A Cantor set of tori with monodromy near a focus-focus singularity

We write down an asymptotic expression for action coordinates in an integrable Hamiltonian system with a focus-focus equilibrium. From the singularity in the actions we deduce that the Arnol'd determinant grows infinitely large near the pinched torus. Moreover, we prove that it is possible to globally parametrise the Liouville tori by their frequencies. If one perturbs this integrable system, then the KAM tori form a Whitney smooth family: they can be smoothly interpolated by a torus bundle that is diffeomorphic to the bundle of Liouville tori of the unperturbed integrable system. As is well-known, this bundle of Liouville tori is not trivial. Our result implies that the KAM tori have monodromy. In semi-classical quantum mechanics, quantisation rules select sequences of KAM tori that correspond to quantum levels. Hence a global labeling of quantum levels by two quantum numbers is not possible.Comment: 11 pages, 2 figure

Differential constraints for the Kaup -- Broer system as a reduction of the 1D Toda lattice

It is shown that some special reduction of infinite 1D Toda lattice gives differential constraints compatible with the Kaup -- Broer system. A family of the travelling wave solutions of the Kaup -- Broer system and its higher version is constructed.Comment: LaTeX, uses IOP styl

Dissipative Boussinesq System of Equations in the B\'enard-Marangoni Phenomenon

By using the long-wave approximation, a system of coupled evolution equations for the bulk velocity and the surface perturbations of a B\'enard-Marangoni system is obtained. It includes nonlinearity, dispersion and dissipation, and it can be interpreted as a dissipative generalization of the usual Boussinesq system of equations. As a particular case, a strictly dissipative version of the Boussinesq system is obtained. Finnaly, some speculations are made on the nature of the physical phenomena described by this system of equations.Comment: 15 Pages, REVTEX (Version 3.0), no figure

Bifurcation curves of subharmonic solutions

We revisit a problem considered by Chow and Hale on the existence of subharmonic solutions for perturbed systems. In the analytic setting, under more general (weaker) conditions, we prove their results on the existence of bifurcation curves from the nonexistence to the existence of subharmonic solutions. In particular our results apply also when one has degeneracy to first order -- i.e. when the subharmonic Melnikov function vanishes identically. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalisations to higher orders of the subharmonic Melnikov function are also identically zero. In general the bifurcation curves are not analytic, and even when they are smooth they can form cusps at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The technique we use is completely different from that of Chow and Hale, and it is essentially based on rigorous perturbation theory.Comment: 29 pages, 2 figure

Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known two-dimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges.Comment: Talk given at the Workshop "Nonlinear Physics: Theory and Experiment. V", Gallipoli (Lecce, Italy), 12-21 June, 200

Semen analysis and reproductive hormones in boys with classical Hodgkin lymphoma treated according to the EuroNet-PHL-C2 protocol

STUDY QUESTION:What is the impact of the EuroNet-PHL-C2 treatment for boys with classical Hodgkin lymphoma (cHL) on semen parameters? SUMMARY ANSWER:More than half of the patients (52%, n = 16/31) had oligozoospermia or azoospermia at 2 years from cHL diagnosis; particularly boys treated for advanced-stage cHL had low sperm counts and motility. WHAT IS KNOWN ALREADY:Chemotherapy and radiotherapy to the inguinal region or testes can impair spermatogenesis and result in reduced fertility. The EuroNet-PHL-C2 trial aims to minimize radiotherapy in standard childhood cHL treatment, by intensifying chemotherapy. The present study aims to assess the (gonadotoxic) impact of this treatment protocol on semen parameters and reproductive hormones in boys aged <= 18 years. STUDY DESIGN, SIZE, DURATION:This international, prospective, multi-centre cohort study was an add-on study to the randomized phase-3 EuroNet-PHL-C2 trial, where the efficacy of standard cHL treatment with OEPA-COPDAC-28 (OEPA: vincristine, etoposide, prednisone, and doxorubicin; COPDAC-28: cyclophosphamide, vincristine, prednisone, and dacarbazine) was compared to intensified OEPA-DECOPDAC-21 chemotherapy (DECOPDAC-21: COPDAC with additional doxorubicin and etoposide and 25% more cyclophosphamide). Patients were recruited between January 2017 and September 2021. PARTICIPANTS/MATERIALS, SETTING, METHODS:Eligibility criteria included male patients, diagnosed with classical HL before or at the age of 18 years, and treated according to the EuroNet-PHL-C2 protocol in any of the 18 participating sites in the Netherlands, Germany, Belgium, Czech Republic, and Austria. Sperm parameters (sperm concentration, progressive motility, sperm volume, and calculated total motile sperm count) were assessed at diagnosis and 2 years after diagnosis in (post)pubertal boys. Laboratory measurements (serum follicle-stimulating hormone (FSH) and inhibin B) were performed in samples drawn at diagnosis, during treatment (2-3 times), and at 2 years post-diagnosis, and (age-adjusted) analyses were conducted separately for pre-pubertal and (post)pubertal boys. Outcomes were compared between the treatment levels (TL1, TL2, and TL3) and consolidation treatment schemes (COPDAC-28 and DECOPDAC-21). MAIN RESULTS AND THE ROLE OF CHANCE:In total, 101 boys were included in the present analysis: 73 were (post)pubertal (median age 15.4 years, (IQR 14.4; 16.6), 10 TL1, 29 TL2, 34 TL3, 62% of TL2/3 patients received COPDAC-28) and 28 boys were pre-pubertal (median age 9.6 years (IQR 6.6; 11.4), 4 TL1, 7 TL2, 17 TL3, 38% of TL2/3 patients received COPDAC-28). The study included six boys who had received pelvic radiotherapy; none were irradiated in the inguinal or testicular area. At diagnosis, 48 (post)pubertal boys delivered semen for cryopreservation; 19 (40%) semen samples were oligospermic and 4 (8%) were azoospermic. Low sperm concentration (<15 mil/ml) appeared to be related to the HL disease itself, with a higher prevalence in boys who presented with B symptoms (76% vs 26%, aOR 2.3 (95% CI 1.0; 3.8), P = 0.001) compared to those without such symptoms. At 2 -years post-diagnosis, 31 boys provided semen samples for analysis, of whom 12 (39%) boys had oligozoospermia and 4 (13%) had azoospermia, while 22 boys (71%) had low total motile sperm counts (TMSC) (<20 mil). Specifically, the eight boys in the TL3 group treated with DECOPDAC-21 consolidation had low sperm counts and low progressive motility after 2 years (i.e. median sperm count 1.4 mil/ml (IQR <0.1; 5.3), n = 7 (88%), low sperm concentration, low median progressive motility 16.5% (IQR 0.0; 51.2), respectively). Age-adjusted serum FSH levels were significantly raised and inhibin B levels (and inhibin B:FSH ratios) were decreased during chemotherapy in (post)pubertal boys, with subsequent normalization in 80% (for FSH) and 60% (for inhibin B) of boys after 2 years. Only 4 out of the 14 (post)pubertal boys (29%) with low sperm concentrations after 2 years had elevated FSH (>7.6 IU/l), while 7 (50%) had low inhibin B levels (<100 ng/l). In pre-pubertal boys, reproductive hormones were low overall and remained relatively stable during chemotherapy. LIMITATIONS, REASONS FOR CAUTION:The present analyses included sperm and laboratory measurements up to 2 years post-diagnosis. Long-term reproductive outcomes and potential recovery of spermatogenesis remain unknown, while recovery was reported up to 5- or even 10-year post-chemotherapy in previous studies. Boys who were pre-pubertal at diagnosis were still too young and/or physically not able to deliver semen after 2 years and we could not assess a potential difference in gonadotoxicity according to pubertal state at the time of treatment. Overall, the statistical power of the analyses on sperm concentration and quality after 2 years was limited. LIMITATIONS, REASONS FOR CAUTION The present analyses included sperm and laboratory measurements up to 2 years post-diagnosis. Long-term reproductive outcomes and potential recovery of spermatogenesis remain unknown, while recovery was reported up to 5- or even 10-year post-chemotherapy in previous studies. Boys who were pre-pubertal at diagnosis were still too young and/or physically not able to deliver semen after 2 years and we could not assess a potential difference in gonadotoxicity according to pubertal state at the time of treatment. Overall, the statistical power of the analyses on sperm concentration and quality after 2 years was limited. WIDER IMPLICATIONS OF THE FINDINGS:Results of the semen analyses conducted among the 31 boys who had provided a semen sample at 2 years post-treatment were generally poor. However, additional long-term and adequately powered data are crucial to assess the potential recovery and clinical impact on fertility. The participating boys will be invited to deliver a semen sample after 5 years. Until these data become available, benefits of intensified chemotherapy in cHL treatment to reduce radiotherapy and lower risk for development of secondary tumours should be carefully weighed against potentially increased risk of other late effects, such as diminished fertility due to the increased chemotherapy burden. Boys with newly diagnosed cHL should be encouraged to deliver sperm for cryopreservation whenever possible. However, patients and clinicians should also realize that the overall state of disease and inflammatory milieu of cHL can negatively affect sperm quality and thereby reduce chance of successful fertility preservation. Furthermore, the measurement of FSH and inhibin B appears to be of low value in predicting low sperm quality at two years from cHL treatment. STUDY FUNDING/COMPETING INTEREST(S):This study was funded by the Dutch charity foundation KiKa (project 257) that funds research on all forms of childhood cancer. C.M.-K., D.K., W.H.W., D.H., MC, A.U., and A.B. were involved in the development of the EuroNet-PHL-C2 regimen. The other authors declare no potential conflict of interest

The impact of treatment for childhood classical Hodgkin lymphoma according to the EuroNet-PHL-C2 protocol on serum anti-Mullerian Hormone

STUDY QUESTION: What is the impact of the EuroNet-PHL-C2 treatment protocol for children with classical Hodgkin lymphoma (cHL) on gonadal function in girls, based on assessment of serum anti-Mullerian € hormone (AMH)? SUMMARY ANSWER: Serum AMH levels decreased after induction chemotherapy and increased during subsequent treatment and 2 years of follow-up, with lowest levels in patients treated for advanced stage cHL. WHAT IS KNOWN ALREADY: Treatment for cHL, particularly alkylating agents and pelvic irradiation, can be gonadotoxic and result in premature reduction of primordial follicles in females. The current EuroNet-PHL-C2 trial aims to reduce the use of radiotherapy in standard childhood cHL treatment, by intensifying chemotherapy. This study aims to assess the gonadotoxic effect of the EuroNet-PHL-C2 protocol. STUDY DESIGN, SIZE, DURATION: This international, prospective, multicenter cohort study is embedded in the EuroNet-PHL-C2 trial, an European phase-3 treatment study evaluating the efficacy of standard cHL treatment with OEPA-COPDAC-28 (OEPA: vincristine, etoposide, prednisone, and doxorubicin; COPDAC-28: cyclophosphamide, vincristine, prednisone, and dacarbazine) versus intensified OEPA-DECOPDAC-21 (DECOPDAC-21: COPDAC with additional doxorubicin and etoposide and 25% more cyclophosphamide) in a randomized setting. Participants were recruited between January 2017 and September 2021. PARTICIPANTS/MATERIALS, SETTING, METHODS: Female patients aged ≤18 years, treated according to the EuroNet-PHL-C2 protocol for cHL were recruited across 18 sites in the Netherlands, Belgium, Germany, Austria, and Czech Republic. All parents and patients (aged ≥12 years old) provided written informed consent. Serum AMH levels and menstrual cycle characteristics were evaluated over time (at diagnosis, one to three times during treatment and 2 up to 5 years post-diagnosis) and compared between treatment-levels (TL1, TL2, and TL3) and treatment-arms (OEPA-COPDAC-28 and OEPA-DECOPDAC-21). Serum samples obtained from patients after receiving pelvic radiotherapy were excluded from the main analyses. MAIN RESULTS AND THE ROLE OF CHANCE:A total of 104 females, with median age at diagnosis of 15.6 years (IQR 13.7; 17.0), were included in the analysis. Ninety-nine were (post)pubertal. Eighteen girls were diagnosed with an early stage of cHL (TL1) and 86 with intermediate or advanced stage disease (50 TL2 and 36 TL3, 66% received COPDAC-28 and 34% DECOPDAC-21). Five patients received pelvic radiotherapy. Median AMH level at diagnosis was 1.7 mg/l (IQR 0.9; 2.7). After two courses of OEPA chemotherapy, AMH levels decreased substantially in all patients (98% &lt;0.5 mg/l), followed by a significant increase during the consolidation treatment and follow-up. After 2 years, 68% of patients reached their baseline AMH value, with overall median recovery of 129% (IQR 75.0; 208.9) compared to baseline measurement. Five patients (7%) had AMH &lt;0.5 mg/l. In patients treated for advanced stage disease, AMH levels remained significantly lower compared to early- or intermediate stage disease, with median serum AMH of 1.3 mg/l (IQR 0.8; 2.1) after 2 years. Patients who received DECOPDAC-21 consolidation had lower AMH levels during treatment than patients receiving COPDAC-28, but the difference was no longer statistically significant at 2 years post-diagnosis. Of the 35 postmenarchal girls who did not receive hormonal co-treatment, 19 (54%) experienced treatment-induced amenorrhea, two girls had persisting amenorrhea after 2 years. LIMITATIONS, REASONS FOR CAUTION: The studied population comprises young girls with diagnosis of cHL often concurring with pubertal transition, during which AMH levels naturally rise. There was no control population, while the interpretation of AMH as a biomarker during childhood is complex. The state of cHL disease may affect AMH levels at diagnosis, potentially complicating assessment of AMH recovery as a comparison with baseline AMH. The current analysis included data up to 2–5 years post-diagnosis. WIDER IMPLICATIONS OF THE FINDINGS: The current PANCARE guideline advises to use the cyclophosphamide-equivalent dose score (CED-score, as an estimation of cumulative alkylating agent exposure) with a cut-off of 6000 mg/m2 to identify females aged &lt;25 years at high risk of infertility. All treatment-arms of the EuroNet-PHL-C2 protocol remain below this cut-off, and based on this guideline, girls treated for cHL should therefore be considered low-risk of infertility. However, although we observed an increase in AMH after chemotherapy, it should be noted that not all girls recovered to pre-treatment AMH levels, particularly those treated for advanced stages of cHL. It remains unclear how our measurements relate to age-specific expected AMH levels and patterns. Additional (long-term) data are needed to explore clinical reproductive outcomes of survivors treated according to the EuroNet-PHLC2 protocol.</p

Clinical and self-reported markers of reproductive function in female survivors of childhood Hodgkin lymphoma

Purpose: To evaluate the impact of treatment for Hodgkin lymphoma (HL) on clinical reproductive markers and pregnancy outcomes.Methods: This study was embedded within the DCOG LATER-VEVO study; a Dutch, multicenter, retrospective cohort study between 2004 and 2014. Serum anti-Müllerian hormone (AMH), follicle stimulating hormone (FSH), inhibin B, antral follicle count (AFC), and self-reported (first) pregnancy outcomes were evaluated in female childhood HL survivors and controls.Results: 84 HL survivors and 798 controls were included, aged 29.6 and 32.7 years old at time of assessment. Median age at HL diagnosis was 13.4 years. Cyclophosphamide equivalent dose (CED-score) exceeded 6000 mg/m2 in 56 women and 14 survivors received pelvic irradiation.All clinical markers were significantly deteriorated in survivors (odds-ratio for low AMH (&lt; p10) 10.1 [95% CI 4.9; 20.6]; low AFC (&lt; p10) 4.6 [95% CI 2.1; 9.9]; elevated FSH (&gt; 10 IU/l) 15.3 [95% CI 5.7; 41.1], low Inhibin B (&lt; 20 ng/l) 3.6 [95% CI 1.7; 7.7], p &lt; 0.001). Pregnancy outcomes were comparable between survivors and controls (± 80% live birth, ± 20% miscarriage). However, survivors were significantly younger at first pregnancy (27.0 years vs 29.0 years, P = 0.04). Adjusted odds-ratio for time to pregnancy &gt; 12 months was 2.5 [95% CI 1.1; 5.6] in survivors, p = 0.031. Adverse outcomes were specifically present after treatment with procarbazine and higher CED-score.Conclusion: HL survivors appear to have an impaired ovarian reserve. However, chance to achieve pregnancy seems reassuring at a young age. Additional follow-up studies are needed to assess fertile life span and reproductive potential of HL survivors, in particular for current HL treatments that are hypothesized to be less gonadotoxic.</p