A summary of appropriate patient preparation for WBS from reference by Ma et al.
\r\n\tIt will cover, but it will not be limited to, various aspects of iPSCs including: their characteristic features, sources, induction, programming and maintenance, therapeutic indications and clinical trials on the use of iPSCs, epigenetic modification and control, special requirements and quality control, and complications related to the use of iPSCs.
\r\n\tAuthors and scientists from all over the world are encouraged to submit their chapters. The book chapters can be in the form of review articles, clinical trials (human or animal studies), systematic reviews and meta-analyses, prospective or retrospective studies and even valuable case reports with updated literature reviews. It is recommended that authors make balanced reviews without having bias and that they include not only the most recent but also the most valuable related scientific literature.
Serum thyroglobulin (Tg) is a tissue-specific 660 kDa protein that serves as a precursor in thyroid hormone biosynthesis . It is synthesized by both thyroid follicular cells and differentiated cancer cells. Monitoring of serum Tg and Tg antibody (TgAb) levels, together with neck ultrasonography and 131I whole-body scintigraphy (WBS), is used as a diagnostic tool in postoperative follow-up for patients with differentiated thyroid carcinoma (DTC) . Generally, good correlation is seen between Tg and WBS in follow-up studies for DTC after thyroid remnant ablation . Undetectable serum Tg with negative WBS results suggests complete remission, whereas detectable, or elevated, serum Tg is associated with radioiodine uptake in local or distant metastases. Patients with thyroid cancer cells lacking radioiodine uptake despite their elevated serum Tg level have been referred to as WBS-negative, Tg-positive patients . The possible explanations and management for the discordant finding are discussed in this chapter.\n
Tg is a thyroid tissue-specific antigen produced by thyroid follicular cells. Its measurement is the best sign of detecting thyroid tissue, including metastasis of DTC. After a total thyroidectomy and radioiodine ablation, any detactable Tg is interpreted as recurrent disease. Although it is a highly sensitive and specific marker of recurrence, Tg measurement cannot locate the recurrent DTC [4, 5]. Imaging technologies, including WBS, ultrasonography, computed tomography (CT), and magnetic resonance imaging (MRI), play great role in locating the DTC metastases. Tg is a very sensitive marker for thyroid malignancy, and it is not uncommon to encounter patients who show at initial follow-up, detectable Tg levels with negative imaging studies . Patients with thyroid cancer cells lacking radioiodine uptake on WBS despite their elevated serum Tg level have been referred to as WBS-negative, Tg-positive patients; they represent 10–15% of patients with DTC at follow-up. Due to its inferior sensitivity, the routine WBS has been supplanted by serum Tg and neck ultrasonography, CT and/or MRI. The reasons for raised Tg and negative scan results have been summarized previously . The elevated Tg level and negative WBS are classified into “true-negative WBS with false-positive Tg” and “false-negative WBS and true-positive Tg.” Because of the low sensitivity of WBS, cervical ultrasonography plays more important role in the follow-up of DTC patients. Therefore, a new challenging scenario has emerged: the ultrasonography-negative, Tg-positive patient .\n
Tg assays interference\n
The possibility of a false-positive serum Tg because of assay interference is rare but should be considered. And serum Tg has a lower false-negative rate than WBS after stimulation of thyroid stimulating hormone (TSH) either by thyroid hormone withdrawal or by recombinant human thyroid stimulating hormone (rhTSH) [9–13]. Optimal follow-up requires remnant ablation, and TSH-stimulated Tg testing . The sensitivities and specificities of various Tg assays vary widely between laboratories, even with the use of an international standard (CRM 457) [14, 15], which have potential to disrupt serial monitoring and prompt inappropriate clinical decisions . Additionally, undetectable serum Tg became detectable in a significant percentage of DTC patients by changing assays [16–18]. Therefore, Tg should be dynamically monitored using the same assay performed in the same laboratory. If possible, patient’s serum is frozen and saved for recovery test to assess the reliability of Tg when there is a change in Tg assay . As the sensitivity of commercially available Tg assays improves, TSH-stimulated Tg may not be necessary in patients with low and intermediate risk of recurrence .
Circulating TgAb and HAb interferences
A major problem that hampers accurate Tg measurement is the interference by TgAb and HAb resulting in either an under- or overestimation of the serum Tg concentration [19–21]. Depending on the population studied and the assay used, up to 25–30% of patients with DTC have a positive test for TgAb at the time of initial diagnosis [22, 23]. In addition, a small to moderate percentage of patients (in the literature ranging from <1 to >10%) show HAb interference in Tg measurement, an integral tool in the management of DTC patients. These antibodies typically falsely lower the Tg value in immuno-chemiluminometric assays and immuno-radiometric assays, while raising the value in radio-immunoassay.
Therefore, TgAb should be measured in the same serum sample taken for Tg assay [15, 24]. Although for clinical purposes the measurement of Tg and TgAb before thyroidectomy for a suspected or proven DTC is not recommended, a pre-thyroidectomy Tg and TgAb measurement might be used as an “in vivo” recovery test in order to assess the reliability of Tg for use as a postoperative tumor marker [19, 25]. In DTC patients, the limit of quantitation (LoQ) of a given TgAb assay should be regarded as the upper normal limit for the presence of TgAb . Thyroid laboratories should report two reference ranges for TgAb: one based on the presence of TgAb in a population free of thyroid disease, which should be used for the diagnosis of autoimmune thyroid disorders, and the LoQ which should be used as the upper normal limit in DTC patients. A proposed algorithm for follow-up in TgAb-positive patients with DTC was listed in Figure 1 from Verburg et al. .
Persistence of Tg-Ab for more than 1 year after thyroidectomy and 131I ablation probably indicates the presence of residual thyroid tissue and possibly and/or DTC recurrence [22, 24, 26]. A recent study also showed that TSH receptor mRNA accurately predicted disease status in 68% of DTC patients .
Benign sources of Tg secretion
Apart from their ability to interfere with Tg assays, benign lesions (possibly with foci of thyroiditis) in persistent residual thyroid tissue or non-thyroidal tissue producing Tg may also result in false-positive Tg in DTC patients. However, residual occult disease is usually the source of post-operative Tg elevations [28–30]. Rarely, TSH-stimulated thymus may produce Tg .
Rarely, ectopic thyroid tissue may persist at the base of the tongue or, more often, at any other position along the thyroglossal tract, with the potential to elevate serum Tg levels. The thyroglossal tract is the most common location for ectopic thyroid tissue. This tissue retains not only the ability to concentrate iodine, but also to produce Tg and release it into the bloodstream [28, 31, 32]. The iodine metabolism-related proteins such as human sodium/iodide symporter (hNIS) , TSH receptor at both mRNA and protein level  are present in non-thyroidal tissues, including the thymus. Usually, these functions are dormant, but they may be activated by TSH stimulation . Interestingly, these extra-thyroidal foci may be resistant to multiple 131I treatments [28, 31, 32]. In a series of 548 consecutive diagnostic WBS, ectopic thyroid tissue in the tongue or in the upper part of the thyroglossal duct was visualized in five patients (0.9%) . However, in another study of 60 patients, 19/60 (31.7%) had a linear or focal radioactivity at the superior midline of the neck, suggesting thyroglossal duct remnant . The absence of metastases in the thymus despite high Tg levels was confirmed in five cases [33, 35]. Rare cases of thyroid tissue ectopy has been summarized in some locations such as struma ovarii, the heart (struma cordis), the submandibular, parotid and salivary glands, the duodenum, the adrenal glands, the liver and gallbladder, the pancreas, the axilla, and iris of the eye .\n\n
In summary, interference with Tg assays by TgAb and HAb, benign lesions (possibly containing thyroiditis) in persistent residual thyroid tissue or nonthyroidal tissue producing Tg may also result in false-positive Tg in DTC patients.\n
The possible causes of false-negative WBS are mentioned below.\n
Defect of iodine-trapping mechanism such as acquired inactivation mutation of NIS, TPO gene, pendrin, and TSHR
Thyroid hormone synthesis starts with the active uptake of iodine from the circulation via NIS. This process, known as iodine trapping, is stimulated directly by TSH and more circuitously by iodine deficiency. Other proteins, including TPO, TSHR, and pendrin, also play an important role in the thyroid metabolism of iodine. Any defect in NIS, TPO, Tg, and TSHR will contribute to false-negative WBS .
De-differentiation of tumor such that it can still produce Tg but has lost its iodine-trapping ability
Various molecular changes within papillary thyroid cancer cells, such as RET/PTC rearrangements, RAS and BRAF mutations , β-catenin mutations, PAX8/PPARã, histone acetylation factors involved in angiogenesis including overexpression of vascular endothelial growth factor (VEGF) and EGF receptor (EGFR) underlie the loss of iodide uptake ability . The dedifferentiated DTC cells lost the ability to concentrate iodine but may retain Tg synthesizing capability [3, 7], which underlines the phenomenon of Tg-positive and WBS-negative lesions.
Dispersed microscopic metastases, which are too small to be visualized
Improper patient preparation before WBS
When it is determined that an elevation of Tg is real, if WBS is negative, false-negative scan such as stable iodine contamination and inadequate TSH elevation should be considered . TSH levels should be elevated to at least 30 mIU/L before concluding that a negative WBS is meaningful. This can be achieved either by withdrawal of thyroxine or by rhTSH administration. rhTSH is as effective as thyroid hormone withdrawal on 131I thyroid remnant ablation for DTC patients with significant benefits in decreased whole-body radiation exposure and health-related quality of life [38, 39]. A summary of appropriate patient preparation for WBS in the hypothyroid state is presented in Table 1 from Ma et al. .\n\n
|Withdrawal of L-T4 for 4–6 weeks or of triiodothyronine for 2 weeks.\n|
|A strict low-iodine diet (50 g iodine per day) followed for 7–14 days before WBS and continuing throughout period of imaging.\n|
|Avoidance of iodine-containing medications (e.g., iodinated contrast medium, amiodarone, betadine), iodine-rich foods (e.g., kelp), and possible additives of iodine in vitamin and electrolyte supplements.\n|
|TSH > 30 mIU/L.\n|
|A mild laxative sometimes administered on the evening before WBS to simplify image interpretation.\n|
|Information relating to patient’s compliance with low-iodine diet, TSH level, history of thyroid hormone withdrawal, measurement of Tg, history of prior administration of contrast medium or iodine-containing drugs (e.g., amiodarone), menstrual history/pregnancy test, nursing/lactation history, etc.\n|
|Measurement of urinary iodine in doubtful cases to rule out iodine contamination; repeated WBS 4–6 weeks after iodine depletion regimen such as diuretic program.\n|
|Rule out women with pregnancy and breast feeding.\n|
In the clinical setting, the precise location of WBS-negative recurrent DTC is mandatory because surgery is the only curative treatment option and metastases that are unable to concentrate 131I are associated with more aggressive clinical behavior . Cervical ultrasonography, CT and MRI, 124I PET/CT have limited roles in the diagnosis of DTC metastases with positive Tg and negative WBS. Non-iodine imaging agents—such as 201Tl, 99mTc-sestamibi, 99mTc-tetrofosmin, somatostatin receptor (SRS) scan have reasonable accuracy . However, they have been replaced by 18F-FDG in the follow-up algorithm of DTC patients with positive Tg and negative WBS.\n
Cervical ultrasonography has high sensitivity in detecting recurrence in the thyroid bed and nodal metastases of DTC in the neck [42, 43]. It has been used as first-line diagnostic imaging in DTC follow-up [44, 45]. However, neck ultrasonography has limitations: one is that it does not reveal DTC recurrences in other body sites. It is also difficult for cervical ultrasonography to differentiate scar tissue and locally recurring fibrosis and between nonspecific nodal enlargements and nodal metastases . Therefore, the other limitation of ultrasonography is the low specificity in DTC patients of altered anatomy after thyroid surgery.
CT and MRI
In patients with elevated or rising Tg or TgAb and no evidence of disease on neck ultrasonography or WBS (if performed), CT imaging of the neck and chest should be considered . Diagnostic CT scan may complement neck ultrasonography for the detection of macrometastases in the central compartment, in the mediastinum and behind the trachea [48–50], and is the most sensitive tool for the detection of micro-metastases in the lungs. MRI has also been advocated for imaging the neck and the mediastinum. It is performed without and with injection of gadolinium chelate as contrast medium and does not require any injection of iodine contrast medium. Brain and skeletal MRI and/or CT, or abdominal MRI may be performed in high-risk DTC patients with elevated Tg (generally >10 ng/mL) and negative WBS or ultrasonography, who have systemic symptoms related to those organs, or who will have 131I therapy and may be at risk for complications of tumor swelling . MRI is less sensitive than CT scan for the detection of lung micronodules .
18F-FDG-PET/CT or PET/MRI
The iodine-negative DTC lesions were found to have increased expression of the glucose transporter-1, and often have FDG uptake . Therefore, 18F-FDG-PET is particularly useful in the detection of recurrent or metastatic DTC in patients with positive Tg and negative WBS, allowing detection of metastases not detected by other imaging modalities . In a recent meta-analysis, the combined sensitivity and specificity for FDG-PET/CT were 93 and 81%, respectively .
Factors influencing PET/CT sensitivity include tumor de-differentiation, larger tumor burden and to a lesser extent, TSH stimulation . PET is more sensitive in patients with an aggressive histological subtype, including poorly differentiated, tall cell, and Hürthle cell thyroid cancer. The sensitivity of PET (ranging from less than 10–30%) is low in patients with a TSH-stimulated Tg < 10 ng/mL. It is therefore recommended to consider 18F-FDG-PET only in DTC patients with a stimulated Tg level ≥10 ng/mL . A meta-analysis of seven prospective controlled clinical trials indicated that FDG-PET under TSH stimulation either by thyroid hormone withdrawal or by rhTSH had slightly improved diagnostic performance in detecting Tg-positive and WBS-negative DTC lesions. FDG-PET/CT is useful in staging, response assessment after chemotherapy, targeted therapies, or radiotherapy and prognostic assessment for patients with cancer . Therefore, PET/CT imaging should be performed as first-line, with empiric 131I treatment being considered only for those patients with no detectable FDG uptake . PET/CT can also identify lesions with high FDG uptake (SUV) that may be more aggressive and should have multi-targeted kinase inhibitors or close monitoring. A study observed that elevated Tg, but normal PET exists as a definitive entity in DTC. Positive Tg with negative PET was regarded as a favorable prognostic indicator to predict symptom-free status during the follow-up period .
However, false positives occur with PET imaging with or without TSH stimulation . The frequency of false-positive lesions varies among series from 0 to 39%, and this high number justifies a fine-needle aspiration (FNA) biopsy with cytology and Tg measurement in the aspirate fluid in cases where surgery is planned, based on PET results.
FDG-PET/CT is useful in staging, response assessment after chemotherapy, targeted therapies, or radiotherapy and prognostic assessment for patients with cancer . PET/CT imaging is more sensitive and should be performed as first-line, with empiric 131I treatment being considered only for those patients with no detectable FDG uptake . PET/CT can also identify lesions with high FDG uptake (SUV) that may be more aggressive and should have multi-targeted kinase inhibitors or close monitoring. A study observed that elevated Tg, but normal PET exists as a definitive entity in DTC. Positive Tg with negative PET was regarded as a favorable prognostic indicator to predict symptom-free status during the follow-up period .
124I emits positrons, allowing PET/CT imaging in DTC patients. It is used as for dosimetry and also as a diagnostic tool to localize DTC metastases. 124I PET/CT accurately measures the volume, uptake, and half-life of 124I in each DTC lesion, therefore permitting a reliable individual dosimetric assessment for DTC metastases . 124I-PET has higher sensitivity in detecting the residual thyroid tissue and/or DTC metastases than that of WBS (99% vs. 66%) [57–61]. The combination of 124I and FDG-PET/CT affords a valuable diagnostic method that can be used to make therapeutic decisions in patients with positive Tg and negative WBS [57, 61]. 124I-PET/CT with thyroid hormone withdrawal was found to detect significantly more foci of metastases of DTC . However, it is unclear whether and to what extent patient preparation with rhTSH rather than thyroid hormone withdrawal affects the diagnostic accuracy of 124I PET/CT . 124I is not yet widely available for clinical use and is primarily a research tool at this time.
Somatostatin receptor scan (SRS)
Thyroid tumors are known to express SRS, and therefore, 111In-pentetreotide (somatostatin analog) can visualize non-iodine avid DTC metastases with high concentration of SRS. A case of negative WBS recurrent metastatic papillary thyroid carcinoma with positive 111In-pentetreotide scan was reported . Technetium-99m labeled somatostatin analog, 99mTc-Hynic-TOC scintigraphy had a sensitivity of 88.46% (23/26), specificity of 100% (2/2), and an accuracy of 89.2% (25/28) . SRS scintigraphy may be useful both in the staging and monitoring of patients with WBS-negative DTC metastases. 68Gallium-somatostatin analogs PET/CT is currently a promising method to study well-differentiated neuroendocrine tumor which has a better sensitivity and therefore is superior to 99mTc or 111In labeled SRS [63, 64]. SRS scan positive patients are potential candidates for SRS-targeted therapy.
In addition, 68Gallium-somatostatin analogs PET/CT is currently a promising method to study well-differentiated neuroendocrine tumor which has a better sensitivity and therefore is superior to 99mTc or 111In labeled SRS [63, 64]. 18F-FLT and 11C-MET may also have a diagnostic roles in this clinical setting .
Fine-needle aspiration (FNA)
FNA biopsy for cytology and Tg measurement in the aspirate fluid is performed for suspicious lymph nodes >8–10 mm in their smallest diameter. Non-suspicious and small nodes (<8–10 mm in the smallest diameter) can be monitored with neck ultrasonography . Ultrasonography guidance aspiration may improve the results of FNA biopsy, in particular for small lymph nodes and those located deep in the neck. The measurement of Tg in the FNA biopsy washout fluid (FNAB-Tg) is the more accurate tool to detect DTC recurrences and metastases in the neck [66, 67]. However, the application of FNA biopsy Tg is currently hindered by the absence of methodological standardization, a lack of definite cutoff points, and the ongoing debate regarding its accuracy in nonthyroidectomized patients, those with elevated serum Tg, and those with circulating TgAb [66, 67]. A Tg concentration in the aspirate fluid between 1 and 10 ng/mL is moderately suspicious for malignancy ; above 10 ng/mL are highly suspicious of DTC metastases [68–70].
In summary, in patients with elevated or rising Tg (>10 ng/mL) or TgAb and no evidence of disease on neck ultrasonography or WBS (if performed), CT imaging of the neck and chest, MRI of the neck and abdomen may be considered. 18F-FDG-PET/CT also plays an important role in the detecting DTC metastases with positive Tg (>10 ng/mL) and negative WBS, and negative conventional imaging. The result of 18F-FDG-PET/CT is helpful in guiding the treatment strategy. FNA biopsy and Tg measurement in washout fluid are helpful in the confirmation of foci detected by 18F-FDG-PET/CT.\n
Thyroid hormone withdrawal induces substantial short- and long-term morbidity, decreased quality of life due to associated hypothyroidism.131I therapy may cause early and late sialoadenitis in up to 30% which can lead to xerostomia, dental caries, and stomatitis [71, 72], with a majority of patients suffering from significant changes in physical, psychological, and social well-being [73, 74]. Therefore, the pros and cons of empiric 131I treatment should be well-balanced justified.\n
The management of elevated serum Tg and radioiodine-negative scans was outlined by Ma et al. . Of 438 patients from 16 studies who were treated empirically with 131I for iodine-negative and Tg-positive DTC disease, 267 (62%) displayed pathological uptakes in the thyroid bed, lungs, bone, mediastinum and lymph nodes. In studies in which data were available for serum Tg levels during TSH suppression therapy or TSH withdrawal, 56% (188/337) patients showed decreased Tg. Of 242 patients from 5 studies who received no specific treatment for iodine-negative and Tg-positive DTC disease, 44% (106/242) showed spontaneous normalization and a significant decrease in serum Tg. Thus, high doses of 131I have therapeutic effects if the Tg level is considered an index of tumor burden, at least in the short term, and could also localize previously undiagnosed recurrences. Therefore, empiric131I treatment may be justified in high-risk patients with serum Tg > 10 ng/mL and a negative WBS and FDG-PET scan results [3, 75, 76]. Pulmonary metastases may be found only on post-therapy WBS . In a study of 283 DTC patients treated with 100mCi (3.7 GBq) of 131I, 6.4% had lung and bone metastases detected after treatment that had been suspected based on high serum Tg alone but had not been detected after 2mCi (74 MBq) 131I WBS .\n
However, most studies in this area have limited reliability as they lacked control groups and an adequate follow-up period . Still missing from our knowledge are long-term survival rates, changes in mass sizes on post-therapy imaging, and radiation-induced side-effects of 131I therapy. Although the tumor burden may be diminished, most patients with negative WBS and positive Tg are not rendered disease free by 131I therapy . Nearly half of patients with Tg-positive and WBS-negative DTC show spontaneous normalization and significant reduction in serum Tg without any specific treatment, 131I therapy should be individualized according to the clinical characteristics and imaging features. A five-year follow-up of 29 patients with elevated Tg (>2 ng/mL) and negative 131I WBS found that 24/29 patients showed Tg decreasing trend without 131I therapy, of whom only one patient recurred; the other 5/29 patients showed a rising trend and all recurred .\n
Therefore, additional diagnostic techniques are strongly recommended for patients with Tg-positive and WBS-negative metastases. If these diagnostic results are positive, treatment options such as surgery, external radiotherapy and tumor embolization can be considered. Empiric 131I therapy is more commonly considered for those with distant metastases or inoperable local disease. If FDG-PET result is negative, one course of 131I treatment may be considered in high-risk patients with. Repeated 131I therapy may be given to patients who had persistent non-resectable DTC metastases and iodine uptake, and there are significant therapeutic benefits until the lesion has been eradicated or the lesion no longer responds to treatment. The risk of repeated therapeutic doses of RAI must be balanced against uncertain long-term benefits. In the case of negative post-therapy WBS, the patient should be considered to have radioiodine-refractory disease and no further 131I therapy should be administered.\n
Retinoic acids (RA) and lithium
RA are active metabolites of vitamin A able to regulate growth and differentiation of many cell types by binding to specific nuclear receptors, the RA receptors, and the retinoid X receptors (RXR) . Lithium increases the residence time of 131I in the thyroid tissue [37, 81, 82]. RA and lithium  were used to redifferentiate metastatic DTC and render them responsive to 131I therapy. However, they only yielded a limited clinical benefit.
Iodine-trapping-related gene transfection
hNIS protein is a membrane glycoprotein that transports iodide ions into thyroid cells. This process, known as iodine trapping, is stimulated directly by TSH. Other proteins, including thyroperoxidase (TPO) and pendrin, also play an important role in the thyroid metabolism of iodine . Strategies of gene transfection focused on NIS; TPO has been studied to enhance tumor uptake iodine [84, 85]. Co-transfection of the hNIS and hTPO genes can lead to longer retention of radio iodine . Targeted NIS gene transfer, by viral and non-viral vectors, followed by radionuclide 131I, 188Re, 211At therapy, has been recently suggested for the treatment of advanced or WBS-negative DTC metastases. In thyroid cells, TSH stimulates NIS synthesis . Therefore, hTSHR transfection was investigated in FTC-133 thyroid cells, which improved the expression of thyroid-specific molecules including TSHR, NIS, TPO, and Tg and radioiodide uptake [87, 88]. Iodine-trapping-related gene transfection has not been used clinically yet.
MAPK kinase inhibitor
Mitogen-activated protein kinase (MAPK) signaling inhibits the expression of thyroid hormone biosynthesis genes, including the NIS and TPO, which facilitate iodine uptake and organification, respectively [89, 90]. Inhibition of the MAPK pathway may renew the therapeutic efficacy of 131I by enhancing uptake in patients with thyroid cancer that is refractory to 131I . MAPK1-2 inhibitor selumetinib (AZD6244, ARRY-142886), orally administered at a dose of 75 mg twice daily increased the uptake of 124I in 12 of 20 patients. Selumetinib enhanced 131I uptake in eight patients with advanced DTC. After 131I treatment, partial responses were achieved in 5, stable disease in 3. No severe adverse events were observed .
In summary, strategy of re-differentiation of iodine-negative DTC metastases by RA has limited clinical benefit. Iodine-trapping-related protein transfection remains experimental. MAPK kinase inhibitor needs to be confirmed in large population.\n
Both sorafenib (BAY 43-9006) and lenvatinib are multi-kinase inhibitors with potent activity against RAF, VEGF receptors, fibroblast growth factor receptors, PDGF receptor, c-KIT and RET kinases [37, 88, 91]. Sorafenib and lenvatinib are both FDA approved for iodine refractory DTC metastases . They achieved clinical benefits in terms of partial response of 12.5–38%, progression-free survival from 9 to 24 months in radioiodine-refractory DTC metastases [37, 91]. The therapeutic effects of other tyrosine kinase inhibitors including sunitinib, imatinib, vandetanib were also summarized  and a dozen ongoing trials currently listed in the ClinicalTrials.gov database, evaluating 12 kinase-inhibiting drugs .\n
Adverse effects occurred in 98.6% patients receiving sorafenib: the most frequent were hand-foot skin reactions, diarrhea, alopecia, and rash or desquamation .\n
Selection of a targeted agent should depend on disease trajectory, side effect profile, and goals of therapy. Kinase inhibitor therapy should be considered in radioiodine-refractory DTC metastases, rapidly progressive, symptomatic and/or imminently threatening disease not otherwise amenable to local control using other approaches. Patients who are candidates for kinase inhibitor therapy should be thoroughly counseled on the potential risks and benefits of this therapy as well as alternative therapeutic approaches including best supportive care .\n
TSH suppression is considered essential in the treatment of patients with positive Tg and negative WBS, because TSH is a trophic hormone that can stimulate the growth of cells derived from thyroid follicular epithelium [45, 95, 96]. Therefore, the recommended TSH level is below 0.1 mU/L, or slightly below or slightly above the lower reference range .
Surgery and stereotactic radiotherapy (SBRT)
Most recurrent DTCs respond well to surgery and SBRT [45, 97]. The isolated skeletal metastasis of DTC is recommended for surgery or SBRT . Neurosurgery or SBRT is preferred treatments for solitary brain metastases of DTC [98, 99]. SBRT is considered for loco-regional recurrence that is not surgically resectable, or with extra-nodal extension or involvement of soft tissues, in particular in patients with no evidence of distant disease, but has no role in most patients with resectable lymph node metastases .
Systematic chemotherapy can be considered for DTC lesions with positive Tg and negative WBS that are not surgically resectable, not responsive to 131I, not amenable to EBRT treatment, or not responsive to multi-targeted kinase inhibitors, and have clinically significant structural disease progression during the last 6–12 months. Two of 49 (3%) patients with DTC metastases had a response to five chemotherapy protocols . In a review by Ahuja et al., 38% of patients with thyroid cancer had reduction in tumor mass to doxorubicin . Combination chemotherapy does not show clear superiority to doxorubicin therapy alone . Therefore, the traditional chemotherapy has limited effects on iodine refractory DTC metastases [9, 103].
Other treatments include percutaneous ethanol injection (PEI), radiofrequency, or laser ablation.
PEI for patients with metastatic DTC in lymph node is promising as a nonsurgical-directed therapy [104, 105]. Most of the studies limited PEI to patients who had undergone previous neck dissections and 131I treatment, those who had FNA-proven DTC in the lymph node and those with no known distant metastases . A general consensus from studies and reviews is that PEI could be considered in patients who are poor surgical candidates . Radiofrequency ablation has been associated with a mean volume reduction that ranges between approximately 55–95% [107, 108], and 40–60% complete disappearance of the DTC metastases in the treatment of recurrent thyroid cancer [108, 109]. More recently, preliminary findings using ultrasonography-guided laser ablation for treatment of cervical lymph node metastases have been reported .\n
In summary, true-negative WBS with positive Tg may be due to benign thyroid remnants (possibly containing thyroiditis) or, rarely, nonthyroidal tissue producing Tg. False-negative WBS with positive Tg can be caused by a defective of acquired iodine-trapping inactivation; dedifferentiation of tumor which can still produce Tg but has lost its iodine-trapping ability; small dispersed microscopic metastases. Other radioisotopes and additional diagnostic options play an important role in the ascertainment of patients with negative WBS and Tg-positive DTC metastases. FDG-PET/CT should be considered in high-risk DTC patients with negative WBS and positive Tg. If FDG-PET diagnostic results are negative, one course of 131I treatment may be considered in high-risk patients and individualized. No further 131I therapy is indicated for patients with a negative post-therapy WBS. The preferred hierarchy of treatment for Tg-positive and WBS-negative metastases is surgical excision of loco-regional disease in potentially curable patients, 131I therapy for residual radioiodine-responsive disease, external beam radiation or other directed treatment modalities such as thermal ablation, TSH suppression for patients with stable or slowly progressive asymptomatic disease, and systemic therapy with multi-kinase inhibitors, especially for patients with significantly progressive macroscopic refractory disease.\n
Mathematical modeling (simulation) of physical processes is an important tool for the study of the environment.\n
Mathematical modeling is a means of studying the real objects, processes, or systems by replacing the real objects on the mathematical models, which are more comfortable to study with the aid of computers.\n
The mathematical model is an approximate representation of real-world objects, processes, and systems, expressed in mathematical terms. In this case, a significant feature of original are saved from researcher’s point of view.\n
First of all, some definitions and some concepts are given for the convenience of exposition.\n
A mathematical model of a real object will be called as mathematical dependencies and connections between the elements of a mathematical model. These elements are chosen on the basis of the interests of the researcher himself and the ultimate goals of study of the object. Usually, dependencies and relationships have forms of differential equations, integral equations, algebraic connections, etc.\n
The functions of external influences and external loads that are present in the mathematical model of the object in the form of symbols will be called as models of external loads.\n
The initial conditions, boundary conditions, and other conditions for the mathematical model will be called as additional conditions.\n
The totality of the mathematical model of the object, models of external influences, and additional conditions will be called as a mathematical description of the object.\n
The study of the behavior of the mathematical model of an object under the influence of models of external loads and additional conditions will be called as mathematical modeling or mathematical simulation.\n
The practical significance of the results of mathematical modeling or simulation of physical processes depends on the degree of coincidence of the results of mathematical modeling of the selected mathematical description of the real process with experimental data . Such property of a mathematical description of a physical process is usually called adequacy.\n
A mathematical description will be called as an adequate mathematical description (AMD) of the process under study if the results of mathematical modeling (simulation) using this description coincide with experimental data with the accuracy of experimental measurements.\n
The definition of adequacy will be clarified later for some types of mathematical models.\n
If the coincidence of the results of mathematical modeling with experiment is bad, then further use of these mathematical descriptions is problematic.\n
It is note that authors of works on mathematical modeling concern seldom questions of adequacy of the constructed mathematical description of process to real measurements [2, 3, 4, 5]. Sometimes, such adequacy is proved by the real facts; sometimes, authors refer to results of other authors; and sometimes, there have not been any arguments.\n
The considered situation requires formation of some uniform approach to this problem, common methodological approach, general algorithms, and common criteria of estimation of adequacy degree.
Currently, there are two main approaches to the problem of constructing an adequate mathematical description [1, 6, 7, 8]: for a mathematical model with a priori chosen structure and inaccurate parameters, a model of external influence is determined, which together with the mathematical model of the process provide the adequacy condition (coincidence with experiment);
a model of external loads is given a priori and then parameters of mathematical model or of its structure are selected, in such a way that results of mathematical simulation match up with experiment.
Having a comparison of the results of mathematical modeling with experimental data in definition of adequate mathematical description ensures the objectivity of the results of the synthesis of a mathematical description. In the literature, this approach is called as an identification method: estimation of the parameters of an adequate mathematical description based on the results of measurements of the characteristics of the state of the physical process [9, 10].\n
Mathematical models of physical processes can be presented as systems of ordinary differential equations, systems of partial differential equations, algebraic relations, integral equations, etc.\n
It should be noted that in many works, the accuracy of the results of mathematical modeling is several times lower than the accuracy of experimental data.\n
In the given work, the mathematical models of physical processes described only by the system of the ordinary differential equations will be examined [2, 3]. Such idealization of real processes or dynamic systems is widely used in various areas for the description of control systems , as well as of mechanical systems with the concentrated parameters [5, 12], economic processes , biological processes , ecological processes , etc. In some works with the help of such systems, human emotions are simulated .\n
Many problems investigated in the given work, have place for other types of mathematical models of physical processes, for example, for mathematical models in the form of the partial differential equations.\n
The chapter proposes several criteria for checking the adequacy of the constructed mathematical descriptions for cases when the mathematical model of the physical process is represented by a system of differential equations.\n
The author hopes that the offered criteria of adequacy will be useful in a construction of the adequate mathematical descriptions of real physical processes.\n
Consider the specified criteria for mathematical descriptions in the form of a system of differential equations.\n
where \n\n—matrices with constant coefficients, which are given approximately, \n\n—vector-function variables, characterized the state of process (\n\n—a mark of transposition), \n\n—vector-function of external load; and \n\n are normalized functional spaces.\n
We assume that state variables \n\n of system (1) correspond to some real characteristics of process which is under investigation \n\n.\n
By mathematical description of the physical process, we mean the set of the system of Eq. (1), the vector of the external loads functions \n\n and the initial conditions \n\n. In other words, a mathematical description is a collection of mathematical models, models of external influences, and initial conditions.\n
The process of solving the system of differential Eq. (1) under the influence of selected models of external loads \n\n, taking into account, the initial conditions \n\n, is usually called mathematical modeling or mathematical simulation.\n
An adequate mathematical description of a physical process of such type with respect to all variables \n\n of quantitative type will be called as the mathematical description for which the results of mathematical simulation of variables \n\n coincide with the results of experimental measurements \n\n of the characteristic \n\n with the accuracy of the experiments \n\n:
In practice, the measurement of the characteristics of state variables is limited to only one or two components. We formulate a refined definition of the adequacy of a mathematical description for the case of a single variable.\n
An adequate mathematical description of a physical process of such type with respect to the variable \n\n (\n\n) of quantitative type will be called as the mathematical description for which the results of mathematical simulation of a variable \n\n coincide with the results of experimental measurements \n\n of the characteristic \n\n with the accuracy of the experiment \n\n:
For the rest of the variables, coincidence with experiment is not determined. Adequate mathematical descriptions are similarly determined in the case of several measurements of state variables. The metrics of comparison in this case is determined by the objectives of specific studies.\n
The criteria of mathematical description adequacy of quantitative type, which are offered in the given chapter, can be used for other types of mathematical descriptions of physical processes, for example, for mathematical descriptions in the form of the partial differential equations . They have many common features.\n
It can be shown that there are an infinite set adequate mathematical descriptions for the same physical experiment.\n
In addition, qualitatively, different physical processes can have adequate mathematical descriptions for the same experiment.\n
Mathematical model of process of type (1) is given a priori with inexact parameters and then the models of external loads were determined for which the results of simulation coincide with experiment [22, 23];
Now, we will consider the synthesis of adequate mathematical description of quantitative type in the frame of first approach analyzing the process with the concentrated parameters, for which the motion is described by ordinary differential equations of n-order (1).\n
We assume that some functions of state \n\n in system (1) are obtained from experiment and presented by graphs. Besides, we suppose that some functions of external loads, for example, \n\n are unknown. According to first approach, it is necessary to develop the construction of such model of external load component, which is characterized by the functions of state \n\n of mathematical model (1), and will coincide with experimental measurements \n\n with inaccuracy of initial data. Such mathematical model of process behavior together with obtained model of external load can be considered as adequate mathematical description of quantitative type of process.\n
Such method of obtaining of mathematical models of external loads (functions \n\n) is determined in literature as a method of identification [9, 10]. By the way, physical reasons of occurrence of such external loads are not being taken into account. They are only functions, which in combination with mathematical model (1) provide results of modeling, which coincide with experiment with the given accuracy.\n
Consider an example of a mathematical description that satisfies the criterion of the adequacy of a quantitative type for all variables \n\n.\n
Now, we consider in detail, the problem in which the dynamics of the main mechanical lines of rolling mills is investigated [24, 25]. One variant of the kinematic scheme of it is presented in Figure 1. (a) where the engine is marked by label (1), the coupling is marked by label (2), gears is marked by label (3), driving shafts is marked by label (4), operational barrels is marked by label (5).\n\n
The four-mass model with weightless elastic connections is chosen as mathematical model of dynamic system of the main mechanical line of the rolling mill [24, 25]. The system of vibrations equations is obtained from the Lagrang?s equations of second kind:
Here, the following designations were accepted: \n\n—moment of engine, \n\n—moments of inertia of the concentrated weights, \n\n—rigidity of the appropriate elastic connection, \n\n—moments of technological resistance put to the upper and lower operational barrels accordingly, and \n\n—moments of elasticity forces, which are applied to shafts between mass \n\n and \n\n; \n\n\n
Actually, the constructed mathematical model may correspond to real process and may not. It is necessary to check up correctness of the constructed mathematical model. For this purpose, the data of experiment are used. If the results of mathematical modeling coincide with results of experiment (with accuracy of measurements), then mathematical description of process is considered as adequate to a reality in the quantitative sense. In other words, the mathematical description corresponds to real process.\n
The information related to the real motion of the main mechanical line of rolling mill was obtained by an experimental way [23, 24, 26]. Such information is being understood as availability of functions \n\n. The records of functions \n\n of a given process are shown in Figure 2.\n\n
It is obvious, that the results of mathematical modeling of system (4) depend directly on character of change of external loads, which is applied to operational barrels of the rolling mill and external impact of the engine \n\n, \n\n. Sometimes, it is possible to pick up such loadings \n\n, \n\n in which the results of mathematical modeling \n\n coincide with experiment (Figure 2).\n
If such choice is possible, then mathematical model (4) combined with the found loads \n\n, \n\n will give adequate mathematical description of real process. It is necessary to note that in many papers analyzing the problem of mathematical modeling with the use of system of a differential Eq. (4) together with functions \n\n, \n\n are determined as mathematical model of process. Coincidence is understood as coincidence with the accuracy of experimental measurements.\n
According to this approach, it is necessary to construct such models of external loads \n\n on system (4), for which the functions \n\n of elastic moments in the links of the model (solution of the system(4)), coincide with the corresponding experimental functions of the moments of elastic forces in the links of the main line of the rolling mill (Figure 2).\n
Consider the construction of an adequate mathematical description within the framework of the first approach. To construct, for example, a model \n\n, which corresponds to the moment of the external load to the upper work roll, consider the second equation in the system (4). The solution of this equation has the form
We will assume that function \n\n in (5) belongs to the normalized space \n\n (\n\n is the period of time at which the function \n\n is studied) and the solution \n\n of Eq. (5) belongs to the normalized functional space \n\n.\n
Let us rewrite the equation (5) in the more compact form
where z is the searched element, uδ is the given element, which belong, respectively, to the functional spaces \n\n and \n\n, \n\n is the integral operator. In this case, \n\n.\n
Since the right-hand side \n\n of the integral Eq. (5) is determined from the experiment, it is natural to assume that instead of the exact right-hand side \n\n of the Eq. (6), some approximation of it is given \n\n:\n
\n\n\n, δ is given.\n
The set of possible solutions \n\n of Eq. (6) consists of elements that correspond to the equation with given accuracy:
Each function in the set \n\n together with the given mathematical model (6) provides an adequate mathematical description of the physical process.\n
In this case, the problem of identifying model of external load in the rolling mill is considered as the inverse of the synthesis problem .\n
In this chapter, oscillograms of the moments of the forces of elasticity in the links of the main line of rolling mill 1150, obtained in [24, 25], are used. A copy of this oscillogram is shown in Figure 2. The value of \n\n is chosen equal to 0.48 s.\n
When synthesizing the model of external load on the lower work roll of the state, it is necessary to use the last differential equation in the system (4).\n
Thus, models of external loads \n\n were obtained that together with the mathematical model (4) and the initial conditions yield simulation results that coincide with the experimental measurements with the accuracy of the experiment. In other words, the mathematical model (4), models of external loads and initial conditions give an adequate mathematical description of quantitative type for all variables of physical process.\n
Now, we consider another example of astrodynamical processes mathematical description, which has the property of adequacy of quantitative type in only one variable.\n
Based on theoretical analysis of mathematical vortex model of planetary systems, the analytical expression for planetary distances in the prevailing planetary systems was obtained. These distances are functions of the coordinates of the centers of vortical rings of primary planetary vortex. Comparison of theoretical and real distances planets of the Solar system show their good agreement.\n
Known in the cosmogonic theories of the solar system, the law of Titsius-Bode (1772) of planetary distances \n\n\n
is a successful empirical approximation of the real sequence of distances \n\n of planets with number \n\n from to Sun. In this case, the first planet (Mercury) corresponds to the value \n\n, Venus—\n\n, the Earth—\n\n, etc., and the conditional unformed planet between Mars and Jupiter must be attributed to the value of \n\n. Despite the excellent conformance of this law to the average number of planets, the law (7) for the first and distant planets of Neptune and Pluto is not fulfilled .\n
In the twentieth century, some attempts were made [28, 29] to theoretically obtain the law of planetary distances, but in the basis of these theories, the authors had to impose new arbitrary hypotheses. Schmidt  introduces a hypothetical function of the distribution of kinetic moments in the masses of the primary nebula, and for the simplest functions it receives a quadratic law, a geometric progression, and others. Kuiper  deduces his law on the basis of the theory of tidal stability using the concept of “critical density of Rosh.” However, the law it received give the distance between planets, which differ on several orders from the real distance.\n
Below, based on the mathematical vortical model of the formation of planetary systems [30, 31], the analytical law of planetary distances for any planetary systems was obtained, which gives a good agreement with real distances in the solar system, which has another form compared with (7).\n
The general picture and the basic relations in the primary vortex explosion, which creates stars and their planetary systems, is constructed in  on the basis of a separate exact solution of the Euler hydrodynamic equations for spherical eddy currents . The main physical feature of this axisymmetric spatial flow, called the planetary vortex , is the presence of a vortex dipole in the center of the vortex dipole, which flows through a moving, twisted stream of outer space, and the interaction of these motions generates vortical flow of a planetary vortex .\n
Using the method of integrating the complete nonlinear system of Euler’s hydrodynamic equations was introduced and flow functions \n\n constructed. The function of flow in spherical coordinates (r, θ, φ)  is constructed.\n
The planetary vortex described above as a complicated vortex flow is the initial stage of the formation of a star planetary system from the primary nebula that has fallen into the vortex region. Further prolonged evolution of this vortex to the state of the planetary system is characterized by a variety of complex physical processes such as: collision, accretion, accumulation of massive bodies, and their gravitation; the formation of a massive star and its light and gravitational action; mutual gravitational and resonant influence of system structures, etc. .\n
Since the forces of gravitation, collision, and others acting between parts of a single vortex are internal, they do not change its integral physical invariants.\n
In [30, 31], the modern planetary distances were calculated and they are shown in the graphs (Figure 4). As we can see, the theoretical curve in the entire range of distances is almost equidistant from the curve of real distributions of distances in the solar system with deviations in both directions of the order of 20%.\n\n
Finally, the technique developed here for the calculation of the primary vortex parameters can be applied in the reverse direction to determine according to the data of several open planets of the main parameter of the planetary system and the establishment of the method of work  of the complete structure of new exoplanetary systems: the number of vortex planets, their distances from the star, angular velocities, etc. This will give astronomers-observers reasoned data for the search for new, yet open exoplanets in stellar planetary systems, which have already opened 2–6 planets [32, 33].\n
Thus, the mathematical model of the process of formation of planetary systems, which describes interplanetary distances well, is constructed.\n
The development of this mathematical model does not take into account important physical factors that have a significant effect on the behavior of interplanetary matter. To such factors, it is necessary to attribute, first of all, gravitational interaction and heat flows. Because of this, one should not expect a good coincidence of the real characteristics of the physical process with the results of mathematical modeling. This situation occurs when the criterion of adequacy of the qualitative type is not met (see Section 2). Consequently, the coincidence of the results of mathematical modeling (with the adequate mathematical description constructed only one or two of the system’s variables) with all the main characteristics of the physical process is an exception, as a rule, and it can take place only if the adequacy of a quality type is performed.\n
We will consider what prospects of adequate descriptions are valid for further use and what goals should be selected as the creation of adequate mathematical descriptions.\n
It will be useful to address to classical works in this area. In work , the following statement was done: “…the imitation modeling is the creation of experimental and applied methodology which aimed at the use of it for a prediction of the future behavior of system.”\n
So, the adequate mathematical descriptions are intended for the forecast of behavior of real process at first. It is possible with the aid of adequate mathematical modeling to predict behavior of real process in new conditions of operation. For example, it is possible to test more intensive mode of operations of the real machine without risk of its destruction. Such tool (adequate mathematical description) allows determining the optimum parameters of real process.\n
Let us now consider the conditions under which it is possible to further use adequate mathematical descriptions for “…a prediction of the future behavior of system.”\n
Obviously, the structure of system (1), its parameters, and the specific type of external influences are determined by the properties of a real physical process.\n
Let the selected structure of the mathematical model of the physical process include parameters \n\n (e.g., the mass of the elements, the stiffness of the elastic elements, etc.), which are reflecting the actual physical characteristics. The structure of the mathematical model also includes dependencies that reflected real physical patterns and dependencies of the process under study.\n
For the purpose of further substantiated use of mathematical descriptions, it is necessary to require that there is a one-to-one correspondence between the components of the vector parameter \n\n of mathematical description and the actual physical elements. In addition, it is necessary to require that the interconnections between the parameters of a mathematical model comply with the physical laws of the process being studied, and the main external loads had been included. This important correspondence will be called the main correspondence (MC). The execution of the MC believed the fulfillment of the criterion of adequacy of the qualitative type. In other words, a mathematical description of a physical process satisfies the criterion of adequacy of a qualitative type if the main correspondence is fulfilled.\n
An additional requirement for the implementation of the MC is explained firstly by the fact that the quantitative agreement of the results of mathematical modeling with a specific experiment is possible for mathematical descriptions of qualitatively different physical processes due to the selection of parameters of mathematical descriptions.\n
The implementation of the MC leads to the fact that the models of external influences obtained by the method of identification will correspond to the real external influences on the physical process. At least, these models will not contradict the physical meaning. If we return to the example of the synthesis of an adequate mathematical description of the process of mechanical oscillations in the main line of the rolling mill, then it can be argued that the MC is being executed. By virtue of this, the obtained models of external influences have a reasonable physical interpretation (do not contradict the physical meaning). The external load smoothly increases from zero to a steady-state value (see Figure 3).\n
When fulfilling the adequacy of a mathematical description of a qualitative type, it becomes possible to argue that a mathematical description that satisfies two criteria of adequacy will retain its useful properties for other experiments in the future under small changes in the conditions of the physical process. In other words, this description can be used for “a prediction of the future behavior of system.” An example of such a successful application could be further mathematical modeling using an adequate mathematical description for the rolling mill .\n
In the second example, given in Section 1, the main correspondence is not fulfilled, and therefore, the application of the obtained results in the new conditions will not be justified.\n
The algorithm for constructing an adequate mathematical description of a qualitative type cannot be formalized, as in the case of the adequacy of a mathematical description of a quantitative type. The process of constructing such a description mainly depends on subjective factors, such as the scientific tasks of studying the physical process using mathematical modeling methods.\n
In some cases, it is impossible to perform a check of mathematical description adequacy of a quantitative type due to lack of experimental data in principle. Let us give an example of a mathematical description that satisfies only the criterion of adequacy of a qualitative type.\n
An important and relatively new field of applications of methods of mathematical modeling is a tectonic processes study . This work presents a complete algorithm required for the successful application of mathematical modeling methods of operations, which does not include the verification operation. Consider this algorithm in more detail.\n
It is widely known that earthquakes predicting is challenging and unsolved still (but it can be happen in future). Even where earthquakes have unambiguously occurred within the parameters of a prediction, statistical analysis has generally shown these to be not better than lucky guesses. Now, there are hundreds of well-known earthquakes precursors and a number of theories to explain their origin. However, the problem of earthquake prediction in many of its aspects still remains open.\n
Utsu studied theoretically the relation between the size of aftershock activity and the magnitude of the main shock . Independence of the occurrence of main shocks has been assumed in many models, some chapters discuss the migration of large earthquakes and casual relationship between seismic activities in different geophysical regions. Trigger models assume a series of primary events (main shocks) distributed completely random in time. Each of these primary events may generate secondary series of events. Epidemic-type model can be considered as birth and death process.\n
Utsu proposes a new model that takes into account the influence of strain solitary waves as a “trigger” of some shocks and appropriate methods of forecasting. Authors analyze the 2011 Japan earthquake. These studies show that solitary waves can be generated as aftershocks hypocenters at the Moho surface.\n
Significant amount of works have been devoted to research of solitons in solids. For example, considering structural-phenomenological approach one distinguishes damaged environment with microstructure, Kosser’s continuum with limited traffic, Leru’s continuum, environment with deformities , grainy environment, which has the soliton solutions of motion equations. It is known that the pulse perturbations in rocks are different from seismic waves of harmonic type.\n
Let us consider an anisotropic elastic medium. By , it follows that relations between stress and strain in Hooke’s law contain 21 free coefficients. The system of motion equations in this case has the form:
where \n\n, \n\n, and \n\n are the displacements along the respective axes of a Cartesian coordinate system, \n\n is the density,\n
\n\n\n, \n\n is matrix of elastic constants.\n
where \n\n, \n\n is the class of positive-definite, unimodal, twice continuously differentiated functions with a minimum at zero, and for which the second derivative is not a constant, \n\n, \n\n, \n\n are the functions which determined the amplitude of the relevant perturbations, \n\n are constants which determined the localization perturbations, \n\n, \n\n, \n\n are the functions which determined the trajectory of solitary waves.\n
In , authors got the necessary and sufficient conditions of existent solutions of the system (8)–(10) in the form (11). In , the various crystal systems for the existence of the appropriate type of motion equations solutions are studied.\n
The main hypothesis of method of earthquakes forecast is that a single shock causes the appearance of one or more solitary waves that move from the hypocenter of the earthquake. Each wave passing through the zone of accumulation of seismic energy, causing a new earthquake can in turn generate new solitons. The method of prediction involves the separation from the total population of earthquake subsequences caused by the same soliton and the construction of a hypothetical trajectory of the solitons. Knowing the distance between individual impulses along the trajectory of the soliton can estimate its speed. Knowing some point of its trajectory, it is possible to make an assessment of the trajectory. With the rate and trajectory of each soliton, one can estimate its position at any time. Having information about the position of the soliton at some time can determine the “soliton component” of shock probability at this time.\n
As initial data, we consider a sequence of the form: \n\n, \n\n, …, \n\n, where \n\n are earthquake hypocenters, \n\n are magnitudes, and \n\n are times of shocks.\n
Let a trajectory of the soliton be described parametrically: \n\n, \n\n is speed. Then, we have:
where \n\n is the length of the curve corresponding to time interval \n\n.\n
Assume that the speed of the soliton is monotone—decreasing function. If the motion occur in the region of constant density, the ratio will be implemented: \n\n. In the field of variable density: \n\n.\n
Then, we consider the approximate speed \n\n \n\n\n
If a density is constant, then \n\n, where \n\n is the curvature parameters. If a density is variable, then: \n\n. This is criteria for the identification a subsequence of individual solitons trajectory.\n
In Figure 5, the results of seismic process analysis that occurred on the Japanese islands for 3 days before the earthquake of magnitude 8.9 (occurred on March 11, 2011) are shown. Here, the numbers from 0 to 12 mark epicenters of the foreshock, the epicenter of the main shock indicates maximum circle radius (it is near to the epicenter of the foreshock number 1). Curves and straight lines marked the soliton trajectory, which are calculated using a special software. The calculations take into account the hypothetical rate of the solitary waves and their possible reflection from the areas with a high density of rocks.\n\n
As you can see, the foreshock is arranged so that a large number of possible waves pass through the region, where there was a maximum magnitude shock. Clearly, traced kind of a soliton with a focusing effect is at the point where there was the main shock.\n
Thus, here it proposed the mathematical model of the process of earthquake sequences, taking into account the impact of slow solitary wave soliton type as a “trigger” to some shocks. The proposed theory allows us to construct forecasts when geophysics seismic process is similar to that which occurred on Japanese islands in 2011.\n
The considered example of a mathematical description cannot be checked for the adequacy of a quantitative type due to objective reasons. But this description meets the criterion of adequacy of the qualitative type and so the results of mathematical modeling do not contradict the physical meaning.\n
Further, consider an example of a mathematical description, the adequacy of which cannot be fundamentally assessed.\n
Methods of mathematical modeling penetrate recently into many nontraditional areas of human activity such as the study and modeling of emotions . Let us consider in more detail the peculiarities of the application of methods of mathematical modeling in this field.\n
The mathematician Rinaldi investigated as first Petrarch’s emotional cycle and established an ODE model, starting point for the investigations in two directions: mapping the mathematical model to a suitable modeling concept, and trying to extend the model for love dynamics in modern times.\n
A control-oriented approach observes emotions and inspiration as states fading over time-behaving like a transfer function approaching a steady state. This observation suggests a modeling approach by transfer functions. Both model approaches allow an easy extension to modern times.\n
In literature, two special contributions can be found:
Love affairs and differential equations by Strogatz , —harmonic oscillators making reference to Romeo and Juliet;
Laura and Petrarch: an Intriguing Case of Cyclical Love Dynamics by Rinaldi —presenting a nonlinear ODE with cyclic solutions.
Both contributions start directly with nonlinear oscillations, observing a certain historic emotional behavior of prominent couples. Laura group at Vienna University of Technology tries to consider general modeling concepts for emotional relations, which cover or coincide with Petrarch’s emotional cycle, in case of appropriate parameterization.\n
Following a suggestion of Strogatz  here examines a sequence of dynamical models involving coupled ordinary differential equations describing the time-variation of the love or hate displayed by individuals in a romantic relationship. The models start with a linear system of two individuals and advance to love triangles, and finally to include the effect of nonlinearities, which are shown to produce chaos.\n
An obvious difficulty in any model of love is defining what is meant by love and quantifying it in some meaningful way.\n
Strogatz  considers a love affair between Romeo and Juliet, where R(t) is Romeo’s love (or hate if negative) for Juliet at time t and J(t) is Juliet’s love for Romeo.\n
The simplest model is linear with
where \n\n and \n\n specify Romeo’s “romantic style,” and \n\n and \n\n specify Juliet’s style. The parameter \n\n describes the extent to which Romeo is encouraged by his own feelings, and \n\n is the extent to which he is encouraged by Juliet’s feelings. The resulting dynamics are two-dimensional, governed by the initial conditions and the four parameters, which may be positive or negative.\n
A similar linear model has been proposed by Rinaldi  in which a constant term is added to each of the derivatives in (12) to account for the appeal (or repulsion if negative) that each partner presents to the other in the absence of other feelings. Such a model is more realistic since it allows feelings to grow from a state of indifference and provides an equilibrium not characterized by complete apathy. However, it does so at the expense of introducing two additional parameters. While the existence of a non-apathetic equilibrium may be very important to the individuals involved, it does not alter the dynamics other than to move the origin of the RJ state space.\n
Romeo can exhibit one of four romantic styles depending on the signs of a and b, with names adapted from those suggested by Strogatz  and his students:
Eager beaver: \n\n, \n\n (Romeo is encouraged by his own feelings as well as Juliet’s).
Narcissistic nerd: \n\n, \n\n (Romeo wants more of what he feels but retreats from Juliet’s feelings).
Cautious (or secure) lover: \n\n, \n\n (Romeo retreats from his own feelings but is encouraged by Juliet’s).
Hermit: \n\n, \n\n (Romeo retreats from his own feelings as well as Juliet’s).
Note that for the mathematical description (12), the criterion of the adequacy of the quantitative type and the criterion of the adequacy of the qualitative type cannot be checked due to the specificity of the process under study. Therefore, the further use of the results of mathematical modeling is unreasonable. However, there are no obstacles to nontraditional interpretations of the results of mathematical modeling of emotional processes.\n
The proposed adequacy criteria for mathematical descriptions in the form of ordinary differential equations make it possible to reasonably use the results of mathematical modeling to optimize and predict the behavior of physical processes.\n
The proposed criteria are easily transferred on mathematical descriptions in algebraic form .\n
Criteria for the adequacy of mathematical descriptions in the form of partial differential equations are currently missing in the literature. However, some criteria for the adequacy of mathematical descriptions can be transferred to the specified descriptions. For example, the criterion of adequacy of the qualitative type can be transferred almost unchanged.\n