MD Tatjana S Potpara

MD, FESC, FACC Gregory Y H Lip

Lone Atrial Fibrillation: What is known and what is to come

Atrial fibrillation (AF) is the most prevalent sustained cardiac arrhythmia in adults, affecting >1% of general population. AF is commonly associated with structural heart disease and is a major cause of significant cardiovascular morbidity and mortality. AF sometimes develops in a subset of young patients (e.g. aged ≤60 years), with no evidence of associated cardiopulmonary or other comorbid disease (including hypertension), and has been referred to as 'lone AF'. The latter generally has a favorable prognosis; the prognostic and therapeutic implications of an accurate identification of patients with truly lone AF (that is, truly at low risk of complications), if any, would be of the utmost importance. The true prevalence of lone AF is unknown, varying between 1.6% and 30%, depending on the particular study population.

Nonetheless, novel risk factors for AF, including obesity, metabolic syndrome, sleep apnea, alcohol consumption, endurance sports, anger, hostility, subclinical atherosclerosis and others, have been increasingly recognized. Also, various underlying pathophysiologic mechanisms predisposing to AF, including increased atrial stretch, structural and electrophysiological alterations, autonomic imbalance, systemic inflammation, oxidative stress and genetic predisposition, have been proposed. The growing evidence of these diverse (and numerous) pathogenic mechanisms and factors related to AF inevitably raises the question of whether 'lone AF' does exist at all.

In this review article we summarize the current knowledge of the epidemiology, pathophysiology, clinical course and treatment of patients with so-called 'lone AF' and outline emerging insights into its pathogenesis and the potential therapeutic implications of a diagnosis of lone AF.

Introduction

Atrial fibrillation (AF) is the most prevalent sustained cardiac arrhythmia in adults, affecting >1% of general population. The incidence of AF is projected to rise over the next few decades and AF has therefore been referred to as a "non-contagious epidemic" [START_REF] Heeringa | Prevalence, incidence and lifetime risk of atrial fibrillation: the Rotterdam study[END_REF][START_REF] Stewart | Population prevalence, incidence, and predictors of atrial fibrillation in the Renfrew/Paisley study[END_REF][START_REF] Go | Prevalence of diagnosed atrial fibrillation in adults: national implications for rhythm management and stroke prevention: the AnTicoagulation and Risk Factors in Atrial Fibrillation (ATRIA) Study[END_REF][START_REF] Furberg | Prevalence of atrial fibrillation in elderly subjects (the Cardiovascular Health Study)[END_REF][START_REF] Braunwald | Cardiovascular medicine at the turn of the millennium: triumphs, concerns, and opportunities[END_REF].

AF is associated with significant cardiovascular morbidity and mortality, primarily due to ischemic stroke, congestive heart failure (CHF) and impaired quality of life, with the requirement for chronic use of medication [START_REF] Benjamin | Impact of atrial fibrillation on the risk of death: The Framingham heart study[END_REF][START_REF] Wolf | Atrial fibrillation as an independent risk factor for stroke: the Framingham Study[END_REF][START_REF] Menke | Thromboembolism in atrial fibrillation[END_REF][START_REF] Chugh | Epidemiology and natural history of atrial fibrillation: clinical implications[END_REF][START_REF] Wang | Temporal relations of atrial fibrillation and congestive heart failure and their joint influence on mortality: the Framingham Heart Study[END_REF][START_REF] Savelieva | Atrial fibrillation and heart failure: natural history and pharmacological treatment[END_REF]. Well-defined risk factors for AF include advanced age, hypertension, structural heart disease, CHF, diabetes mellitus and thyroid disease [START_REF] Benjamin | Independent risk factors for atrial fibrillation in a population-based cohort. The Framingham Heart Study[END_REF][START_REF] Ryder | Epidemiology and significance of atrial fibrillation[END_REF]. In turn, the risk of AF-related complications is strongly influenced by the nature and extent of underlying co-morbid disease and age [START_REF] Chugh | Epidemiology and natural history of atrial fibrillation: clinical implications[END_REF][START_REF] Ryder | Epidemiology and significance of atrial fibrillation[END_REF][START_REF] Kopecky | Lone atrial fibrillation in elderly persons. A marker for cardiovascular risk[END_REF].

AF sometimes develops in a subset of young patients (e.g. aged ≤60 years), with no evidence of associated cardiopulmonary or other comorbid disease (including hypertension), and has been referred to as 'lone AF', which generally has a favorable prognosis [START_REF] Fuster | ACC/AHA/ESC 2006 guidelines for the management of patients with atrial fibrillation: a report of the American College of Cardiology / American Heart Association Task Force on Practice Guidelines and the European Society of Cardiology Committee for Practice Guidelines (Writing Committee to Revise the 2001 Guidelines for the Management of Patients With Atrial Fibrillation)[END_REF][START_REF] Brand | Characteristics and prognosis of lone atrial fibrillation. 30-year follow-up in the Framingham Study[END_REF][START_REF] Kopecky | The natural history of lone atrial fibrillation. A population-based study over three decades[END_REF][START_REF] Davidson | Diagnosis and characteristics of lone atrial fibrillation[END_REF][START_REF] Rostagno | Clinical course of lone atrial fibrillation since first symptomatic arrhythmic episode[END_REF][START_REF] Scardi | Lone atrial fibrillation: prognostic differences between paroxysmal and chronic forms after 10 years of follow-up[END_REF][START_REF] Jahangir | Long-term progression and outcomes with aging in patients with lone atrial fibrillation. A 30-year follow-up study[END_REF][START_REF] Potpara | Mortality of patients with lone and idiopathic atrial fibrillation is similar to mortality in general population of Serbia[END_REF][START_REF] Jouven | Idiopathic atrial fibrillation as a risk factor for mortality[END_REF]. The term "lone" AF was introduced ~60 years ago and it is still a diagnosis of exclusion, which should be made only after careful assessment including detailed medical history, physical examination, laboratory testing with thyroid status, electrocardiogram, echocardiography and, possibly, chest x-ray and exercise testing [START_REF] Fuster | ACC/AHA/ESC 2006 guidelines for the management of patients with atrial fibrillation: a report of the American College of Cardiology / American Heart Association Task Force on Practice Guidelines and the European Society of Cardiology Committee for Practice Guidelines (Writing Committee to Revise the 2001 Guidelines for the Management of Patients With Atrial Fibrillation)[END_REF][START_REF] Evans | Lone auricular fibrillation[END_REF]. Nonetheless, novel risk factors for AF, including obesity [START_REF] Wang | Obesity and the risk of new-onset atrial fibrillation[END_REF][START_REF] Frost | Overweight and obesity as risk factors for atrial fibrillation or flutter: the Danish diet, cancer, and health study[END_REF][START_REF] Dublin | Risk of new-onset atrial fibrillation in relation to body-mass index[END_REF][START_REF] Wanahita | Atrial fibrillation and obesity -results of a meta-analysis[END_REF], metabolic syndrome [START_REF] Watanabe | Metabolic syndrome and risk of development of atrial fibrillation. The Niigata Preventive Medicine Study[END_REF][START_REF] Chamberlain | Metabolic syndrome and incidence of atrial fibrillation among blacks and whites in the Atherosclerotic Risk in Communities (ARIC) Study[END_REF],

sleep apnea [START_REF] Roche | Relationship among the severity of sleep apnea syndrome, cardiac arrhythmias, and autonomic imbalance[END_REF][START_REF] Porthan | Prevalence of sleep apnea syndrome in lone atrial fibrillation: a case-control study[END_REF][START_REF] Kanagala | Obstructive sleep apnea and the recurrence of atrial fibrillation[END_REF][START_REF] Gami | Obstructive sleep apnea, obesity and the risk of incident atrial fibrillation[END_REF][START_REF] Rupprecht | Causality in the relationship between central sleep apnea and paroxysmal atrial fibrillation[END_REF], increased alcohol consumption [START_REF] Ettinger | Arrhythmias and the "Holiday Heart": alcohol-associated cardiac rhythm disorders[END_REF][START_REF] Koksinen | Alcohol and new onset atrial fibrillation: a case-control study of a current series[END_REF][START_REF] Djousse | Long-term alcohol consumption and the risk of atrial fibrillation in the Framingham Study[END_REF][START_REF] Frost | Alcohol and risk of atrial fibrillation or flutter: a cohort study[END_REF][START_REF] Mukamal | Alcohol consumption and risk of atrial fibrillation in men and women: the Copenhagen City Heart Study[END_REF][START_REF] Mukamal | Alcohol consumption and risk and prognosis of atrial fibrillation among older adults: the Cardiovascular Health Study[END_REF], endurance sport practice [START_REF] Mont | Long-lasting sport practice and lone atrial fibrillation[END_REF][START_REF] Elosua | Sport practice and the risk of lone atrial fibrillation: a case-control study[END_REF][START_REF] Mont | the GIRAFA (Grup Integrat de Recerca en Fibril-lacio Auricular) Investigators. Physical activity, height, and left atrial size are independent risk factors for lone atrial fibrillation in middle-aged healthy individuals[END_REF][START_REF] Molina | Long-term endurance sport practice increases the incidence of lone atrial fibrillation in men: a follow-up study[END_REF][START_REF] Heidbuchel | Endurance sports is a risk factor for atrial fibrillation after ablation for atrial flutter[END_REF], increased pulse pressure [START_REF] Mitchell | Pulse pressure and the risk of new-onset atrial fibrillation[END_REF], anger and hostility [START_REF] Eaker | Anger and hostility predict the development of atrial fibrillation in men in the Framingham Offspring Study[END_REF][START_REF] Mattioli | The relationship between personality, socioeconomic factors, acute life stress and the development, spontaneous conversion and recurrences of acute lone atrial fibrillation[END_REF], subclinical atherosclerosis [START_REF] Fox | Mitral annular calcification is a predictor for incident atrial fibrillation[END_REF][START_REF] Lorenz | Prediction of clinical cardiovascular events with carotid intima-media thickness[END_REF] and others, are increasingly recognized. Also, various underlying pathophysiologic mechanisms predisposing to AF, including increased atrial stretch, structural and electrophysiological alterations, autonomic imbalance, systemic inflammation, oxidative stress and genetic predisposition, have been proposed [START_REF] Allessie | Electrical, contractile and structural remodeling during atrial fibrillation[END_REF][START_REF] Boos | Is atrial fibrillation an inflammatory disorder?[END_REF][START_REF] Sitges | Is there an anatomical substrate for idiopathic paroxysmal atrial fibrillation? A case-control echocardiographic study[END_REF][START_REF] Kourliouros | Current concepts in the pathogenesis of atrial fibrillation[END_REF][START_REF] Roberts | Impact of genetic discoveries on the classification of lone atrial fibrillation[END_REF]. The growing evidence of these diverse (and numerous) pathogenic mechanisms and factors related to AF inevitably raises the question of whether lone AF does exist at all [START_REF] Schoonderwoerd | New risk factors for atrial fibrillation: causes of "not-so-lone atrial fibrillation[END_REF][START_REF] Kozlowski | Lone atrial fibrillation: what do we know?[END_REF]. The prognostic and therapeutic implications of an accurate identification of patients with truly lone AF (that is, truly at low risk of complications), if any, would be of the utmost importance [START_REF] Lip | Refining clinical risk stratification for predcting stroke and thromboembolism using a novel risk factor-based approach[END_REF].

In this review article we summarize the current knowledge of the epidemiology, pathophysiology, clinical course and treatment of patients with so-called 'lone AF' and outline emerging insights into its pathogenesis and the potential therapeutic implications of making a diagnosis of 'lone AF'.

Epidemiology of lone AF

The overall prevalence of AF increases with age: <1% of individuals younger than 60 and ~10% of those >80 years old have AF [START_REF] Go | Prevalence of diagnosed atrial fibrillation in adults: national implications for rhythm management and stroke prevention: the AnTicoagulation and Risk Factors in Atrial Fibrillation (ATRIA) Study[END_REF]. The true prevalence of lone AF is unknown, varying between 1.6% and 30%, depending on the patients' age and criteria used in the study [START_REF] Brand | Characteristics and prognosis of lone atrial fibrillation. 30-year follow-up in the Framingham Study[END_REF][START_REF] Kopecky | The natural history of lone atrial fibrillation. A population-based study over three decades[END_REF][START_REF] Davidson | Diagnosis and characteristics of lone atrial fibrillation[END_REF][START_REF] Scardi | Lone atrial fibrillation: prognostic differences between paroxysmal and chronic forms after 10 years of follow-up[END_REF][START_REF] Levy | on behalf of the College of French Cardiologists. Characterization of different subsets of atrial fibrillation in general practice in France. The ALFA Study[END_REF].

The lifetime risk for development of AF, in subjects aged above 40, is 1:4 both for males and females [START_REF] Wang | Lifetime risk for development of atrial fibrillation. The Framingham heart study[END_REF]. Nevertheless, males have a higher incidence of AF in all age groups. Indeed, AF develops in males at 1.5-fold greater rate than the rate in females, when adjusted for other risk factors [START_REF] Benjamin | Independent risk factors for atrial fibrillation in a population-based cohort. The Framingham Heart Study[END_REF][START_REF] Kannel | Prevalence, incidence, prognosis, and predisposing conditions for atrial fibrillation: population based estimates[END_REF]. There is a male preponderance in lone AF, with a male-to-female ratio of 3-4:1 [START_REF] Brand | Characteristics and prognosis of lone atrial fibrillation. 30-year follow-up in the Framingham Study[END_REF][START_REF] Kopecky | The natural history of lone atrial fibrillation. A population-based study over three decades[END_REF][START_REF] Scardi | Lone atrial fibrillation: prognostic differences between paroxysmal and chronic forms after 10 years of follow-up[END_REF]. However, male predominance is less evident among confirmed familial AF probands or affected relatives, when compared with nonfamilial (sporadic) lone AF patients [START_REF] Chen | Lone atrial fibrillation: influence of familial disease on gender predilection[END_REF]. To explain the increased frequency of sporadic lone AF in males, the possibility of an unrecognized X-linked recessive AF in males with negative family history and apparently sporadic AF (and mothers and sisters as the healthy carriers) has been raised [START_REF] Chen | Lone atrial fibrillation: influence of familial disease on gender predilection[END_REF]. Of note, familial and sporadic lone AF share similar phenotypes and are clinically indistinguishable [START_REF] Chen | Lone atrial fibrillation: influence of familial disease on gender predilection[END_REF].

A genetic contribution to AF was recognized nearly 70 years ago, when Wolff documented an autosomal dominant transmission of lone AF in one family [START_REF] Wolff | Familial auricular fibrillation[END_REF]. Since then, the familial aggregation of lone AF has been increasingly reported and certain genetic mutations (often confined to the particular family, and not found outside the family) have been described [56,65- [START_REF] Fox | Parental atrial fibrillation as a risk factor for atrial fibrillation in offspring[END_REF][START_REF] Elinor | Familial aggregation in lone atrial fibrillation[END_REF][START_REF] Christophersen | Familial aggregation of atrial fibrillation. A Study of Danish Twins[END_REF][START_REF] Arnar | Familial aggregation of atrial fibrillation in Iceland[END_REF]. For example, in the Framingham Study, parental AF increased the risk for offspring AF with an Odds Ratio of 3.2 (95%CI: 1.7-5,9, p<0.01), when the sample was limited to those age <75 years old without antecedent cardiovascular morbidity [START_REF] Fox | Parental atrial fibrillation as a risk factor for atrial fibrillation in offspring[END_REF].

In the past decade, an increasing number of genetic loci and causal mutations of genes (primarily those encoding components of potassium, sodium and L-type calcium ion channels) have been identified in lone AF patients, and significant overlap of AF with congenital long QT, short QT and Brugada syndrome, and with dilated cardiomyopathy has been observed (Table 1) [START_REF] Roberts | Impact of genetic discoveries on the classification of lone atrial fibrillation[END_REF][START_REF] Brugada | Identification of a genetic locus for familial atrial fibrillation[END_REF][START_REF] Ellinor | Locus for atrial fibrillation maps to chromosome 6q14-16[END_REF][START_REF] Chen | KCNQ1 gain-of-funkcion mutation in familial atrial fibrillation[END_REF][START_REF] Oberti | Genome-wide linkage scan identifies a novel genetic locus on chromosome 5p13 for neonatal atrial fibrillation associated with sudden death and variable cardiomyopathy[END_REF][START_REF] Volders | Mapping a novel locus for familial atrial fibrillation on chromosome 10p11-q21[END_REF][START_REF] Yang | Identification of a KCNE2 gain-of-function mutation in patients with familial atrial fibrillation[END_REF][START_REF] Olson | Kv1.5 channelopathy due to KCNA5 loss-of-function mutation causes human atrial fibrillation[END_REF][START_REF] Darbar | Cardiac sodium channel (SCNA5) variants associated with atrial fibrillation[END_REF][START_REF] Makiyama | A novel SCN5A gain-of-function mutation M1875T associated with familial atrial fibrillation[END_REF][START_REF] Fox | Parental atrial fibrillation as a risk factor for atrial fibrillation in offspring[END_REF][START_REF] Elinor | Familial aggregation in lone atrial fibrillation[END_REF][START_REF] Christophersen | Familial aggregation of atrial fibrillation. A Study of Danish Twins[END_REF][START_REF] Arnar | Familial aggregation of atrial fibrillation in Iceland[END_REF][START_REF] Xia | A Kir2.1 gain-of-function mutation underlies familial atrial fibrillation[END_REF][START_REF] Ravn | Gain-of-function in I Ks secondary to a mutation in KCNE5 associated with atrial fibrillation[END_REF][START_REF] Ellinor | Cardiac sodium channel mutation in atrial fibrillation[END_REF][START_REF] Antzelevitch | Loss-of-function mutations in the cardiac calcium channel underlie a new clinical entity characterized by ST-segment elevation, short QT intervals, and sudden cardiac death[END_REF][START_REF] Chaldoupi | The role of connexin40 in atrial fibrillation[END_REF][START_REF] Sabeh | The genetics of atrial fibrillation[END_REF]. However, large-scale sequencing in lone AF cohorts suggests that these channel mutations are not a major cause of AF [START_REF] Roberts | Impact of genetic discoveries on the classification of lone atrial fibrillation[END_REF][START_REF] Sabeh | The genetics of atrial fibrillation[END_REF].

Nevertheless, patients at comparable risk for AF often do not exhibit uniform susceptibility to the arrhythmia, which could possibly be attributed to a genetically determined "AF diathesis" [START_REF] Roberts | Impact of genetic discoveries on the classification of lone atrial fibrillation[END_REF][START_REF] Sabeh | The genetics of atrial fibrillation[END_REF][START_REF] Parvez | Lone AF -Etiologic factors and genetic insights into pathophysiology[END_REF]. The introduction of genome-wide association studies, which compare non-familial AF cases to age-related and gender-matched controls, has enabled some insight into the genetics of acquired AF, and several common gene or single nucleotide polymorphisms including those encoding the renin-angiotensin-aldosterone system (RAAS), calcium handling, neurohumoral and lipoprotein pathways, gap junction proteins, ion channels, interleukins, signaling molecules and mediators of other molecular pathways have been examined [START_REF] Roberts | Impact of genetic discoveries on the classification of lone atrial fibrillation[END_REF][START_REF] Sabeh | The genetics of atrial fibrillation[END_REF][START_REF] Parvez | Lone AF -Etiologic factors and genetic insights into pathophysiology[END_REF]. However, common polymorphisms identified thus far perhaps explain only <10% of the inherited contribution to AF [START_REF] Sabeh | The genetics of atrial fibrillation[END_REF][START_REF] Parvez | Lone AF -Etiologic factors and genetic insights into pathophysiology[END_REF][START_REF] Brauch | Comprehensive mutation scanning of LMNA in 268 patients with lone atrial fibrillation[END_REF][START_REF] Yang | A novel KCNA5 mutation implicates tyrosine kinase signaling in human atrial fibrillation[END_REF].

To date, genetic studies have revealed diverse mechanisms of susceptibility to AF (Table 1).

Single-gene disorders provide an insight into the electrophysiological mechanisms which predispose to AF, but it is very likely that vulnerability to AF primarily arises from the influence of multiple genes which, either alone or in combination, alter the atrial structural and functional properties. The recognition of those genes and further insight into their effect on atrial electrophysiology could facilitate the implementation of a pharmacogenomic treatment approach for treatment of AF in the future [START_REF] Roberts | Impact of genetic discoveries on the classification of lone atrial fibrillation[END_REF].

Pathophysiology and pathogenesis of lone AF

The occurrence of AF requires a trigger and appropriate atrial substrate to sustain reentry of electrical impulses. Various triggers can create propagating wavelets, which may lead to reentrant circuits and AF in the presence of functionally and structurally altered atrial myocardium (Figure 1) [START_REF] Allessie | Pathophysiology and prevention of atrial fibrillation[END_REF][START_REF] Haissaguerre | Spontaneous initiation of atrial fibrillation by ectopic beats originating in the pulmonary veins[END_REF]. The electrophysiological hallmarks of an atrial pro-fibrillatory state are a reduced atrial effective refractory period (ERP), attenuated ERP rate adaptation and reduced impulse conduction velocity [START_REF] Allessie | Pathophysiology and prevention of atrial fibrillation[END_REF].

The condition of the atrial substrate is strongly influenced by the underlying disease process and ageing. Furthermore, AF itself initiates the complex processes of atrial electrical, contractile and structural remodeling, which further contributes to the AF persistence and progression (that is, "AF begets AF") (Table 1) [START_REF] Allessie | Electrical, contractile and structural remodeling during atrial fibrillation[END_REF][START_REF] Wijffels | Atrial fibrillation begets atrial fibrillation. A study in awake chronically instrumented goats[END_REF]. Such atrial electrical and contractile remodeling occurs rapidly (within hours and days) and is reversible, whilst structural changes occur after weeks or months of persistent AF and are usually irreversible [START_REF] Allessie | Electrical, contractile and structural remodeling during atrial fibrillation[END_REF]. Thus, the early restoration of sinus rhythm (SR) should, at least theoretically, prevent further progression of AF. Unfortunately, a large body of evidence suggests that "SR does not beget SR" and that, in fact, vulnerability to AF and the risk of AF perpetuation and stability often extend beyond the time course of electrical remodeling reversal, which led to the hypothesis that "a second factor" (that is, some structural changes) is responsible for AF recurrence [START_REF] Fynn | Clinical evaluation of a policy of early repeated internal cardioversion for reccurrence of atrial fibrillation[END_REF][START_REF] Garratt | Repetitive electrical remodeling by paroxysms of atrial fibrillation in the goat: no cumulative effect on inducibility or stability of atrial fibrillation[END_REF].

In the setting of lone AF, whereby structural disease is absent by definition, AF may be viewed as a primary electrophysiological disorder with ectopic focal discharges from the pulmonary veins and posterior wall of the left atrium playing an important role in both the initiation and maintenance of AF [START_REF] Haissaguerre | Spontaneous initiation of atrial fibrillation by ectopic beats originating in the pulmonary veins[END_REF][START_REF] Jais | Distinctive electrophysiological properties of pulmonary veins in patients with atrial fibrillation[END_REF][START_REF] Chen | Catheter ablation of atrial fibrillation originating from the nonpulmonary vein foci[END_REF]. However, atrial fibrosis and loss of muscle mass have been observed in lone AF patients, along with isolated atrial myocardial perfusion abnormalities and coronary flow reserve impairment indicative of microvascular dysfunction [START_REF] Frustaci | Histological substrate of atrial biopsies in patients with lone atrial fibrillation[END_REF][START_REF] Skalidis | Isolated atrial microvascular dysfunction in patients with lone recurrent atrial fibrillation[END_REF]. For example, in a recent study of 25 patients with paroxysmal lone AF, bi-atrial abnormalities remote from arrhythmia (and thus very unlikely to result from AF-induced remodeling) were documented, including conduction abnormalities, structural changes and sinus node dysfunction [START_REF] Stiles | Paroxysmal lone atrial fibrillation is associated with abnormal atrial substrate. Characterizing the "Second Factor[END_REF]. More recently, one study of 58 lone AF patients also documented distinct abnormalities in atrial electrophysiological properties [START_REF] Centurion | Electrophysiological changes of the atrium in patients with lone paroxysmal atrial fibrillation[END_REF].

After the initial description of isolated atrial structural changes consistent with myocarditis in lone AF patients [START_REF] Frustaci | Histological substrate of atrial biopsies in patients with lone atrial fibrillation[END_REF], many studies have documented a relationship between circulating markers of inflammation (including C-reactive protein and interleukins) and AF. Indeed, inflammation plays a significant role in the initiation and perpetuation of AF and the AF-related prothrombotic state [START_REF] Boos | Is atrial fibrillation an inflammatory disorder?[END_REF][START_REF] Engelmann | Inflammation in the genesis and perpetuation of atrial fibrillation[END_REF][START_REF] Patel | Update on the association of inflammation and atrial fibrillation[END_REF][100]. Inflammatory biomarkers are associated with future development, recurrence and burden of AF and the likelihood of successful cardioversion [101-105]. However, inflammation may be associated not with AF per se, but rather with underlying cardiovascular disease [106]. Whether the initiation of AF activates direct inflammatory reactions or the presence of pre-existing inflammation promotes AF is not completely clarified, and both mechanisms may be operating. For example, rapid atrial activation accompanied with calcium accumulation may result in calcium overload and apoptotic loss of atrial myocytes, with the consequent low-grade inflammatory response as a part of structural remodeling [107].

Alternatively, in patients with triggering atrial foci, systemic inflammation with increased circulating C-reactive protein may predispose to AF via the classic complement pathway activation and atrial tissue damage, or by binding to phosphocholine which may contribute to membrane dysfunction by the alterations in sodium and calcium handling [START_REF] Engelmann | Inflammation in the genesis and perpetuation of atrial fibrillation[END_REF].

Other potential biomarkers of a predisposition to apparently lone AF have also been described.

Atrial natriuretic peptide (ANP) and brain natriuretic peptide (BNP) are sensitive biomarkers of cardiac contractile dysfunction, and both peptides are elevated in AF patients with underlying structural heart disease [108]. In lone AF patients, a discordant pattern of elevated N-terminalpro-BNP and normal N-terminal-pro-ANP levels can be observed, but the precise mechanism of ANP and BNP uncoupling is not clarified [109]. The question of whether the increased BNP levels are the result of AF or simply a sign of subclinical cardiac alterations which predispose to AF still remains unanswered.

Apelin is an endogenous peptide hormone, which is normally involved in the counter-regulation of the angiotensin and vasopressin systems. A significant reduction of mean apelin levels has Variations in autonomic nervous system tone are also involved in AF. Sympathetic stimulation often initiates AF in patients with structural heart disease, whilst lone AF patients more frequently have paroxysmal AF in the setting of increased vagal tone [START_REF] Allessie | Pathophysiology and prevention of atrial fibrillation[END_REF]111]. Although both sympathetic and parasympathetic components play a role in AF, the cholinergic component appears to be important for spontaneous initiation of AF; in the experimental setting, for example, electrical stimulation of the left atrial ganglionic plexi or the autonomic nerve endings with retrograde activation of the ganglia induces spontaneous firing from pulmonary veins followed by AF [112].

Ultrastructural alterations and cellular electrophysiology in lone AF

A decreased duration of action potential (AP), depressed AP plateau and attenuated response of AP duration to depolarization rate are the principal electrophysiological findings at the atrial cellular level [113,114]. These alterations are mediated through changes in sodium, potassium and calcium channels, which are not likely to be the principal cause of AF, but rather a nonspecific response to variety of stresses such as an altered oxidative state in the setting of increased metabolic demand, elevated sympathetic tone and increased levels of circulating cytokines. The redox state and metabolic reserve are also age-dependent, and regulation of channels is largely genetically determined [START_REF] Allessie | Pathophysiology and prevention of atrial fibrillation[END_REF].

Connexins are protein subunits of gap junctions -the clusters of transmembrane channels that link adjacent cells and mediate cell-to-cell electrical coupling and communication. Decreased numbers and altered spatial arrangement of connexin40 in AF have been documented, both in the atrial myocardium and thoracic veins [115][116][117]. These findings correlate with facilitated trigger activity in the pulmonary veins and discontinuous and slowed conduction due to intercellular uncoupling in the atrial myocardium [118,119]. In addition, connexin40 gene mutations or polymorphisms underlie an inherited predisposition to AF [START_REF] Chaldoupi | The role of connexin40 in atrial fibrillation[END_REF]. Atrial ultrastructural changes in AF patients are identified at the level of cardiomyocytes and extracellular matrix. Whilst fibrosis and myocyte degeneration could be attributed to the underlying structural disease, AF itself induces de-differentiation of cardiomyocytes and regression toward a more immature developmental stage with depleted contractile apparatus and fetal actin and myosin forms [START_REF] Kourliouros | Current concepts in the pathogenesis of atrial fibrillation[END_REF][START_REF] Allessie | Pathophysiology and prevention of atrial fibrillation[END_REF]. Apoptotic cell death is also more pronounced in the fibrillating atria [START_REF] Kourliouros | Current concepts in the pathogenesis of atrial fibrillation[END_REF]. An increased accumulation of various proteins in the extracellular matrix, including collagen, fibronectin-1 and fibrillin-1, has also been documented, along with the upregulation of matrix metalloproteinases, transforming growth factor ß1 and AT-1 angiotensin II receptor [START_REF] Allessie | Electrical, contractile and structural remodeling during atrial fibrillation[END_REF][START_REF] Kourliouros | Current concepts in the pathogenesis of atrial fibrillation[END_REF][START_REF] Allessie | Pathophysiology and prevention of atrial fibrillation[END_REF]. These changes may significantly alter the atrial micro-architecture (which is already complex under normal conditions) and increase vulnerability to AF in the setting of an acute stretch, which promptly activates the immediate gene program for cell hypertrophy, augments the synthesis of angiotensin II, influences the stretch-activated and other ion channels, receptors and enzymes with cytoskeletal connections, thus creating a fertile electrophysiological and anatomical environment for initiation and propagation of AF [START_REF] Allessie | Pathophysiology and prevention of atrial fibrillation[END_REF]. Reduced calcium systolic release and myolysis underlie contractile remodeling, which could trigger thrombus formation and atrial dilatation [120].

Risk factors for lone AF

Apart from the traditional AF risk factors, there are numerous conditions which have been associated with AF in apparently healthy individuals. Independent of other cardiovascular risk factors, obesity confers an increased risk of new-onset AF (3%-8% with each unit of increase in body mass index), which has been attributed to diastolic dysfunction due to myocardial thickening, elevated plasma volume and increased neurohormonal activation resulting in left atrial dilatation [START_REF] Wang | Obesity and the risk of new-onset atrial fibrillation[END_REF][START_REF] Frost | Overweight and obesity as risk factors for atrial fibrillation or flutter: the Danish diet, cancer, and health study[END_REF][START_REF] Dublin | Risk of new-onset atrial fibrillation in relation to body-mass index[END_REF]. Lone AF patients have also been reported to be taller and leaner than other AF patients [START_REF] Wanahita | Atrial fibrillation and obesity -results of a meta-analysis[END_REF]. A "holiday heart syndrome" of paroxysmal AF after occasional excessive alcohol intake was described over 30 years ago. Alcohol has a direct toxic effect on cardiomyocytes, causes a hyperadrenergic state with impaired vagal tone and may increase the intra-atrial conduction time.

However, moderate alcohol intake (2-3 drinks/week) was not related to the increased risk of AF in the Cardiovascular Health Study [START_REF] Ettinger | Arrhythmias and the "Holiday Heart": alcohol-associated cardiac rhythm disorders[END_REF][START_REF] Koksinen | Alcohol and new onset atrial fibrillation: a case-control study of a current series[END_REF][START_REF] Djousse | Long-term alcohol consumption and the risk of atrial fibrillation in the Framingham Study[END_REF][START_REF] Frost | Alcohol and risk of atrial fibrillation or flutter: a cohort study[END_REF][START_REF] Mukamal | Alcohol consumption and risk of atrial fibrillation in men and women: the Copenhagen City Heart Study[END_REF][START_REF] Mukamal | Alcohol consumption and risk and prognosis of atrial fibrillation among older adults: the Cardiovascular Health Study[END_REF]. Other stimulants including caffeine and nicotine have also been associated with AF [START_REF] Schoonderwoerd | New risk factors for atrial fibrillation: causes of "not-so-lone atrial fibrillation[END_REF].

In contrast to mild regular exercise, endurance sport practice is associated with a higher prevalence of AF, and >1500 lifetime hours of practice seems to be the threshold for the association. Enlargement of the cardiac chambers, an increase in left ventricular mass and left atrial dilatation, along with increased vagal tone resulting in bradycardia and shortening of the atrial ERP, transient hypovolaemia and changes in electrolytes due to sweating have been proposed to explain the vulnerability to AF in sportsmen. There is also anecdotal evidence of AF development after the use of anabolic steroids, and changes in the autonomic function and the baro-reflex could be the underlying mechanism [START_REF] Mont | Long-lasting sport practice and lone atrial fibrillation[END_REF][START_REF] Elosua | Sport practice and the risk of lone atrial fibrillation: a case-control study[END_REF][START_REF] Mont | the GIRAFA (Grup Integrat de Recerca en Fibril-lacio Auricular) Investigators. Physical activity, height, and left atrial size are independent risk factors for lone atrial fibrillation in middle-aged healthy individuals[END_REF][START_REF] Molina | Long-term endurance sport practice increases the incidence of lone atrial fibrillation in men: a follow-up study[END_REF][START_REF] Heidbuchel | Endurance sports is a risk factor for atrial fibrillation after ablation for atrial flutter[END_REF][START_REF] Schoonderwoerd | New risk factors for atrial fibrillation: causes of "not-so-lone atrial fibrillation[END_REF].

Various social characteristics, including the type of personality, anger and hostility and acute life stress have been shown to increase the vulnerability to AF [START_REF] Eaker | Anger and hostility predict the development of atrial fibrillation in men in the Framingham Offspring Study[END_REF][START_REF] Mattioli | The relationship between personality, socioeconomic factors, acute life stress and the development, spontaneous conversion and recurrences of acute lone atrial fibrillation[END_REF]. Drug-induced AF should be considered when inotropic agents (dopamine), cholinergics (acetylcholine), adenosine, broncodilatators (especially sympathicomimetic inhalants), corticosteroids, cytostatics, central nervous system drugs (anticholinergics, dopamine agonists, antidepressants, antipsychotics, anaestetics) and others are administered [121]. [MAC]). In the Framingham Study, for example, MAC was an independent predictor for incident AF (hazard ratio 1.6, 95%CI: 1.1-2.2) [START_REF] Fox | Mitral annular calcification is a predictor for incident atrial fibrillation[END_REF][START_REF] Lorenz | Prediction of clinical cardiovascular events with carotid intima-media thickness[END_REF].

Clinical course of lone AF

In some studies, lone AF patients may have a similar risk of thromboembolism, CHF and mortality as the general population [START_REF] Kopecky | The natural history of lone atrial fibrillation. A population-based study over three decades[END_REF][START_REF] Jahangir | Long-term progression and outcomes with aging in patients with lone atrial fibrillation. A 30-year follow-up study[END_REF][START_REF] Potpara | Mortality of patients with lone and idiopathic atrial fibrillation is similar to mortality in general population of Serbia[END_REF]. However, other studies suggest otherwise. For example, in the Paris Prospective Study I, lone AF was associated with increased mortality in middle aged working men [START_REF] Brand | Characteristics and prognosis of lone atrial fibrillation. 30-year follow-up in the Framingham Study[END_REF][START_REF] Jouven | Idiopathic atrial fibrillation as a risk factor for mortality[END_REF]. Also, ageing and development of underlying heart disease strongly influence the long-term prognosis of (what was originally) lone AF [START_REF] Kopecky | Lone atrial fibrillation in elderly persons. A marker for cardiovascular risk[END_REF][START_REF] Jahangir | Long-term progression and outcomes with aging in patients with lone atrial fibrillation. A 30-year follow-up study[END_REF]. Therefore, regular follow-up with evaluation of cardiovascular risk factors is necessary in patients diagnosed as 'lone AF'. This is supported by the observation that as many as 44% of apparently It has also been suggested that paroxysmal lone AF carries a better prognosis than chronic lone AF, regarding thromboembolism and mortality [START_REF] Scardi | Lone atrial fibrillation: prognostic differences between paroxysmal and chronic forms after 10 years of follow-up[END_REF]. The vast majority of lone AF patients present with paroxysmal arrhythmia and have a relatively low rate of progression to permanent AF during the follow-up, with an estimated risk of progression of approximately 30% over 30 years, being significantly influenced by ageing and development of comorbidities [START_REF] Jahangir | Long-term progression and outcomes with aging in patients with lone atrial fibrillation. A 30-year follow-up study[END_REF]. Clinical AF types are summarized in Figure 2. The principal aims of AF treatment are to reduce symptoms and to prevent AF-related complications, using antithrombotic therapy, ventricular rate control and treatment of concomitant cardiovascular risk factors and/or comorbidities. Symptom relief may additionally require rhythm control including cardioversion(s), long-term use of antiarrhythmic drugs and/or catheter-ablation (Figure 3) [124].

Management of lone AF

Stroke and thromboembolism is a severe complication of AF. More recently, greater efforts have been directed towards the identification of "truly low risk" AF patients who are at low risk of stroke and, therefore, do not need any antithrombotic therapy; all other patients with ≥1 stroke risk factors merit oral anticoagulation therapy (Figure 3) [START_REF] Lip | Refining clinical risk stratification for predcting stroke and thromboembolism using a novel risk factor-based approach[END_REF]. This approach is acknowledged in the latest 2010 European Society of Cardiology Guidelines for AF [124].

The cumulative risk of stroke is very low in true 'lone AF' patients (1.3% over 15 years) and no therapy (or aspirin) is advised [START_REF] Jahangir | Long-term progression and outcomes with aging in patients with lone atrial fibrillation. A 30-year follow-up study[END_REF]124]. Of note, the thromboprophylactic effect of aspirin in lone AF patients is questionable. In the Japan AF Stroke trial, aspirin150-200mg daily was worse than no therapy (the annual incidence of primary outcomes was 3.1% vs. 2.4%, respectively), and the use of aspirin was associated with non-significant increased risk of major bleeds (1.6% with aspirin vs. 0.4% in the control group) [125]. Meta-analysis of 7 trials comparing aspirin with placebo in AF patients showed a non-significant 19%-reduction in stroke incidence (95%CI -1% to 35%), which could easily be attributed to an effect of aspirin on atherosclerosis and Moreover, new oral anticoagulants are being developed for stroke prevention in AF. Those are the oral direct thrombin inhibitors (dabigatran etexilate, AZD0837) and oral factor Xa inhibitors (rivaroxaban, apixaban, endoxaban, betrixaban, YM150 and others). In contrast to the oral vitamin-K-antagonists (VKA), such as warfarin, these new agents have a rapid onset of action, a low propensity for food and drug interactions and a predictable anticoagulant effect after fixeddose administration and coagulation monitoring is unnecessary. The efficacy and safety of dabigatran compared with warfarin for the prevention of stroke and systemic embolism in AF patients was investigated in the Randomized Evaluation of Long-term anticoagulant therapy with dabigatran etexilate (RE-LY) trial [128]. The Apixaban VERsus acetylsalicylic acid to pRevent strOkES (AVERROES) study was stopped early because of clearly evident superiority of apixaban 5mg bid over aspirin in patients intolerant of or unsuitable for VKA, with similar rates of major bleeding with aspirin and the oral anticoagulant, and aspirin was less well tolerated [124].

Rhythm control may be particularly successful in patients with paroxysmal lone AF, primarily due to a relatively low propensity to AF progression, and various strategies could be implemented including no therapy, a "pill-in-the-pocket" strategy or long-term use of antiarrhythmic drugs (Figure 3). In general, available antiarrhythmic drugs may not be particularly effective in acute cardioversion of AF, and are only moderately successful in longterm rhythm control. Should ß-blockers be ineffective, flecainide, propafenone, sotalol or dronedarone could be tried. Although amiodarone has a superior efficacy compared with other antiarrhythmic drugs, it should be reserved as an alternative because of its potential toxicity with long-term use [124,129]. Catheter ablation may be applied in symptomatic lone AF patients in whom antiarrhythmic drugs were not effective or at patient's preference

Conclusions

Accumulating insights into the pathogenesis of apparently lone AF suggest that the entity comprises heterogeneous subsets of AF patients who should not a priori be classified as having a benign rhythm disorder. Growing evidence of numerous common variations within the genome and atrial ultrastructural alterations that predispose to AF makes the classification of AF as secondary to structural disease or as a 'lone' arrhythmia increasingly meaningless. Detailed assessment of lone AF patients is warranted, both at presentation and regularly during the follow-up. Emerging risk factors and pathophysiological mechanisms predisposing to AF should also be considered and, clearly, there is a need for further research regarding stroke risk stratification and proper antiarrhythmic drug treatment in so-called lone AF patients.

Funding and Acknowledgements

None. KCNQ1 [67] KCNE2 [70] KCNJ2 [78] KCNE5 [79] KCNH2 [56] IKs IKs CACNA1C, CACNB2 [81] ICaL enhanced heterogenity of repolarization multiple reentry wavelets Associated both with Brugada syndrome and Short QT syndrome Gen. loci [65-69]

  The heritability of AF in the general population has been documented in several populationbased studies

  patients with lone AF, indicating certain alterations of the cardiac humoral axis [110].

11 Obstructive

 11 sleep apnea may cause various hemodynamic alterations and autonomic imbalance predisposing to AF, and, conversely, a paroxysm of AF may result in central sleep apnea caused by an acute decrease in the left ventricular function accompanied by an increase in pulmonary wedge pressure and consequent stimulation of pulmonary vagal receptors[START_REF] Roche | Relationship among the severity of sleep apnea syndrome, cardiac arrhythmias, and autonomic imbalance[END_REF][START_REF] Porthan | Prevalence of sleep apnea syndrome in lone atrial fibrillation: a case-control study[END_REF][START_REF] Kanagala | Obstructive sleep apnea and the recurrence of atrial fibrillation[END_REF][START_REF] Gami | Obstructive sleep apnea, obesity and the risk of incident atrial fibrillation[END_REF][START_REF] Rupprecht | Causality in the relationship between central sleep apnea and paroxysmal atrial fibrillation[END_REF].

Finally, what seems

  to be apparently 'lone AF' may occur in the setting of subclinical atherosclerosis (determined by carotid intima-media thickness and/or mitral annular calcification

  lone AF patients may actually have occult arterial hypertension [122]. Patients originally diagnosed with lone AF may follow divergent clinical courses based on the left atrial volume. In contrast to those with normal atria, patients with increased left atrial volume (either at diagnosis or during the follow-up) subsequently experienced adverse cardiovascular events including stroke, which occurred at left atrial volumes >32mL/m 2 [123].

  atherothrombosis[126].Dedicated re-assessment of thromboembolic risk factors during regular follow-ups is mandatory in lone AF patients, since the stroke risk increases with ageing or development of comorbidities (including hypertension). However, various atrial ultrastructural changes, contractile dysfunction and endothelial perturbation have been documented in apparently lone AF[101-105,108- 110,120,127]. With growing insights into these alterations, all of which could influence the stroke risk, and increasing number of potentially useful biomarkers, further refinement of thromboembolic risk assessment should be expected.

  [124]. Ablation strategies clearly offer better rhythm control during the short-term follow-up, but late AF recurrences are not uncommon [130-132]. Classic surgical 'maze procedure' and pulmonary vein isolation are highly effective in rhythm control, but novel procedures including thoracoscopic access and video assistance are promising [124,133,134]. Most of the currently available antiarrhythmic drugs exert their anti-AF effects via inhibition of cardiac ion channels, at the expense of an increased risk of life-threatening ventricular of left ventricular systolic dysfunction or organotoxicity [135]. Until recently, large clinical trials evaluating the ability of antiarrhythmic drugs to improve all-cause mortality have yielded neutral outcomes at best [120,135]. However, there appears to be significant reduction in both cardiovascular morbidity and mortality in AF patients treated with dronedarone, which has encouraged an essential change of focus of rhythm control management [129,136]. Nevertheless, there was no significant reduction in the all-cause mortality in AF patients treated with dronedarone [136]. In the prevention of recurrent AF, dronedarone is better than placebo [137] and less successful, but with significantly fewer adverse effects, compared with amiodarone [129]. However, the use of dronedarone is not safe in patients with significantly reduced left ventricular systolic function (ejection fraction ≤35%), in whom dronedarone has been shown to increase mortality, predominantly due to cardiovascular causes (most notably CHF) [138]. Novel pharmacological investigational approaches to rhythm control in AF include atrialselective multiple ion channel blockers which are expected to reduce the risk of ventricular arrhythmias (vernakalant, ranolazine, AVE0118, GsMTx4, tertiapin-Q, NIP-141, NIP-142, etc), improved derivatives of existing drugs (AZD1305, AVE1231, SSR149744C, ATI2042), gap junction modifiers (rotigaptide, AAP10, GAP-134, etc), and upstream therapy, which targets arrhythmogenic structural remodeling in the atria and/or factors that promote such remodeling including hypertension, heart failure and inflammation (angiotensin-converting enzyme inhibitors [ACEIs], angiotensin receptor blockers [ARBs], statins and omega-3 polyunsaturated fatty acids [PUFAs]) [120,124,135]. Vernakalant hydrochloride is a relatively atrial-selective, early-activating K + and frequency-dependent Na + channel blocker with a half-life of 2 to 3 hours. In a randomized clinical trial, vernakalant demonstrated rapid conversion of short-duration AF compared with placebo and was well tolerated [139]. Upstream therapy could prevent the development of AF (primary prevention) or recurrence and progression to permanent AF (secondary prevention). At present, upstream therapies are not recommended for primary prevention of AF in patients without cardiovascular disease [124].However, the use of ACEIs and ARBs may be considered in patients with recurrent AF who are taking antiarrhythmic drug therapy. There is good experimental evidence of antifibrillatory and these drugs in AF patients, but the results of clinical studies are conflicting[124,140,141]. In animal models of AF, statins attenuate atrial electrical and structural remodeling and reduced inducibility of AF. However, clinical evidence in support of using statins in the secondary AF prevention, except for post-operative AF, is insufficient[124, 142,143].

Figure 1 . 29 Figure 2 . 30 Figure 3 .

 1292303 Figure 1. Pathophysiology of Atrial Fibrillation.

Page 12 of 30 International Journal of Clinical Practice International Journal of Clinical Practice

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	13

Page 16 of 30 International Journal of Clinical Practice International Journal of Clinical Practice

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

International Journal of Clinical Practice International Journal of Clinical Practice

	F o F o F o F o F o F o F o F o F o
	r r r r r r r r r
	P P P P P P P P P
	e e r e e r e e r e e r e e r e e r e e r e e r e e r
	R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w
	O n O n O n O n O n O n O n O n O n
	l y l y l y l y l y l y l y l y l y
	20

Table 1 . The genetics of atrial fibrillation.

 1

	Chromosome	Gene	Ion channel/	Electrophysiological effects of the	Mechanism of AF	Associated phenotype/comment
			current/other	mutation
	11p15.5
	21q22.1
	17q23-24
	X chrom.
	7q36.1

candidate genes) [56,68,82,83]

  AP=action potential, ERP=effective refractory period, DCM=dilated cardiomyopathy, ANP=atrial natriuretic peptide, RAAS=rennin-angiotensin-aldosterone system.

	10q22-24 6q14-16 10p11-q21 5p11 Chrom. 3 5p13 Chrom. 1 1p36-p35 4q25 16q22 Other mutations (1q21 Documented, but rare causes of AF AF=atrial fibrillation, Page unknown unknown unknown unknown SCN5A NUP155 GJA5 NPPA PITX2 ZFHX3 KCNN3	? ? ? ? Nav1.5 Nuclear pore protein Connexin40 ANP RAAS (?) ? ?	? ? ? ? cell hyperexcitability enhanced automaticity of cardiomyocytes ? heterogenity of conduction velocity enhanced vulnerability to electrical reentry shortening of atrial AP, reduction of ERP ? ? ?	? ? ? ? electrical reentry with "mother-rotor" and spiral reentry waves ? multiple reentry wavelets multiple reentry wavelets (in vitro) ? ? ?	Overlaps with locus for DCM Overlaps with locus for DCM Unknown Associated with prolonged P-wave Not associated with Long QT syndrome Associated with Emery-Dreifuss muscular dystrophy syndromes Connexins: transmembrane proteins that form gap-junctions (intercellular pores) which serve as low-resistance conduction pathways between adjacent cells As a mediator of inflammation, ANP may cause atrial fibrosis Suppresses pacemaker cells from forming outside the sinus node during cardiogenesis Associated with both ischemic and cardioembolic strokes -

27 of 30 International Journal of Clinical Practice International Journal of Clinical Practice

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	28

International Journal of Clinical PracticeInternational Journal of Clinical Practice


Francesco Dell'isola

Luca Placidi

Variational principles are a powerful tool also for formulating field theories

Variational principles and calculus of variations have always been an important tool for formulating mathematical models for physical phenomena. Variational methods give an efficient and elegant way to formulate and solve mathematical problems that are of interest for scientists and engineers and are the main tool for the axiomatization of physical theories.

1 Introduction and historical background

Metrodoron and his followers

The ideas we want to evocate in this lecture are very old and were put forward already in the hellenistic period: for a detailed discussion about this point the reader is referred to the beautiful book by Lucio [START_REF] Russo | The Forgotten Revolution[END_REF]. In that book it is established that "modern" science actually was born in the hellenistic era, when Metrodoron lived. Metrodoron was a pupil of a famous greek philosopher, Epicurus, and, in our opinion, the following Metrodoron's sentence is a statement (the first?) belonging to the modern philosophy of science:

)*%#&,&#&'6.

"Always remember that you were born mortal and such is your nature and you were given a limited time: but by means of your reasonings about Nature you could rise to infinity and to eternity and you indeed contemplate "the things that were, and that were to be, and that had been before"". Metrodoron Gnomologium Epicureum Vaticanum X (fr.37 Alfred Körte, Metrodori Epicurei Fragmenta, " Jahrbücher für classiche Philologie", Suppl. XVII, 1890, p. 557).

This dictum, following Körte, comes from a lost letter or book by Metrodoron (the Epicurean philosopher) addressed to Menestratos who was presumably one of his pupils. The words quoted in bold are a citation from Iliad, I 70 (the translation into English of the sentence in boldface is ours; except for this citation the translation has been taken from Homer by Murray (1924), see the ref. [START_REF] Homer | The Iliad. Translated[END_REF]).

In different words, Metrodoron states that by using (the right!) equations you can forecast future behavior of physical systems.

Why Variational Principles and Calculus of Variation?

In recent time, a lost Archimedes' book [START_REF] Netz | The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity's Greatest Scientist[END_REF] has been rediscovered. Some authors claim that Archimedes seems to have solved, in this book and using a variational principle, the technological problem of finding the optimal shape of a boat. Archimedes seems to have chosen, as optimality criterion, that the vertical position must be a "very" stable configuration (see [START_REF] Rorres | Completing Book II of Archimedes's On Floating Bodies[END_REF]). In the book of Russo [START_REF] Russo | The Forgotten Revolution[END_REF] it is demonstrated in even a more convincing way that many optimization techniques were well-known in hellenistic science. In particular Russo proves that the problem of the determination of the regular polygon having maximal area has been solved in that period. Thus, the use of a variational principle and optimization methods to solve technological problems is less recent than it is usually believed. In general, variational formulation of the governing equations of solid and fluid mechanics is a classical but very challenging topic. This kind of formulation allows for an easier proof of the well-posedness of mathematical problems, for an easier investigation of the study of stability of particular solutions, and for a simpler implementation of numerical methods. Often (but one who believes in Russo's vision about the birth of science could say instead "always"), mechanical problems are more naturally posed by means of variational methods. Hamilton's principle of stationary (or least) Action is the conceptual basis of practically all models in physics. The variational formulation is also useful for obtaining simpler approximate asymptotical models as it is done in the theory of homogenization.

We want simply to state here that the Principle of Virtual Works and the Principle of Least Action have roots much deeper than many scientists believe (see Vailati, 1897). Although many histories of science claim dif-ferently, most likely the majority of physical theories were first formulated in terms of these Principles, and only subsequently they were reconsidered from other points of view. In our opinion the Principle of Least Action, which supplies a "geometric" version of mechanics was indeed the tool used by the true founders of mechanics (i.e. the scientists of the hellenistic period) to establish it. As argued also by Colonnetti (5) and Netz and Noel (19)) surely also Archimedes and ancient greek scientists were accepting such a point of view.

The epigones of the hellenistic science, who were not able to understand the delicate mathematical arguments used by the first scientists, however could understand the minimality conditions obtained by their "maitres" (i.e. the conditions corresponding to those which we call now Euler-Lagrange equations and boundary conditions) and could grasp the "physical" arguments used to interpret them. This phenomenon is perfectly clear to everyone who is ready to consider carefully -for instance-the evolution of the theory of Euler-Bernoulli Beam (a useful reference about this point is the book of [START_REF] Benvenuto | La scienza delle costruzioni e il suo sviluppo storico[END_REF]). Euler postulated a Principle of Least Action for the Elastica, and gets the celebrated equilibrium differential equation and boundary conditions for the equilibrium of beams by using the calculation procedure due to Lagrange (which is the departing idea of Calculus of Variations). Then Navier prepared his lectures for the Ecole Polytechnique and resumed the results obtained by Euler deciding to "spare" to the (engineering) students the difficulties of the calculus of variations. He started directly from the equilibrium equation, obtained by means of an "ad hoc" principle of balance of force and couple, and imposed boundary conditions based on "physical" assumptions. As a consequence, for a long while, generations of engineers believed that the beam equations were to be obtained in this way. Only when numerical simulations became popular, then they (actually, some of them) became aware of variational "principles". However these principle were proven as theorems starting from "balance postulates" and were considered simply as a mathematical (rather abstruse) tool and not as a fundamental heuristic concept. And this attitude is not changed even when it became clear that every serious advancement of mechanical science has been obtained using variational principles. Indeed the so called "physical sense" (a gift that many claim to posses but which nobody can claim to be able to master or to teach) is not very useful to postulate the right "balance principles" when one is in "terra incognita". For instance, when Lagrange and Sophie Germain wanted to find the plate equations they needed to employ a variational principle (and they could find the (right!) natural boundary conditions). Again when Cosserat brothers wanted to improve Cauchy Continuum Mechanics they "rediscovered" the right tech-nique: i.e. the Principle of Least Action. Also Quantum Mechanics has been developed starting from a Variational Principle (see e.g. the references of Feynman [START_REF] Feynman | The Feynman Lectures on Physics: The Definitive and Extended Edition[END_REF], Lagrange [START_REF] Lagrange | Mécanique Analytique[END_REF] and Lanczos [START_REF] Lanczos | The Variational Principles of Mechanics[END_REF]).

Therefore an important warning is due to young researcher: refrain from trying to extend available models by means of "ad hoc" adaptations of available theories: always look for the right Action functional to be minimized!

The problem of including dissipation

One useful tool for handling complicated situations is used in Continuum Mechanics by Paul Germain when formulating second gradient theories: the Principle of Virtual Powers. Again, as remarked always in the history of the development of ideas, when this history can be reconstructed, the effective way to be used to proceed is that which starts from a Principle of Least Action, eventually generalized into a Principle of Virtual Powers. For a long time the opponents to Second Gradient Theories argued about its lack of consistency, due to the difficulties they claim to find in "getting" boundary conditions. This is a really odd statement. Indeed variational principles easily produce mathematically correct boundary conditions. So maybe what those opponents want to say is that as they are not so clever as Navier, they do not manage to interpret physically the boundary conditions found via a (correct and meaningful) variational principle. Of course if one refuses to use the Principle of Least Action he can find very difficult the job of determining some set of boundary conditions which are compatible with the (independently postulated!) bulk evolution equations. If instead one accepts the Archimedean (the reader will allow us to dream, without definitive evidence that such was the point of view of Archimedes) approach to mechanics then all these problems of well-posedness of mathematical models completely disappear.

Variational Principles always produce intrinsically well-posed mathematical problems, if the Action functional is well behaving. Of course passing from Lagragian systems (the evolution of which are governed by a Least Action functional) to non-Lagragian systems (for which such a functional may not exist) maybe difficult. This problem is related (but is not completely equivalent) to the problem of modelling dissipative phenomena. It is often stated that dissipation cannot be described by means of a Least Action Principle. This is not exactly true, as it is possible to find some Action functionals for a large class of dissipative systems. However it is true that not every conceived system can be regarded as a Lagragian one. This point is delicate and will be only evocated here. In general a non-Lagragian system can be regarded as Lagragian in two different ways: i) because it is an "approximation" of a Lagrangian system (see the case of Cattaneo equation for heat propagation), and this approximation leads to "cancel" the lacking part of the "true" Action Functional ii) because the considered system is simply a subsystem of a larger one which is truly Lagrangian. The assumption that variational principle can be used only for non-dissipative systems is contradicted by, e.g., the work presented in this book by Prof. Frankfort [START_REF] Francfort | The chapter in this book by Prof[END_REF], where you find variational principles modelling dissipative systems. Indeed it is often stated that a limit of the modelling procedure based on variational principles consists in their impossibility of encompassing "nonconservative" phenomena. We do not believe that this is the case: however in order to avoid to be involved in a problem which is very difficult to treat, when dealing with dissipative systems, we will assume a slightly different point of view, usually attributed to Hamilton and Rayleigh.

2 Finding a mathematical model for natural phenomena

Principle of Least Action

We want to discuss here about the problem of finding a mathematical model for natural phenomena. We start with an epistemological Principle: "The Principle of Least Action tells us how to construct a mathematical model to be used for describing the world and for predicting the evolution of the phenomena occurring in it".

In the following modeling scheme, we give the right heuristic strategy to be used for finding an effective model using the Principle of Least Action. The recipe includes the following ingredients:

1. Establish the right kinematics needed to describe the physical phenomena of interest, i.e. the kinematical descriptors modeling the state of considered physical systems. 2. Establish the set of admissible motions for the system under description, i.e. establish the correct model for the admissible evolution of the system. 3. Employ the "physical intuition" to find the right Action functional to be minimized, i.e. modeling what Nature wants to minimize.

We start by finding the kinematical descriptors, because of the need of modeling the states of the considered system. Then we introduce motion, in such a way we model the evolution of the system to be described. Finally we ask Nature what is the quantity to minimize. Keeping this quantity in mind, we introduce the Action functional. To start with, it is necessary to focus the attention on a specific class of systems and on phenomena occurring to them. A configuration is the mathematical object used to model the state of considered systems: the set of possible configurations will be denoted by C. The motion is the mathematical model describing the evolution of considered systems: it is a C-valued function defined on time interval (t 0 , t f ); the set of all admissible motions will be denoted by M. The Action is a real-valued function, defined on M, which models the "preferences" of nature.

Finally, to use the Principle of Least Action one needs three steps further, 4. Find the Euler-Lagrange conditions which are consequence of the postulated Least Action Principle 5. Interpret these condition on a physical ground 6. Determine, in terms of the postulated Action functional, the numerical integration scheme to be used to get the previsions needed to drive, by means of our theory, our experimental, technological or engineering activity.

The Rayleigh-Hamilton principle

When postulating an extended Rayleigh-Hamilton principle, the point 4 of the previously presented heuristic strategy will be further divided into two steps as follows: 4a. Once the quantities which expend power on the considered velocity fields are known in terms of postulated Action, introduce a suitable definite positive Rayleigh dissipation functional 4b. Equate the first variation of Action functional to the Rayleigh dissipation functional and get the evolution equations (including boundary conditions) which govern the motion of the system Although in the literature the choice of including a Rayleigh-Hamilton principle in the class of variational principles is sometimes considered inappropriate, we will follow what seems to us the preference of the majority of the authors: therefore we do call "variational" also the strategy which we just described, not limiting the use of this adjective to the models using exclusively the Least Action Principle.

La Cinèmatique d'Abord !

According to Metrodoron, mathematical and physical objects are two different concepts. Indeed, equations are necessary for modeling physical systems but they refer to mathematical objects. When one solves the equations formulated in the framework of his model then he has to transform the obtained equations into previsions valid for the physical system he is studying. A good modeling procedure uses mathematics for finding the motion which minimizes Action. If this mathematics gives a reasonable forecasting of the observed evolution, then the model is valid. However, not everything is described by a given model. A model is always focused on a set of phenomena.

The set of phenomena that are focused by a model is established by the kinematics:

La Cinèmatique d'Abord !

In the previous scheme it is clear that the most "fundamental" step concerns the choice of the set of configurations used for characterizing the "accessible" states of the system. When constructing a mathematical model using the discussed epistemological principle, one must start with a precise and clear determination of the set C. The second step concerns the determination of admissible motions which clearly depend on the evolutionary phenomena one wants to model. A correct modeling process always starts specifying "admissible" kinematics.

La Nature agit toujours par les voies les plus simples

After having specified the admissible kinematics, one can wonder about the desire of Nature. The utility of Nature is a real-valued function defined on M. Following Maupertuis, we will call Action this "utility". Also Nature must consider which is the contingent situation: not all admissible motions are accessible by a physical system under given specific conditions. Therefore we must specify a subset M A of the set M: the set of accessible motions. The real motion will be chosen by the system minimizing the Action in the subset M A . Indeed: La Nature agit toujours par les voies les plus simples.

Two possible choices for the set of admissible and accessible motions

In the famous textbook of Arnold (1) the author, following the tradition, does not "try" to explain Maupertuis' Principle of Least Action. We instead dare to try to deal with this. In the process of minimization of the Action, we need to specify the set of motions among which we look for minima. The choice of Lagrange is that of isochronous motions. Two motions are isochronous when they both start, at the given instant t 0 , from a given configuration C 0 and arrive, at the same instant t f , at the same final configuration C f . On the other hand, the choice of Maupertuis is to focus on the set of motions with a "fixed energy content" and which are starting from the same configuration C 0 and ending (the instants of start and stop are not specified!) at the configuration C f . In the set of admissible motions an "energy" functional must then be introduced: i.e. a functional which associates an energy content to any motion and any time instant t. The set of accessible motions is constituted by all motions from C 0 to C f which have a constant energy content. The choice of Maupertuis, if not suitably modified, seems to limit the range of applicability of variational principles to non dissipative phenomena.

Further famous quotes

Many books in Calculus of Variations and/or Variational Principles, see e.g. that of [START_REF] Lanczos | The Variational Principles of Mechanics[END_REF], start with a preface, introduction or introductory chapter dealing with historical prolegomena and sometimes end with a philosophical chapter. In presenting this lecture notes, we did not dare to break with tradition.

"For this would be agreed by all: that Nature does nothing in vain nor labours in vain". Olympiodorus, Commentary on Aristo- tle's TM Meteora translated by Ivor Thomas in the Greek Mathematical Works Loeb Classical Library "La nature, dans la production de ses effets, agit toujours par les voies les plus simples". Pierre de Fermat.

Now, the problem is:

What is utility?

3 In other words: How to find "Real Motions"?

Up to now no mathematical structure has been assumed for M A . Indeed, Action functional is simply a real-valued map defined on M A . "Practical" problems require the calculation of real motions by means of introduced model. Following Lagrange [START_REF] Lagrange | Mécanique Analytique[END_REF], we introduce a particular class of Action functionals in terms of a Lagrangian Action density function: so constructing in a particular way Action functional to obtain so called "Lagrangian functionals".

We need to introduce a topological structure in M A , i.e. we need to clearly define what we mean when we say that "two motions are close". If we want to find minima of a real-valued function, then we need to estimate derivatives and equate these derivatives to zero. Action is a function defined in the set of motions (not real numbers!). Thus, we need

• to understand what is an infinitesimal variation of motion,

• to find a differential of a functional and • to estimate the order of infinitesimal of its remainder.

In other words, we need to learn how to find a first order Taylor expansion for a Lagrangian functional by establishing the meaning of the expressions :

• Infinitesimal variation of motion.

• Differential of a functional.

• Order of infinitesimals for remainders. This implies the need of Frechèt and Gateaux derivatives in manifolds with charts in Banach spaces. This is the right mathematical frame for studying this subject. However, Lagrange did not know that he was using such a mathematical frame and did not know anything about Frechèt and Gateaux derivatives. Thus, in this notes we try to go around the related mathematical difficulties and follow the original approach of Lagrange.

The motion minimizing Action will be searched among the motions for which the first variation of Action vanishes.

For Lagrangian functionals this condition is equivalent to a partial differential equation which is called Euler-Lagrange condition relative to the given Action functional. This procedure generalizes the corresponding one used for real-valued functions of several real variables. One serious problem with papers that start from balance equations and "play" with forces is that they do not "find" boundary conditions. In these references ((7; 8; 9)) one can find examples of modelling procedures in which one finds simultaneously bulk and boundary conditions.

From an historical point of view, in the theory of beam we deal with contact actions (normal and shear forces and momenta) because Navier has written lecture notes for l'Ecole Polytechique, trying to produce a text for students that was as simple as possible. He wrote final equations and explain not only bulk but also boundary conditions with the aid of "physical sense". However, it is very difficult in general to find evolution equations and boundary conditions with physical sense. On the other hand, variational principles give boundary conditions automatically and without the help of any physical sense.

Thus, Variational Principles allow Science to unveil Nature and for unveiling Nature you need a Lagrangian functional.

Lagrangian Action Functionals: technical details

We follow [START_REF] Landau | Quantum Mechanics: Non-Relativistic Theory[END_REF] and [START_REF] Moiseiwitsch | Variational Principles[END_REF].

Let Ψ σ (x µ ) be any set of n tensor fields defined on R m , (σ being a multi-index and µ = 1, 2, ..., m). We define the Lagrangian density as:

L x µ , Ψ σ , ∂Ψ σ ∂x µ . (1)
We can then introduce the Action functional as

A = T L x µ , Ψ σ , ∂Ψ σ ∂x µ ( 2
)
Where T is a hyper-volume in the m-th dimensional space determined by the coordinates x µ . When we will want to derive the theory of second gradient materials, this approach will not be appropriate, because we would need to add the dependence on the second gradient of Ψ σ in (1).

Variation of the Action Functional

We now consider small variations εη σ (x µ ) of the considered fields Ψ σ (x µ ):

Ψσ (x µ ) = Ψ σ (x µ ) + εη σ (x µ ), (3)
where the η σ (x µ ) are any set of linearly independent functions of the x µ which vanish on the part ∂ d T (∂ d T ⊆ ∂T ) of the boundary ∂T of the hypervolume T , on which the kinematical condition are prescribed. The variation of the Action functional can then be computed as:

∆A = T L x µ , Ψσ , ∂ Ψσ ∂x µ - T L x µ , Ψ σ , ∂Ψ σ ∂x µ , ( 4
)
where T is a hyper-volume in the m-th dimensional space determined by the x µ . The computation of the variation of the Action functional now proceeds as follows:

∆A = T L x µ , Ψ σ + εη σ , ∂Ψ σ ∂x µ + ε ∂η σ ∂x µ - T L x µ , Ψ σ , ∂Ψ σ ∂x µ + O ε 2
(5) which, with a slight abuse of notations, can be written at the first order in ε as:

δA = ε T σ ∂L ∂Ψ σ η σ + m µ=1 ∂L ∂ (∂Ψ σ /∂x µ ) ∂η σ ∂x µ (6)
Integrating by parts and recalling that η σ vanish on ∂ d T it is easy to get:

δA = ε T σ η σ ∂L ∂Ψ σ - m µ=1 ∂ ∂x µ ∂L ∂ (∂Ψ σ /∂x µ ) (7)
+ε

∂T /∂ d T σ η σ m µ=1 ∂L ∂ (∂Ψ σ /∂x µ ) N µ ,
where ∂T /∂ d T is the difference between ∂T and ∂ d T and N µ is the external unit normal of ∂T /∂ d T . Imposing δA = 0, the arbitrariness of η σ gives, for any σ:

∂L ∂Ψ σ - m µ=1 ∂ ∂x µ ∂L ∂ (∂Ψ σ /∂x µ ) = 0, ∀x µ ∈ T, (8)
m µ=1 ∂L ∂ (∂Ψ σ /∂x µ ) N µ = 0, ∀x µ ∈ ∂T /∂ d T. (9)
In the case of a discontinuity material surface Σ (with unit normal N µ ) the ( 9) have to be completed by

m µ=1 [| ∂L ∂ (∂Ψ σ /∂x µ ) |]N µ = 0, ∀x µ ∈ Σ, (10)
where [| (•) |] is the jump of (•) across the surface Σ. These equations are known as the Euler-Lagrange equations corresponding to the considered Lagrangian density.

The Space-Time Case (m = 4)

Let us now consider the particular case in which m = 4. This case corresponds, for instance, to the case x µ = (x 1 , x 2 , x 3 , t). We have that η σ (x µ ) are any set of linearly independent functions of the x µ which vanish on the boundary of time type domain,

η σ (x 1 , x 2 , x 3 , t 0 ) = η σ (x 1 , x 2 , x 3 , t 1 ) = 0
and on the part ∂ d V of the boundary ∂V of the volume V , on which the kinematical conditions are prescribed,

η σ (x 1 , x 2 , x 3 , t) = 0, ∀(x 1 , x 2 , x 3 ) ∈ ∂ d V, ∀t ∈ [t 0 , t 1 ] .
It is easy to show that in this particular case eq. ( 7) yields

δA = ε t1 t0 dt V σ η σ [ ∂L ∂Ψ σ - 3 k=1 ∂ ∂x k ∂L ∂ (∂Ψ σ /∂x k ) (11) - ∂ ∂t ∂L ∂ (∂Ψ σ /∂t) ] +ε ∂V /∂ d V σ η σ 3 k=1 ∂L ∂ (∂Ψ σ /∂x k ) N k + Σ σ η σ 3 k=1 ∂L ∂ (∂Ψ σ /∂x k ) N k
The stationarity δA = 0 of the Action implies, for any σ = 1, 2, ..., n, ∂L ∂Ψ σ -

3 k=1 ∂ ∂x k ∂L ∂ (∂Ψ σ /∂x k ) - ∂ ∂t ∂L ∂ (∂Ψ σ /∂t) = 0, ∀x k ∈ V, (12)
3 k=1 ∂L ∂ (∂Ψ σ /∂x k ) N k = 0, ∀x µ ∈ ∂V /∂ d V, (13) 3 k=1
∂L ∂ (∂Ψ σ /∂x k ) N k = 0, ∀x µ ∈ Σ. (14)
Which are the standard Euler-Lagrange equations. We will see in the next chapters of this book how to generalize [START_REF] Homer | The Iliad. Translated[END_REF] when Σ can move freely.

Principle of Virtual Power and Principle of Least Action

The principle of least Action, when formulated for Action functionals admitting first differentials, can be regarded as a particular form of the principle of virtual powers. Indeed, if

A = A int + A ext + A ine (15)
then δA = 0 ⇐⇒ δA int + δA ext + δA ine = 0.

Identifying

δA int = P int δA ext = P ext , δA ine = P ine , (17)
we get

P int + P ext + P ine = 0. ( 18
)
Which is the standard form of principle of virtual powers. Is the principle of virtual power more general than principle of least Action? First answer: the principle of virtual powers involves differentials which are not exact, in general. Therefore, once fixed the kinematics, the principle of virtual power is actually more general. In both P int and P ext one can include dissipative terms, which cannot, in general, be derived from an Action functional. However, it is not clear if, suitably extending the space of configurations and the set of admissible motions, one can introduce an Action functional also for systems which, in a restricted kinematics, appear as dissipative. Controversies in the literature about this subject are not yet solved.

Hamilton-Rayleigh Approach

We propose to use the Hamilton-Rayleigh compromise. We introduce an Action functional and a Dissipation Rayleigh functional and, by means of them, we formulate the Principle of Virtual Work. Rayleigh dissipation functional R is defined as a linear functional on the set of velocities, not on the set of motions as A. Therefore, δR is defined as a linear functional of the variation δ ṁ. The principle of virtual works formulated following Hamilton-Rayleigh takes the form: (the lack of the upper dot on RHS is not a mistake!) δA (δm) = R (δm) .

(

) 19

Conclusions

We recall an ancient and useful recipe for building theories for describing effectively physical phenomena:

"In Nomina est Natura Rerum". Anonymous

This statement (passed to us by the middle age tradition) is formulated for defending mathematical formalism. This sentence claims that it is impossible to talk about any mathematical model without using the appropriate language. So, for instance, it is impossible to say clearly what is the first variation of Action using simply "words" from natural language, i.e. without writing integrals on T and, to proceed, we need to give "precisely" names to things. Therefore, to specify precisely how our models are constructed we need to introduce symbols and formulas.

However, we can also say that "Nomina sunt Consequentia Rerum." Iustinianus, Institutiones Liber II,7,3

This because we are not blindly building our mathematical model. We get informations about physics and from these informations we actually formulate our models.

We can finally state that the "old" method of basing the formulation of mathematical models on the variational approach works: indeed it works very well.