MD Vaneja Velenik
email: vvelenik@onko-i.si

J Ocvirk

I Oblak

F Anderluh

A phase II study of cetuximab, capecitabine and radiotherapy in neoadjuvant treatment of patients with locally advanced resectable rectal cancer

Keywords: rectal cancer, neoadjuvant chemoradiotherapy, capecitabine, cetuximab

Background

Neoadjuvant chemoradiotherapy (CRT) reduces local tumor recurrence in locally advanced rectal cancer (LARC). This phase II study assessed neoadjuvant cetuximab with capecitabine-based CRT in LARC.

Methods

Patients with stage II/III LARC received capecitabine 1250 mg/m 2 twice daily for 2 weeks followed by intravenous cetuximab 400 mg/m 2 at week 3, then weekly intravenous 250 mg/m 2 cetuximab plus CRT including capecitabine 825 mg/m 2 twice daily (including weekends during radiotherapy) with radiotherapy of 45 Gy (25 x 1.8 Gy), 5 days a week for 5 weeks. Total mesorectal excision was scheduled 4-6 weeks following completion of CRT.

The primary endpoint was pathological complete response (pCR).

Results

Thirty-seven patients were eligible for safety and efficacy. TMN staging at baseline was: T4N2, 11%; T3N2, 40%; T2N2, 3%; T3N1, 35%; T2N1, 3% and T3N0 8%. The most common adverse events included, grade 1/2 acneiform skin rash (86%), and grade 3 radiodermatitis, (16%), diarrhea (11%) and hypersensitivity (5%). pCR was achieved in 3 patients (8%). Overall-, T-and N-downstaging rates were 73%, 57% and 81% respectively.

Total sphincter preservation rate was 76%, and 53% in 17 patients whose tumors were located within 5 cm from the anal verge. Non-fatal perioperative complications occurred in 13 patients (35%) with delayed wound healing occurring in 6 patients (16%). One death was recorded due to sepsis following colonic necrosis.

Conclusion

Neoadjuvant cetuximab with capecitabine-based CRT is tolerable in patients with resectable LARC. Whilst the pCR rate was similar to recent reports, a high pathological downstaging rate was achieved.

Introduction

Multimodal treatment strategies aim to reduce tumor recurrence rates and improve survival in patients with resectable locally advanced rectal cancer (LARC). Pre-and postoperative radiotherapy are reported to decrease the risk of local relapse in this setting. [START_REF] Kapiteijn | Preoperative radiotherapy combined with total mesorectal excision for resectable rectal cancer[END_REF][START_REF] Camma | Preoperative radiotherapy for resectable rectal cancer: A meta-analysis[END_REF][START_REF]Adjuvant radiotherapy for rectal cancer: a systematic overview of 8,507 patients from 22 randomised trials[END_REF] Chemotherapy in combination with radiotherapy can act as a radiosensitizing agent, potentially eradicating micrometastases. The combination of postoperative 5-fluorouracil (5-FU)-based chemotherapy with radiation in the treatment of LARC is reported to improve patient disease-free survival (DFS) and overall survival. [START_REF] O'connell | Improving adjuvant therapy for rectal cancer by combining protracted-infusion fluorouracil with radiation therapy after curative surgery[END_REF] Lower rates of local regional failure (13% vs. 6%) in patients receiving preoperative 5-FU-based chemoradiotherapy (CRT) compared with postoperative 5-FU-based CRT have also been demonstrated. [START_REF] Sauer | Preoperative versus postoperative chemoradiotherapy for rectal cancer[END_REF] The addition of continuous infusion (CI) 5-FU-based chemotherapy concurrently to preoperative long-term fractionation radiation is now considered by many to be the standard of care for LARC patients in Europe, following data from prospective randomized studies. [START_REF] Gerard | Preoperative radiotherapy with or without concurrent fluorouracil and leucovorin in T3-4 rectal cancers: results of FFCD 9203[END_REF][START_REF] Bosset | Chemotherapy with preoperative radiotherapy in rectal cancer[END_REF] However, whilst rates of pathological complete response (pCR) and local control were found to be higher in the CRT arms than radiotherapy alone arms, no significant improvements to DFS or overall survival were found, with the occurrence of metastatic disease remaining a problem. [START_REF] Gerard | Preoperative radiotherapy with or without concurrent fluorouracil and leucovorin in T3-4 rectal cancers: results of FFCD 9203[END_REF][START_REF] Bosset | Chemotherapy with preoperative radiotherapy in rectal cancer[END_REF][START_REF] Bujko | Sphincter preservation following preoperative radiotherapy for rectal cancer: report of a randomised trial comparing short-term radiotherapy vs. conventionally fractionated radiochemotherapy[END_REF] Thus new combinations of chemotherapeutic agents for CRT in LARC patients are urgently sought.

Capecitabine (Xeloda, Hoffmann-La Roche Ltd, Basel Switzerland) an oral fluoropyrimidine prodrug is readily absorbed into the gastrointestinal tract and activated primarily in tumor cells displaying high levels of thymidine phosphorylase. [START_REF] Schuller | Preferential activation of capecitabine in tumor following oral administration to colorectal cancer patients[END_REF] Capecitabine has the same efficacy as CI 5-FU but with less of the associated toxic side effects, [START_REF] Schuller | Preferential activation of capecitabine in tumor following oral administration to colorectal cancer patients[END_REF][START_REF] Costanzo | Capecitabine, a new oral fluoropyrimidine for the treatment of colorectal cancer[END_REF] and has demonstrated radiosensitizing properties in vivo. [START_REF] Sawada | X-ray irradiation induces thymidine phosphorylase and enhances the efficacy of capecitabine (Xeloda) in human cancer xenografts[END_REF] Furthermore when capecitabine was administered concomitantly with radiotherapy to LARC patients in phase II studies, low toxicity profiles, tumor downstaging and pCR rates ranging from 4-31% were reported. [START_REF] Kim | Preoperative concurrent radiotherapy with capecitabine before total mesorectal excision in locally advanced rectal cancer[END_REF][START_REF] Velenik | Capecitabine as a radiosensitizing agent in neoadjuvant treatment of locally advanced resectable rectal cancer: prospective phase II trial[END_REF] Cetuximab (Erbitux developed by Merck KGaA Darmstadt, Germany [under license from Imclone, NY, USA]) an immunoglobulin G1 monoclonal antibody specifically targets the epidermal growth factor receptor (EGFR), competitively inhibiting ligand binding and liganddependent downstream signaling. [START_REF] Li | Structural basis for inhibition of the epidermal growth factor receptor by cetuximab[END_REF][START_REF] Goldstein | Biological efficacy of a chimeric antibody to the epidermal growth factor receptor in a human tumor xenograft model[END_REF] Cetuximab in combination with chemotherapy has demonstrated improved efficacy in the first-line treatment of patients with EGFR expressing metastatic CRC (mCRC) compared with chemotherapy alone arms. [START_REF] Van Cutsem | Cetuximab and chemotherapy as initial treatment for metastatic colorectal cancer[END_REF][START_REF] Bokemeyer | Fluorouracil, leucovorin, and oxaliplatin with and without cetuximab in the first-line treatment of metastatic colorectal cancer[END_REF] Furthermore cetuximab has been shown to be safely administered with radiotherapy, improving survival in patients with locally advanced squamous cell carcinoma of the head and neck (SCCHN). [START_REF] Bonner | Radiotherapy plus cetuximab for squamous-cell carcinoma of the head and neck[END_REF]

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Page 4 of 14 EGFR expression has been associated with reduced DFS in patients with LARC following preoperative CRT. [START_REF] Li | Epidermal growth factor receptor as a prognostic factor in locally advanced rectal-cancer patients treated with preoperative chemoradiation[END_REF][START_REF] Bertolini | Prognostic and predictive value of baseline and posttreatment molecular marker expression in locally advanced rectal cancer treated with neoadjuvant chemoradiotherapy[END_REF] Thus EGFR-targeted agents appear to be attractive candidates for use with CRT in this setting. Recent phase I/II studies have reported acceptable toxicity and tumor downstaging when cetuximab was administered in combination with fluoropyrimidinebased CRT in rectal cancer patients. [START_REF] Bertolini | Neoadjuvant treatment with single-agent cetuximab followed by 5-FU, cetuximab, and pelvic radiotherapy: a phase II study in locally advanced rectal cancer[END_REF][START_REF] Machiels | Phase I/II study of preoperative cetuximab, capecitabine, and external beam radiotherapy in patients with rectal cancer[END_REF] In the present study the addition of cetuximab to capecitabine-based CRT was investigated in the neoadjuvant treatment of resectable LARC patients.

Patients and Methods

Study design

Main eligibility criteria

Eligible patients had a histologically verified stage II or III adenocarcinoma of the rectum, (International Union against Cancer [UICC] TNM classification 2002). Other inclusion criteria were; ≥18 years of age at diagnosis; World Health Organization (WHO) performance status ≤2; adequate bone marrow, liver, renal and cardiac function (no history of ischemic heart disease); no prior radiotherapy and/or chemotherapy; signed informed consent. Exclusion criteria included; patients with a history of prior malignancy other than non-melanoma skin cancer or in situ carcinoma of the cervix; a known hypersensitivity to biological agents; pregnant or lactating patients.

Pretreatment evaluation

Patient pretreatment work-up comprised a complete history, physical examination, full blood count, serum biochemistry, carcinoembryonic antigen, chest radiography, ultrasonography (US) and/or a computed tomography (CT) scan of the whole abdomen and colonoscopy with biopsy. The extent of locoregional disease was determined by magnetic resonance imaging (MRI) of the pelvis (100%); 56% of patients also had endoscopic US and 7% had a CT scan of the pelvis.

M A N U S C R I P T A C C E P T E D

Patient treatment

A summary of the study treatment is shown in Figure 1. Patients received capecitabine 1250 mg/m² twice daily for 2 weeks. Cetuximab 400 mg/m 2 was intravenously administered on week 3, followed by 250 mg/m 2 /week during radiotherapy. Radiotherapy started on week 4 and was delivered once a day using 15 MV photon beams and a four-field box technique, 5 days a week. The small pelvis received 45 Gy in 25 fractions of 1.8 Gy over 5 weeks. Threedimensional conformal CT-based treatment planning was performed. The clinical target volume (CTV) encompassed the primary tumor, entire mesorectal tissue, and internal iliac and presacral lymph nodes up to the L5/S1 junction and 5 cm distal to the primary tumor. An additional 1 cm in all directions was added to the CTV to obtain the planning target volume (PTV). During irradiation, patients were treated in the prone position, with a full bladder, and a belly-board immobilization device was used. A multileaf collimator was used for shaping the fields and for the protection of normal tissues. Chemotherapy was administered concomitantly with radiotherapy and consisted of capecitabine administered orally at a daily dose of 1650 mg/m 2 , divided into two equal doses given 12 hours apart, one administered an hour prior to irradiation. Chemotherapy was started on the first day and finished on the last day of radiotherapy (including weekends).

Patient and tumor assessments

During treatment, patients were evaluated weekly. Clinical examinations and complete blood counts were performed and body weight was measured. Toxic side effects were assessed according to National Cancer Institute Common Toxicity Criteria (NCI-CTC version 3.0).

Following re-evaluation of the primary tumor with pelvic MRI and an assessment of tumor response as defined by RECIST criteria, definitive surgery was scheduled for 6 weeks after the completion of CRT. Surgical management included a sphincter preservation approach whenever possible, using the total mesorectal excision (TME) technique. The option for a temporary colostomy during surgery was left to the surgeon's discretion.

Postoperative, pathological evaluation of the surgical specimen was performed. Pathological complete response was defined as the complete disappearance of all tumor cells.

Histological regression of the primary tumor was semi-quantitatively determined according to a 5 scale tumor regression grading (TRG) [START_REF] Dworak | Pathological features of rectal cancer after preoperative radiochemotherapy[END_REF]

Postoperative chemotherapy

Postoperative adjuvant treatment with capecitabine for 3 cycles starting within 6 weeks after surgery was recommended. A three-week cycle consisted of 2 weeks of capecitabine treatment (1250 mg/m 2 twice-daily) followed by a 1 week treatment break.

Statistical considerations

The primary endpoint of the study was the pCR rate. At the time of the study design the reported pCR for neoadjuvant capecitabine-based CRT in LARC ranged from 4-31%, including 9% from our previously reported phase II trial. [START_REF] Velenik | Capecitabine as a radiosensitizing agent in neoadjuvant treatment of locally advanced resectable rectal cancer: prospective phase II trial[END_REF] The aim was to evaluate whether a 26% pCR rate was achievable by adding cetuximab to standard preoperative treatment. The lowest pCR rate of interest was set at 10%, with an alpha error of 5% and a power of 80%, a minimum of 29 evaluable patients were needed (calculated using power sample calculation, for single sample percentages α=5%, 1-β=20%). Assuming that 10% or more patients would not be evaluable the aim was to enroll at least 35 patients.

Secondary endpoints included: the sphincter preservation rate (SPR) in low-sited tumors, overall tumor downstaging rate, the rate of radical (R0) resections and toxicity. The effect of preoperative CRT on tumor downstaging was assessed by comparing the pretreatment radiologically determined TNM stage with the postoperative pathologic TNM stage. Statistics were descriptive and analyzed using the SPSS statistical software package, version 13, (SPSS Inc., Chicago, IL, USA). radiotherapy (50.4 Gy) and with two 5-day cycles of 5-FU (425 mg/m²/day) and leucovorin (20 mg/m²/day) in protracted infusion but did not receive cetuximab. Thus 37 patients received cetuximab and completed the preoperative treatment protocol and comprised the safety population who were also evaluable for efficacy. Patient and disease characteristics are summarized in Table 1. The most frequent MRI staging was uT3N+ (75%). In 18 patients the primary tumor was sited 5 cm or less from the anal verge.

Results

Patient characteristics

Acute toxicity and treatment compliance

Neoadjuvant treatment with cetuximab was discontinued after the first administration in 4/37 patients due to hypersensitivity reactions. These patients were subsequently treated with capecitabine and radiation only. For one patient 2 doses of cetuximab were omitted due to grade 3 hepatotoxicity. All patients received the planned dose of radiation. The median duration for radiotherapy treatment was 33 days (range 33-43 days). The median time of interruption to radiotherapy treatment was 2 days (range 1-7 days) and occurred in 30 patients, due to treatment-related toxicity (4 patients) or mechanical failure/ holiday period (26 patients). The median duration of chemotherapy with capecitabine during radiotherapy was 33 days (range 33-43 days) and the median time of chemotherapy interruption was 6 days (range 1-11 days). Only two patients received less than 90% of the planned capecitabine dose due to grade 3 hepatotoxicity, (one patient) and grade 3 diarrhea, (one patient). The relative dose intensity for cetuximab, capecitabine and radiotherapy was 91%, 98% and 100% respectively.

The frequency and grade of treatment-related acute toxicities are summarized in Table 2.

The most frequent side-effect of CRT was grade 1/2 acneiform rash. The most frequent grade 3 adverse event was radiodermatitis. During the treatment period, 22 patients lost weight compared with the beginning of treatment. The maximum body weight loss was 19% (median 5%). Of the remaining patients, eight succeeded in maintaining a constant weight, and nine patients experienced a weight increase of up to 6% (median 1%).

Surgery and perioperative toxicity

A summary of patient surgery and perioperative toxicity is shown in Table 3. All patients underwent definitive surgery following neoadjuvant therapy and were operated on between 31 and 59 days (median 41 days) after the last day of CRT. The median hospital stay for surgery was 10 days (range; 7-32 days), and the most common operation was low anterior resection. A temporary stoma was created in 22/28 patients undergoing sphincter preservation surgery. As determined by histopathological examination of surgical specimens,

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Page 8 of 14 resection with negative circumferential margins at the primary tumor site was achieved in all patients. For one patient undergoing a low anterior resection, microscopic foci of cancer cells were found in the distal surgical margin (R1 resection) and this patient subsequently underwent an abdominoperineal resection, with no tumor cells detected on histopathological inspection of the resected tissue.

No perioperative deaths were recorded within one month following surgery. Non-lethal perioperative complications were recorded in 13 patients, with the most common cause due to wound healing problems (Table 3). Two patients required further operations (one patient for anastomotic leakage and abdominal abscess, and one patient for an incarcerated transversostoma).

Efficacy

Pathological TNM stages, as assessed on histopathological examination of resected specimens, in relation to preoperative TNM status, are presented in Table 4. Three patients demonstrated pCR (TRG4; 8.1%, 95% CI -8.8 to 8.8%). TRG 3, 2 and 1 were found in seven (19%), 19 (51%), and seven patients (19%) respectively. Overall downstaging occurred in 27/37 patients (73%) with decreases in T-and N-stages observed in 21 (57%) and 30 patients (81%), respectively. Increases in T-and/or N-stages (upstaging) were not recorded.

Tumor response as defined by RECIST on MRI imaging was evaluated in 34/37 patients by independent surgeons and radiologists blinded to patient clinical data. In three patients the control MRI before surgery was not performed (due to mechanical failure for two patients and for unknown reasons for one patient). Objective complete response (CR) was observed in four patients, partial response (PR) in 20 patients, stable disease in nine patients and progressive disease (PD) in one patient (3%). The disease control rate (CR+PR+SD) was 97%.

Preservation of the anal sphincter

Prior to therapy, abdominoperineal resection was planned in 11/37 patients who had definitive surgery. Following completion of CRT, sphincter preserving surgery was successfully performed in two of these patients. The SPR was 76% (28/37 patients) and in the subgroup of 17 patients with tumors located within 5 cm of the anal verge, sphincter preservation was possible in nine patients.

Postoperative chemotherapy

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Postoperative chemotherapy was administered to 34 patients. Reasons for not administering adjuvant chemotherapy were: death due to sepsis after colonic necrosis and perforation 6 weeks after the operation (one patient); a >8 week interval between the operation and adjuvant therapy (one patient); postoperative complications (one patient).

After a median follow-up of 13.5 months (range; 4.8-25.6 months), three patients had suffered relapse due to systemic progression (one pulmonary spread, one pulmonary and retroperitoneal and one suprarenal). Re-examination of the pretreatment MRI of the pelvis revealed bone metastases in another patient. No local recurrences had occurred.

Discussion

Neoadjuvant cetuximab plus CRT on tumor response in LARC

In this phase II study the addition of cetuximab to capecitabine and radiotherapy in the neoadjuvant treatment of LARC patients led to a pCR rate of 8% and a high rate (73%) of pathological downstaging of tumors. The pCR rate is similar to that reported in a phase I/II study in Belgian LARC patients, (5%), where a lower rate of pathological tumor downstaging (38%) was recorded. [START_REF] Machiels | Phase I/II study of preoperative cetuximab, capecitabine, and external beam radiotherapy in patients with rectal cancer[END_REF] Similar doses of cetuximab and CRT were used in the two studies, however in the current study capecitabine was administered twice daily at 1250 mg/m 2 for two weeks, three weeks prior to radiotherapy whereas in the Belgian study a dose escalation of capecitabine was administered twice daily (650 mg/m 2 or 850 mg/m 2 ) concomitantly with cetuximab and radiotherapy. [START_REF] Machiels | Phase I/II study of preoperative cetuximab, capecitabine, and external beam radiotherapy in patients with rectal cancer[END_REF] Patient baseline and tumor characteristics were comparable and whether the administration of capecitabine three weeks prior to cetuximab plus CRT leads to higher tumor downstaging in this setting requires further clinical investigation.

More recently a similar pCR (7.7%) was reported in LARC patients receiving neoadjuvant cetuximab given as a single dose and then combined with 5-FU and radiotherapy. [START_REF] Bertolini | Neoadjuvant treatment with single-agent cetuximab followed by 5-FU, cetuximab, and pelvic radiotherapy: a phase II study in locally advanced rectal cancer[END_REF] The addition of a second chemotherapeutic agent to neoadjuvant capecitabine-based CRT with cetuximab has not demonstrated a marked improvement to the rates of pCR or tumor downstaging in LARC, [START_REF] Horisberger | Cetuximab in Combination with Capecitabine, Irinotecan, and Radiotherapy for Patients with Locally Advanced Rectal Cancer: Results of a Phase II MARGIT Trial[END_REF] which remain similar to those reported from studies of neoadjuvant capecitabine-based CRT, [START_REF] Kim | Preoperative concurrent radiotherapy with capecitabine before total mesorectal excision in locally advanced rectal cancer[END_REF][START_REF] Velenik | Capecitabine as a radiosensitizing agent in neoadjuvant treatment of locally advanced resectable rectal cancer: prospective phase II trial[END_REF][START_REF] Dunst | Phase I trial evaluating the concurrent combination of radiotherapy and capecitabine in rectal cancer[END_REF] or capecitabine-based CRT with irinotecan or oxaliplatin. [START_REF] Willeke | A phase II study of capecitabine and irinotecan in combination with concurrent pelvic radiotherapy (CapIri-RT) as neoadjuvant treatment of locally advanced rectal cancer[END_REF][START_REF] Carlomagno | Neo-adjuvant treatment of rectal cancer with capecitabine and oxaliplatin in combination with radiotherapy: a phase II study[END_REF] In the present study an assessment of radiological tumor response according to RECIST criteria was made. An overall response rate of 71% was recorded, which was similar to the that reported for patients with locally advanced SCCHN receiving cetuximab in combination with radiotherapy. [START_REF] Bonner | Radiotherapy plus cetuximab for squamous-cell carcinoma of the head and neck[END_REF] The rates of tumor downstaging as determined by MRI and histopathological techniques were similar and reflected the findings from the MERCURY study where the two techniques were equivalent to 0.5 mm in the assessment of tumor spread in rectal cancer patients. [START_REF]Extramural depth of tumor invasion at thin-section MR in patients with rectal cancer: Results of the Mercury study[END_REF] Recent studies have focused on the identification of predictive biomarkers for response to treatment with cetuximab. Data from randomized studies in mCRC patients suggest that tumor KRAS mutational status is predictive for response to cetuximab in combination with chemotherapy. [START_REF] Van Cutsem | Cetuximab and chemotherapy as initial treatment for metastatic colorectal cancer[END_REF][START_REF] Bokemeyer | Fluorouracil, leucovorin, and oxaliplatin with and without cetuximab in the first-line treatment of metastatic colorectal cancer[END_REF] In rectal cancer patients treated with cetuximab-based therapy, the frequency of TRG3-4 was reported to be higher (58% vs. 7.7%) in patients whose KRAS wild-type tumors had a high EGFR gene copy number compared with those tumors harboring a low EGFR gene copy number. [START_REF] Bengala | Epidermal growth factor receptor gene copy number, K-ras mutation and pathological response to preoperative cetuximab, 5-FU and radiation therapy in locally advanced rectal cancer[END_REF] Debucquoy and colleagues suggest that decreased tumor proliferation observed in patients receiving an initial dose of cetuximab prior to CRT might affect the subsequent efficacy of capecitabine and RT leading to the low pCR reported. [START_REF] Machiels | Phase I/II study of preoperative cetuximab, capecitabine, and external beam radiotherapy in patients with rectal cancer[END_REF] Upregulated EGFR expression following the initial dose of cetuximab, and the presence of pro-inflammatory changes in tumors sampled at resection was associated with improved patient DFS, suggesting these to be potentially novel predictive markers of cetuximab-based treatment in this setting. [START_REF] Debucquoy | Molecular response to cetuximab and efficacy of preoperative cetuximab-based chemoradiation in rectal cancer[END_REF]

Neoadjuvant cetuximab plus CRT on sphincter preservation rate

A secondary endpoint of this study was the SPR. All patients received definitive surgery, and 76% underwent sphincter preserving surgery, including 2/11 patients planned to receive abdominoperineal resection prior to neoadjuvant CRT. Within the group of patients receiving sphincter preserving surgery 53% had tumors located within 5 cm of the anal verge. Similar SPR have been reported in patients receiving capecitabine-based neoadjuvant CRT. [START_REF] Kim | Preoperative concurrent radiotherapy with capecitabine before total mesorectal excision in locally advanced rectal cancer[END_REF][START_REF] Velenik | Capecitabine as a radiosensitizing agent in neoadjuvant treatment of locally advanced resectable rectal cancer: prospective phase II trial[END_REF] However an improvement in the SPR in the preoperative CRT arm compared to radiotherapy alone arm has not been conclusively demonstrated in randomized studies. [START_REF] Sauer | Preoperative versus postoperative chemoradiotherapy for rectal cancer[END_REF][START_REF] Bujko | Sphincter preservation following preoperative radiotherapy for rectal cancer: report of a randomised trial comparing short-term radiotherapy vs. conventionally fractionated radiochemotherapy[END_REF]

Toxicity and postoperative complications of cetuximab plus CRT

The combination of cetuximab with capecitabine-based CRT was generally well tolerated with the majority of patients completing the treatment protocol. The most common adverse event was grade 1/2 skin rash, which has also been described in other studies in mCRC investigating cetuximab, [START_REF] Van Cutsem | Cetuximab and chemotherapy as initial treatment for metastatic colorectal cancer[END_REF][START_REF] Bokemeyer | Fluorouracil, leucovorin, and oxaliplatin with and without cetuximab in the first-line treatment of metastatic colorectal cancer[END_REF] and was similar to that previously reported in LARC patients receiving neoadjuvant cetuximab plus CRT as was the frequency of grade 3 diarrhea (11%). [START_REF] Machiels | Phase I/II study of preoperative cetuximab, capecitabine, and external beam radiotherapy in patients with rectal cancer[END_REF] Radiodermatitis occurred in 16% of patients which was similar to that reported in locally advanced SSCHN where the addition of cetuximab to radiotherapy was found not to significantly increase the incidence of grade 3 radiodermatitis. [START_REF] Bonner | Radiotherapy plus cetuximab for squamous-cell carcinoma of the head and neck[END_REF] There were no immediate

Conclusion

In summary, neoadjuvant cetuximab in combination with capecitabine-based CRT in LARC was well tolerated and whilst pCR rates were not high, a high frequency of tumor downstaging was observed. To date, although preoperative CRT significantly improves local control in LARC, there is no improvement on overall survival compared with adjuvant therapy. To assess the value of EGFR-targeted agents in this setting, future randomized control trials should maybe evaluate the efficacy and safety of these agents with CRT on these endpoints and perhaps in patients assessed for potential predictive biomarkers of efficacy.

Conflicts of interest

There are no conflicts of interest to declare.

M A N U S C R I P T A C C E P T E D

  This was a prospective, open-label single center phase II study. The protocol was approved by the Ethical Committee of Slovenia, Agency for Medicinal Products and Medical Devices, Ministry of Health and by the Independent Ethical Committee of the Institute of Oncology, Ljubljana, Slovenia. The trial was registered at CinicalTrials.gov. (NCT00689702). The study was conducted in accordance with the principles of the Declaration of Helsinki and the note for guidance on good clinical practice.

  : TRG 0, no regression; TRG 1, minimal regression (dominant tumor mass with obvious fibrosis and/or vasculopathy); TRG 2, moderate regression (predominantly fibrotic changes with few tumor cells or groups); TRG 3, M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS Page 6 of 14 good regression (very few tumor cells in fibrotic tissue); TRG 4, total regression (no tumor cells, only fibrotic mass).

Forty-three patients

  with resectable primary LARC were recruited between February 2007 and September 2008. One patient did not receive study treatment due to the detection of metastatic disease during the pretreatment work-up. One patient withdrew consent following three weeks of treatment, and another was excluded due to protocol violation. One patient was discontinued from chemotherapy on day 4 of treatment following the development of cardiac ischemia demonstrated on electrocardiogram (ECG), and was subsequently treated with radiation (50.4 Gy) only. Two patients were withdrawn from capecitabine treatment after 4 and 6 days due to chest pain of unknown origin, these patients were treated with M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS Page 7 of 14

  M A N U S C R I P T A C C E P T E Dpostoperative deaths and the majority of the complications were associated with delayed wound healing.

Figure 1 .Figure 1

 11 Figure legend

Table 1 .

 1 Patient and disease characteristics at baseline

	Characteristics

a

As planned before the start of preoperative chemoradiotherapy. MRI, magnetic nuclear imaging; N, node; T, tumor; WHO, World Health Organization.

Acknowledgement

The study was sponsored by the Institute of Oncology, Ljubljana and was cosponsored by Merck KGaA (Darmstadt, Germany) who also provided the study drug cetuximab and some editorial assistance.

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Conflicts of interest

There are no conflicts of interest to declare.

M A N U S C R I P T A C C E P T E D


Surgical Trainee D N Robson

Rew

Consultant Surgeon

Collective wisdom and decision making in Surgical Oncology. A review paper for the EJSO

Keywords:

Aim: To describe systems for capturing and optimising collective knowledge and insight in areas of complexity and uncertainty in surgical oncology, with particular reference to the Delphi process and related systems.

.

Methods:

Internet search engines (Google, Google Scholar) and four databases (SCOPUS, PubMed, Medline and Embase) were searched to find English language articles on the use of The Delphi Process and related systems in surgical oncology, using a variety of search terms.

Findings:

There are a number of established systems for co-opting group knowledge and facilitating collective decision-making. These find applications in commerce, industry, government and defence. They have also been applied to problems in surgical oncology, for example using the Delphi process to optimise the management of colorectal cancers and metastases.

Conclusions:

Collective decision making tools find practical applications in the allocation of resources and in clinical decision making in fields of surgical oncology practice where there is a wide range of evidence and expert opinion. Such methodologies set new standards for the collating of professional expertise and for the writing of "best clinical practice" guidelines in the cancer subspecialities.

Introduction

There are many areas of uncertainty in clinical and oncological practice. The breadth and extent of the published literature in any one subject area is now huge, and accruing at an enormous rate. The "cloud" experience of huge numbers of individual clinicians treating individual patients with their infinite variety of presentations and responses to treatment across the world often goes uncaptured and unrecorded. There is no prospect of any one person collating, analysing and distilling this mass of published and unpublished knowledge in a realistic time frame, even with the help of modern search engines.

Digital technologies and the opening up of the Internet have driven forward the science of understanding mass behaviours, and have provided means for the efficient dissemination and collation of responses and opinions from large numbers of people.

We see these processes in daily use in dissemination by scientific search engines such as SCOPUS, Google Scholar and Web of Science, and in communal interactive systems such as Facebook, Twitter and MySpace.

Individual genius, insight and intellectual courage drives human society forward in many ways, as illustrated by the work of Galileo, Newton, Einstein and Darwin.

However, many challenges in medicine and public health require a collective approach. They need systems which tap into many different sources of wisdom and experience if resources are to be allocated most efficiently, and if clinical outcomes are to be optimised on an individual and a population basis.

In recent times, we have seen a major shift in medical decision making from the "wise individual" to the multidisciplinary team. Our horizons have risen from the local and parochial to the regional, national and supranational. The Internet has brought worldwide and near instantaneous visibility to the key literature and as email has radically simplified cross border communications. Internet development has also brought a whole new thinking and language of terminologies which recognises "the wisdom of crowds" and which captures this collective knowledge through processes such as "crowdsourcing", and "collaborative filtering". Perhaps the best known of these techniques in Medicine is the Delphi process. This has already found a number

The Delphi system

The Delphi System is an approach used to gain consensus among a panel of nominated voting experts. It was originally developed in the 1950s by the Research and Development Corporation, RAND, for forecasting future warfare after World War II (1, 2). In 1959, Helmer and Rescher published a paper describing a tool for pooling the predictions of a panel of experts for questions that could not be answered as yet by exact science (3).

In addressing major organisational problems, group decision-making is open to bias from single experts and 'follow thy leader' tendencies. There is often a reluctance to abandon previously stated opinions and groups may be subject to a variety of pressures. The Delphi System offers anonymity, and asynchronicity in time and place, thus allowing individual beliefs to be expressed independently in contributing to the conclusions of the whole group.

The Delphi System has been widely applied outside healthcare in strategic military and political planning and in business decision making (4-7). However, there is as yet

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no level 1 evidence to prove the value of the process as an alternative to group discussion for clinical decision-making.

The Delphi system lends itself to a range of applications in cancer management, both in the allocation of resources and in interpreting a complex clinical evidence base, so as to aid multidisciplinary decision making across diverse disciplines. Professional guidance to interpreting the literature using the Delphi technique may also assist clinical MDT decision making and hence optimise practice in a rapidly changing practice and technical environment.

Components of the Delphi Process

The Delphi system is a structured process with four core principles of anonymity, asynchronicity, controlled feedback and statistical analysis. A question or problem is identified and defined by the leading body. An extensive factual information search should take place and evidence-based statements provided to potential members of the panel. The panel should be experts in their field and be adept at structural, critical analysis of information. Debriefing sessions can be used to ensure panel experts are aware of the educational resources. Information is provided as a list of statements for submission to the Delphi experts.

A group moderator is then nominated. The moderator does not vote but provides information, delegates tasks, submits statements to the panel, collects response, analyses data, submits the response for further panel scrutiny in repeated rounds and collaborates the result for presentation to the panel.

A series of voting rounds take place which may be separated in time and place, and which may be conducted through a variety of media, including email. Early voting rounds help to define study objectives. Subsequent rounds rank these objectives in order of importance and develop criteria for further consideration, which are then ranked. The process identifies areas where there are differences in opinion or agreement, which may or may not lead to a consensus view on each of the questions posed.

The Delphi rounds

The facilitator submits a questionnaire or list of factual statements to the Delphi experts. The experts respond anonymously. A list of goals and criteria are developed from the result analysis of this questionnaire and processed by the facilitator to formulate a second questionnaire, which goes to a second voting round. The results are analysed and criteria developed for ranking in order of importance by the Delphi panel.

In the third Delphi round, the ranked criteria are analysed and weighted again. The unimportant are discarded and the moderately important may be rephrased and resubmitted, new criteria for panel response may be developed. The revised list of weighted criteria is submitted to the panel and for further ranking. The Delphi process can run to numerous rounds, but to avoid panel fatigue it has been recommended that 4 rounds are optimal. The panel rank the criteria for the final time in the consensus round.

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In the consensus round, the results are processed by the facilitator and presented to the group. Criteria are ranked and recommendations for practice are made. A consensus may or may not be secured on each question posed. The Delphi document was available to all Committee members as a web-based document customized for the consensus process. Eight chapters were identified: epidemiology, diagnostics, pathology, surgery, radiotherapy and chemotherapy, treatment toxicity and quality of life, follow-up, and research questions. For each chapter, and a series of statements were developed. Each member commented and voted three times.

Examples of the

The votes were anonymous and rounds one and two were asynchronous, with three weeks available for voting. Statements upon which an agreement was not reached after voting round 1 (24 th September to 20 th October) and voting round 2 (10 th November to 1 st December) were openly debated during the Consensus Conference in Perugia in Italy from 11 th -13 th December 2008. A hand-held televoting system collected the opinions of both the Committee members and the audience after each debate.

A final voting round took place by email after the conference between 20 th January and 10 th February 2009. There were 207 voted sentences. Of these, 86% achieved large consensus, 13% achieved moderate consensus, and only 3 resulted in minimum consensus. The results were summarised and published [START_REF] Valentini | Multidisciplinary Rectal Cancer Management: 2 nd European Rectal Cancer Consensus Conference (EURECA-CC2)[END_REF]. These indicators included items of structure and organisation, including hospital volumes, the establishment of HPB MDTs, equipment and review systems; process and service targets for specialist referral, investigation and treatment; and outcome indicators [START_REF] Dixon | Development of Quality Indicators of Care for Patients Undergoing Hepatic Resection for Metastatic Colorectal Cancer Using a Delphi Process[END_REF].

The Research Agenda of the American Society of Colon and Rectal Surgeons

The Delphi method can be extended to the polling of the entire membership of a specialist Society, as illustrated by the following exercise conducted by the American The authors reported that the research agenda produced by this study reflects the clinical issues of greatest importance to the ASCRS members, and would help direct future research [START_REF] Burt | Developing a research agenda for the American Society of Colon and Rectal Surgeons: Results of a Delphi approach[END_REF].

Modifications of the Delphi Process: Oncosurge

In 2005, an international panel of HPB surgeons set out to create a therapeutic decision model identifying appropriate procedure sequences for the management of colorectal metastases. The RAND Corporation/University of California, Los Angeles Appropriateness Method (RAM) was used to assess strategies of resection, local ablation and chemotherapy. After a comprehensive literature review, an expert panel rated the appropriateness of each treatment option for a total of 1,872 ratings decisions in 252 cases. A decision model was constructed, consensus measured and results validated using 48 virtual cases, and 34 real cases with known outcomes. The group described its methodology as follows:

"The RAM methodology is a modified Delphi process which invites experts to apply their judgments on the appropriateness of treatments for an average patient within each category, treated by the average provider in the average hospital. The term "average" is clearly defined. "Appropriateness" describes the relative weight of the benefits and risks of a medical procedure, such that a procedure is "appropriate" when the expected benefits outweigh the possible negative consequences by a sufficient margin. The definition of a "sufficient margin" recognises the subjectivity of the expectations and risk perceptions of the individual patient, and their clinician(s) [START_REF] Fitch | The RAND/UCLA Appropriateness User's Manual[END_REF]. Therefore, to be of clinical use, the decision model must rank and stratify benefit/risk and benefits ratios for each treatment strategy in each clinical presentation. A procedure is judged to be "uncertain" when benefits and risks are similar, or if there is disagreement within the expert panel".

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In this study, the appropriateness of different treatments was scored on a scale of 1 (highly inappropriate) to 9 (clearly appropriate) against each clinical presentation. For the purpose of analysis, ratings from 1-3 for any given treatment were considered inappropriate, 4 to 6 represented uncertainty, and 7 to 9 indicated appropriateness.

The group rating was taken as the median judgment of the panelists. Uncertainty could indicate that the experts were uncertain, or that there was disagreement among the experts, which would indicate a need for further clinical research. Bimodal disagreement (1 to 3 v 7 to 9) led to a discussion preceding the second round of ratings.

An international panel of 16 oncologists, surgeons and radiologists with wide experience in the diagnosis and treatment of liver metastases were provided with a systematic review of the indications for surgical resection of colorectal liver metastases prepared by two of the authors of this study and funded by the British Cancer Research Campaign (now CRUK). Each model patient was assigned to one indication, but considered for each possible treatment.

A preliminary panel meeting established three treatment groups for consideration: liver resection, local ablation, and chemotherapy. For each treatment segment, all relevant possible patient characteristics were included in the list of indications.

A second meeting reviewed the results of the first round of ratings, and modifications were made to the list of treatment options and indications. Panelists then privately rated each of the three treatment groups separately. The strategies that emerged from the second round of ratings were incorporated into a decision matrix in the form of a computer program, the OncoSurge decision model. Validation of the model by the expert panel was accomplished using 48 representative virtual theoretical cases using the real decision characteristics identified by the appropriateness analysis. These 48 cases were created specifically to ensure that the expert panel had considered the full range of all potential clinical presentations, making the validation process span the full range of potential patients. The validation process using the 48 virtual theoretical cases was reinforced by the panelists using 34 real cases from their panelists' own clinical practices with known outcomes.

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Overall consensus rates for a series of statements ranged from 93.4 to 99.1%. For example, absolute contraindications to liver resection were judged to be unresectable extrahepatic disease, more than 70% liver involvement, liver failure, and lack of surgically fitness. Recommendations were also made in respect of factors which did not influence treatment strategy; indications for immediate resection and after chemotherapy, and in respect of surgery in relation to other ablation techniques.

The model identified criteria for resectability for individual patients; made recommendations for optimal treatment strategies; and provided opportunities for medical education.

The authors concluded that the OncoSurge decision model for colorectal liver metastases combines the best available scientific evidence with the collective judgment of worldwide experts. It allows a clinician to optimise treatment in the context of local skills, resources for each patient. The methodology can be applied efficiently across health care systems worldwide, to decision making in multimodality therapy for other cancers, such as rectal cancer or adjuvant therapy for breast, lung, and pancreatic cancer [START_REF] Gj | OncoSurge: A Strategy for M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 16 Improving Resectability With Curative Intent in Metastatic Colorectal Cancer[END_REF].

Development of quality indicators for patients undergoing colorectal cancer surgery.

McGory et al reported using a modification of the RAND/UCLA Appropriateness Methodology to update National Cancer Institute-sponsored consensus panel guidelines for colorectal cancer surgery from 2000. They used structured interviews with experienced colorectal cancer surgeons and systematic reviews of the literature to identify candidate quality indicators addressing perioperative care for patients undergoing surgery for colorectal cancer. A panel of 14 colorectal surgeons, general surgeons, and surgical oncologists then evaluated and formally rated the indicators using the modified Delphi method.142 candidate indicators were identified in six broad domains: privileging (which addresses surgical credentials), preoperative evaluation, patient-provider discussions, medication use, intraoperative care, and postoperative management. The expert panel rated 92 indicators as valid, which address all domains of perioperative care and which could be used as quality performance measures and for quality-improvement programs [START_REF] Mcgory | Development of quality indicators for patients undergoing colorectal cancer surgery[END_REF].

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The Delphi approach to attain consensus in methodology of local regional therapy for peritoneal surface malignancy At the Fifth International Workshop on Peritoneal Surface Malignancy (PSM) in Milan, December 2006, the Delphi process was used to seek a consensus on technical aspects of cytoreductive surgery (CRS) and hyperthermic intraperitoneal chemotherapy (HIPEC). Topics discussed included pre-operative workup; eligibility to CRS + HIPEC; intra-operative staging system; technical aspects of surgery; residual disease classification systems; HIPEC: nomenclature and modalities; drugs, carrier solution and optimal temperature; morbidity grading systems. Conflicting points were voted upon in two rounds by an international panel in local-regional therapy.

Peritoneal surface malignancies (PSM) have been traditionally regarded as uniformly terminal conditions. The combination of cyto-reductive surgery and perioperative intraperitoneal chemotherapy has changed PSM management from palliation to possible cure. Due to the inherent differences in biological and clinical behavior, the optimal adaptation of comprehensive treatment to each PSM is still a matter of debate.

A session of The Fifth International Workshop on Peritoneal Surface Malignancy (Milan, Italy, December 4-6, 2006) was committed to reach a consensus pertaining to conceptual and technical aspects of the loco-regional treatment of each PSM. The consensus developing process was based on principles of the Delphi method. A total of 103 international experts from 17 countries were included in six Working Groups for each of the following PSM: peritoneal mesothelioma, abdominal sarcomatosis, carcinomatosis of gastric, colo-rectal, appendiceal, and ovarian origin. The respective working groups wrote evidence reports. The main conflicting points regarding preoperative evaluation, patient eligibility, combined treatment methodology, postoperative follow-up and future investigational perspectives were listed as multiple-choice questions. 160 Conflicting Points were identified. A consensus of more than 50% of voters favoring one option was reached in 143 of these [START_REF] Baratti | The Fifth International Workshop on Peritoneal surface malignancy (Milan, Italy, December 4-6, 2006): Methodology of disease-specific consensus[END_REF][START_REF] Baratti | The Delphi approach to Attain consensus in methodology of local regional therapy for peritoneal surface malignancy: Shigeki Kusamura[END_REF].

Other Oncological Applications of the Delphi process

The Delphi process has been used on an occasional basis to identify priorities in cancer nursing service provision and cancer educational priorities for GPs [START_REF] Broomfield | Using the Delphi technique to identify the cancer education requirements of general practitioners[END_REF] [START_REF] Hall | A model for making project funding decisions at the National Cancer Institute[END_REF], while Brook et al reported its use in assessing medical technologies [START_REF] Brook | A method for the detailed assessment of the appropriateness of medical technologies[END_REF].

Other Collective Knowledge Systems

The Internet is driving new commercial systems to capture the power of collective intelligence embodied in the conscious and subconscious choices and preferences that individuals make whenever they use the Web. While we are a long way from capturing the collective decision making of clinicians and MDTs on a case-by-case basis and hence from learning how to optimise individual cancer outcomes, we can at least start to see how this might be possible in a wired and networked world.

The wisdom of crowds is a concept which describes the aggregation of information in groups, resulting in decisions that may be better than could have been made by any single member of the group. The most meaningful outputs are derived from groups and "crowds" where there is diversity of opinion; independence of opinion; decentralisation and specialisation; and a method of aggregating the collective judgements into a collective decision [START_REF] Surowiecki | The Wisdom of Crowds: Why the Many Are Smarter Than the Few and How Collective Wisdom Shapes Business, Economies, Societies and Nations[END_REF]. Groups and crowds are also capable of irrational behaviour and of collective failures of intelligence and common sense, where the groups are too homogeneous, centralised, bureaucratised or divided.

Crowdsourcing is the study of information inherent in the actions of large numbers of independently acting individuals, leading to measurable collective outputs (for example in marketing or politics).

Discussion

Collective decision-making systems have various advantages over the committee work and open group discussions in decision-making and strategic planning. However, they must be applied and interpreted with care, as they have a number of vulnerabilities, according to the composition of the expert voting panel, and the reliability of the evidence base upon which each expert's opinion is based.

Limitations of Collective Decision Making System

The Delphi model and related systems have a number of practical limitations. They require a considerable amount of organisation to produce recommendations that are a snapshot of knowledge and opinion at one point in time. Given the rapid rate of The quality of any consensus achieved is determined by the quality of the information available to the panel. In many subject areas, there is a "perceived wisdom" which may or may not reflect ground truth or a correct reading of the entirety of the literature. While in theory all evidence must be made available to support the decision making process, in practice no one expert is likely to be able to afford all of the time needed to rehearse all of the literature or to modify all personal prejudices.

The process may also be biased by the selection of panel members themselves. The panel will be a "coalition of the willing", and is more likely to reflect cooperating opinion than adversarial opinion. The panel may suffer from illusory expertise: some of the experts' knowledge may be confined to a narrow field. The experts may be designated from a small collective of colleagues or an imbalance of subspecialities, thus inherently biasing the interpretation of the available data in the context of their own experience. Hence, the larger and wider the panel of decision makers, the more truly representative is the output likely to be.

The Delphi process has other potential sources of bias. The participants may vote or score a statement or opinion without being fully informed, because the information provided to them, or the range of questions posed, reflects the prejudices of the originating group or organisation. They are also vulnerable to the subjectivity of the moderators and facilitators. Thus, the revised criteria can be resubmitted to the panel in a way to promote a desired direction. Format bias in a questionnaire may produce the same result. The questionnaire may be ambiguous and although ambiguous statements should be discounted, they may be included in results data depending upon the scrutiny of the monitor and facilitator [START_REF] Downs | The feasibility of creating a checklist for the assessment of the methodological quality both of randomised and non-randomised studies of health care interventions[END_REF]. The conduct of a Delphi process may be of interest and may reflect a very accurate reading of the available evidence base, but there is as yet no obligation on any individual or group to act upon the product. Moreover, the passage of time will negate much of the value of the process, unless there is a built in review and reiteration process at regular intervals, which has not as yet been reported in surgical oncology subject areas.

Conclusions

In conclusion, the Delphi process and related group decision making systems provide a meritorious and democratic means by which a large number of individuals can express a collective opinion on a wide range of topics and issues on any one subject area. These processes represent a generational shift from decision making by individuals and committees to anonymous and democratic collaborative working and combining of expertise, where all contributors have a voice and a "secret ballot".

The enabling technologies for collective decision making are now well established.

Among other applications, they suggest a route to a much more robust format for Guidelines to "best clinical practice" in various of the cancer subspecialities, such as are published from time to time in the EJSO and elsewhere. We commend a trial of expert panel voting systems in preparing the next generation of such guidelines and in other areas of uncertainty as to future directions for research, resource allocation and technology implementation in surgical oncology.

5

 5 The four key principles of the Delphi System:Anonymity: This ensures that the personalities and status of the voting experts cannot influence group behaviour. Opinions perceived as unpopular or maverick can be freely expressed. Experts may change their opinions without pressure to match preconceived expectations.Asynchronicity: This allows the process to move forward without depending upon participants being together in time and place. This allows time for reflection and for geographically widely dispersed experts to contribute effectively.Controlled feedback: This allows the results of each subsequent round formulate the next, under direction of the moderator.Statistical output; The Delphi process produces quantitative results from the qualitative beliefs of the panel. The Delphi group can then assign a level of confidence in the results and gauge satisfaction with the outcome.

  Delphi process in surgical oncology in practice Colorectal Cancer Management (Europe) The International Conference on 'Multidisciplinary Rectal Cancer Treatment: Looking for a European Consensus" (EURECA-CC2) provides an example of the Delphi process in action. The Conference was organised in Italy and endorsed by the European Society of Medical Oncology (ESMO), European Society of Surgical Oncology (ESSO), and the European Society of Therapeutic Radiation Oncology (ESTRO), and 34 specialists with a range of expertise were invited to form the Panel.

7

 7 M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT Hepatic resection for metastatic colorectal cancer (Canada) The Candian Hepatico-Pancreatico-Biliary (HPB) Society nominated a Delphi panel of experts to provide quality of care guidelines for the management of patients undergoing hepatic resection for metastatic colorectal cancer (Ref). The CHPBS group of 16 experts included two medical oncologists, one anaesthetist, one hepatologist, one pathologist, one palliative care physicians and nine surgeons. A literature search produced a list of evidence-based statements as possible quality indicators. The voting took place by post.In each Delphi round, each statement was scored on a 5-point Likert scale as follows:(1) do not include, (2) little reason to include, (3) could include, (4) should include, (5) must include. Replies were anonymous and asynchronous. The response was compiled as an Excel spreadsheet and those statements with a mean score of >4 were kept as quality indicators. Those with an average score of >3 but <4 were re-sent to participants to score again with any new statements developed from round 1.Statements scoring <3 were discarded. The process was repeated for the final round 3, with participants being asked to add a weighting score to the established quality indicators. A 100% response was received for all rounds. The panel identified 18 quality indicators for care of the patient undergoing hepatic resection from metastatic colorectal cancer.

  Society of Colon and RectalSurgeons (ASCRS) to reach a consensus on the research questions of highest importance in terms of clinical care. A modified Delphi process was used. 203 responding members in Round 1 submitted M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 8 colorectal surgical practice. In Rounds 2 and 3, 399 and 360 respondents, respectively, further prioritised and refined the list of questions. The final 20 items included 14 questions related to colorectal cancer, and six on benign disease topics, which were scored and tabulated.

  Hallet al reported the use of the Delphi method to prioritise funding decisions at the

  and the increase in available information, they can become rapidly outdated unless mechanisms are in place to revisit and reappraise the decisions.

Fortunately, the

 the Internet and modern communication systems come to our assistance here. There is no reason why large numbers of individuals, such as the entire membership of a subspeciality association, or the subscribers of a journal, should not M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 14 be open to canvassing quickly and cost effectively, and their views readily collated in a Delphi type process on one or more subject areas.

Acknowledgements

We are grateful to Professors Irving Taylor and Cornelis Van der Velde for drawing our attention to the EURECA-CC2 Conference. Their editorial formed the starting point for our further researches on this subject (21)


Weini Huang

Arne Traulsen

Fixation probabilities of random mutants under frequency dependent selection

Keywords: Moran process, evolutionary game theory, fixation probabilities

Evolutionary game dynamics describes frequency dependent selection in asexual, haploid populations. It typically considers predefined strategies and fixed payoff matrices. Mutations occur between these known types only. Here, we consider a situation in which a mutation has produced an entirely new type which is characterized by a random payoff matrix that does not change during the fixation or extinction of the mutant. Based on the probability distribution underlying the payoff values, we address the fixation probability of the new mutant. It turns out that for weak selection, only the first moments of the distribution matter. For strong selection, the probability that a new payoff entry is larger than the wild type's payoff against itself is the crucial quantity.

Introduction

Evolutionary game theory is a method to study frequency dependent selection in asexual populations. [START_REF] Smith | Evolution and the Theory of Games[END_REF][START_REF] Weibull | Evolutionary Game Theory[END_REF]Hofbauer andSigmund, 1998, 2003;[START_REF] Cressman | Evolutionary Games and Extensive Form Games[END_REF][START_REF] Nowak | Evolutionary dynamics of biological games[END_REF]. Whenever the fitness of the individuals depends on the composition of the population, the dynamics of the evolving population can be described based on an evolutionary game. Constant selection, where the fitness of a type is fixed, can be considered as a special case in this context, where the payoff depends only on the strategy, but not on the frequencies of other types in the population.

The standard approach to evolutionary game dynamics is the replicator dynamics [START_REF] Taylor | Evolutionary stable strategies and game dynamics[END_REF][START_REF] Hofbauer | A note on evolutionary stable strategies and game dynamics[END_REF][START_REF] Zeeman | Population dynamics from game theory[END_REF]. It describes the change in frequency x i of strategy i as ẋi = x i (π i -π ), where π i is the

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payoff of strategy i and π is the average payoff in the population. More recently, the focus of research has turned to finite populations. The most popular model for evolutionary game dynamics in finite populations is the frequency dependent Moran process introduced by Nowak et al. (2004). Although, mutations are often disregarded, they can be incorporated without any problems. Typically, the mutations produce types that are pre-defined in the payoff matrix and one considers the mutation-selection equilibria of the system [START_REF] Bomze | Stability by mutation in evolutionary games[END_REF]Nowak et al., 2004;[START_REF] Imhof | Evolutionary cycles of cooperation and defection[END_REF][START_REF] Imhof | Imitation process with small mutations[END_REF][START_REF] Traulsen | Exploration dynamics in evolutionary games[END_REF]Antal et al., 2009a,b,c;[START_REF] Van Segbroeck | Reacting differently to adverse ties promotes cooperation in social networks[END_REF].

Here, we consider a different possibility in which mutants are characterized by a new payoff matrix game with randomly chosen entries. For low mutation rates, only one mutant is present at a time. The average fixation time under neutral selection is N generations. Thus, it is unlikely that several mutants are present at the same time when the mutation rate μ fulfills μ N -2 . This estimate holds for situations in which the mutant is advantageous or disadvantageous for all abundances or if it is first disadvantageous and becomes advantageous at high abundances [START_REF] Antal | Fixation of strategies for an evolutionary game in finite populations[END_REF]. When there is a stable coexistence between the types, however, the average fixation time diverges exponentially with the intensity of selection and the population size. In this case, our approach is only valid when the mutation rates go to zero. When the mutation rate is low, the crucial quantity of the population dynamics is the fixation probability φ 1 , the probability that a new mutant takes over the population. We address the fixation probability for the simple case of a 2 × 2 game. The entries of the payoff matrix, however, are chosen from a probability distribution, excluding the interaction of the wild type with itself, which should not be affected by the mutation. But during the course of evolution, the payoff matrix remains fixed. Due to the probabilistic payoff matrix, the fixation probability φ itself becomes a random number. Since [START_REF] Kimura | Evolutionary rate at the molecular level[END_REF] introduced the neutral theory, many evolutionary biologists believe that changes in evolutionary confer only small or even vanishing selective advantages. It is very unlikely that a new mutation leads to a large selective advantage in a well adapted population. Thus, small intensities of selection seem to be biologically highly relevant [START_REF] Ohta | Near-neutrality in evolution of genes and gene regulation[END_REF]. It turns out that in our case, weak selection approximation corresponds to a moment expansion of the probability distribution, such that only the first few moments of the probability distribution of the payoff values matter. On the other hand, for strong selection the fixation probability is governed by the probability that a random mutant can invade the population.

The remainder of this paper is organized as follows: In Sec. 2, we recall the-Moran process as a standard model of frequency dependent evolutionary dynam-

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ics in finite populations. In Sec. 3, we address payoff matrices with Gaussian distributed entries and generalize the findings to arbitrary distributions in Sec. 4. Finally, in Sec. 5 we discuss how our findings relate to fitness distributions found empirically, based on frequency independent selection, and relate our results to this scenario.

The Moran Process and fixation probabilities

To model frequency dependence, we consider two player games, which can be represented by the payoff matrix

A B A a b B c d .
When A interacts with another A, it obtains a, and when it interacts with any B, it obtains b. Similarly, B obtains c or d, when interacting with A or another B respectively. Assuming there are i type A individuals and Ni type B individuals, the average payoffs of A and B in a mixed population are

π A = i -1 N -1 a + N -i N -1 b, ( 1a
)
π B = i N -1 c + N -i -1 N -1 d. (1b)
Here, we have explicitly assumed that there are no self-interactions. Note that the payoff difference is always a linear function in i,

Δπ = π B -π A = u i + v, (2)
where u = -a+b+c-d

N-1 and v = a-Nb+Nd-d N-1
. The impact of the game on fitness is determined by the intensity of selection β. While commonly it is assumed that fitness f is a linear function for the payoffs, it is often mathematically more convenient to choose f as an exponential function of the payoff. Both approaches can be justified by mathematical simplicity. But the exponential function allows to address a strong selection limit, that can be relevant in specific biological situations where a certain trait is necessary for survival, e.g. resistance towards toxins. Moreover, the exponential function guarantees that that fitness is always positive, even when the payoffs π are negative [START_REF] Traulsen | Analytical results for individual and group selection of any intensity[END_REF]. This leads to

f A = e +βπ A , (3a) f B = e +βπ B . (3b)

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We consider this game in the context of a frequency dependent Moran process (Nowak et al., 2004;[START_REF] Taylor | Evolutionary game dynamics in finite populations[END_REF][START_REF] Antal | Fixation of strategies for an evolutionary game in finite populations[END_REF]. Suppose there is a finite population of N individuals. One individual is chosen at random, but proportional to fitness, to give birth to a new individual of the same type. Before the new offspring is added, one individual chosen at random is removed to keep the population size constant. The probability to increase the number of type A individuals from i to i + 1 is T + i , and the probability to decrease the number of type A individuals from i to i -1 is T - i . For the Moran process, we have

T + i = i f A i f A + (N -i) f B N -i N , (4a)
T - i = (N -i) f B i f A + (N -i) f B i N . ( 4b
)
Due to the choice of an exponential function as payoff to fitness mapping, the ratio of the transition probabilities becomes particularly simple,

T - i T + i = f B f A = e +β(π B -π A ) . (5)
In the absence of mutations, we have T + 0 = 0 and T - N = 0, cf. Eqs. 4. Thus, there are two absorbing states, the state with all A, and the state with all B. The fixation probability φ i describes the probability of i type A individuals to take over the entire population. Obviously, the fixation probabilities fulfill the equation [START_REF] Goel | Stochastic Models in Biology[END_REF]) for a full derivation. Solving this recursion with the boundary conditions φ 0 = 0 and φ N = 1 leads to [START_REF] Nowak | Evolutionary Dynamics[END_REF][START_REF] Antal | Fixation of strategies for an evolutionary game in finite populations[END_REF]

φ i = T - i φ i-1 + T + i φ i+1 + (1 -T - i -T + i )φ i , see e.g.
φ i = 1 + i-1 k=1 k i=1 T - i T + i 1 + N-1 k=1 k i=1 T - i T + i . (6)
In particular, we are interested in φ 1 , for which the enumerator is simply one. Together with Eq. ( 5), we can write φ 1 as

φ 1 = 1 1 + N-1 k=1 exp +β k i=1 (π B -π A ) . ( 7
)
The two sums in Eq. ( 7) can be solved analytically, leading to closed expressions for the fixation probabilities (Traulsen et al., 2007a). However, for our numerical and analytical considerations, we consider Eq. ( 7) in the form given above.

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Fixation of Random Mutants

We consider type B as the wild type and type A as the mutant type. Typically, one is interested in the fixation probability of a mutant with fixed values in the payoff matrix. But sometimes, the payoff of the mutant may not be fixed or even unpredictable. For this reason, we focus on payoff matrices with random entries. First, we analyze the case in which the payoff values a, b, and c that describe the mutant's interactions are Gaussian random variables with mean μ and standard deviation σ. Alternative scenarios are discussed below. Note that the payoff matrix is constant, in contrast to the work of [START_REF] Fudenberg | Evolutionary dynamics with aggregate shocks[END_REF], where the payoff is subject to noise.

No matter which kind of randomness we consider in the payoff matrix, the fixation probability of a mutant in the population is the expectation value of φ 1 . Thus, we have to calculate the expectation value of the right hand side of Eq. ( 7).

When the new payoff values a, b, c are continuous random variables with probability density functions p(a), p(b), and p(c) respectively, we can write the expectation of the fixation probability, Eq. ( 7), as follows

E(φ 1 ) = 1 1 + N-1 k=1 exp +β k i=1 (π B -π A ) p(a)p(b)p(c) da db dc. (8)
For a given population size N, and given probability density functions, this equation can be solved by numerical integration in three dimensions, see Fig. 1. However, the asymptotic for strong selection, β → ∞, can be inferred directly, because for strong selection the fixation probability is either zero or one Altrock and Traulsen (2009a). For weak selection, β 1, we obtain an analytical approximation for the solution of the integrals.

Weak selection approximation

To address the case of weak selection, we expand the fixation probability φ 1 , Eq. ( 7) at β = 0. First, we expand the exponential function in Eq. ( 7) up to second order for β 1, exp

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ +β k i=1 (π B -π A ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ≈ 1 + β k i=1 (π B -π A ) + β 2 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k i=1 (π B -π A ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 2 . (9)

A c c e p t e d m a n u s c r i p t

Hence, we find for the fixation probability

φ 1 ≈ 1 1 + N-1 k=1 1 + β k i=1 (π B -π A ) + 1 2 β 2 k i=1 (π B -π A ) 2 (10) = 1 N + β N-1 k=1 k i=1 (π B -π A ) C 1 +β 2 1 2 N-1 k=1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k i=1 (π B -π A ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 2 C 2 .
After another expansion for β 1 we obtain

φ 1 ≈ 1 N -β C 1 N 2 + β 2 C 2 1 N 3 - C 2 N 2 . ( 11
)
Note that this expansion is valid for any payoff difference π B -π A . For example, let us consider the situation in which Δπ = π B -π A is constant. This occurs for frequency independent selection, but also for a + d = b + c, which is often referred to as "equal gains from switching" [START_REF] Nowak | The evolution of stochastic strategies in the prisoner's dilemma[END_REF]. In this case, we have C 1 = N(N -1)Δπ/2 and C 2 = N(N -1)(2N -1)Δπ 2 /6. This leads to

φ 1 ≈ 1 N 1 -β(N -1) Δπ 2 -β 2 (N 2 -1) Δπ 2 12 . ( 12
)
For a first order expansion to be meaningful, β 1 is not enough. Instead, we have to ensure βNΔπ 1.

In principle, we could assume any function for π B -π A . The most important case, however, are 2 × 2 games, which lead to a linear dependence of the payoff difference Δπ = π B -π A = u i + v, cf. Eq. ( 2). In this case, we have

C 1 = N-1 k=1 k i=1 (u i + v) (13) = N-1 k=1 u k 2 2 + (u + 2v)k 2 = u N N 2 -1 6 + v N(N -1) 2 .

A c c e p t e d m a n u s c r i p t

For the second order term in Eq. ( 10), we obtain

C 2 = 1 2 N-1 k=1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k i=1 (u i + v) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 2 (14) = 1 2 N-1 k=1 uk 2 2 + (u + 2v)k 2 2 = u 2 8 N-1 k=1 k 4 + u 2 4 + uv 2 N-1 k=1 k 3 + u 2 8 + uv 2 + v 2 2 N-1 k=1 k 2 = N(N -1) u 2 3N 3 + 3N 2 -2N -2 120 + uv 3N 2 + N -2 24 + v 2 2N -1 12 .
Thus, we find for the expectation value of φ 1 under weak selection

E(φ 1 ) ≈ 1 N -β E(C 1 ) N 2 -β 2 E(C 2 ) N 2 + β 2 E(C 2 1 ) N 3 . ( 15
)
Note that C 1 is linear in the payoffs, whereas C 2 is quadratic in the payoffs. So far, this equation is valid for any distribution of payoffs p(x). Next, we focus on the case in which a, b, and c follow Gaussian distributions with mean μ and standard deviation σ 2 , p(x) = 1 σ √ 2π exp -(x-μ) 2 2σ 2 . In our expectation value Eq. ( 15), we have only terms involving the first and the second moment of the distribution. Using E(x) = μ and E(x 2 ) = μ 2 + σ 2 , we find

E(u) = - d -μ N -1 (16a) E(v) = d -μ (16b) E(u 2 ) = (d -μ) 2 (N -1) 2 + 3σ 2 (N -1) 2 (16c) E(v 2 ) = (d -μ) 2 + N 2 + 1 (N -1) 2 σ 2 (16d) E(uv) = - (d -μ) 2 N -1 - N + 1 (N -1) 2 σ 2 .
(16e)

A c c e p t e d m a n u s c r i p t

With these terms, Eq. ( 15) reduces to

E(φ 1 ) ≈ 1 N (17) -β N -2 3N (d -μ) +β 2 N -2 N -1 16N 2 -57N + 47 360N (d -μ) 2 +β 2 N -2 N -1 6N 2 -7N + 7 120N σ 2 .
In the simplest case, the average payoff entry associated with the mutant is identical to the wild type's payoff interacting with himself, such that we have μ = d. Then, the linear term in Eq. ( 17) vanishes and only a second order weak selection approximation will lead to deviations from the neutral case. We obtain

E(φ 1 ) ≈ 1 N + β 2 N -2 N -1 6N 2 -7N + 7 120N σ 2 . ( 18
)
Now, the variance σ can be absorbed into the selection intensity. In other words, changing the variance of the Gaussian distribution is equivalent to changing the intensity of selection. In Fig. 1, the quadratic approximation Eq. ( 18) is compared to the numerical solution of the integrals in Eq. ( 8) and to individual based simulations. For the third order weak selection approximation, we refer to the Appendix.

Strong selection limit

For strong selection, β → ∞, only those mutants that are advantageous from the beginning of their invasion until the time they finally reach fixation take over the population. This means the mutant type A will reach fixation only if a > c and b > d, see Eq. 1. In this case, the fixation probability is 1 for β → ∞.

However, the fixation probability is not only one for a > c and b > d. It also converges to one with β → ∞ in coexistence games with a < c and b > d if we have a + b > c + d in addition. This situation occurs for μ = d with probability 1/8. However, for such games the average fixation times diverge rapidly with population size and intensity of selection [START_REF] Antal | Fixation of strategies for an evolutionary game in finite populations[END_REF]Traulsen et al., 2007a). Thus, for practical purposes we can neglect the fixation in coexistence games under strong selection, as it can hardly ever be observed.

Thus, we only consider a > c and b > d here for the fixation probability. The probability for this payoff ranking is the fixation probability for strong selection.

A c c e p t e d m a n u s c r i p t

First, let us consider the situation when a, b, and c follow a Gaussian distribution with mean μ = d and standard deviation σ. The probability that a is larger than c is 1/2. The probability that b is larger than d is also 1/2. Therefore, the fixation probability is simply 1/4, see Fig. 1.

In frequency independent mutant scenarios, the payoff of both type does not depend on its interaction partner, i.e. a = b and c = d. In this case, the fixation probability of a random mutant is 1/2 for μ = d.

Computer simulations

We simulate the population dynamics exactly as described in Sec. 2. In each time step, each individual interacts with all others in the population and obtains a payoff. However, our results for weak selection would not change significantly if they interact only with a random subset of the population (Traulsen et al., 2007b;[START_REF] Woelfing | Stochastic sampling of interaction partners versus deterministic payoff assignment[END_REF]. Then, an individual is selected with probability proportional to its fitness and produces identical offspring. Another individual chosen at random is removed. To compute fixation probabilities, we simulate this birth-death process many times, each time with a new, different payoff matrix. Note that we are combining two sources of randomness, as the fixation process itself and the payoff matrix are stochastic.

When we compare our analytical results to computer simulations, a further difficulty appears: Formally, we are always considering a Markov chain with two absorbing states i = 0 and i = N. Eventually, we will end up in one of them. But the time until we reach these states diverges with the intensity of selection and the population size if a < c and d < b [START_REF] Antal | Fixation of strategies for an evolutionary game in finite populations[END_REF]Traulsen et al., 2007a;Altrock and Traulsen, 2009b), see above. Thus, we have two choices: Either, we include coexistence games and say that no fixation has occurred if we have waited for a very long time and still both types are present. However, then we cannot expect that our numerical results coincide with the analytical theory, because only the latter approach takes the possibility of fixation in coexistence games into account. This approach is appropriate when selection is not too strong. For strong selection, Alternatively, we can exclude coexistence games from the beginning, both in our numerical solution of the integrals in Eq. ( 8) and in our simulations. If we do this and assume that fixation never occurs in coexistence games, we find a convergence for strong selection towards a fixation probability of 1/4.

As shown in Fig. 1, the simulations and the numerical results agree nicely both under strong selection and weak selection if we take these complications into account. However, under strong selection the fixation probability from simulations temporarily exceeds the strong selection limit. The reason is that in those games in which wild type only slightly dominates the mutant, the latter may still occasionally reach fixation. This makes the fixation probability of the mutant type slightly higher than it should be theoretically. For β → ∞, however, the fixation probability converges to 1/4, as expected.

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Alternative fitness distributions

So far, we have concentrated on Gaussian distributed payoff entries. Next, we relax this restriction and consider more general distributions. It turns out that for weak selection, only the first moments of the distribution matter, whereas for strong selection, fixation is governed by the probability that a payoff value is larger than the average. Thus, our results from above generalize easily to general distributions. If we approximate up to second order, only d -μ, the difference between mean μ and the wildtype's payoff d against itself and the standard deviation σ affect the fixation probability. For strong selection, the fixation probability of mutant type is determined by the product of the probability that the mutant is advantageous when it invades, b > d and the probability that it is advantageous when it is frequent, a > c. The latter probability is 1 2 and the former probability is given by the shaded part of the distribution in the figure.

Weak selection approximation

The weak selection approximation Eq. ( 17) corresponds to a moment expansion of the probability distribution: 8 For the linear term, only the difference between average μ and the wild type's payoff against himself d matters. For the quadratic term, we have to take into account the second moment as well. Thus, Eq. ( 17) holds for any distribution with mean μ and standard deviation σ. The same reasoning holds when we take higher order terms in β into account, see Appendix. When μ = d, the moments of the distribution matter up to the order of our approximation in β, see Fig. 2 .

Strong selection limit

For strong selection, β → ∞, only those mutants whose fitness always exceeds that of the wild type, will eventually reach fixation. In game theory, such types

A c c e p t e d m a n u s c r i p t

are called dominant. The probability of the mutant type to be dominant is based on the fitness distribution. If a, b, and c follow a symmetric distribution p with mean μ and standard deviation σ, and μ = d, we obtain the same strong selection limit as for the Gaussian distribution with μ = d, see Sec. 3.2. This is because in our strong selection argument, we only have to consider the probability that a certain payoff entry is larger than another one or larger than the mean. If the fitness distribution is asymmetric or μ d, this argument no longer holds and different strong selection limits will be reached under special fitness distributions. Since both a and c are chosen from the same distribution, the probability p 1 that a is larger than c, and thus the probability that the new mutant dominates when it has high abundance, is simply p 1 = ∞ -∞ p(x)p(y)Θ(xy)dxdy = 0.5, where Θ(xy) is the step function. Therefore, the only influence of the details of the distribution occurs through the payoff entry b. The mutant will be dominant if in addition b is larger than d, which is the condition that the mutant is advantageous when it enters the population. This occurs with probability p 2 = ∞ d p(x)dx. For d = μ and distributions in which the median is equal to the mean, we obtain p 2 = 1/2. But in general, the value of p 2 depends on the precise shape of the distribution, see Fig. 2, and the fixation probability is given by p 1 p 2 .

For example, let us assume a, b, and c follow an exponential distribution with mean λ, e.g. p(a) = 1 λ exp -a λ , such that a, b, c ≥ 0. Thus, we obtain

p 2 = λ -1 ∞ d exp -x λ dx = exp -d λ .
Even for λ = d, we have p 2 = exp[-1] ≈ 0.368. Thus, the probability that the mutant dominates the wild type is p 1 p 2 = exp [-1] /2 ≈ 0.184. The asymptotic limit of the fixation probability for strong selection, β → ∞, is also p 1 p 2 . For λ = d/ ln 2, we would obtain the same asymptotic limit as for the Gaussian distribution.

Discussion

We have introduced a model in which a new mutant in asexual population is characterized by a new payoff matrix. We have calculated the probability that such a mutant interacting with the wild type in a novel, unpredictable way can take over a population. This depends on the details of the interactions, which are in our case based on the distribution of payoff values. How does this relate to the usual approaches of population genetics that discuss fitness distributions? One simplifying assumption of evolutionary game theory is that individuals are haploid and reproduction is asexual. So we should first aim at e.g. comparing to experimental data from bacteria.

A c c e p t e d m a n u s c r i p t

Recent works in population genetics have attempted to measure fitness distributions experimentally [START_REF] Zeyl | Estimates of the rate and distribution of fitness effects of spontaneous mutation in Saccharomyces cerevisiae[END_REF][START_REF] Cowperthwaite | Disributions of beneficial fitness effects in RNA[END_REF][START_REF] Kassen | Distribution of fitness effects among beneficial mutations before selection in experimental populations of bacteria[END_REF]. Also in our case, we have a distribution of fitness values, reflected by the entries of a payoff matrix. However, our model is looking at the invasion and fixation of mutants from a different perspective. The first and most important point is the way we define fitness. Mutations bring variety, and fitness is to describe the advantages of one type over others. In the traditional population genetic view, fitness is typically constant, but our model considers fitness under frequency dependent selection. The evolutionary dynamics under these two approaches will be quite different, especially when selection is not weak, see Fig. 3. For constant fitness, a mutant in an asexual population is either advantageous or disadvantageous compared to the wild type. Therefore, the fixation or extinction of the mutant type is usually fast for strong selection. But for frequency dependent fitness, mutant and wild type may coexist with each other for a long time. This occurs when a mutant performs better than the wild type when it is rare, but the wild type has a fitness advantage when the mutant is frequent. One would expect that frequency dependence is the rule rather than the exception, because the success of a strategy typically depends on the actions and abundance of others.

The distribution of fitness values is a central concept in population genetics. [START_REF] Gillespie | Some properties of finite populations experiencing strong selection and weak mutation[END_REF] and [START_REF] Orr | The population genetics of adaptation: the adaptation of DNA sequences[END_REF][START_REF] Orr | The distribution of fitness effects among beneficial mutations[END_REF] have proposed that the fitness distribution of beneficial mutants would be approximately exponential. Alternative distributions like gamma distribution, L-shaped distribution and slightly bell-shaped distribution are also considered to be possible. The fitness distribution is a function of the environment and thus it is influenced by many factors, such as the adaption of the wild type to the environment. It also makes a difference if only single-step mutations are considered or if also mutants with several mutations are taken into account. However, these attempts are typically based on the assumption of fixed fitness values. If fitness is frequency dependent, as in our model, the selective advantage of a novel type depends on its frequency. A new mutation may be able to invade, but not to take over the population. Alternatively, new mutants may be disadvantageous, but turn highly successful when they have crossed a certain threshold.

Payoff matrices with random entries have been considered before: In a seminal paper, [START_REF] Fudenberg | Evolutionary dynamics with aggregate shocks[END_REF] have shown that a game with time-dependent random payoff matrix can be described by a stochastic form of the replicator equation. They have argued that the system spends most time in the vicinity of the risk dominant equilibrium. This is the strategy with the larger basin of attraction under Here, wild type's payoff against itself d is larger than mean μ of the Gaussian random variables in the payoff matrix (d = 2, μ = 1). This is consistent with the notion in population genetics that deleterious mutants are more common than beneficial ones. For weak selection, the dynamics under both approaches are similar. However, the scenarios under strong selection are quite different. In the inset, we show the weak selection approximations of the fixation probability under frequency dependent fitness. As the probability to be deleterious is higher, the fixation probability of a random mutant decreases first with the intensity selection, before it starts to increase again. It is obviously that the approximation becomes more accurate when higher order terms are considered. In particular, the linear term cannot capture the fixation probability when the intensity of selection is increased here (population size N = 100, averages over 10 5 runs).

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positive frequency dependent selection. The approach of Fudenberg and Harris leads to much faster convergence to the risk dominant equilibrium than the approach of [START_REF] Kandori | Learning, mutation, and long run equilibria in games[END_REF], looking at mutations only under strong selection.

Our model assumes that the nature of interactions, i.e. the payoff matrix, does not change in time, whereas [START_REF] Fudenberg | Evolutionary dynamics with aggregate shocks[END_REF] consider a situation in which fitness is not only frequency, but also time dependent, such that fixation probabilities are not meaningful. [START_REF] Berg | Matrix games, mixed strategies, and statistical mechanics[END_REF] as well as [START_REF] Galla | Two-population replicator dynamics and number of Nash equilibria in random matrix games[END_REF] have considered random bimatrix games with a large number of strategies to address the number of Nash equilibria and the fraction of strategies contributing to mixed Nash equilibria.

A c c e p t e d m

a n u s c r i p t [START_REF] Eriksson | Evolution of strategies in repeated stochastic games with full information of the payoff matrix[END_REF] have asked what kind of strategies is most successful if the payoff matrix is chosen at random and change in every round of game. These papers either consider large numbers of strategies, sophisticated decision processes or temporal changes of the payoff matrix that do not affect the identity of the strategies. Our approach is different in many aspects: Players do not switch between strategies or choose a particular way to play in each game. Rather, they play a fixed strategy that they also pass on to their offspring. We consider an ensemble of fixed games and explore how the probability that a mutant can take over a population depends on that ensemble. It turns out that the dependence on the underlying distribution of interaction parameters corresponds to a moment expansion for weak selection and to an integral of a part of the fitness distribution for strong selection.

Acknowledgement

W.H. gratefully acknowledges funding by the Chinese Scholarship Council and the American Association of University Women. A.T. is funded through the Emmy-Noether program of the German Research Foundation (DFG).

A. Third order weak selection

The third order expansion of the fixation probability of a single mutant under weak selection is calculated here. We consider the fitness as frequency dependent. First, we expand the exponential function in Eq. ( 7) for β 1, exp

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ +β k i=1 (π B -π A ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ≈ 1+β k i=1 (π B -π A )+ β 2 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k i=1 (π B -π A ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 2 + β 3 6 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k i=1 (π B -π A ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 3 ( 19
) Then, Eq. ( 7) can be writen as

φ 1 ≈ 1 N + βC 1 + β 2 C 2 + β 3 1 6 N-1 k=1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k i=1 (π B -π A ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 3 C 3 , ( 20
)
where C 1 and C 2 are defined as Eq. ( 13) and Eq. ( 14) in the main text. Expanding Eq. ( 20) for β 1, we have

φ 1 ≈ 1 N -β C 1 N 2 + β 2 C 2 1 N 3 - C 2 N 2 -β 3 C 3 1 N 4 - 2C 1 C 2 N 3 + C 3 N 2 (21)

A c c e p t e d m a n u s c r i p t

As π B -π A = u i + v, cf. Eq. ( 2), we have

C 3 = 1 6 N-1 k=1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k i=1 (u i + v) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 3 (22) = N u 3 15N 6 -42N 4 + 35N 2 -8 630 + u 2 v 5N 5 -3N 4 -10N 3 + 5N 2 + 5N -2 240 +N uv 2 12N 4 -15N 3 -10N 2 + 15N -2 240 + v 3 N(N -1) 2 24
When we assume that a, b, c follow an arbitrary distribution f (x) with mean μ, variance σ 2 and skewness γ, the fixation probability under weak selection is the expectation value of φ 1 ,

E(φ 1 ) ≈ 1 N -β E(C 1 ) N 2 + β 2 E(C 2 1 ) N 3 - E(C 2 ) N 2 -β 3 E(C 3 1 ) N 4 - 2E(C 1 C 2 ) N 3 + E(C 3 ) N 2 .
(23) Compared with the second order approximation where only the first and the second moment of the distribution are involved, we need to include the third moment of the distribution E(x 3 ) = μ 3 + 3μσ 2 + γ. Then, we obtain four new terms beside those in Eq.( 16)

E(u 3 ) = - (d -μ) 3
(N -1) 3 -9(d -μ) σ 2 (N -1) 3 + γ (N -1) 3 (24a)

E(v 3 ) = (d -μ) 3 + 3(N 2 + 1) (N -1) 2 (d -μ) σ 2 - N 3 -1 (N -1) 3 γ (24b) E(u 2 v) = (d -μ) 3 (N -1) 2 + 5N -1 (N -1) 3 (d -μ) σ 2 - γ (N -1) 2 (24c) E(uv 2 ) = - (d -μ) 3 N -1 - 3N 2 -1 (N -1) 3 (d -μ) σ 2 + (N 2 -1) (N -1) 3 γ (24d)

A c c e p t e d m a n u s c r i p t

With all the items in Eq. ( 16) and Eq. ( 24), the fixation probability for a mutant in Eq.( 23) becomes

E(φ 1 ) ≈ 1 N (25) -β N -2 3N (d -μ) +β 2 N -2 N -1 16N 2 -57N + 47 360N (d -μ) 2 +β 2 N -2 N -1 6N 2 -7N + 7 120N σ 2 -β 3 N -2 (N -1) 2 32N 4 -345N 3 + 1106N 2 -1347N + 530 15120N (d -μ) 3 -β 3 N -2 (N -1) 2 12N 4 -293N 3 + 546N 2 -541N + 288 5040N (d -μ) σ 2 +β 3 N -2 (N -1) 2 32N 4 -177N 3 + 182N 2 -255N + 194 15120N γ.

Figure 1 :

 1 Figure 1: Fixation probability of a single mutant with random payoff values under different selection intensities. The payoff values are Gaussian distributed with the mean equal to the wild type individual's payoff. We start from a single mutant and wait until it is either lost or it takes over the whole population. The fixation probability is the fraction of runs in which the mutants took over. Coexistence games are only taken into account for weak selection, see text. The black box represent the region between strong selection and weak selection (population size N = 100, averages over 10 5 runs).

Figure 2 :

 2 Figure2: The probability of fixation of a random mutant for general payoff value distributions is determined by different properties of the distribution. For weak selection, the fixation probability is determined by the first moments of the distribution of the new payoff values a, b and c. If we approximate up to second order, only d -μ, the difference between mean μ and the wildtype's payoff d against itself and the standard deviation σ affect the fixation probability. For strong selection, the fixation probability of mutant type is determined by the product of the probability that the mutant is advantageous when it invades, b > d and the probability that it is advantageous when it is frequent, a > c. The latter probability is 1 2 and the former probability is given by the shaded part of the distribution in the figure.

Figure 3 :

 3 Figure3: In the main panel, we compare the probability of fixation under constant fitness and frequency dependent fitness. Here, wild type's payoff against itself d is larger than mean μ of the Gaussian random variables in the payoff matrix (d = 2, μ = 1). This is consistent with the notion in population genetics that deleterious mutants are more common than beneficial ones. For weak selection, the dynamics under both approaches are similar. However, the scenarios under strong selection are quite different. In the inset, we show the weak selection approximations of the fixation probability under frequency dependent fitness. As the probability to be deleterious is higher, the fixation probability of a random mutant decreases first with the intensity selection, before it starts to increase again. It is obviously that the approximation becomes more accurate when higher order terms are considered. In particular, the linear term cannot capture the fixation probability when the intensity of selection is increased here (population size N = 100, averages over 10 5 runs).