Median MMP-8 concentrations on admission were 96 ng/mL (IQR 44–188 ng/mL) and 147 ng/mL (IQR 53–214 ng/mL) for the whole cohort and the subgroup, respectively. Corresponding median TIMP-1 concentrations were 265 ng/mL (IQR 154–466 ng/mL) and 348 ng/mL (IQR 214–663 ng/mL), respectively.
Association of MMP-8 and TIMP-1 with 90-Day Mortality
Of the 563 patients, 132 (23.4%) died within 30 and 152 (27.0%) within 90 days of hospital admission. ICU mortality was 9.2% (N = 52). A comparison of demographics, severity scores, and admission MMP-8, TIMP-1, and MMP-8/TIMP-1 molar ratio between 90-day survivors and nonsurvivors is shown in Table 2.
The 90-day nonsurvivors were older and had more severe disease on admission by SAPS II, SOFA at 24 hours from admission, and maximal SOFA score. Among the 90-day nonsurvivors, there was significantly more septic shock, chronic cardiac disease, ARDS, and fewer postoperative patients than among 90-day survivors. The groups were not different with respect to the presence of infection or sepsis. A significant difference emerged in admission TIMP-1 levels, WMWodds 1.68 (95% CI, 1.43–2.08), and possibly in admission MMP-8, WMWodds 1.20 (95% CI, 1.01–1.43) but not in the MMP-8/TIMP-1 molar ratio, WMWodds 0.83 (95% CI, 0.67–1.04), between 90-day survivors and nonsurvivors.
To test the ability of MMP-8 and TIMP-1 to differentiate survivors from nonsurvivors, we constructed ROC curves and calculated AUCs for MMP-8A and TIMP-1A and TIMP-1H. Figure 2 demonstrates the ROC curves for MMP-8A and TIMP-1A. The AUCs for MMP-8A, TIMP-1A, and TIMP-1H were 0.55 (95% CI, 0.49–0.60), 0.63 (95% CI, 0.58–0.69), and 0.61 (95% CI, 0.55–0.67), respectively. In backward logistic regression analysis, TIMP-1A was independently associated (P = 0.004) with 90-day mortality when MMP-8, age, SAPS II without age points, SOFA at 24 hours from admission, and chronic cardiac disease were included in the model. The log odds were linearly related to each independent variable. The model also included interaction terms between (1) SAPS II without age points and age and (2) SOFA score and age. The Hosmer-Lemeshow χ2 of the final model was 11.14, P = 0.194.
At the optimal cutoff level of TIMP-1A 458.6 ng/mL (95% CI, 362.1–736.7 ng/mL), the sensitivity and specificity of the test to discriminate 90-day nonsurvivors and 90-day survivors were 0.41 and 0.80, respectively. The positive likelihood ratios for TIMP-1A and TIMP-1H at the cutoff level of 458.6 ng/mL were 2.08 (95% CI, 1.6–2.8) and 1.63 (95% CI, 1.25–2.13), respectively. At the cutoff level of 127.3 ng/mL (95% CI, 31.3–330.5 ng/mL), MMP-8A had a positive likelihood ratio of 1.26 (95% CI, 1.00–1.60).
To further investigate the association of TIMP-1 with mortality, we calculated 90-day mortality in different quartiles of TIMP-1A. The Kaplan-Meier survival curves are shown in Figure 3. The difference in 90-day survival between groups based on TIMP-1A levels divided into quartiles was statistically significant. In patients with TIMP-1A levels exceeding 458.6 ng/mL, mortality was significantly higher (P < 0.001) than in patients with levels below the cutoff.
Associations of MMP-8 and TIMP-1 levels with severity of respiratory failure were significant (P = 0.002 and P< 0.001, respectively). In the quartile with the lowest PaO2/FIO2 on admission (lowest value during the preceding 6 hours of mechanical ventilation), MMP-8 (P = 0.002) and TIMP-1 (P < 0.001) levels were higher than in the highest PaO2/FIO2 quartile, and the levels decreased sequentially with increasing PaO2/FIO2 ratio.
Correlations of MMP-8 and TIMP-1 with Infection Parameters and Severity of Disease
MMP-8A correlated significantly with TIMP-1A (P < 0.001, Rho = 0.247), white blood cell (WBC) count (P < 0.001, Rho=0.241), and C-reactive protein (CRP) (P< 0.001, Rho=0.409). MMP-8A was negatively correlated with PaO2/FIO2 ratio (P < 0.001, Rho = −0.162). TIMP-1A correlated with MMP-8A, CRP (P < 0.001, Rho = 0.409), SOFA score at 24 hours from ICU admission (P < 0.001, Rho = 0.323), and SAPS II score (P < 0.001, Rho = 0.162). TIMP-1A was negatively correlated with PaO2/FIO2 ratio (P < 0.001, Rho = −0.260).
MMP-8 and TIMP-1 in Different Modes of Mechanical Ventilation
Most patients with an admission blood sample (N =426, 81.6%) received invasive mechanical ventilation during the study, whereas 63 patients (11.2%) received only noninvasive ventilatory support by either conventional continuous positive airway pressure (CPAP) or CPAP with pressure support. In 18 patients (3.4%), noninvasive ventilation was converted to invasive ventilation before inclusion in the study. In 17 patients (3.3%), noninvasive ventilation was converted to invasive ventilation after inclusion in the study. Admission MMP-8 levels were lowest in the invasively ventilated patients (P < 0.001). There were no differences in TIMP-1A levels among the 4 groups (P = 0.630). SAPS II score and SOFA score at 24 hours were higher in the invasively ventilated patients, but we detected no differences in mortality among the groups. MMP-8A was not different between survivors and nonsurvivors in any of the groups. TIMP-1A was significantly higher, WMWodds 1.85 (95% CI, 1.44–2.42), in nonsurvivors only in the group of invasively ventilated patients, which represented the largest group of patients.
MMP-8 and TIMP-1 in Patients with ARDS
The ARDS subgroup comprised 43 patients. Twenty-four (55.8%) of these patients were classified as ALI/non-ARDS by the AECC criteria. Admission MMP-8 was not different in the ARDS subgroup than in the other patients, median 147 ng/mL (IQR 53–214 ng/mL) vs 94 ng/mL (IQR 43–179 ng/mL), WMWodds 1.24 (95% CI, 0.85–1.84). TIMP-1A was higher in the ARDS subgroup than in other patients, median 348 ng/mL (IQR 214–663 ng/mL) vs 254 ng/mL (IQR 151–458 ng/mL), WMWodds 1.65 (95% CI, 1.15–2.44). In the ARDS subgroup, the differences in MMP-8A, WMWodds 0.80 (95% CI, 0.37–1.64) or TIMP-1A, WMW odds 1.88 (95% CI, 0.93–4.71), levels between 90-day survivors and nonsurvivors were not significant. AUC for TIMP-1A in ROC analysis was 0.69 (95% CI, 0.52–0.85) (P = 0.03). At the cutoff level of 392.4 ng/mL, the sensitivity and specificity for differentiating 90-day nonsurvivors from 90-day survivors were 0.62 and 0.76, respectively, and the positive likelihood ratio was 2.60 (95% CI, 1.13–6.00). Survival in groups divided into quartiles, according to TIMP-1A concentrations, may have differed among groups (log-rank, P = 0.02).
MMP-8 and TIMP-1 in Patients with Sepsis
Because MMP-8 and TIMP-1 are upregulated in septic patients,12 we performed a subgroup analysis on patients diagnosed as having sepsis within 48 hours of admission. Of the 522 patients with an admission blood sample, 68 (13%) were diagnosed as having sepsis (information missing in 2 patients).
Median MMP-8A levels in nonseptic (N = 452) and septic (N = 68) patients were 89.27 ng/mL (IQR 41.57–166.52 ng/mL) and 170.82 ng/mL (IQR 81.79–287.81 ng/mL), respectively, WMWodds 2.06 (95% CI, 1.52–2.90). The corresponding median TIMP-1A levels were 233.44 ng/mL (IQR 142.12–406.31 ng/mL) and 573.98 ng/mL (IQR 372.32–750.64 ng/mL), respectively, WMWodds 5.06 (95% CI, 3.74–7.33). MMP-8A levels did not differ significantly between 90-day survivors and nonsurvivors in either nonseptic patients, WMWodds 1.19 (95% CI, 0.93–1.53) or septic patients, WMWodds 0.88 (95% CI, 0.49–1.57). TIMP-1A levels were higher in 90-day nonsurvivors than in 90-day survivors in patients with no diagnosed sepsis, WMWodds 1.67 (95% CI, 1.31–2.16). In septic patients, the difference was not significant, WMWodds 1.60 (95% CI, 0.89–3.18).
MMP-8 and TIMP-1 in Elective and Emergency Admissions
Ninety-five patients (17%) were admitted to an ICU after an elective operation. Of these, 59 (62%) were operated on for cardiac disease, 7 (7%) for respiratory neoplasms, 7 (7%) for gastrointestinal neoplasms, 6 (6%) for abdominal aortic aneurysms, and 16 (17%) for other diseases. To control for possible bias from this patient group, we analyzed TIMP-1A and MMP-8A separately in these 2 groups. Median MMP-8A and TIMP-1A in elective patients were 61 ng/mL (IQR 37–120 ng/mL) and 254 ng/mL (IQR 164–449 ng/mL), respectively. In patients with emergency admission, median MMP-8A and TIMP-1A were 105 ng/mL (IQR 46–205 ng/mL) and 269 ng/mL (IQR 144–472 ng/mL), respectively. The difference between these groups was significant in MMP-8A, WMWodds 1.54 (95% CI, 1.21–1.99) but not in TIMP-1A, WMWodds 0.99 (95% CI, 0.78–1.26). MMP-8 performed poorly in predicting 90-day mortality in both groups (AUC 0.63 [95% CI, 0.45–0.81], [ns] for elective patients and AUC 0.52 [95% CI, 0.46–0.58] for emergency patients). By contrast, TIMP-1A showed some predictive ability in the elective patient group (AUC 0.79 [95% CI, 0.65–0.92]) but little in the emergency group (AUC 0.62 [95% CI, 0.56–0.68]) in predicting 90-day mortality.
We demonstrated that serum levels of TIMP-1 and possibly MMP-8 are associated with increased mortality in a large unselected group of patients treated with mechanical ventilation for >6 hours. Admission TIMP-1 was found to be an independent predictor of mortality, although TIMP-1 performed better in the ARDS subgroup than in the whole cohort in predicting 90-day mortality.
To the best of our knowledge, this is the first study to show an association of TIMP-1 with increased mortality in a large population of critically ill patients with ARF. We and others have reported an association of elevated TIMP-1 with increased mortality in septic patients.11–13 In 1 study of 192 septic patients, elevated TIMP-1 and lower MMP-9 were associated with increased mortality.13 The AUC values for TIMP-1 were comparable with our findings. However, the optimal cutoff value was slightly higher (531 ng/mL) than the 458.6 ng/mL in our study.13 The most likely explanation is the uncertainty (wide 95% CI) of the cutoff in our study. In a smaller study of patients with severe sepsis, plasma levels of TIMP-1 performed well in discriminating survivors from nonsurvivors: AUC for TIMP-1 was 0.78, and patients with TIMP-1 levels >3200 ng/mL had a 4.5 times higher risk of dying than others.11 The cutoff level was higher than in our study, which may reflect differences in patient populations. In septic patients, systemic TIMP-1 levels may be higher generally than in our more heterogeneous patient population. Of our patients, only 13% (N = 68) had sepsis within 48 hours of admission. In our subgroup analysis, patients with sepsis had higher MMP-8A and TIMP-1A levels than nonseptic patients, but TIMP-1A did not differ between 90-day survivors and 90-day nonsurvivors. This may have been due to the relatively low number of patients with sepsis in the cohort. Notably, the ranges of the TIMP-1A levels in surviving and nonsurviving patients overlapped. Thus, TIMP-1 on its own may not be a reliable predictor of survival.
In the ARDS subgroup, TIMP-1 was elevated, and high levels were associated with increased mortality. In ARDS patients, an association of TIMP-1 with mortality has been reported.21 Decreased MMP-9/TIMP-1 ratio in bronchoalveolar lavage fluid has been associated with prolonged ARDS and worse prognosis.21,22 In these studies, the main focus was on MMPs, and elevated TIMP-1 had been interpreted in the context of MMP inhibition. TIMP-1 inhibits both MMP-8 and MMP-9 by noncovalently binding to the active part of MMP. In normal physiology, this balance between proteinase and inhibitor is strictly regulated, and an excess of uninhibited proteinase may cause tissue damage.8 In addition to the MMP-inhibiting activity, other biological functions of TIMP-1 have been described. TIMP-1 is reported to promote fibroblast and other cell proliferation, and it possesses antiapoptotic and proinflammatory effects.23–26 TIMP-1 is expressed in a variety of cells that participate in acute inflammatory reactions.23,24 Expression of TIMP-1 is stimulated by various substances, including platelet-derived growth factors, basic fibroblast growth factor, epidermal growth factor, and interleukins −1, −6, and −10,23–27 and it is elevated in hyperoxia-induced lung injury.28,29 Platelet-derived growth factors and basic fibroblast growth factor have a profibrotic effect in acute lung injury.30,31
Interestingly, in our study, TIMP-1 levels were associated with severity of oxygenation disturbance. This may be explained by an increased oxidative stress in the groups of worse gas exchange. TIMP-1 has been shown to be elevated in connection with hyperoxia in the lung28,29,32 and in oxidative damage to the venous endothelium.33 Unfortunately, measuring the level of oxidative stress or a surrogate other than the PaO2/FIO2 ratio was not possible with our data. Another explanation may lie in the functions of TIMP-1 that are independent of MMP inhibitory activity.23–26 Hypothetically, the stimulation of fibrotic processes, simultaneous inhibition of proteolysis, and promotion of proinflammatory processes by TIMP-1 could worsen the outcome in ARDS. Also, the antiapoptotic effect could inhibit the clearance of inflammatory cells from the lung. The biological significance of TIMP-1 in ARDS is a subject for further studies.
In our study, MMP-8 levels may have differed between survivors and nonsurvivors. However, the values were largely overlapping, and MMP-8 predicted outcome poorly. This is in broad accordance with Fligiel et al.,34 who noted in a smaller study that MMP-8 and MMP-9 were not associated with disease severity or mortality of ARDS patients. To the best of our knowledge, there are no previous studies on an association of systemic MMP-8 with mortality in a large cohort of ARF patients. MMP-8 and MMP-9 have been investigated in diverse diagnostic groups, and they tend to be elevated in acute illness10–12,21,22,34–37 and chronic lung diseases.38–40
MMPs play a role in many stages of acute inflammatory reactions. Their synthesis is induced, and their degranulation from leukocytes is stimulated by proinflammatory mediators. Local production of the enzyme has been demonstrated as well.7 Preclinical studies provide evidence of both harmful and beneficial roles in lung injury.14 In a clinical study of pediatric patients, elevated MMP-8 in tracheal aspirate samples was associated with increased duration of mechanical ventilation, but the patient population was too small to evaluate mortality.10
Fligiel et al.34 analyzed different MMPs in the bronchoalveolar lavage fluids of 28 ARDS patients. They found elevated levels of neutrophil-derived MMP-8 and MMP-9 and TIMP-1, and their strong correlation with neutrophil count was seen. Similarly, in our study, MMP-8 and TIMP-1 levels correlated with CRP and WBC count. MMP-8 levels were similar in the ARDS subgroup to those in the whole cohort. MMP-8 performed poorly as a predictive marker also in this subgroup. According to our results, systemic MMP-8 is not useful as a biomarker in ARF or ARDS.
There are several explanations for why MMP-8 and TIMP-1 performed differently in predicting mortality, although TIMP-1 is a MMP-8 inhibitor. First, systemic MMP-8 may not reflect the activity or concentration of the enzyme at the tissue level. In most studies, MMP-8 has been measured in bronchoalveolar lavage and tracheal aspirate samples, and it is possible that systemic levels reflect local processes poorly. MMP-8 levels were considerably higher in peritoneal fluid than in serum in a study of patients with peritonitis, and the levels did not correlate.37 MMP-8 levels are likely highest in the compartment where neutrophils accumulate because neutrophils are the principal source of MMP-8. TIMP-1, in turn, is released by several cell types, with the exception of neutrophils, and is found in most body fluids.24
The regulation of MMP-8 and TIMP-1 is not identical. First, depending on cell type, some cytokines may either upregulate or downregulate TIMP-1.41,42 Second, in serum, MMP-8 is inhibited by nonspecific plasma antiproteinases, for example, α2-macroglobulin.43 Methods used for measuring MMP-8, such as ELISA and IFMA, do not differentiate between the enzyme bound to its inhibitor and the free active enzyme. Therefore, measuring systemic MMP-8 levels by these methods may not reflect the true amount of biologically active enzyme. MMP-8 may also avoid its inhibitors at the tissue level when bound to the cell membrane of activated neutrophils.44 Third, TIMP-1 may have several functions independent of MMP inhibition.23–26 Finally, MMP-8 may have a dual role in inflammation, possibly initially harmful but later important in healing processes.14 This would explain why expression of high levels of the enzyme on admission is not associated with long-term outcome.
Mechanical ventilation per se may affect MMP-845 and MMP-9.46 In two-thirds of our patients, MMP-8 levels increased from admission to 48 hours later (data not shown). Due to different courses in the baseline diseases, our data were insufficient to analyze the relationship between mechanical ventilation and MMP-8 levels.
In this large, prospective, multicenter study, we included a clinically relevant group of critically ill patients, which may be seen as a strength of the study. Because different diagnostic groups were included, our results are generalizable. Our findings do not have direct clinical implications. However, biomarker panels for predictive purposes have recently been tested in a clinical study.47 According to our study, TIMP-1 is an independent predictor of poor outcome and may thus have potential in clinical outcome prediction as a component of a panel of selected biomarkers. This is a subject for future research.
Some limitations of the study must be addressed. First, we included patients who received invasive and noninvasive mechanical ventilation. The severity of disease may have been different between the groups. However, most patients received invasive ventilation, and this was the only group in which TIMP-1 levels differed between survivors and nonsurvivors. Our results may be generalizable for invasively ventilated patients, but they should be interpreted with caution for noninvasively ventilated patients. The large unselected group of patients also include patients not fitting the description “critically ill” despite the need for prolonged mechanical ventilation. Inclusion of elective postoperative patients may have biased our results. However, in a subgroup analysis of elective and emergency patients, the results were similar to those for the entire cohort. TIMP-1 had some predictive ability in discriminating 90-day survivors from nonsurvivors in patients needing mechanical ventilation after an elective operation. A relatively large proportion of cardiac surgical patients in the electively operated patient group may explain why TIMP-1 also had some predictive ability in this group. TIMP-1 has been shown to predict adverse cardiac events in patients with cardiac disease.48,49
The second limitation is that our data lacked information of admission lactate levels, which are important in outcome prediction.50 Furthermore, an extended analysis of cytokine levels and MMP-9 would have facilitated interpretation of the inflammatory status of different patient groups. In the subgroup, serial measurements of our markers and other inflammatory variables could have enhanced our knowledge about the time course of these markers throughout the disease.
The last limitation is the study size. According to a post hoc sample size calculation, based on 95% CIs for AUC as in a previous study,51 and the number of 90-day deaths in this study population, a sample size of at least 500 patients would yield clinically precise AUCs with 95% CIs <0.15 with 90-day mortality. The sample size calculation was only an approximate estimation due to an unknown distribution of measured laboratory variables. Nevertheless, the current study is one of the largest in this field.
In conclusion, high levels of TIMP-1 were independently associated with increased 90-day mortality in a large group of critically ill, mechanically ventilated patients. However, serum MMP-8 levels did not predict outcome in ARF or ARDS patients.
Appendix. Participants in the FINNALI Study
Central Finland Central Hospital: Dr. Raili Laru-Sompa, Tiina Kirkhope, Sirpa Nykänen; East Savo Central Hospital: Dr. Markku Suvela, Sirpa Kauppinen, Anne-Marja Turkulainen; Helsinki University Central Hospital (HUCH)-Jorvi Hospital: Dr. Tero Varpula, Mira Rahkonen; HUCH-Meilahti Hospital ICU: Dr. Anne Kuitunen, Dr. Anna-Maija Korhonen, Dr. Rita Linko, Dr. Marjatta Okkonen, Janne Myller, Jarmo Pekkola, Leena Pettilä, Sari Sutinen; HUCH-Meilahti Hospital, Cardiac Surgical ICU: Dr. Raili Suojaranta-Ylinen, Dr. Sinikka Kukkonen, Elina Nurmi-Toivonen; HUCH-Meilahti Hospital, Department of Medicine: Dr. Tom Bäcklund, Dr. Juhani Rossinen, Riina Mäkelä; HUCH-Töölö Hospital: Dr. Janne Reitala, Dr. Jyrki Vuola, Raija Niemi, Marja-Leena Pihlajamaa, Aira Uusipaavalniemi; HUCH-Surgical Hospital: Dr. Anna-Maria Koivusalo, Pasi Kyllönen; Kanta-Häme Central Hospital: Dr. Risto Puolakka, Tarja Heikkilä, Piia Laitinen; Keski-Pohjanmaa Central Hospital: Dr. Samuli Forsström, Dr. Tadeusz Kaminski, Tuija Kuusela; Kuopio University Hospital: Dr. Esko Ruokonen, Dr. Ilkka Parviainen, Elina Halonen, Sari Rahikainen; Kymenlaakso Central Hospital: Dr. Jussi Pentti, Dr. Seija Alila, Reija Koskinen; Lappi Central Hospital: Dr. Outi Kiviniemi, Tarja Laurila, Tiina Pikkuhookana; North Karelia Central Hospital: Dr. Matti Reinikainen, Tanja Eiserbeck, Helena Jyrkönen; Oulu University Hospital: Dr. Tero Ala-Kokko, Tarja Lamberg, Sinikka Sälkiö; Päijät-Häme Central Hospital: Dr. Pekka Loisa; Satakunta Central Hospital: Dr. Vesa Lund, Pauliina Perkola, Päivi Tuominen; Seinäjoki Central Hospital: Dr. Kari Saarinen, Dr. Matti Viitanen, Niina Siirilä, Johanna Soini; South Karelia Central Hospital: Dr. Seppo Hovilehto, Dr. Anne Kirsi, Dr. Pekka Tiainen, Sanna Asikainen; South Savo Central Hospital: Dr. Heikki Laine, Pekka Kettunen, Kirsi Reponen; Tampere University Hospital: Dr. Sari Karlsson, Dr. Jyrki Tenhunen, Anna-Liina Korkala, Samuli Kortelainen, Sanna Mäkinen, Minna-Liisa Peltola; Turku University Hospital: Dr. Juha Perttilä, Dr. Olli Arola, Dr. Outi Inkinen, Dr. Erkki Kentala, Jutta Kotamäki; Vaasa Central Hospital: Dr. Pentti Kairi.
Name: Johanna Hästbacka, MD, PhD.
Contribution: This author contributed to the study concept, conducted the statistical analysis and interpretation of the data, and prepared the manuscript.
Attestation: Johanna Hästbacka has reviewed the original study data and the data analysis. Johanna Hästbacka is the archival author.
Name: Rita Linko, MD, PhD.
Contribution: This author helped contributing to the study concept, collecting the clinical data, and drafting the manuscript.
Attestation: Rita Linko has approved the final manuscript.
Name: Taina Tervahartiala, DDS, PhD.
Contribution: This author contributed to the laboratory analysis of the samples and drafting the manuscript.
Attestation: Taina Tervahartiala has approved the final manuscript.
Name: Tero Varpula, MD, PhD.
Contribution: This author helped contributing to the study concept, collecting the data, and drafting the manuscript.
Attestation: Tero Varpula has approved the final manuscript.
Name: Seppo Hovilehto, MD.
Contribution: This author contributed to collecting the clinical data and laboratory samples and drafting the manuscript.
Attestation: Seppo Hovilehto has approved the final manuscript.
Name: Ilkka Parviainen, MD, PhD.
Contribution: This author contributed to collecting the clinical data and laboratory samples and drafting the manuscript.
Attestation: Ilkka Parviainen has approved the final manuscript.
Name: Suvi T Vaara, MD, PhD.
Contribution: This author contributed to analyzing the results and drafting the manuscript.
Attestation: Suvi Vaara has approved the final manuscript.
Name: Timo Sorsa, DDS, PhD.
Contribution: This author contributed to the analysis of the laboratory samples and drafting the manuscript.
Attestation: Timo Sorsa has approved the final manuscript.
Name: Ville Pettilä, MD, PhD.
Contribution: This author helped contributing to the study concept, collecting the clinical data, analyzing the results, and drafting the manuscript.
Attestation: Ville Pettilä attests to having reviewed the original study data and data analysis and having approved the final manuscript.
This manuscript was handled by: Steven L. Shafer, MD.
We thank Carol Pelli for expert language editing and Professor Antti Nevanlinna for statistical advice.
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