Joseph R Fitchett
email: joseph.fitchett@doctors.org.uk2207871572

The Right to Health in Practice

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The origin of human rights Rights are moral and legal entitlements. Human rights have foundations in the theory of natural law and by definition human rights "belong justifiably" to all persons [6]. Several core notions play essential roles in the realization of human rights, namely the concept of a right, a duty, an entitlement and an obligation. For descriptive purposes, rights are classified as negative rights and positive rights. Negative rights imply freedoms, for instance the right to be free from forced medical experimentation or the right to be free from torture and ill treatment. Positive rights imply entitlements, for instance the right to access essential medicines and vaccines.

The philosophical basis for human rights is not restricted to the 20 th Century and the United Nations Universal Declaration of Human Rights [7]. Core principles shape several religious and ancient legal texts, such as the Babylonian Code of Hammurabi, the Hindu Laws of Manu, and the Analects of Confucius. The origins of the modern human rights movement arguably stems from the end of the 18 th Century at a time where the relationship between government and the governed was evolving rapidly and redefining itself, highlighted by treatises in political philosophy on the Social Contract. Two influential revolutions in the United States of America, 1776, and in France, 1789, generated the Declaration of Independence and the Declaration of the Rights of Man and the Citizen, respectively, upholding the concept of universal natural rights. However it is the genocidal atrocities and medical experimentation of an unparalleled evil committed by the Nazis, directed primarily at millions of Jews throughout Europe, from which the Universal Declaration of Human Rights (UDHR) was born. The inhumanity of the Nazi regime and gross disregard for human rights of all human beings left people and their leaders questioning the morality of the human race.

In the present day, the International Covenant on Economic, Social and Cultural Rights (ICESCR) [8] and the International Covenant on Civil and Political Rights (ICCPR) [9] are legally binding instruments in international human rights law upholding, enshrining and protecting universal human rights. They are part of what is referred to as the Bill of Human Rights, which is composed of the ICESCR, ICCPR and UDHR. These instruments are vital in ensuring that governments are held accountable for their action, or inaction, for the sustained improvement of health among their society.

The right to health

Human rights are undeniably interdependent, indivisible and interrelated [10]. This concept is significant when referring to health. The World Health Organization (WHO) definition of health is considerably cited and defines health as "a state of complete physical, mental and social well-being and not merely the absence of disease" [11]. Both the underlying determinants of health and a functioning health system accessible to all without discrimination are fundamental to the realisation of the right to the highest attainable standard of physical and mental health ('the right to health' or 'the right to physical and mental health'). The right to water, to education and information, to adequate housing and shelter, freedom from human rights violations such as torture and inhumane treatment all inextricably impact upon health (directly and indirectly) [12]. The right to health however is not about a right to be healthy as many aspects, including genetic predisposition for instance, are outside the direct control of States.

Beyond the WHO Constitution and article 25 of the Universal Declaration of Human Rights, the right to health is articulated in the 1978 Declaration of Alma-Ata, regional treaties, article 12 of the ICESCR and further clarified in "General Comment No.14: The Right to the Highest Attainable Standard of Health". Every State in the world has ratified at least one international human rights treaty upholding the right to health and over 115 national constitutions recognize the right to health [13], therefore governments and affiliated institutions have committed themselves to respecting, protecting and realizing the right in national law and domestic policy.

In practice, health systems and services operating with a human rights-based approach must be appropriately available, accessible, acceptable and of good quality. The principles of participation, non-discrimination, transparency and accountability on all levels and equality are paramount. Naturally, it is counterproductive to expect countries without the financial means to provide their citizens with a utopian healthcare system and implement policies to redress the underlying determinants of health all at once. Nevertheless, the United Nations Human Rights Council monitors States and their legal obligations to ensure the appropriate measures are in place to realize the right to health, and indeed the role of civil society is absolutely essential.

A human rights framework in practice

Healthcare workers have unique access to individuals at times of greatest need and are in a prime position to report injustices and violations as they happen. The fundamental principles of human rights provide a practical framework to guide healthcare practitioners and public health professionals to realize of the inextricable right to the highest attainable standard of physical and mental health for all.

There is a moral imperative and obligation for the State to address injustices with concerted action and healthcare professionals have a responsibility to act when appropriate. States are accountable and obliged to report to the United Nations. States must exert their influence to ensure public and private institutions are accountable and act in the best interests of the communities and individuals they serve. Just actions based on respective duties and obligations must ensure that progressive realization of the right to health and fair distribution of limited resources occurs without delay. Ethical principles have the compelling power to influence action. The following principles are central to realizing the right to health and guiding day-to-day practice by healthcare professionals:

Respect

Due regard, civility and non-discrimination for all women, men, boys and girls regardless of background or ethnicity.

Equality A sense of egalitarianism, sympathy and empathy. An appreciation of the injustices in situations others may be confronted with.

Dignity

Honouring the values and individuality of all persons.

Autonomy

Respecting the freedom of competent human beings.

Empowering the individuals who carry the resultant outcomes.

Participation

The opportunity, by men and women, to be involved in the process and discuss subjective issues on both sides in an effort to understand, not presume, the optimal way forward.

Fairness

Facilitating accountability for unfair processes and unfair outcomes.

These principles are not novel. In fact these principles complement the traditional principles of medical ethics and the Hippocratic Oath. "First do no harm" is a fundamental moral concept that relies on an underlying respect between healthcare professionals and their autonomous patients, equality in treatment based on need, and fairness. However the principles presented above are inconsistently applied at present to resource allocation for health, which favour instead utilitarian and consequentialist models. The universality of human rights act as a reminder that individuals are born into unequal circumstances with a tremendous influence on the rest of their lives and may remain trapped, vulnerable and destined to fight hard for social justice. To state the trivial but indisputable, we do not choose where we are born.

Redistributing societal benefits and burdens for the benefit of the marginalised is not uncommon in many states worldwide, as observed with social security policies. Translation of this principle beyond national borders is rare. Health is influenced by both the forces of the underlying determinants of health and access to healthcare. The right to health upholds the rights of all individuals, including those most vulnerable that may be forgotten in times of limited resources, those on the edges or the extremes. A right to health approach moves beyond the mere "cost-effectiveness" analysis. Naturally, in the words of Paul Hunt, past-UN Special Rapporteur on the Right to Health, human rights cannot answer all the difficult practical problems a health system faces on a daily basis, any more than ethics or health economics can [14]. However they provide a morally assertive and legally binding approach, ensuring an environment of universal respect, egalitarianism and accountability for unfair violations and participatory discussion to improve justice for all.

Evidence from the United Kingdom Department of Health looking at human rights in healthcare evaluated that: "a human rights based approach to health and social care can, and will increasingly in the future, have a tangible impact on the treatment and care of service users, and that it is one way for achieving good practice within the sector". [15] Furthermore, Gunilla Backman et al. recently published a landmark study in The Lancet assessing health systems and the right to health in 194 countries, identifying key features of the right to health and proposing 72 indicators to monitor and analyse just health systems worldwide [16]. These indicators are essential to manage the progressive realisation of the right to health, which is not merely efficient management or good in a humanitarian sense, but an obligation under international law. Facts and figures are important, however not exclusively. Values, although complex to define, difficult to quantify and judged culturally relative, must be taken into consideration concurrently in all healthcare-related decisions.

Healthcare workers must lead the way

Methods to strengthen and improve the reporting system at the United Nations are urgently required to uphold universal human rights. The WHO alone cannot efficiently redress the inequalities in health and resources across the globe. A variety of players in a globalized world have direct and indirect impacts on health and resource availability for fair distribution. Institutions require clear communication and a bridge to connect with individuals they serve. Conflicting policies and priorities, such as the TRIPS agreement and access to essential medicines, restrict improvement in conditions for millions worldwide. To realize human rights and the right to health, the moral principles underlying the legal obligations must be applied as a framework to practical issues in daily life. Healthcare professionals must lead the way. Without this translation from theory to practice, conflict between what we say and what we do will remain.

"Each person possesses an inviolability founded on justice that even the welfare of society as a whole cannot override. For this reason justice denies that the loss of freedom for some is made right by a greater good shared by others." John Rawls, A Theory of Justice [17] In the spirit of John Rawls, this principle must be applied to health to ensure the loss of health by some does not justify or make right the greater health of others. Health is global. For social justice on a global level, consistency in applying the human rights framework is urgently required for the realization of the right to health and redressing injustices worldwide.

Respect

Due regard, civility and non-discrimination for all women, men, boys and girls.

Equality

A sense of egalitarianism, sympathy and empathy. An appreciation of the injustices in situations others may be confronted with.

Dignity

Honouring the values and individuality of all persons.

Autonomy

Respecting the freedom of competent human beings.

Empowering the individuals who carry the resultant outcomes.

Participation

The opportunity, by men and women, to be involved in the process and discuss subjective issues on both sides in an effort to understand, not presume, the optimal way forward.

Fairness

Facilitating accountability for unfair processes and unfair outcomes.

Context

In the current era of globalization, the world is diversifying as never before. Inequalities in economic, social, spiritual, political and civil matters characterise daily life. Estimates consider 80% of global disease burden lies in "developing" or low-income countries, based on crude calculations by disability-adjusted life years (DALYs) [1]. And measures do not seem to be in place to redress these inequalities. For instance, the Commission on Health Research Development estimated that 90% of all global research and development expenditure is dedicated to 10% of the world's disease burden, primarily concentrated in wealthier countries [2]. Today, there may be a new climate of awareness maturing. Governments representing "developed" or high-income countries often discuss the urgent need to help the world's poorest or rescue the "bottom billion" from devastating illness [3][4]. However the optimistic rhetoric is not always matched by foreign policy and international trade agreements (consider TRIPS [5], the World Trade Organization's Trade-Related Aspects of International Property Rights Agreement consolidating strict patent rules worldwide with significant impact on access to essential medicines). The following perspective provides a comprehensive overview of the right to health and proposes a human rights-based approach to health as a sustainable framework that transcends borders for justice in healthcare.

The origin of human rights

Rights are moral and legal entitlements. Human rights have foundations in the theory of natural law and by definition human rights "belong justifiably" to all persons [6]. Several core notions play essential roles in the realization of human rights, namely the concept of a right, a duty, an entitlement and an obligation. For descriptive purposes, rights are classified as negative rights and positive rights. Negative rights imply freedoms, for instance the right to be free from forced medical experimentation or the right to be free from torture and ill treatment. Positive rights imply entitlements, for instance the right to access essential medicines and vaccines. The philosophical basis for human rights is not restricted to the 20 th Century and the United Nations Universal Declaration of Human Rights [7]. Core principles shape several religious and ancient legal texts, such as the Babylonian Code of Hammurabi, the Hindu Laws of Manu, and the Analects of Confucius. The origins of the modern human rights movement arguably stems from the end of the 18 th Century at a time where the relationship between government and the governed was evolving rapidly and redefining itself, highlighted by treatises in political philosophy on the Social Contract. Two influential revolutions in the United States of America, 1776, and in France, 1789, generated the Declaration of Independence and the Declaration of the Rights of Man and the Citizen, respectively, upholding the concept of universal natural rights. However it is the genocidal atrocities and medical experimentation of an unparalleled evil committed by the Nazis, directed primarily at millions of Jews throughout Europe, from which the Universal Declaration of Human Rights (UDHR) was born. The inhumanity of the Nazi regime and gross disregard for human rights of all human beings left people and their leaders questioning the morality of the human race.

In the present day, the International Covenant on Economic, Social and Cultural Rights (ICESCR) [8] and the International Covenant on Civil and Political Rights (ICCPR) [9] are legally binding instruments in international human rights law upholding, enshrining and protecting universal human rights. They are part of what is referred to as the Bill of Human Rights, which is composed of the ICESCR, ICCPR and UDHR.

The right to health

Human rights are undeniably interdependent, indivisible and interrelated [10]. This concept is significant when referring to health. The World Health Organization (WHO) definition of health is considerably cited and defines health as "a state of complete physical, mental and social well-being and not merely the absence of disease" [11]. Both the underlying determinants of health and a functioning health system accessible to all without discrimination are fundamental to the realisation of the right to the highest attainable standard of physical and mental health ('the right to health' or 'the right to physical and mental health'). The right to water, to education and information, to adequate housing and shelter, freedom from human rights violations such as torture and inhumane treatment all inextricably impact upon health (directly and indirectly) [12]. The right to health however is not about a right to be healthy as many aspects, including genetic predisposition for instance, are outside the direct control of States.

Beyond the WHO Constitution and article 25 of the Universal Declaration of Human Rights, the right to health is articulated in the 1978 Declaration of Alma-Ata, regional treaties, article 12 of the ICESCR and further clarified in "General Comment No.14: The Right to the Highest Attainable Standard of Health". Every State in the world has ratified at least one international human rights treaty upholding the right to health and over 115 national constitutions recognize the right to health [13], therefore governments and affiliated institutions have committed themselves to respecting, protecting and realizing the right in national law and domestic policy.

In practice, health systems and services operating with a human rights-based approach must be appropriately available, accessible, acceptable and of good quality. The principles of participation, non-discrimination, transparency and accountability on all levels and equality are paramount. Naturally, it is counterproductive to expect countries without the financial means to provide their citizens with a utopian healthcare system and implement policies to redress the underlying determinants of health all at once. Nevertheless, the United Nations Human Rights Council monitors States and their legal obligations to ensure the appropriate measures are in place to realize the right to health, and indeed the role of civil society is absolutely essential.

A human rights framework in practice

Healthcare workers have unique access to individuals at times of greatest need and are in a prime position to report injustices and violations as they happen. The fundamental principles of human rights provide a practical framework to guide healthcare practitioners and public There is a moral imperative and obligation for the State to address injustices with concerted action and healthcare professionals have a responsibility to act when appropriate. States are accountable and obliged to report to the United Nations. States must exert their influence to ensure public and private institutions are accountable and act in the best interests of the communities and individuals they serve. Just actions based on respective duties and obligations must ensure that progressive realization of the right to health and fair distribution of limited resources occurs without delay. Ethical principles have the compelling power to influence action. The following principles are central to realizing the right to health and guiding day-to-day practice by healthcare professionals:

Respect

Due regard, civility and non-discrimination for all women, men, boys and girls.

Equality A sense of egalitarianism, sympathy and empathy. An appreciation of the injustices in situations others may be confronted with.

Dignity

Honouring the values and individuality of all persons.

Autonomy

Respecting the freedom of competent human beings.

Empowering the individuals who carry the resultant outcomes.

Participation

The opportunity, by men and women, to be involved in the process and discuss subjective issues on both sides in an effort to understand, not presume, the optimal way forward.

Fairness

Facilitating accountability for unfair processes and unfair outcomes.

These principles are not novel. However they are inconsistently applied at present to health and resource allocation, which favour instead utilitarian and consequentialist models. The universality of human rights act as a reminder that individuals are born into unequal circumstances with a tremendous influence on the rest of their lives and may remain trapped, vulnerable and destined to fight hard for social justice. To state the trivial but indisputable, we do not choose where we are born.

Redistributing societal benefits and burdens for the benefit of the marginalised is not uncommon in many states worldwide, as observed with social security policies. Translation of this principle beyond national borders is rare. Health is influenced by both the forces of the underlying determinants of health and access to healthcare. The right to health upholds the rights of all individuals, including those most vulnerable that may be forgotten in times of limited resources, those on the edges or the extremes. A right to health approach moves beyond the mere "cost-effectiveness" analysis. Naturally, in the words of Paul Hunt, past-UN Special Rapporteur on the Right to Health, human rights cannot answer all the difficult practical problems a health system faces on a daily basis, any more than ethics or health economics can [14]. However they provide a morally assertive and legally binding approach, ensuring an environment of universal respect, egalitarianism and accountability for unfair violations and participatory discussion to improve justice for all.

Evidence from the United Kingdom Department of Health looking at human rights in healthcare evaluated that: "a human rights based approach to health and social care can, and will increasingly in the future, have a tangible impact on the treatment and care of service users, and that it is one way for achieving good practice within the sector". [15] Furthermore, Gunilla Backman et al. recently published a landmark study in The Lancet assessing health systems and the right to health in 194 countries, identifying key features of the right to health and proposing 72 indicators to monitor and analyse just health systems worldwide [16]. These indicators are essential to manage the progressive realisation of the right to health, which is not merely efficient management or good in a humanitarian sense, but an obligation under international law. Facts and figures are important, however not exclusively. Values, although complex to define, difficult to quantify and judged culturally relative, must be taken into consideration concurrently in all healthcare-related decisions.

Healthcare workers must lead the way

Methods to strengthen and improve the reporting system at the United Nations are urgently required to uphold universal human rights. The WHO alone cannot efficiently redress the inequalities in health and resources across the globe. A variety of players in a globalized world have direct and indirect impacts on health and resource availability for fair distribution. Institutions require clear communication and a bridge to connect with individuals they serve. Conflicting policies and priorities, such as the TRIPS agreement and access to essential medicines, restrict improvement in conditions for millions worldwide. To realize human rights and the right to health, the moral principles underlying the legal obligations must be applied as a framework to practical issues in daily life. Healthcare professionals must lead the way. Without this translation from theory to practice, conflict between what we say and what we do will remain.

"Each person possesses an inviolability founded on justice that even the welfare of society as a whole cannot override. For this reason justice denies that the loss of freedom for some is made right by a greater good shared by others." John Rawls, A Theory of Justice [17] In the spirit of John Rawls, this principle must be applied to health to ensure the loss of health by some does not justify or make right the greater health of others. Health is global. For social justice on a global level, consistency in applying the human rights framework is urgently required for the realization of the right to health and redressing injustices worldwide.

  realize of the inextricable right to the highest attainable standard of physical and mental health for all.

Table 1

 1 Core principles for the realization of the right to health

	The Right to Health in Practice
	Joseph R. Fitchett
	Department of Infectious and Tropical Diseases
	London School of Hygiene & Tropical Medicine
	Keppel Street, London WC1E 7HT, United Kingdom
	Joseph.fitchett@doctors.org.uk
	+2207871572
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	1


PhD, FRCP Clifford J Bailey
email: c.j.bailey@aston.ac.uk

Michael Kodack

Patient adherence to medication requirements for therapy of type 2 diabetes

Keywords: diabetes, cardiovascular risk factors, adherence, compliance, persistence, clinical inertia, polypharmacy, and multifactorial intervention

Type 2 diabetes is a complex and progressive endocrine and metabolic disease that typically requires substantial lifestyle changes and multiple medications to lower blood glucose, reduce cardiovascular risk and address comorbidities. Despite an extensive range of available and effective treatments, < 50% of patients achieve a glycemic target of HbA 1c < 7.0%, and about two thirds die of premature cardiovascular disease. Adherence to prescribed therapies is an important factor in the management of type 2 diabetes that is often overlooked. Inadequate adherence to oral antidiabetes agents, defined as collecting <80% of prescribed medication, is variously estimated to apply to between 36% and 93% of patients. All studies affirm that a significant proportion of type 2 diabetes patients exhibit poor adherence that will contribute to less than desired control. Identified factors that impede adherence include complex dosing regimens, clinical inertia, safety concerns, socioeconomic issues, ethnicity, patient education and beliefs, social support, and polypharmacy. This review explores these factors and potential strategies to improve adherence in patients with type 2 diabetes.

Introduction

The global diabetes epidemic is predicted to increase from 285 million in 2010 to 439 million by 2030 [START_REF]Diabetes Federation -Diabetes Atlas. Diabetes and impaired glucose tolerance. Global burden: prevalence and projections[END_REF]. About 90% to 95% of these patients will have type 2 diabetes, the onset of which will occur at younger ages [START_REF]Diabetes Federation -Diabetes Atlas. Diabetes and impaired glucose tolerance. Global burden: prevalence and projections[END_REF]. In the United States, approximately 24 million people (8% of the population) have diabetes with about 18 million diagnosed and 6 million undiagnosed [START_REF] Huang | Projecting the future diabetes population size and related costs for the US[END_REF][START_REF]National diabetes fact sheet[END_REF]. It is anticipated that by 2034 the prevalence of diabetes in the United States will increase to 44 million [START_REF] Huang | Projecting the future diabetes population size and related costs for the US[END_REF].

The long-term complications of diabetes, particularly type 2 diabetes, present a formidable threat that requires comprehensive management of glycemia, and a myriad of cardiovascular (CV) risk factors and comorbidities [START_REF] Diabetes | Standards of medical care in diabetes-2010[END_REF]. Although an extensive and effective range of therapies is available to address these issues [START_REF] Diabetes | Standards of medical care in diabetes-2010[END_REF][START_REF] Nathan | On behalf of the American Diabetes Association; European Association for Study of Diabetes. Medical management of hyperglycemia in type 2 diabetes: a consensus algorithm for the initiation and adjustment of therapy: a consensus statement of the American Diabetes Association and the European Association for the Study of Diabetes[END_REF][START_REF] Krentz | New drugs for type 2 diabetes mellitus: what is their place in therapy?[END_REF], only slightly more than half of patients achieve a HbA 1c target of < 7.0%, and about two thirds of patients die of CV disease [START_REF]National diabetes fact sheet[END_REF][START_REF] Ong | International Journal of Clinical Practice International Journal of Clinical Practice Partnership for Effective Diabetes Management. What are the practical implications for treating diabetes in light of recent evidence? Updated recommendations from the Global Partnership for Effective Diabetes Management[END_REF]. See Table 1 for a list of available drugs with selected properties highlighted. There are also established protocols, guidelines, and algorithms to accommodate the needs of most patients under the majority of circumstances [START_REF] Diabetes | Standards of medical care in diabetes-2010[END_REF][START_REF] Nathan | On behalf of the American Diabetes Association; European Association for Study of Diabetes. Medical management of hyperglycemia in type 2 diabetes: a consensus algorithm for the initiation and adjustment of therapy: a consensus statement of the American Diabetes Association and the European Association for the Study of Diabetes[END_REF]8).

However, an often neglected issue concerns the number of patients who do (and do not) take their prescribed medications.

This narrative review explores the important subject of adherence in patients with Adherence, however, concerns the extent to which the patient achieves an agreed upon treatment without close supervision. Patients achieving > 80% of their prescribed medication (typically calculated as days of medication collected divided by days of medication prescribed) is often accepted as a measure of adherence. Persistence describes the duration of time without default, that is, the time that a patient continues to maintain therapy as a proportion of total time of follow-up. [START_REF] Dimatteo | Variations in patients' adherence to medical recommendations quantitative review of 50 years of research[END_REF]. This supports the view that conditions perceived as imminently life threatening, uncomfortable, or painful are more likely to receive better adherence.

Extensive evidence indicates that intensive control of blood glucose is associated with reduced long-term micro-and macrovascular complications in patients with type 2 diabetes [START_REF] Ray | Effect of intensive control of glucose on cardiovascular outcomes and death in patients with diabetes mellitus: a meta-analysis controlled trials[END_REF][START_REF] Kelly | Systematic review: glucose control and cardiovascular disease in type 2 diabetes[END_REF][START_REF] Holman | 10-year follow-up of intensive glucose control in type 2 diabetes[END_REF]. One might postulate therefore that greater adherence with treatment regimens should improve metabolic control outcomes. Indeed, nonadherent patients are at increased risk for the development of micro-and macrovascular complications, hospitalizations, and death [START_REF] Ho | Effect of medication nonadherence on hospitalizations and mortality among patients with diabetes mellitus[END_REF]. Also a study of adherence in an indigent population with type 2 diabetes noted that each 10% decrease in adherence was accompanied by a +0.14% increase in HbA 1c [START_REF] Pladevall | Page 19 of 33 International Journal of Clinical Practice International Journal of Clinical Practice pharmaceutical adherence in a diabetic population[END_REF]. Additionally, the benefits of improved adherence in patients with type 2 diabetes have been linked to fewer emergency department visits and fewer inpatient admissions (16).

Clinical inertia

For patients with type 2 diabetes and some other chronic disorders, treatment goals are well defined, practice guidelines are widely disseminated, and effective treatments are available [START_REF] Nathan | On behalf of the American Diabetes Association; European Association for Study of Diabetes. Medical management of hyperglycemia in type 2 diabetes: a consensus algorithm for the initiation and adjustment of therapy: a consensus statement of the American Diabetes Association and the European Association for the Study of Diabetes[END_REF][START_REF] Rodbard | Statement by an American Association of Clinical Endocrinologists/American College of Endocrinology consensus panel on type 2 diabetes mellitus: an algorithm for glycemic control[END_REF][START_REF]International Diabetes Federation Global Guideline for Type 2 Diabetes[END_REF]. Nevertheless, initiation and escalation of therapy are often delayed [START_REF] Phillips | Clinical inertia[END_REF][START_REF] Brown | The burden of treatment failure in type 2 diabetes[END_REF]. This is clinical inertia. Data reported in 1997 from a large hospital diabetes clinic noted that, over a 3-year period, glucose lowering therapy was only intensified in an average of 36% of 1051 visits of patients who met the criteria for escalation of treatment.

This was despite an agreed protocol for management of patients with type 2 diabetes [START_REF] Phillips | Clinical inertia[END_REF][START_REF] El-Kebbi | Diabetes in urban African-Americans. IX. Provider adherence to management protocols[END_REF]. A review of practice within the Kaiser Permanente organization found that between 1996 and 2003 patients with type 2 diabetes who were inadequately controlled (HbA 1c ≥ 8.0%) on a sulfonylurea were delayed on average for 20 months before additional or alternative therapy was introduced [START_REF] Brown | The burden of treatment failure in type 2 diabetes[END_REF]. Recently, the problem of clinical inertia has been raised by guidelines and standards (4). In a 2009 report, 41% of patients with sustained hyperglycemia did not receive appropriate care within 6 months of identification, and 25% had not received appropriate care after 1 year. Interestingly, appropriate care was often deferred until the HbA 1c reached 9.0% or greater [START_REF] Lafata | UK: comparison of published quality-of-care reports with results of the Quality and Outcomes Framework for Diabetes[END_REF].

Factors that contribute to clinical inertia include a perceived lack of training and confusion or lack of focus on glycemic goals [START_REF] Phillips | Clinical inertia[END_REF]. However, in the United Kingdom, the introduction of financial incentives for general practitioners to have more patients achieve specific HbA 1c targets (the Quality and Outcomes Framework) resulted in rapid and substantial improvements in glycemic control for patients with type 2 diabetes (23).

While this scheme has its limitations, it is continuing to provide improvements in glycemic control [START_REF] Calvert | Effect of the quality and outcomes framework on diabetes care in the UK: retrospective cohort study[END_REF].

Although clinical inertia is customarily directed to physicians, patient considerations are also relevant [START_REF] Grant | Relationship between patient medication adherence and subsequent clinical inertia in type 2 diabetes glycemic management[END_REF]. Patients with type 2 diabetes are often required to make significant changes to their behavior and lifestyle to achieve improved glycemic control. Patient adherence to their medication regimen is strongly associated with the willingness of the physician to intensify treatment [START_REF] Grant | Relationship between patient medication adherence and subsequent clinical inertia in type 2 diabetes glycemic management[END_REF]. However; physicians are less likely to intensify therapy in a poorly adherent patient despite an elevated HbA 1c . The reasons for this are not fully appreciated, but it is recognized that patients generally follow medical recommendations in 2 ways: (a) they return for office/clinic visits; and, (b) they agree to take medicines as recommended, although they may not take every dose Bailey 8 [START_REF] Phillips | Clinical inertia[END_REF]. Therefore, issues around the level of trust in the relationship impact its influence on motivating the patient to engage in health-promoting behaviors, and the contribution of providers to promote it [START_REF] Grant | Relationship between patient medication adherence and subsequent clinical inertia in type 2 diabetes glycemic management[END_REF].

Another reason for clinical inertia may be the heavy publicity associated with recent trials of intensive glucose control in patients with type 2 diabetes, notably the Action to Control Cardiovascular Risk in Diabetes (ACCORD) study and the Veterans Administration Diabetes Trial (VADT) [START_REF] Accord Study | Effects of intensive glucose lowering in type 2 diabetes[END_REF][START_REF] Duckworth | On behalf of the VADT Investigators. Glucose control and vascular complications in veterans with type 2 diabetes[END_REF]. These trials did not show significant reduction in CV event mortality, which could engender reluctance by some physicians and patients to strive for low glycemic targets.

Polypharmacy

While there is no consensus definition of polypharmacy [START_REF] Ibrahim | Polypharmacy and possible drug-drug interactions among diabetic patients receiving home health care services[END_REF], it is well recognized that the treatment of patients with type 2 diabetes is associated with the use of multiple medications, which include not only glucose-lowering therapies but also treatments for accompanying disorders such as dyslipidemia, hypertension, and depression [START_REF] Good | Polypharmacy in elderly patients with diabetes[END_REF][START_REF] Austin | Polypharmacy as a risk factor in the treatment of type 2 diabetes[END_REF][START_REF] Emslie-Smith | Page 21 of 33 International Journal of Clinical Practice International Journal of Clinical Practice adherence in patients with type 2 diabetes[END_REF].

Indeed, several studies have suggested that a patient with type 2 diabetes can often expect to take as many as 5 or more medications daily [START_REF] Ibrahim | Polypharmacy and possible drug-drug interactions among diabetic patients receiving home health care services[END_REF]32,[START_REF] Dunning | Medication knowledge and self-management by people with type 2 diabetes[END_REF]. Irrespective of whether all of their medicines are essential or desirable, it is acknowledged that multiple medications contribute to poor adherence [START_REF] Good | Polypharmacy in elderly patients with diabetes[END_REF][START_REF] Austin | Polypharmacy as a risk factor in the treatment of type 2 diabetes[END_REF][START_REF] Emslie-Smith | Page 21 of 33 International Journal of Clinical Practice International Journal of Clinical Practice adherence in patients with type 2 diabetes[END_REF]. Multiple medications are also associated with more adverse events (AEs), drug interactions, and duplications of therapy, as well as extra treatments to address the side effects of some medications [START_REF] Austin | Polypharmacy as a risk factor in the treatment of type 2 diabetes[END_REF].

Polypharmacy is commonly considered a marker of poor glycemic control and should serve as a 'red flag' leading to assessment of the adequacy of therapy and adherence [START_REF] Willey | Polypharmacy with oral antidiabetic agents: an indicator of poor glycemic control[END_REF]. The ACCORD study has recently noted that individuals who fail to achieve acceptable glycemic control despite extensive multiple medications are likely to be more vulnerable to AEs [START_REF] Accord Study | Effects of intensive glucose lowering in type 2 diabetes[END_REF].

Other medication-related factors

Several medication-related factors can influence adherence in patients with type 2 diabetes. Overall, regimen complexity, including more frequent dosing, impairs adherence. Improved adherence has also been reported with respect to once-and twicedaily dosing of oral glucose-lowering agents [START_REF] Dezii | Effect of once-daily and twice-daily dosing on adherence with prescribed glipizide oral therapy for type 2 diabetes[END_REF][START_REF] Donnan | Adherence to prescribed oral hypoglycemic medication in a population of patients with type 2 diabetes: a retrospective cohort study[END_REF]. In addition, the rate of persistence with oral therapy was improved by a once-daily schedule. Since patients treated with once-daily oral antidiabetes drug therapy required more tablets, the authors of the study concluded that dosing frequency has a greater impact on adherence than tablet numbers.

Patient understanding of the drug dosing regimen and its therapeutic value are important for good adherence. In a study that used pharmacist interventions to improve adherence to diabetes care, confusion about dose timing or frequency was a predictor of poor adherence [START_REF] Ingersoll | The impact of medication regimen factors on adherence to chronic treatment: a review of the literature[END_REF]. Patients were more likely to adhere to a regimen if they believed that their medications were justifiably indicated, safe, and effective (32). An investigation of polypharmacy and adherence in patients with type 2 diabetes, found that patients were poorly adherent if they were not convinced that the medicine was effective or indicateda situation common in asymptomatic diseases (32). Among a type 2 diabetes cohort receiving a median of 7 medications, patients reported that they had very limited knowledge about indications and virtually no appreciation of treatment risks [START_REF] Harder | Appropriateness and surveillance of medication in a cohort of diabetic patients on polypharmacy[END_REF].

Other factors impairing adherence

Depression significantly impairs adherence to treatment in patients with type 2 diabetes and the degree of adherence worsens with the depth of the depression [START_REF] Odegard | Medication taking and diabetes: a systematic review of the literature[END_REF][START_REF] Chao | The mediating role of health beliefs in the relationship between depressive symptoms and medication adherence in persons with diabetes[END_REF]. Since depressed patients are less likely to report that they have not been adherent, impaired glycemic control might be falsely attributed to ineffective treatment and additional drugs unnecessarily added to the regimen [START_REF] Odegard | Medication taking and diabetes: a systematic review of the literature[END_REF]. However, forgetting to take medications is not solely a feature of affective disorders. In a study of medication usage in a real-world, cross-sectional population of patients with type 2 diabetes treated for ≥ 10 years, 20% of participants reported that they regularly forgot to take their medications without any particular explanation [START_REF] Dunning | Medication knowledge and self-management by people with type 2 diabetes[END_REF].

Racial, ethnic, and socioeconomic factors can also significantly affect adherence.

Among patients with type 2 diabetes in the United States, HbA 1c appears to be significantly higher in non-Hispanic black women and Mexican American men (40). This may be due to a number of race-identified factors such as difficulties with language, dietary management, failure to self-monitor blood glucose, and lack of regular exercise [START_REF] Misra | Ethnic and gender differences in psychosocial factors, glycemic control, and quality of life among adult type 2 diabetic patients[END_REF]. In addition, there are significant black-Caucasian differences in glycemic control with higher mean HbA 1c in the former that are not due to differences in adherence and may be related to genetic or environmental factors (p < 0.0001) [START_REF] Adams | Medication adherence and racial differences in A1C control[END_REF]. These issues, including cultural factors, increase the importance of adherence in these racial/ethnic groups because minority ethnic groups in the United States have a higher rate of diabetic complications, even after adjusting for differences in glycemic control (40). indicate that they are at higher risk of difficulties in achieving glycemic control than their Caucasian counterparts [START_REF] Delamater | Risk for metabolic problems in minority youth with diabetes[END_REF]. This appears to be significantly related to the effect of lower adherence to dietary recommendations and blood glucose monitoring [START_REF] Auslander | Disparity in glycemic control and adherence between African American and Caucasian youths with diabetes. Family and community contexts[END_REF].

Other factors that may impact the adherence in all patients, as well as specific racial/ethnic groups, include paying for medications, obtaining refills, and fear of needles [START_REF] Willey | Polypharmacy with oral antidiabetic agents: an indicator of poor glycemic control[END_REF][START_REF] Odegard | Barriers to medication adherence in poorly controlled diabetes mellitus[END_REF].

Social support impacts mental and physical health and outcomes by mediating health-related behaviors [START_REF] Glasgow | Social environment and regimen adherence among type II diabetic patients[END_REF]. This is established in families and children with type 1

diabetes. In the Diabetes Control and Complications Trial (DCCT), one of the key elements of successful intensive therapy was the availability of support provided to patients by the healthcare team [START_REF]Diabetes Control and Complications Trial/Epidemiology of Diabetes Interventions and Complications (DCCT/EDIC) Study Research Group. Resource utilization and costs of care in the diabetes control and complications trial[END_REF]. Social support is also considered to be an important element of adherence in patients with type 2 diabetes. For example, regimen-specific measures of family support for glucose testing, medication-type, diet and exercise have been found to impact the potential for adherence [START_REF] Glasgow | Social environment and regimen adherence among type II diabetic patients[END_REF]. While the number of variables that impact adherence can seem daunting, many can probably be modified by making minor changes in clinical practice.

What clinicians can do

Physicians can institute several elements into their practice to improve adherence outcomes in patients with type 2 diabetes and other chronic diseases. These include improving communication, addressing costs, managing dosing issues, and enhancing education about the potential for AEs and the appropriate responses to perceived AEs.

Improving communications

The most commonly overlooked element to ensure adherence is appropriate evaluation of patient comprehension regarding the need for treatment and the value of the therapeutic regimen (49). In a study of 408 English-and Spanish-speaking adults with type 2 diabetes in public hospitals in San Francisco, > 50% of patients had inadequate or marginal health literacy [START_REF] Schillinger | Association of health literacy with diabetes outcomes[END_REF]. Inadequate health literacy was characterized as an inability to read common medical items such as prescription bottles, nutrition labels or appointment slips. Marginal health literacy pertained to difficulties with more complex materials such as educational brochures or informed consent documents [START_REF] Schillinger | Association of health literacy with diabetes outcomes[END_REF]. Taking extra time to ensure that patients have adequate recall and comprehension of the treatment regimens can alleviate this issue (49). A study based on new concepts (e.g. change in medication) found that type 2 diabetes patients of physicians who assessed comprehension and recall had significantly lower HbA 1c (odds ratio 8.96; 95% confidence interval, 1.1 to 74.9; p = 0.02) [START_REF] Schillinger | Closing the loop: physician communication with diabetic patients who have low health literacy[END_REF].

Clarifying the benefits of treatment can also improve adherence. This is particularly important in patients with seemingly asymptomatic diseases such as type 2 diabetes and those with increased CV risk factors. An unambiguous appreciation of the long-term implications of inadequate blood glucose control and the value of therapies to reduce CV risk is paramount. Helping patients recognize the subtle symptoms of chronic hyperglycemia such as fatigue, difficulty sleeping, nocturia, infections and missed school or workdays is also important (49). Adherence was significantly lower when patients did not understand that a treatment will improve their current or future health. Moreover, presentation of a new medicine or alteration in treatment, such as the introduction of insulin therapy, should be communicated in a manner that is portrayed to be a benefit rather than a punishment for perceived poor adherence. Thus, creating a 'trusted' relationship with the patient, fostering shared ownership of the patient's condition, setting realistic goals in a constructive manner, and creating a continuum of self-management are all accepted features of best practice that are believed to assist adherence [START_REF] Korff | Collaborative management of chronic illness[END_REF][START_REF] Ciechanowski | The patient-provider relationship: attachment theory and adherence to treatment in diabetes[END_REF].

Addressing costs

Cost of treatment is a significant concern to both patients and providers and for private health care this requires consideration on an individual basis with every patient. In a survey of adults with type 2 diabetes in the United States, 11% reported that they had limited their medications in the previous 12 months due to costs [START_REF] Piette | Problems paying out-of-pocket medication costs among older adults with diabetes[END_REF]. Signals that may indicate potential cost-related adherence issues include an income < $20,000/year and medication costs in excess of $50/month [START_REF] Kurlander | Cost-related non-adherence to medications among patients with diabetes and chronic pain: factors beyond finances[END_REF]. In the United States, patients without health insurance are particularly vulnerable to costs of care. In an attempt to provide benefits to patients, pharmacies of some corporate entities provide 30-day supplies of common generic drugs used to treat type 2 diabetes, cardiac risk factors, and comorbidities of type 2 diabetes for a nominal fee [START_REF]Wal-Mart $4 Prescription Program[END_REF]. These medications include metformin, statins, angiotensin-converting enzyme inhibitors and other antihypertensives, antidepressants, and common antibiotics.

Financial issues may alter adherence even in patients with adequate insurance.

Employers and insurers tier drugs and use copays to control drug expenditures [START_REF] Kessler | Page 24 of 33 International Journal of Clinical Practice International Journal of Clinical Practice with first-fill adherence rates for diabetic medications: a cohort study[END_REF].

Thus, when a drug is not on a formulary or a cheaper alternative is unavailable, the patient's out-of-pocket expenses can be significantly higher. It has been demonstrated that copayments of < $10 significantly improve first-prescription refills and that higher copayments predict poor adherence and potential treatment failure (58,59). Research also indicates that the effect of high copays on adherence is greatest during early phases of treatment and that higher copays are associated with early termination of medication usage [START_REF] Kessler | Page 24 of 33 International Journal of Clinical Practice International Journal of Clinical Practice with first-fill adherence rates for diabetic medications: a cohort study[END_REF]. Furthermore, a strategy of increasing copayments after the first few refills of a prescription does not promote persistence.

In the United Kingdom, the National Institute for Health and Clinical Excellence (NICE) and various regional formularies are becoming very cost conscious, and while this influences treatment choices, there is no evidence of its effect on adherence.

Managing dosing issues

Because of comorbidities and CV risk factors, many patients with type 2 diabetes are prescribed complex regimens with multiple medications. Due to the relationship between polypharmacy and poor adherence, it is preferable that treatment is accomplished with a regimen that includes the fewest medications and fewest doses per week (49). A major factor affecting adherence with antidiabetes agents is the daily dosing frequency [START_REF] Guillausseau | Impact of compliance with oral antihyperglycemic agents on health outcomes in type 2 diabetes: a focus on frequency of administration[END_REF][START_REF] Eisen | The effect of prescribed daily dose frequency on patient medication compliance[END_REF].

Adherence and persistence rates improve if goals can be achieved with the fewest number of tablets per day [START_REF] Dezii | Effect of once-daily and twice-daily dosing on adherence with prescribed glipizide oral therapy for type 2 diabetes[END_REF][START_REF] Donnan | Adherence to prescribed oral hypoglycemic medication in a population of patients with type 2 diabetes: a retrospective cohort study[END_REF][START_REF] Dailey | Patient compliance and persistence with antihyperglycemic drug regimens: evaluation of a population of type 2 diabetic patients[END_REF][START_REF] Paes | Impact of dosing frequency on patient compliance[END_REF]. Also, electronic monitoring indicates that once-daily dosing is accompanied by a decrease in the number of missed doses [START_REF] Kardas | The DIACOM study (effect of Dosing frequency of oral Antidiabetic agents on the COMpliance and biochemical control of type 2 diabetes)[END_REF].

One approach that integrates the need to treat multiple comorbidities while optimizing dosing is to select a fixed dose combination of the necessary medications that treat more than one pathophysiologic element [START_REF] Seufert | International Journal of Clinical Practice International Journal of Clinical Practice antidiabetic combination tablet: design, evaluation, and clinical benefits for patients with type 2 diabetes[END_REF](66)[START_REF] Bailey | Avandamet: combined MET-rosiglitazone treatment for insulin resistance in type 2 diabetes[END_REF]. When administered once daily, fixed dose combination therapy results in significantly better adherence than 2-tablet Bailey 15 regimens (68). This can be important when patients who have been managed with monotherapy require a second agent or for patients already receiving 2 separate agents.

Currently available fixed dose single tablet combinations include metformin plus a sulfonylurea, thiazolidinedione, dipeptidyl peptidase-4 inhibitor, or meglitinide. Other commercial formulations include a variety of combination doses of thiazolidinedione plus a sulfonylurea [START_REF] Bailey | Fixed-dose single tablet antidiabetic complications[END_REF]. In studies of patients with type 2 diabetes who previously received monotherapy with an oral glucose-lowering drug, adherence was greater when they were treated with a fixed dose combination versus a combination of separate tablets [START_REF] Melikian | Adherence to oral antidiabetic therapy in a managed care organization: a comparison of monotherapy, combination therapy, and fixed-dose combination therapy[END_REF][START_REF] Vanderpoel | Adherence to fixed-dose combination of rosiglitazone maleate/MET hydrochloride in subjects with type 2 diabetes mellitus: a retrospective database analysis[END_REF].

The improved adherence with fixed dose combination therapy is accompanied by better glycemic control than with separate tablet combinations at similar or greater doses [START_REF] Duckworth | Improvements in glycemic control in type 2 diabetes patients switched from sulfonylurea coadministered with MET to glyburide-MET tablets[END_REF][START_REF] Bailey | Rosiglitazone/MET fixed-dose combination compared with uptitrated MET alone in type 2 diabetes mellitus: a 24-week, multicenter, randomized, parallel-group study[END_REF].

Improving education about the potential for AEs

Since potential AEs can negatively impact adherence (32,74), it is necessary to frame this within the context of the benefits and risk of medications, and of supporting patient preference where possible (49). However, patients with type 2 diabetes may be willing to accept some AEs in exchange for improved glycemic control, but the potential for specific AEs is not weighted equally. In a survey of 407 patients with type 2 diabetes in the United States and the United Kingdom, glycemic control was selected as the most important property of the medication. Patients preferred to avoid medications that produced weight gain (75). The potential for weight gain of 9.0 kg decreased likely adherence by 30%, while the potential to increase the risk of heart attack by 1% decreased potential adherence by 16.5% (75). Mild and transient gastric distress did not significantly influence medication preference, but persistent stomach problems were identified as a negative medication feature. Interestingly, the potential for mild to moderate hypoglycemia did not affect potential adherence unless this was likely to occur more than twice a month (75).

Latinos and African Americans were significantly more likely than Caucasians to worry about the effect of potential AEs on quality of life and potential medication dependency [START_REF] Huang | Racial/ethnic differences in concerns about current and future medications among patients with type 2 diabetes[END_REF]. In addition, certain ethnic minorities may be candidates for fixed dose combination therapy as initial treatment since they are more likely to express a reluctance to add medications to their current regimen [START_REF] Huang | Racial/ethnic differences in concerns about current and future medications among patients with type 2 diabetes[END_REF]. The potential for drug interactions should also be considered as a factor that impacts adherence in patients with type 2 diabetes. In a study of 139 patients (average age 74 years) with polypharmacy discharged from the hospital and receiving home health care services, 38.8% of the subjects were considered to be at potential risk for drug interactions [START_REF] Ibrahim | Polypharmacy and possible drug-drug interactions among diabetic patients receiving home health care services[END_REF].

Conclusions

Type 2 diabetes is a chronic condition and typically requires varied and complex treatment programs. Substantial evidence indicates adherence and persistence with therapy are limiting factors in the drive to achieve and maintain desirable management goals. Factors that impede adherence include treatment choices, administration regimens, clinical inertia, communication deficits, and barriers of trust and belief. Therefore, it is important that these issues are addressed within the process of disease management.

  Patient adherence to medication requirements for therapy of type 2 diabetes

type 2

 2 diabetes. Information was obtained from a search of the PubMed and MEDLINE databases for English-language articles published from January 1985 to February 2010 with the following keywords used in the search: type 2 diabetes, cardiovascular risk factors, adherence, compliance, persistence, clinical inertia, polypharmacy, and The review examines the extent and clinical impact of poor adherence and potential strategies to address the problem.Factors affecting adherence to medication regimens for the management of type 2 diabetesOverviewCompliance, adherence, and persistence are terms commonly used to describe the patient's response to medical advice or instruction. Compliance is associated with conformity and describes willingness to follow a prescribed course of treatment.

Type 2

 2 diabetes has been described as a growing epidemic among children and adolescents throughout the world[START_REF] Vivian | Type 2 diabetes in children and adolescents--the next epidemic?[END_REF] and studies of minority youths in the United States

*

  Consult full prescribing information for individual agents for complete details on indications, contraindications, warnings and precautions. Most agents have rarely caused hypersensitivity reactions. a Efficacy requires presence of circulating insulin. b Efficacy requires presence of a functional β-cell mass. c Taken with meals prandial, less severe hypoglycemia. d Taken with meals rich in complex carbohydrate. e Subcutaneous injection. f Pramlintide is an amylin analogue. DPP-4 = dipeptidyl peptidase-4; GI = gastrointestinal; GLP-1, glucagon-like peptide-1; K+-ATP = Kir 6.2 inwardly rectifying potassium channel; PPARγ = peroxisome proliferator-activated receptor gamma; ↑ increase; ↓ decrease; ~ approximately; -no change. Developed from the American Diabetes Association, Nathan et al, Bailey et al. (4,5,8)

Table 2

 2

	Bailey 6
	viral infections (11.7%), arthritis (18.8%), gastrointestinal disorders (19.6%), and cancer
	(20.9%)
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

lists several well-recognized factors that can impact adherence

[START_REF] Odegard | Medication taking and diabetes: a systematic review of the literature[END_REF]

.

Studies to identify the factors affecting adherence have used questionnaires, patient diaries, pill counts at follow-up appointments, prescriptions filled, counting unopened spaces in blister packs, and electronic monitoring. While each of these methods has limitations, it is clear that poor adherence is a common and serious problem among patients with chronic diseases. In a meta-analysis of 569 studies of adherence across a range of medical disorders, the average nonadherence rate was estimated at 24.8%

[START_REF] Dimatteo | Variations in patients' adherence to medical recommendations quantitative review of 50 years of research[END_REF]

.

Nonadherence was highest for patients with sleep disorders (34.5%), diabetes (32.5%), and pulmonary disease (31.2%), and lowest for patients with human immunodeficiency

Page 8 of 33 International Journal of Clinical Practice International Journal of Clinical Practice

	Bailey 9
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

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	Bailey 12
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

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	Bailey 14
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

Page 14 of 33 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

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	International Journal of Clinical Practice	Page 28 of 33
	Bailey 17 Bailey 18
	F o F o F o F o 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 r r r r
	P P P P 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 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 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 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 l y l y l y l y
	International Journal of Clinical Practice

Table 1

 1 Drugs available for the management of patients with type 2 diabetes with selected properties highlighted*

			Decrease			Warnings
		Main mode	in	Body		and
	Type of agent	of action	HBA 1C	weight	Problems	precautions
	Oral
	Metformin	• Decrease	~1-2%		Lactic	Renal
		insulin			acidosis (rare)	impairment,
		resistance a			GI intolerance	any
		• Reduce hepatic				hypoxemic condition
		glucose		↓/-
		output
		• Increase
		peripheral
		glucose
		utilization
	Sulfonylureas	• Increase	~1-2%		Hypoglycemia Selection
		insulin				restricted by
		secretion b				severe liver
		• Stimulate pancreatic β-cells by		↑		or renal disease, or porphyria
		closure of
		K+-ATP
		channels
	Meglitinides	• Increase	~0.5-		Lesser risk of	Liver or
		insulin	1.5%		hypoglycemia	severe renal
		secretion b,c			(fewer and	disease
		• Usually administere d pre-meals:			less severe than sulfonylureas)
		rapid onset,
		short		↑/-
		duration of
		action
		• Stimulate
		pancreatic
		β-cells by
		closure of
		K+-ATP

Page 29 of 33 International Journal of Clinical Practice International Journal of Clinical Practice

		International Journal of Clinical Practice	Page 30 of 33
		channels
	Gliptins (DPP-4 inhibitors) Type of agent	• Increase insulin secretion b Main mode of action	~0.5-1.5% Decrease in HBA 1C	Body weight	Small risk of hypoglycemia (seldom Problems	Substantial Warnings and renal or liver disease precautions
	F o r P e e r R e v i e w severe), Parenteral • Inhibit DPP-4, allowing increased t½ for incretins, which potentiate nutrient-induced insulin secretion -mostly when GLP-1 ~0.5-Risk of • Increase used with other receptor 2.0% hypoglycemia insulin agonists e when used secretion b antidiabetic agents Thiazolidinediones (glitazones) • Increase insulin action a • Stimulate PPARγ • Increase adipogene-sis • Alter glucose-fatty and cycle ~1.0-1.5% ↑ Heart failure, edema, anemia, fractures α-glucosidase inhibitors • Slow carbohydrate ~0.5-1.0% --F with other • Resistant to antidiabetic degradation by DPP-4 ↓ agents, nausea o r • Potentiate nutrient-induced insulin P secretion e Pramlintide e,f • Decrease gastric ~0.3-0.6% Risk of hypoglycemia e emptying when used r R with insulin • Decrease glucagon, ↓ satiety e • Indicated e only as add-v on to insulin i therapy e Insulins e • Decrease hepatic Variable, as Hypoglycemia Substantial Not to be used in severe renal or Cardiac disease, fluid retention, severe liver or renal disease Intestinal diseases, severe gastrointestinal disease (e.g. gastroparesis) Discontinue if pancreatitis is suspected Contraindicated in gastroparesis or hypoglycemia unawareness lifestyle w glucose required adjustments and
	Bromocriptine	digestion d • Not established a production glucose peripheral • Increase	~0.5-0.8%	↑	-	O n Fibrotic reactions, n O	kidney disease Psychotic disorders glucose monitoring
	Colesevelam	• Uncertain, may uptake, storage, and utilization		~0.5-0.8%			l y hypotension Bile sequestrant Intestinal diseases l y
		increase • Decrease				-
		GLP-1 lipolysis
		secretion b
		International Journal of Clinical Practice

Page 31 of 33 International Journal of Clinical Practice International Journal of Clinical Practice

Table 2

 2 Barriers to medication adherence

	Patient factors	Treatment regimen factors
	Fear	Complexity of regimen
	Knowledge and skill	Frequency of dosing
	Self-reliance	Cost
	Health beliefs	Adverse events
		Interference with lifestyle
	Depression
	Lack of confidence in immediate or
	future benefits of the medication
	Remembering doses
	Adapted with permission from Odegard et al. (9)

Page 33 of 33 International Journal of Clinical Practice International Journal of Clinical Practice

Acknowledgements

The authors thank Amylin Pharmaceuticals, Inc. and Eli Lilly and Company for their support of this manuscript. The funders had no role in the literature review, data analysis, or conclusions. Amylin Pharmaceuticals reviewed the manuscript for technical accuracy only.

Author contributions

CJ Bailey and M Kodack contributed to drafting, editing, data analysis/interpretation, and critically reviewing this manuscript.


Jacques Lévy Véhel
email: jacques.levy-vehel@inria.fr

Franklin Mendivil
email: franklin.mendivil@acadiau.ca

Local complex dimensions of a fractal string

We investigate in this work a local version of the theory of fractal strings and associated geometric zeta functions. Such a generalization allows to describe the asymptotic behaviour of a "fractal" set in the neighborhood of any of its points. We give basic properties and several examples illustrating the possible range of situations concerning in particular the evolution of the local complex dimensions along the set and the relation between local and global zeta functions.

Background and Motivations

The theory of Fractal Strings has been developed over the past years by M. Lapidus and co-workers in a series of papers, including [START_REF] Lapidus | Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the WeylBerry conjecture[END_REF][START_REF] Lapidus | Spectral and fractal geometry: From the WeylBerry conjecture for the vibrations of fractal drums to the Riemann zetafunction[END_REF][START_REF] Lapidus | Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the WeylBerry conjecture[END_REF][START_REF] Lapidus | The Riemann hypothesis and inverse spectral problems for fractal strings[END_REF][START_REF] Lapidus | A Tube formula for the Koch snowflake curve with applications to complex dimensions[END_REF][START_REF] Lapidus | The Riemann zeta-function and the one-dimensional WeylBerry conjecture for fractal drums[END_REF][START_REF] Hambly | Random Fractal Strings: Their Zeta Functions, Complex Dimensions and Spectral Asymptotics[END_REF]. See also the book [START_REF] Lapidus | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] and the more recent works [START_REF] Lapidus | Fractal strings and multifractal zeta functions[END_REF][START_REF] Lapidus | Toward zeta functions and complex dimensions of multifractals, Complex Variables and Elliptic Equations (special issue dedicated to[END_REF][START_REF] Véhel | Multifractal and higher dimensional zeta functions[END_REF].

A fractal string L is simply a bounded open subset Ω of R. Ω may be written as a disjoint union of intervals I j = (a j , b j ), i.e. Ω = ∞ j=1 I j . For our purpose, no distinction needs to be made between the open set and the sequence of lengths of the intervals that define it. By abuse of notation, we will thus speak of the fractal string as L = { j } ∞ j=1 , where j are the lengths of the I j . Thus, a possible definition of a fractal string is: Definition 1 A fractal string L is an at-most countable, non-increasing sequence of lengths whose sum is finite.

To a fractal string is associated a complex function, the (geometric) zeta function of L: The interest of ζ L is that it provides rich information about the fractal structure of the boundary ∂Ω of Ω (or of L for our purpose), as is developed in the papers mentioned above. The most basic fact is that the critical exponent for ζ L is related to the upper Minkowski dimension (or upper box dimension) of ∂Ω 1 . Recall the following definition of the upper Minkowski dimension:

The one-sided volume of the tubular neighborhood of radius ε of ∂Ω is

V (ε) = vol 1 {x ∈ Ω| d(x, ∂Ω) < ε}.
The Minkowski dimension of ∂Ω is

D = D L := inf{α ≥ 0| V (ε) = O(ε 1-α ) as ε → 0 + }.
It is a remarkable fact that only the lengths of Ω (that is, of the associated fractal string L) affect its value. It is thus relevant to speak of the Minkowski dimension of ∂L. By abuse of notation, we will often speak of the dimension of L instead of ∂L.

If D is the Minkowski dimension and lim ε→0 + V (ε)ε D-1 exists L is said to be Minkowski measurable, in which case

D = inf{α ≥ 0| lim ε→0 + V (ε)ε α-1 < ∞}.
The Minkowski content of L is then defined as M(D, L) = lim ε→0 + V (ε)ε D-1 .

The following theorem describes the relation between the Minkowski dimension of a fractal string L and the sum of its lengths to the exponent σ ∈ R.

Theorem 1 [10]

D = inf{σ ∈ R| ∞ j=1 σ j < ∞}.
This sum is monotonically decreasing (as a function of σ), so there is a unique such D ∈ R. We see that D is the abscissa of convergence of ζ L (s), where s ∈ C.

A crucial assumption in all that follows is that the zeta function ζ L has a meromorphic extension to an open region of the complex plane which extends strictly to the left of line R(z) = D. By an abuse of notation, we use the same notation for the zeta function and its meromorphic extension.

Thus, the number D is a real pole of ζ L and it is the Minkowski dimension of L. This suggests to extend the notion of dimension to complex numbers by considering the poles of the meromorphic extension of ζ L :

Definition 3 The set of complex dimensions of a fractal string L contained in some region R ⊆ C is D(R) = {ω ∈ R| ζ L has a pole at ω}.
(See [START_REF] Lapidus | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] for a more general definition.) The relevance of considering complex dimensions is illustrated in particular by the following theorem :

Theorem 2 [START_REF] Lapidus | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] If a fractal string satisfies certain mild conditions, the following are equivalent:

1. L has only one complex dimension with real part equal to D L .

∂L is Minkowski measurable.

See [START_REF] Lapidus | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] for more details. More refined results exist, that allow for instance to characterize the asymptotic behavior of V (ε) (as ε tends to 0) in terms of the residues of ζ L . Example: A simple example will best illuminate and motivate these definitions. The example we choose is the standard middle-1/3 Cantor subset of [0, 1]. See reference [START_REF] Lapidus | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] pages 13-16 for a more complete discussion of this example. The lengths for this fractal string are 1/3, 1/9, 1/9, 1/27, 1/27, 1/27, 1/27, 1/81, . . . , 1/81, 1/243, . . .

with 2 n-1 copies of 3 -n for n ≥ 1. It is easy to see that ζ L (s) = 1/3 1 -2 • 3 -s , so that ζ L
In this particular case, we have

V ( ) = 2 •(2 n -1)+(2/3) n -2 = (2 ) 1-D (1/2) {-log 3 (2 )} + (3/2) {-log 3 (2 )} -2 ,
where n = [-log 3 (2 )] and x = [x] + {x} is the decomposition of a positive real number into its integer and fractional parts and D = ln(2)/ ln(3) is the dimension of the Cantor set. This means that

V ( ) = 1-D G( ) -2 (2)
where G is a non-constant multiplicatively periodic function of multiplicative period 2π/ ln(3) (see [START_REF] Lapidus | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], page 16). Notice the connection between the interpole spacing of ζ L (s) and the oscillations of V ( ) (that is, there is the quantity 2π/ ln(3) in formulas ( 1) and ( 2)).

The theory of fractal strings has been generalized in various directions: a random setting is considered in [START_REF] Hambly | Random Fractal Strings: Their Zeta Functions, Complex Dimensions and Spectral Asymptotics[END_REF]; higher-dimensional extensions are proposed in [START_REF] Lapidus | A Tube formula for the Koch snowflake curve with applications to complex dimensions[END_REF][START_REF] Véhel | Multifractal and higher dimensional zeta functions[END_REF]; and multifractal zeta functions are investigated in [START_REF] Lapidus | Fractal strings and multifractal zeta functions[END_REF][START_REF] Lapidus | Toward zeta functions and complex dimensions of multifractals, Complex Variables and Elliptic Equations (special issue dedicated to[END_REF][START_REF] Véhel | Multifractal and higher dimensional zeta functions[END_REF].

The aim of this article is to study a localized version of the geometric zeta function tailored to measure the dimensional properties of a fractal set in the neighbourhood of any of its points. By definition, the local behaviour of a self-similar fractal E is the same everywhere (for example, dim H (E ∩ B (x)) = dim H (E), as long as this intersection is non-empty, where B (x) is the open ball of radius centered at x). Thus if the local dimension differs at different points we know that the set cannot be self-similar and thus may exhibit different scaling behaviour at different points. It is natural to wonder whether localizing the formalism of fractal strings and geometric zeta functions may help in characterizing the fine pointwise behaviour of non-self-similar fractal sets, as for instance local Minkowski measurability. This is what we develop in the following. Note that this work is meant as a preliminary investigation: we content ourselves with defining local fractal strings in a meaningful way (section 2), proving basic properties (section 3), and giving a variety of examples that illustrate the range of possible behaviours (section 4).

Local fractal strings

In all the sequel, C ⊂ [0, 1] will denote a compact set, so that [0, 1] \ C = i l i is a union of open intervals. By abuse of language, we will occasionally refer to C as a string, instead of the bounded components of its complement. We assume that ζ (x, s) has a meromorphic extension to a neighborhood of {z : Re(z) ≥ D (x)} for all sufficiently small . We continue the tradition of abusing notation by denoting both the original zeta function and its meromorphic extension by ζ (x, s).

Definition 5

The -localization of the complex dimensions of the fractal string {l n } is the set of poles of this meromorphic extension of ζ (x, s) on the critical line. We denote this set of poles by D (x).

We will be interested in the set D(x) = lim D (x) and when this limit exists. D (x) is a collection of points on the line in C with real part D (x), we need to specify in which sense we take this limit.

Definition 6 Let K n be closed subsets of C with K n being on the line with real part s n and similarly K with real part s. We say that K n → K if s n → s and for every compact set A we have A ∩ K n → A ∩ K in the Hausdorff metric.

In particular, notice that this means that any bounded subset of poles converges uniformly.

Definition 7

The local complex dimensions at the point x ∈ C for the fractal string {l n } is the set D(x), when this limit exists.

Remark that we do not define

ζ(x, s) = lim →0 ζ (x, s) = inf >0 ζ (x, s)
since for all but countably many points in C (the endpoints of the l n ) we have that lim

→0 ζ (x, s) = 0
uniformly in s on compact subsets. This is why we first define D (x) and then take the limit of the complex dimensions. This is similar to the usual definition of the local dimension

dim loc (C, x) = lim →0 dim(C ∩ B (x))
and not

dim loc (C, x) = dim(lim →0 C ∩ B (x)) = 0.
In a similar way

D(x) = inf >0 inf{s > 0 : ζ (x, s) < ∞} and not inf{s > 0 : inf >0 ζ (x, s) < ∞} = 0.

Basic properties

We first remark that, for a fractal string {l n } and a point x, it is not usually the case that D(x) ⊂ D, that is the set of local complex dimensions is not, in general, a subset of the set of global complex dimensions (example 3 in the next section is one illustration of this).

Proposition 1 The quantity

D(x) = lim →0 D (x)
always exists and is the local Minkowski dimension (or local upper box dimension) of C at x.

Proof:

The sets {i : l i ∩ B (x)} are increasing as functions of , thus

D(x) = inf >0 D (x)
The fact that D(x) is the local Minkowski dimension is by definition.

Proposition 2 D(x) is an upper semi-continuous function of x ∈ C.

Proof: Take x ∈ C. Then for any > 0 and z so that |z -x| < we have

D(z) ≤ D -|x-z| (z) ≤ D (x)
and so sup

|z-x|< D(z) ≤ sup |z-x|< D -|x-z| (z) ≤ D (x) which means that lim sup z→x D(z) = inf >0 sup |z-x|< D(z) ≤ inf >0 sup |z-x|< D -|x-z| (z) ≤ inf >0 D (x) = D(x).
Theorem 3 in section 4 will provide a converse to proposition 2.

Proposition 3 There exists at least one point x ∈ C with D(x) equal to the global Minkowski dimension of the string.

Proof:

Let D denote the (global) Minkowski dimension. Then clearly we have D(x) ≤ D for all x. Suppose D(x) < D for all x. For any x i , there is some

i > 0 so that D i (x i ) < (D + D(x i ))/2.
As C is compact, we can cover it with finitely many B i (x i ). Let j be such that D j (x j ) is the maximum amongst the D i (x i ). Then as the Minkowski dimension is finitely stable we have that

D ≤ D j (x j ) < (D + D(x j ))/2 < D, a contradiction.
Our final result in this section gives a criterion for the existence of the local complex dimensions: Proposition 4 Let C be a string and suppose that x ∈ C is such that D(x) has a strict local maximum at x. Then the local complex dimensions D(x) of C at x exist.

Proof:

Since D(x) > D(y) for any y sufficiently close to x, this implies that for all small 0 < δ < , we have D (x) = D δ (x). This entails convergence as → 0, as this set is constant.

It is interesting to compare this result with example 8, where the local complex dimensions do not exist at a point because of a lack of convergence. In that example the point in question is a strict local minimum of D(x).

Examples of local fractal strings

Our first example is just to given an example of the simplest type of behaviour that can occur when we localize a fractal string.

Example 1

Take the intervals l n = (1/(n + 1), 1/n) for n ∈ N. For x = 1/n, we have

ζ (x, s) = [n(n -1)] -s + [(n + 1)n] -s
for all < 1 n(n+1) , so this zeta function has no poles and D(x) = 0, as expected. For x = 0, we have that

ζ (0, s) = n>1/ [n(n -1)] -s .
Clearly D = 1/2 in this case. Furthermore, the asymptotic decay rate of ζ (0, s) does not depend on > 0, so D (x) is constant in . For a detailed discussion of this fractal string, see [START_REF] Lapidus | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] Section 6.5.1.

Example 2 Self-similar local fractal string

Take a fractal string generated by an IFS

w i (x) = r i x + a i on [0, 1] with gaps of length g i (that is, [0, 1]\ ∪ i w i ([0, 1]) is a union of open intervals with lengths g i ).
The usual (non-local) zeta function ζ L (s) of this string satisfies the functional equation

ζ L (s) = i g s i + i r s i ζ L (s)
from which we obtain

ζ L (s) = i g s i 1 -i r s i .
Now, for the local fractal string, let x ∈ C, the Cantor set generated by this IFS, and > 0 be given. Then there is an n and some sequence i 1 , i 2 , . . . , i n with

w i1 • w i2 • • • • w in (C) ⊂ B (x) and thus ζ (x, s) = (r i1 r i2 • • • r in ) s ζ L (s) + other terms.
Thus, all the poles of ζ L (s) are also poles of ζ (x, s). We already observed that all the poles of ζ (x, s) are also poles of ζ L (s). Thus, D (x) is simply the set of poles of the non-local zeta function ζ L (s) and is thus constant with respect to . So, as we expect, the behaviour of the local fractal string for a self-similar fractal is constant everywhere and is just measuring the global behaviour of the string.

Example 3 Continuously varying D(x).

Now we construct C ⊂ [0, 1], a Cantor set, where the local Minkowski dimension (in fact, Hausdorff dimension) continuously varies from 0 to log(2)/ log(3) (the reason for the log(2)/ log(3) factor will become a little more clear when we revisit this example in example 5).

To do this, we first discuss a general method of constructing compact, totally disconnected and perfect subsets of [0, 1] (studied also in [START_REF] Cabrelli | On the Hausdorff h-Measure of Cantor Sets[END_REF]).

Let (a n ) be a positive summable sequence, assuming without loss of generality that n a n = 1. We construct a Cantor set C a with gap lengths a n by first removing an interval of length a 1 from the interior of [0, 1], leaving two closed intervals I 1 1 , I 1 2 . Having constructed the kth step, we have 2 k closed intervals I k i and we remove from the interior of I k i an open interval of length a 2 k +i-1 (see the figure below). Because of the way the intervals are removed, there is only one location possible for each "gap" (for example, the gap of length a 1 begins at the point

∞ k=1 2 k-1 -1 i=0 a 2 k +i ,
the sum over the "left half" of the binary tree). Notice that the order of the sequence (a n ) can make a big difference in the resulting Cantor set. Further, the sequence determines the Cantor set but there are many sequences which result in the same Cantor set. We mention that there are relations between the decay rate of (a n ) and the dimension of the associated Cantor set (see [START_REF] Besicovitch | On the complementary intervals of a linear closed set of zero Lebesgue measure[END_REF][START_REF] Cabrelli | On the Hausdorff h-Measure of Cantor Sets[END_REF]).

We will specify a summable sequence a n in a particular order and this will generate the desired Cantor set.

Let let Σ 0 = ∅ and Σ n = {0, 1} n , for n ≥ 1. We also let w(∅) = 1 and

w(σ) = i σ i 2 -i + 2 -n-1 for σ ∈ Σ n . Further, for each σ ∈ Σ n , we define l(σ) = 2 -(n log(3))/(w(σ) log(2)) = 3 -n/w(σ) .
Finally, we let

N (σ) = 2 n + n i=1 σ i 2 i (so N (∅) = 1, N (0) = 2, N (1) = 3, N (00) = 4, N (01) = 5 etc) and notice that N : ∪ n Σ n → N is a bijection.
Using this notation, we define a N (σ) = l(σ). By our construction, each x ∈ C is identified with a unique (infinite) binary sequence σ(x) = σ 1 (x)σ 2 (x)σ 3 (x) . . . . In fact, if we place the product topology on Σ ∞ = {0, 1} ∞ we get that this identification is a homeomorphism between C and Σ ∞ . We extend the function w (given above as w :

Σ n → [0, 1]) to w : Σ ∞ → [0, 1] by w(σ) = n σ n 2 n
(so this is just binary representation of numbers in [0, 1]). We will show that D(x) = log(3)/(log(2)w(σ(x))) for any x ∈ C.

Roughly speaking, we define l(σ) = 2 -n log(3)/(log(2)w(σ)) because we want the decay rate near some x ∈ C to be approximately n -log(3)/(log(2)w(σ(x))) . That is, if σ(x) = σ 1 σ 2 σ 3 . . . σ n . . ., then we see that for large n, w(σ(x)) ≈ w(σ 1 σ 2 . . . σ n ). Let σ n = σ 1 σ 2 . . . σ n be the truncation of σ to n places. Then

l(σ n+1 ) l(σ n ) = 2 -(n+1) log(3)/(log(2)w(σ n+1 )) 2 -n log(3)/(log(2)w(σ n )) ≈ 2 -log(3)/(log(2)w(σ))
so that this definition gives the correct local decay rate. A little more formally, for x ∈ C and > 0, define w min = inf{w(σ(z)) : z ∈ B (x)} and w max = sup{w(σ(z)) : z ∈ B (x)}.

and Ω = σ -1 (B (x)) ⊂ Σ ∞ . Then for any σ ∈ Ω we have

2 -n log(3)/(log(2)wmax) ≤ l(σ n ) ≤ 2 -n log(3)/(log(2)wmin) .
This means that the decay rate of the sequence associated with local Cantor set C ∩ B (x) is bounded between n -log(3)/(log(2)wmax) and n -log(3)/(log(2)wmin) . By the results in [START_REF] Cabrelli | On the Hausdorff h-Measure of Cantor Sets[END_REF] (in particular Proposition 3), this means that

log(3)/(log(2)w max ) ≤ dim H (C∩B (x)) ≤ dim B (C∩B (x)) = D (x) ≤ log(3)/(log(2)w min ).
Since both w and σ are continuous functions, as → 0, we have w min , w max → w(σ(x)) and thus D(x) = log(3)/(log(2)w(σ(x))), as desired. In fact, in this case the "local Hausdorff" dimension is also D(x), as our estimate bounds the Hausdorff dimension of B (x) ∩ C. Example 3 may be somewhat generalized to give a converse to proposition 2:

Theorem 3 Let f : [0, 1] → [0, 1) be an upper semi-continuous function. Let w : Σ → [0, 1] give the binary representation (as above). Then there exists a Cantor set C so that for each x ∈ C, the local Minkowski dimension D(x) at x is f (w(σ)), where σ ∈ Σ is the address of x.

Proof:

The construction is very much the same as in example 3. We only make a very slight modification to the lengths and then the estimates are a little different. For each σ ∈ Σ n , let the interval

I σ = [ n i=1 σ i 2 -i , 2 -n + n i=1 σ i 2 -i ] ⊂ [0, 1],
so that I σ is a dyadic interval of length 2 -n . Further, let f σ = sup{f (x) : x ∈ I σ }. Then if σ ∈ Σ with w(σ) = x, then because f is u.s.c. we know that f (x) = lim n f σ n (recall that σ n is the truncation of σ to the first n places). It is also useful to note that

f σ = max{f α : α ∈ Σ n+k , α n = σ} for any k ∈ N.
Referring to the construction in example 3, we now define l(σ) = 2 -n/fσ and a N (σ) = l(σ) and let C be the resulting Cantor set.

We now make use of the following result from [START_REF] Tricot | Douze définitions de la densité logarithmique[END_REF] to estimate the Minkowski (upper box) dimension D:

D = lim sup ln(n) -ln(a n ) = inf{s > 0 : n a s n < ∞}.
Take x ∈ C and σ ∈ Σ n be such that x ∈ I σ . Then for all k ∈ N, there is some

α k ∈ Σ n+k with f α k = f σ . We have N (α k ) = η2 n+k for some 1 ≤ η < 2 and a N (α k ) = 2 -(n+k)/fσ
. Thus for all k we have that

ln(N (α k )) -ln(a N (α k ) ) = ln(η)f σ (n + k) ln(2) + f σ > f σ
and thus for the lengths in I σ we have that lim sup ln(n) -ln(an) ≥ f σ . From the same argument as in the example we have f σ as an upper bound and thus D Iσ (x) = f σ . Letting n → ∞ we converge to x and to f (w(σ)), as announced.

Example 4

It is possible to modify the construction of the previous example so that all the lengths are powers of any given 0 < r < 1. To do this, simply define

l(σ) = r k(σ) with k(σ) = - n log(2) log(3) w(σ(x)) log(2) log(r) = - n log(3) w(σ(x)) log(r) . (3)
This will give the same values for D(x) for the local zeta functions; the complex dimensions need not be the same.

Example 5 Rearranging the middle third Cantor string Now we construct an example which has continuously varying D(x) (as example 3) but in which the lengths are a subset of the lengths of the standard middle third Cantor string. Take the string from example 3 but modify them as in example 4 by taking r = 1/3. This means that all the lengths are negative powers of 3. Call the lengths of the resulting string l n and the entire string L. We show that it is possible to rearrange the intervals in the middle third Cantor string and obtain the string L plus some "left-over" lengths. We arrange these "extra" lengths as in example 1, thus giving this part of the string trivial local dimension except at one point. Since we are primarily interested in the remainder of the string, we will ignore these "extra" lengths.

The middle third Cantor string has 2 k intervals of length 3 -k-1 . Let N k be the number of lengths in L equal to 3 -k . We estimate that N k ≤ 2 k . Thus there is a re-scaling of L, call it L whose lengths are a subset of the lengths of the middle third Cantor string, as desired.

To estimate N k , we define

S n k = {x ∈ [0, 1] : n k ≤ x < n k -1 },
noting that S n k is roughly the set of σ ∈ Σ n so that k(σ) = k (k(σ) as defined in (3)). The set S n k represents roughly 2 n |S n k | terms of size 3 -k , so we estimate that

N k = k-1 n=1 2 n n k(k -1) = k -2 k(k -1) 2 k + 2 k(k -1)
so that N k ≤ 2 k as desired.

Example 6 A string with different local and global oscillations

To begin this example, we first briefly discuss a self-similar string with a single gap (see Section 2.2.1 in [START_REF] Lapidus | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF]). For our purposes, we take the single gap (0, 3 -n ) and 2 n IFS maps of the form w i (x) = 3 -n x + i/3 n for i = 1, 2, . . . , 2 n . The zeta function of this self-similar string is

ζ n (s) = 3 -ns 1 -2 n 3 -ns
with complex poles at the points ln(2) ln(3) + 2πik n ln(3) , k ∈ Z.

We will call this fractal string the 3 -n string. For each k ∈ N, it contains 2 nk copies of the length 3 -n(k+1) . The length of this string is 3

-n /(1 -2 n 3 -n ) = 1/(3 n -2 n ).
Let L be a disjoint union of a 3 -n string for each n ∈ N, arranged in order on R and going from left to right (the total length of L is less than 3/2).

Consider the length 3 -j for some j. We see that this length occurs 2 j-1 times in the standard middle 1/3 Cantor string. If this length occurs in some 3 -n1 and 3 -n2 string with 2 k1 copies and 2 k2 copies, respectively, then we must have that k 1 = k 2 and both are strictly smaller than j. This implies that the total number of copies of the length 3 -j in L is no greater than 2 j . Thus, the set of lengths of L is a subset of the lengths of the fractal string generated by a standard middle 1/3 Cantor set of total length 1.

This means that we can rearrange the complementary intervals in the standard middle 1/3 Cantor set so that we obtain the above lengths plus some "extra" lengths. We assume that we do this.

As we mentioned, for any point in the 3 -n string, the local complex dimensions are of the form ln(2) ln(3) + 2πik n ln(3) .

Thus, locally we have oscillations with period 2π/(n ln(3)) while globally we have oscillations with period 2π/ ln(3).

Example 7 Globally Minkowski measurable, (almost) everywhere locally nonmeasurable.

We now given an example of a fractal string which globally has no oscillations (so no poles off of the real axis on the critical line) but everywhere locally it has complex poles, except at one point. For each n = 1, 2, . . ., let C n be a standard middle 1/3 Cantor set which has been scaled to have total length n -4/3 . We make our string by placing C 1 starting at the point 0 ∈ R, then leaving a gap of length 1, then placing C 2 , then leaving a gap of length 2 -4/3 , then placing C 3 , etc. We must place one more limit point at the far "end" of the set, call this point ω. The total length of this string is 2 n n -4/3 < 8. The string C n has 2 k copies of the length n -4/3 3 -k-1 , so the zeta function for C is given by

ζ(s) = ∞ n=1 n -4s/3 3 -s ∞ k=0 2 n 3 -ns + ∞ n=1 n -4s/3 = ∞ n=1 n -4s/3 3 -s 1 -2 • 3 -s + ∞ n=1 n -4s/3 = 3 -s 1 -2 • 3 -s ∞ n=1 n -4s/3 + ∞ n=1 n -4s/3 = ζ 1 (s)ζ 2 (s) + ∞ n=1 n -4s/3 = ζ 1 (s)ζ 2 (s) + ζ 2 (s),
where ζ 1 is the zeta function for the standard Cantor set and ζ 2 is the zeta function for the string with lengths n -4/3 . The critical exponent for ζ 1 is ln(2)/ ln(3) ≈ 0.63 while the critical exponent for ζ 2 is 3/4. Thus, the critical exponent for ζ is 3/4, where there are no poles with complex dimensions. Thus, globally the string is Minkowski measurable and of dimension 3/4. However, for any x ∈ C, x = ω, and small enough > 0, there is some n so that B (x) ∩ C ⊂ C n and thus for any such the local zeta function is of the form

ζ (x, s) = φ(s) 1 -2 • 3 -s ,
where φ is holomorphic. Thus

D (x) = { ln(2) ln(3) + 2πik ln(3) : k ∈ N}.
Since this is constant for all small , this is also the set of local complex dimensions at any such point. Thus, locally the string is not Minkowski measurable at any ω = x ∈ C. However, at ω we see that

ζ (ω, s) = 3 -s 1 -2 • 3 -s n≥N n -4s/3 + n≥N n -4s/3
for some N . Thus the local complex dimensions at ω are the same as for the entire string.

The construction in example 7 is interesting to note. If we have two sequences of lengths a n and b n with associated zeta functions ζ a and ζ b , then the string with lengths a n b m has zeta function ζ a (s)ζ b (s), so the zeta function is "multiplicative" in this sense. It is clear that it is "additive" as well.

Constructing a set with the converse property, i.e. globally non-measurable but everywhere locally measurable remains an open question.

Example 8 Nonconvergence of D (x), so a point with no local complex dimensions.

Our final example is one in which there is a point where the local complex dimensions are not defined because of a lack of convergence. Let r k = 1/3 when k is odd and r k = 1/4 when k is even. Furthermore, let n k ∈ N grow sufficiently fast so that the solutions s k > 0 to r s k k + r s k n k k = 1 are strictly decreasing. Notice that it is always possible to set n 2k+2 = n 2k+1 but it is necessary to have n 2k+1 > n 2k . Let C k be a Cantor set with two contractions equal to r k and r n k k and scaled so that |C k | = 2 -k . Notice that the Hausdorff and box dimension of C k is equal to s k . We construct our string C by stacking the C k end-to-end from left-to-right and starting from C 1 and then adding a limit point ω. Then |C| = 1, the Minkowski dimension of C is ln(2)/ ln(3) and C is not Minkowski measurable because it has non-real complex poles on the critical line.

We claim that the local complex dimensions do not exist at ω. For all > 0, the critical exponent for ζ (ω, s) is equal to s k where k = min{n : B (ω) ∩ C n = ∅}. Thus the complex dimensions of ζ (ω, s) lie entirely on the line with real part s k and are of the form

s k + 2πim -ln(r k ) , m ∈ N.
Now we just note that even though s k → 0, the separation between the complex poles oscilates between 2π/ ln(3) and 2π/ ln(4), so does not converge.

Definition 2

 2 The geometric zeta function of a fractal string L is ζ L (s) = s ∈ C such that the series converges.

  has a meromorphic extension to all of C. Thus, the poles of ζ L (s) 0, ±1, ±2, ±3, . . . .

Definition 4

 4 Let x ∈ C and > 0 be given. We define ζ (x, s) = n:ln∩B (x) =∅ |l n | s (where |l n | is the length of the interval l n ) as the -localization of the zeta function around the point x ∈ C. We denote by D (x) = inf{s > 0 : ζ (x, s) < ∞}, the exponent of convergence of ζ (x, s).

.

Since we will only deal with the upper Minkowski dimension in this work, we will often leave out the word "upper" in the name.

Acknowledgments

This work was while FM was visiting JLV at INRIA. FM thanks JLV and INRIA for their hospitality. FM gratefully acknowledges support for this work from the Natural Sciences and Engineering Research Council of Canada (NSERC).