asymptomatic in their DTI study [56]. DeSouza found that the fractional anisotropy (FA) of REZ in patients with trigeminal neuralgia was 22% lower than that of the healthy side and 27% lower than that of healthy controls. Other white matter microstructural measurements of patients such as radial diffusivity (RD), axial diffusivity (AD), and mean diffusivity (MD) were higher than those of healthy controls, indicating demyelinating lesions without axonal injury may be an important factor in the pathogenesis of trigeminal neuralgia [57]. They then compared changes in white matter fiber connections in the REZ region before and after microvascular decompression. It is found that FA, MD, RD, and AD all recovered in the normal direction after treatment and the degree of recovery was proportional to the degree of pain relief [58]. It is suggested that nerve compression by blood vessels is only the inducement of trigeminal neuralgia, and the occurrence and maintenance of pain may be related to abnormal white matter

The task-state fMRI conducted by Moisset initially explored the effect of the trigger point of patients with trigeminal neuralgia on brain functional activity. They found that when the pain was onset, spinal trigeminal nucleus (SpV), thalamus, S1, S2, ACC, insular, premotor cortex, motor cortex, frontal nucleus, putamen, etc. were clearly activated in patients with trigger points. In patients without trigger points, the brain regions abovementioned were also activated except SpV, brainstem, and ACC. It can be considered that the structural or functional changes at these three regions may be related to the formation of the trigger point [59]. There are still a small number of studies on resting-state fMRI in patients with trigeminal neuralgia. Athinoula et al. found a weaker functional connection between amygdala and insula and S2 in a comparative study of migraine and trigeminal neuralgia [60].

**30**

fibers at REZ.

**Figure 2.**

the rostralanterior part of ACC and the basal part of frontal cortex. These two sites are generally in a state of functional inhibition in patients with chronic neuropathic pain. A study of drug therapy for trigeminal neuralgia has shown that lamotrigine can reduce the pain level of patients by reducing the excitability of the prefrontal, parietal, and temporal lobe and inhibit allodynia [79].

As a method for treating neuropathic pain, neuromodulation technology has been applied for decades. However, whether it is invasive spinal cord electrical stimulation, deep brain stimulation, or noninvasive transcranial magnetic stimulation, there is a problem of inefficiency. The reason may be due to individual differences in the effects of disease on the brain functional network. fMRI can accurately capture the brain regions with abnormal functions and combine the analysis of functional connections in the brain to determine the regions of interest for neuromodulation, which is used to guide the target area of transcranial magnetic stimulation or the placement of epidural stimulation electrodes [80, 81]. It is helpful to improve the efficacy of neuromodulation techniques.

## **5. Limitations and future direction of fMRI research in NP**

Most of the previous studies were limited by the fMRI data analysis method. They can only analyze for a single factor, or only focus on brain regions or functional connections with significant differences, while ignoring the complexity and synergy of the brain function network as a whole structure. And the conclusions drawn lack clinical utility and provide limited assistance in the diagnosis and prediction of diseases.

In recent years, with the increasing maturity of machine learning technology, machine learning and pattern recognition technology are being used more and more for fMRI data analysis. By using machine learning technology to process the massive characteristic data generated by fMRI, multiple dimensions such as gray matter volume, diffusion of water molecules, and functional connectivity can be used simultaneously [82]. In the future, this research method of overall analysis of brain function networks can realize pattern recognition of different disease states. It can make fMRI technology better serve the clinic and provide assistance for the diagnosis and prognosis analysis of neuropathic pain.

## **6. Conclusion**

A large number of brain regions associated with neuropathic pain have been discovered by fMRI technology, and there has been some data accumulation for changes in functional connectivity between brain regions. However, how to analyze and process these fMRI data and make a reasonable explanation for better understanding the underlying disease mechanism as well as treatment improvement is the key to further expand the value of fMRI application in the future.

**33**

**Author details**

Beijing, China

Zhi Dou\* and Liqiang Yang

\*Address all correspondence to<EMAIL_ADDRESS>
provided the original work is properly cited.

Department of Pain Management, Xuanwu Hospital, Capital Medical University,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*The Application of Functional Magnetic Resonance Imaging in Neuropathic Pain*

*DOI: http://dx.doi.org/10.5772/intechopen.89200*

*The Application of Functional Magnetic Resonance Imaging in Neuropathic Pain DOI: http://dx.doi.org/10.5772/intechopen.89200*

## **Author details**

*Medical Imaging - Principles and Applications*

parietal, and temporal lobe and inhibit allodynia [79].

improve the efficacy of neuromodulation techniques.

diagnosis and prognosis analysis of neuropathic pain.

key to further expand the value of fMRI application in the future.

prediction of diseases.

**6. Conclusion**

**5. Limitations and future direction of fMRI research in NP**

the rostralanterior part of ACC and the basal part of frontal cortex. These two sites are generally in a state of functional inhibition in patients with chronic neuropathic pain. A study of drug therapy for trigeminal neuralgia has shown that lamotrigine can reduce the pain level of patients by reducing the excitability of the prefrontal,

As a method for treating neuropathic pain, neuromodulation technology has been applied for decades. However, whether it is invasive spinal cord electrical stimulation, deep brain stimulation, or noninvasive transcranial magnetic stimulation, there is a problem of inefficiency. The reason may be due to individual differences in the effects of disease on the brain functional network. fMRI can accurately capture the brain regions with abnormal functions and combine the analysis of functional connections in the brain to determine the regions of interest for neuromodulation, which is used to guide the target area of transcranial magnetic stimulation or the placement of epidural stimulation electrodes [80, 81]. It is helpful to

Most of the previous studies were limited by the fMRI data analysis method. They can only analyze for a single factor, or only focus on brain regions or functional connections with significant differences, while ignoring the complexity and synergy of the brain function network as a whole structure. And the conclusions drawn lack clinical utility and provide limited assistance in the diagnosis and

In recent years, with the increasing maturity of machine learning technology, machine learning and pattern recognition technology are being used more and more for fMRI data analysis. By using machine learning technology to process the massive characteristic data generated by fMRI, multiple dimensions such as gray matter volume, diffusion of water molecules, and functional connectivity can be used simultaneously [82]. In the future, this research method of overall analysis of brain function networks can realize pattern recognition of different disease states. It can make fMRI technology better serve the clinic and provide assistance for the

A large number of brain regions associated with neuropathic pain have been discovered by fMRI technology, and there has been some data accumulation for changes in functional connectivity between brain regions. However, how to analyze and process these fMRI data and make a reasonable explanation for better understanding the underlying disease mechanism as well as treatment improvement is the

**32**

Zhi Dou\* and Liqiang Yang Department of Pain Management, Xuanwu Hospital, Capital Medical University, Beijing, China

\*Address all correspondence to<EMAIL_ADDRESS>
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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**34**

*Medical Imaging - Principles and Applications*

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[10] Chen F, Chen F, Shang Z, et al. White matter microstructure degenerates in patients with

postherpetic neuralgia. Neuroscience

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[15] Lu H, Yuping N. When neuroscience

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2002;**357**(1424):1003-1037

1990;**87**(24):9868-9872

Journal. 1993;**64**(3):803-812

2014;**13**(12):1292-1296

meets network thinking: A new

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Chinese Journal of Biomedical Engineering. 2011;**30**(1):6-10

Letters. 2017;**656**:152-157

1990;**14**(1):68-78

1-7

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Pharmacology. 2001;**429**(1-3):1-11

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2004;**557**(1-3):125-128

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1999;**83**(2):259-273

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reorganization of the rat limbic system following neuropathic injury. Scientific

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2011;**37**:25-63

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Notes. 2014;**7**:739

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2004;**19**(12):3211-3218

Pain. 2005;**114**(3):397-407

996-1006

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**36**

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[52] Montano N, Conforti G, Di Bonaventura R, et al. Advances in diagnosis and treatment of trigeminal neuralgia. Therapeutics and Clinical Risk Management. 2015;**11**:289-299

[53] Maarbjerg S, Wolfram F, Gozalov A, et al. Significance of neurovascular contact in classical trigeminal neuralgia. Brain. 2015;**138**(2):311-319

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Section 2

Nuclear Medicine PET/CT

Imaging and Applications

39

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Section 2


**5. Research on structural thermal modals**

Aeroelastic problems have been the key in vehicle design, which has gradually become a notable obstacle to better vehicle performance with the development of vehicle. Under the effect of aeroheating, the temperature rise of a structure leads to variation in physical parameters of its material. Also, non-uniform temperature field within the structure causes prominent temperature gradients, which produces subsidiary thermal deformation and thermal stress, greatly alters the structural rigidity and thus changes the natural vibration performance of the structure. The variation of natural vibration performance due to thermal load significantly affects the trim, flutter and control characteristics of the vehicle and these effects tend to be unfavorable.

*5.2.1. Thermal modal analysis along the static trajectory*

prominent.

the case with zero wing AOA and that with non-zero wing AOA.

**Figure 16.** The thermal modal analysis strategy based on multi-physics coupling integration.

Set the total cruise flight time to 100 s and the flight environment to be 30 km within standard atmosphere. Calorically perfect gas model is adopted. The thermal modal is analyzed in both

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**Figure 18** presents the first six modal shapes of the hypersonic wing with zero AOA at time *t* = 100 s. It can be seen that the modal shape of each order at time *t* = 100 s has local changes to varying degrees compared to the modal shape at initial time. The first four modes generally retain the original modal shapes, while the fifth and the sixth change a lot in their modal shapes with the trend towards bending-torsion coupling that usually leads to vibration of the wing. Therefore, sustained aeroheating has effect on modal shapes of higher order more easily for wings that are fixed-supported at root. As AOA is gradually increased, the aerodynamic forces and thermal load imposed on lower wing surface becomes larger than those imposed on the upper wing surface, which changes the thermodynamic state within the wing and thus influences the natural vibration characteristics of the wing. **Figure 19** presents the first six modal shapes of the hypersonic wing at time *t* = 100 s under the AOA of 10 deg. It can be seen from the figure that compared with the modal shapes in the zero AOA case, modal shapes of all orders have little change, which indicates that modal shape is not very sensitive to variation of AOA. It is necessary to point out that the analysis above only involves flight time of 100 s. Actually, time of sustained flight of vehicle is an influencing parameter of great importance. The fluid-thermal-structural coupling will finally reach thermodynamic equilibrium. In the process, the modal frequency and modal shape of the wing will continue to change and at the same time the effect of AOA on modal frequency and modal shape will become more

## **5.1. Thermal modal analysis strategy based on multi-physics coupling**

The thermal modal analysis strategy based on multi-physics coupling integration method is shown in **Figure 16**. It can be summarized as: (1) employing the multi-physics coupling integration method HyCCD based on CFD, CTD and CSD, transient temperature field and stress field within the solid structure along its static or dynamic trajectory are obtained by doing hypersonic fluid-thermal-structural coupling analysis according to the coupling analysis strategy described above and (2) thermal rigidity matrix can be constructed by taking parameters of thermodynamic state within the solid structure of each time point for coupling calculation. Then, the thermal modal characteristics at each time point for coupling calculation of the solid structure are obtained by solving the generalized eigenvalue problem by means of conventional mode analysis method.

## **5.2. Thermal modal characteristic analysis of a typical hypersonic wing**

The thermal modal characteristic analysis under sustained flight of a typical three-dimensional low-aspect-ratio hypersonic wing model is presented in **Figure 17**. It is a symmetrical double edge with the leading edge blunted and small thickness of the trailing edge retained to avoid sharp edges. The material has mass density of 4539 kg/m3 and Poisson number of 0.32 with all the other physical properties varying with temperature. The temperature at initial time is 300 K with zero initial stress and the reference temperature of thermal stress is 300 K. Fixed support is adopted at the wing root.

Modeling and Analysis of Fluid-Thermal-Structure Coupling Problems for Hypersonic Vehicles http://dx.doi.org/10.5772/intechopen.70658 127

**Figure 16.** The thermal modal analysis strategy based on multi-physics coupling integration.

#### *5.2.1. Thermal modal analysis along the static trajectory*

In general, the aerodynamic force and thermal load have a great impact on the inlet cowl leading edge, which suffices to cause thermal and dynamic damage to thermal protection structures despite the short imposing time in actual flight, presenting severe challenge for material selection and structure design of thermal protection. As for the air-breathing hypersonic vehicles, the impact of shock interaction is common in the surrounding flowfield. Therefore, thermal protection design of local leading edges (structures such as the tail and rudders) on the windward side wrapped by the nose shocks should be done carefully besides the nose and

Aeroelastic problems have been the key in vehicle design, which has gradually become a notable obstacle to better vehicle performance with the development of vehicle. Under the effect of aeroheating, the temperature rise of a structure leads to variation in physical parameters of its material. Also, non-uniform temperature field within the structure causes prominent temperature gradients, which produces subsidiary thermal deformation and thermal stress, greatly alters the structural rigidity and thus changes the natural vibration performance of the structure. The variation of natural vibration performance due to thermal load significantly affects the trim, flutter and control characteristics of the vehicle and these effects

The thermal modal analysis strategy based on multi-physics coupling integration method is shown in **Figure 16**. It can be summarized as: (1) employing the multi-physics coupling integration method HyCCD based on CFD, CTD and CSD, transient temperature field and stress field within the solid structure along its static or dynamic trajectory are obtained by doing hypersonic fluid-thermal-structural coupling analysis according to the coupling analysis strategy described above and (2) thermal rigidity matrix can be constructed by taking parameters of thermodynamic state within the solid structure of each time point for coupling calculation. Then, the thermal modal characteristics at each time point for coupling calculation of the solid structure are obtained by solving the generalized eigenvalue problem by

The thermal modal characteristic analysis under sustained flight of a typical three-dimensional low-aspect-ratio hypersonic wing model is presented in **Figure 17**. It is a symmetrical double edge with the leading edge blunted and small thickness of the trailing edge retained

0.32 with all the other physical properties varying with temperature. The temperature at initial time is 300 K with zero initial stress and the reference temperature of thermal stress is

and Poisson number of

**5.1. Thermal modal analysis strategy based on multi-physics coupling**

**5.2. Thermal modal characteristic analysis of a typical hypersonic wing**

to avoid sharp edges. The material has mass density of 4539 kg/m3

the engine cowl leading edge.

126 Advances in Some Hypersonic Vehicles Technologies

tend to be unfavorable.

**5. Research on structural thermal modals**

means of conventional mode analysis method.

300 K. Fixed support is adopted at the wing root.

Set the total cruise flight time to 100 s and the flight environment to be 30 km within standard atmosphere. Calorically perfect gas model is adopted. The thermal modal is analyzed in both the case with zero wing AOA and that with non-zero wing AOA.

**Figure 18** presents the first six modal shapes of the hypersonic wing with zero AOA at time *t* = 100 s. It can be seen that the modal shape of each order at time *t* = 100 s has local changes to varying degrees compared to the modal shape at initial time. The first four modes generally retain the original modal shapes, while the fifth and the sixth change a lot in their modal shapes with the trend towards bending-torsion coupling that usually leads to vibration of the wing. Therefore, sustained aeroheating has effect on modal shapes of higher order more easily for wings that are fixed-supported at root. As AOA is gradually increased, the aerodynamic forces and thermal load imposed on lower wing surface becomes larger than those imposed on the upper wing surface, which changes the thermodynamic state within the wing and thus influences the natural vibration characteristics of the wing. **Figure 19** presents the first six modal shapes of the hypersonic wing at time *t* = 100 s under the AOA of 10 deg. It can be seen from the figure that compared with the modal shapes in the zero AOA case, modal shapes of all orders have little change, which indicates that modal shape is not very sensitive to variation of AOA. It is necessary to point out that the analysis above only involves flight time of 100 s. Actually, time of sustained flight of vehicle is an influencing parameter of great importance. The fluid-thermal-structural coupling will finally reach thermodynamic equilibrium. In the process, the modal frequency and modal shape of the wing will continue to change and at the same time the effect of AOA on modal frequency and modal shape will become more prominent.

**Figure 17.** The three-dimensional low-aspect-ratio hypersonic wing.

*5.2.2. Thermal modal analysis along the dynamic trajectory*

6, *ω*6 = 107.44 Hz.

variation with dynamic variation of the flight trajectory.

A simple flight trajectory is assumed referring to **Figure 5** to further analyze variation of thermal modal characteristics in complicated flight. **Figure 20** presents that dynamic variation of flight trajectory leads to the variation of the thermodynamic state within the wing as well as its modal frequency. From the initial time to the end of cruise, the modal frequency of each order gradually declines; the decline tends to be gentle and then shows a recovery as the vehicle descends. It turns out that the first modals have little change along the flight trajectory, generally retaining the original modal shapes while major variation takes place in the fifth/sixth-order mode, especially the sixth-order mode in which case bending-torsion coupling tends to occur. If the climbing, cruise and descending phase, especially cruise, last long enough, the modal frequency and modal shape of each order can have more significant

**Figure 19.** The first six modals of the hypersonic wing at time *t* = 100 s under the AOA of 10 deg. (a) modal 1, *ω*1 = 18.85 Hz, (b) modal 2, *ω*2 = 40.04 Hz, (c) modal 3, *ω*3 = 63.44 Hz, (d) modal 4, *ω*4 = 77.84 Hz, (e) modal 5, *ω*5 = 97.63 Hz, and (f) modal

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The modal frequencies of each order show a downward trend over time and the decreasing rates vary from mode to mode. Modes of all orders get nearer or farther one another to varying degrees over time, which might impose severe impacts on the natural vibration characteristics of the structure. The results indicate that sustained aeroheating has effect more easily on modal shapes of higher order. Therefore, for hypersonic vehicles with large thin-walled

**Figure 18.** The first six modals of the hypersonic wing with zero AOA at time *t* = 100 s. (a) modal 1, *ω*1 = 18.98 Hz, (b) modal 2, *ω*2 = 40.28 Hz, (c) modal 3, *ω*3 = 63.74 Hz, (d) modal 4, *ω*4 = 78.41 Hz, (e) modal 5, *ω*5 = 98.58 Hz, and (f) modal 6, *ω*6 = 108.24 Hz.

Modeling and Analysis of Fluid-Thermal-Structure Coupling Problems for Hypersonic Vehicles http://dx.doi.org/10.5772/intechopen.70658 129

**Figure 19.** The first six modals of the hypersonic wing at time *t* = 100 s under the AOA of 10 deg. (a) modal 1, *ω*1 = 18.85 Hz, (b) modal 2, *ω*2 = 40.04 Hz, (c) modal 3, *ω*3 = 63.44 Hz, (d) modal 4, *ω*4 = 77.84 Hz, (e) modal 5, *ω*5 = 97.63 Hz, and (f) modal 6, *ω*6 = 107.44 Hz.

#### *5.2.2. Thermal modal analysis along the dynamic trajectory*

**Figure 18.** The first six modals of the hypersonic wing with zero AOA at time *t* = 100 s. (a) modal 1, *ω*1 = 18.98 Hz, (b) modal 2, *ω*2 = 40.28 Hz, (c) modal 3, *ω*3 = 63.74 Hz, (d) modal 4, *ω*4 = 78.41 Hz, (e) modal 5, *ω*5 = 98.58 Hz, and (f)

modal 6, *ω*6 = 108.24 Hz.

**Figure 17.** The three-dimensional low-aspect-ratio hypersonic wing.

128 Advances in Some Hypersonic Vehicles Technologies

A simple flight trajectory is assumed referring to **Figure 5** to further analyze variation of thermal modal characteristics in complicated flight. **Figure 20** presents that dynamic variation of flight trajectory leads to the variation of the thermodynamic state within the wing as well as its modal frequency. From the initial time to the end of cruise, the modal frequency of each order gradually declines; the decline tends to be gentle and then shows a recovery as the vehicle descends. It turns out that the first modals have little change along the flight trajectory, generally retaining the original modal shapes while major variation takes place in the fifth/sixth-order mode, especially the sixth-order mode in which case bending-torsion coupling tends to occur. If the climbing, cruise and descending phase, especially cruise, last long enough, the modal frequency and modal shape of each order can have more significant variation with dynamic variation of the flight trajectory.

The modal frequencies of each order show a downward trend over time and the decreasing rates vary from mode to mode. Modes of all orders get nearer or farther one another to varying degrees over time, which might impose severe impacts on the natural vibration characteristics of the structure. The results indicate that sustained aeroheating has effect more easily on modal shapes of higher order. Therefore, for hypersonic vehicles with large thin-walled

**Acknowledgements**

**Author details**

Fang Chen1

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1 Shanghai Jiao Tong University, Shanghai, China

This study was supported by the National Natural Science Foundation of China (Nos.

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**Figure 20.** The first six modal frequencies along the flight trajectory.

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**Author details**

Teimuraz Tsabadze Georgian Technical University, Tbilisi, Georgia

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*A New Approach for Assessing Credit Risks under Uncertainty DOI: http://dx.doi.org/10.5772/intechopen.93285*