Mathematical Model for Antibiotic Distribution and Eradication of Bacteria causing Endocarditis. Marieke Kool

Transcription

1 Mathematical Model for Antibiotic Distribution and Eradication of Bacteria causing Endocarditis Marieke Kool

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3 Mathematical Model for Antibiotic Distribution and Eradication of Bacteria causing Endocarditis Marieke Kool

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5 Mathematical Model for Antibiotic Distribution and Eradication of Bacteria causing Endocarditis Marieke Kool M. Sc. Thesis Enschede, November 6, 2006 University of Twente, the Netherlands Department of Applied Mathematics Group of Applied Analysis and Mathematical Physics Supervising committee: Prof. dr. S.A. van Gils (UT) Dr. C. Neef (azm) Dr. O. Bokhove (UT) Ir. A.W.J. van der Meer (UT) Cover figure: Schematic reproduction of a heart that is injured by endocarditis ( ).

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7 Abstract This report describes a mathematical model that is defined to predict the duration of treatment of a patient suffering from endocarditis. The efficacy of the antibiotics, or the killing of bacteria, is the aim of treatment. By a mathematical model the effect of antibiotics on bacteria is investigated. A result is that continuous administration of antibiotics is most effective for eradication of a bacteria population. Endocarditis is caused by bacteria that grow at the heart valve and form a vegetation. The vegetation may damage or destroy the heart valve that is of crucial importance to the functioning of the heart. The structure of a vegetation is a mix of bacteria, fibrin, other proteins and host defence factors that are interwoven and preserving each other. Antibiotics are required to sterilize the vegetation in time. Nowadays a patient with endocarditis is treated with antibiotics for approximately 6 weeks. In practice it proves that long-term, high-dose therapy is required to eradicate the bacteria. The mathematical model that is used is a combination of a growth-killing model and a diffusion model. The combined model describes the eradication of bacteria as a result of the administered antibiotics. The bacteria are clotted together in a vegetation and the growth-killing equation describes the growth and decrease of bacteria in that vegetation. The diffusion equation describes the penetration of antibiotics into the bacteria vegetation and the antibiotic concentration inside the vegetation. Factors that are relevant for this mathematical model are the antibiotic concentration, the bacteria density, the killing rate and growth rate of bacteria, the diffusion coefficient and the rate of destruction of antibiotics. The combined growth-killing model and diffusion model is solved by a numerical finite difference method. A general diffusion equation is used to calculate the diffusion in a one dimensional vegetation. The model is implemented for vegetations with different shapes, because the shape of a bacteria vegetation can vary widely. The results for the antibiotic concentration are similar for the different shapes, so it is concluded that the shape is not important for the model. The one dimensional diffusion equation can be used as a starting point for any shape. The bacteria density inside a vegetation is calculated by the growthkilling model that depends on the antibiotic concentration. Several variations of our combined model are investigated to fit the model to a realistic treatment period. First of all the administration of antibiotics is continuous or intermittent. This mode of administration can be determined by a doctor. Numerical results show that continuous administration is preferable to intermittent administration, because the concentration reaches a higher level and consequently bacteria are killed faster. In practice it is often not justified to treat a patient with a high dose of antibiotics continuously as this may be toxic. Another expansion to our model is the addition of a rate of destruction, but antibiotics are often not degraded to a significant degree. The initial bacteria density is assumed to depend on the position inside the vegetation. The initial bacteria density hardly influences the results if the diffusion and antibiotic concentration do not depend on the bacteria density. However, it is realistic to assume that the diffusion of antibiotics does depend on the bacteria density, because the diffusion of antibiotics is harder if bacteria are more packed. Bacteria at the core are more packed than bacteria at the boundary of the vegetation. Initially the growth factor for bacteria is assumed constant. A variation to our model is a growth factor that depends on the position inside the vegetation, because bacteria in the core have a smaller growth rate than bacteria at the boundary.

8 iv For now the combination of the growth-killing model and the diffusion model can not be used to predict the period of treatment that is required to cure a patient with endocarditis. The combined model with an initial bacteria density that depends on the position inside the vegetation and a diffusion coefficient that depends on the bacteria density is used to draw conclusions. The results for this model indicate that continuous administration sterilizes a vegetation after one time period of treatment, say six weeks, if the maximum diffusion coefficient is cm 2 /hr. We recommend to look at more expansions to improve the model. A sensitivity analysis of parameters could indicate the parameters that have great influence. Besides, further clinical investigation of the action of antibiotics on a vegetation is recommended to improve modeling.

9 Contents Abstract iii Preface 1 1 Introduction 3 2 Problem description Objective Phrasing of research questions Definitions Assumptions Characteristics of endocarditis Causes, diagnosis, susceptibility Heart The blood flow Heart valves Endothelium Shape and structure of a vegetation Dynamics of vegetation Antibiotics Current curing methods and administration Diffusion of antibiotics Etching Susceptibility of bacteria Formulation of parameters Factors influencing treatment Relevant factors for modeling Parameter values Modeling, mathematical theory Growth-killing model Diffusion model Etch model Combined model Mode of administration Concluding remarks Solution method The system Scaling Numerical solution Effect for bacteria Intermittent administration

10 vi Contents 7.6 Concluding remarks Different shapes of a vegetation Spherical vegetation Layer Cylindrical vegetation Hemisphere Spherical coordinates Cartesian coordinates Bell-shaped vegetation Concluding remarks Variations of model Rate of destruction Initial bacteria density Diffusion coefficient depending on bacteria density Linear dependence D(B) Asymptotic dependence D(B) Growth factor Concluding remarks Results Conclusion and Recommendations Conclusion Recommendations Appendix 75 A Derivation of diffusion equation 75 B Alternative numerical methods 77 B.1 Implicit Method B.2 Crank-Nicolson C Condition for ν 78 C.1 Convergence C.2 Condition for t in case of hemisphere C.3 Diffusion coefficient depending on bacteria D Boundary condition 82 E Thomas algorithm 83 F Etch model 84

11 Preface The study described in this report is a research question from Dr. C. Neef from the Academic Hospital Maastricht (azm). The work has been conducted at the University of Twente, as the final project of the study Applied Mathematics. The starting point of this work was my interest in a medical-orientated assignment. When endocarditis entered the picture I decided to choose this subject for my final project. This disease can have serious consequences and without proper medical treatment surgery is often required and new complications may rise. Nowadays (clinical) experience results in more effective treatment. However, theoretically there is not much research performed yet; for this study the challenge was to compose a mathematical model that predicts the effects of an effective method of treatment. I would like to take this opportunity to thank some of the people who helped me throughout the last eight months of my study. I would like to thank my supervisor Stephan van Gils for the discussions we had, and for being concerned with the research and my plans for the future. I would like to thank Kees Neef for giving me the opportunity to do this research and keeping me enthusiastic about the subject. He was very patient and helpful during the visits I payed. Further acknowledgments I would like to give to Onno Bokhove and Adri van der Meer who commented my report and provided useful advice. Apart from my committee I would like to thank Chris Klaij for his patience when explaining L A TEX. Chris, thank you for your support and comments on my report! Also thanks to the people from the AAMP and NACM group and to my roommates, Renske, Olger, and Sanne, with whom I spend many lunch-breaks and coffee-breaks. I would like to acknowledge the following individuals from the Academic Hospital Maastricht (azm) for the conversations we had: Frank van Tiel, E. Cheriex, Ellen Stobberingh, Frank Stassen. Thank you for the explanation and suggestions. I am grateful to my parents and sisters, for their support during my study. Last but not least I would like to thank Wiebe, for standing by at all times and for his suggestions and encouragement!

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13 1 Introduction This report presents a mathematical model for predicting the required period of treatment for endocarditis. Without proper treatment endocarditis is a deadly disease. A patient can be cured by antibiotic treatment. In general endocarditis is a serious infection of the heart valves or the inner lining of the heart. The infection is caused by bacteria that multiply and form vegetation on the heart valve. This growing vegetation affects the functioning of the heart. Endocarditis can damage or even destroy the heart valves and result in heart failure [12]. Once the heart valves are destroyed or the state has become severe, surgery and replacement of the heart valve is often required. Figure 1.1 shows a simulation of endocarditis on the aortic heart valves. This example illustrates the effect that endocarditis may have. However, if antibiotics are able to kill the bacteria and cure the infection, surgery is not needed. Figure 1.1: Aortic endocarditis 1 Endocarditis is subject to a lot of clinical research. Note that there are numerous kinds of antibiotics, or combination of antibiotics, to treat endocarditis. New kinds of antibiotics are required because bacteria become resistant. By experimental endocarditis one tries to determine the effect of various kinds of antibiotics to a vegetation of a certain kind of bacteria. This study looks at aspects from a microbiological and a mathematical point of view. Microbiological means that we look at the bacteria causing endocarditis and the effect of antibiotics on those bacteria. Mathematical means that we use an abstract model that uses mathematical language to describe the behavior of endocarditis. The treatment of endocarditis is modeled on a, simplified, micro-scale. By investigation of the effect, or diffusion, of antibiotics the intensity and period of treatment is determined. The research problem concerns a determination of the efficacy of antibiotics when killing a bacteria population. The objective of this study is to predict the duration of treatment of a patient suffering from endocarditis by means of a mathematical model. The structure of this report is as follows. In the first section, Section 2, the problem description is formulated for this study. The research questions also indicate the structure of the report. Section 3 describes the characteristics of the disease. Section 4 is about antibiotics and its functioning. Then, in Section 5, the parameters that are relevant for this study are formulated. Section 6 describes possible models that could be used. In Section 7 the solution method for a combined model is given to calculate the antibiotic concentration and bacteria density inside a vegetation. Section 8 analyzes the process for different shapes. Section 1 cardiologie.web-log.nl/.../2005/05/index.html,

14 4 1 Introduction 9 deals with variations on the model. The report finishes with the results, conclusion and recommendations.

15 2 Problem description In this section the problem description is formulated. The first paragraph briefly describes the objective of the study. The second paragraph deals with the research questions that are posed to meet the objective. In the last two paragraphs definitions and assumptions are given that are used in this report. The treatment of endocarditis is discussed in many articles [2, 5, 11, 19, 27]. Clinical research and experiments (experimental endocarditis) lead to useful knowledge for the treatment of patients. Until now less theoretical research is done. This could contribute to improve the efficacy of administration of antibiotics. Therefore this study is initiated. The following objective shows that research is done within a theoretical framework. 2.1 Objective The objective of this study is to predict the duration of treatment of a patient suffering from endocarditis by means of a mathematical model. The efficacy of the antibiotics, or the killing of bacteria, is the aim of treatment. Therefore the kind of treatment and the way of modeling should focus on an efficient way of killing the bacteria. 2.2 Phrasing of research questions The phrasing of research questions will contribute to the development of a mathematical model for the treatment of endocarditis. This research focuses on the modeling of efficacy of antibiotics. Therefore existing theories and models for the functioning of antibiotics in a vegetation, located on the valvular leaflets, are analyzed and possibly improved. The Central questions are indicated by number 1 to 5. Several steps should be taken to conduct the research. Each central question is split into several subquestions. The subquestions are formulated to treat in more detail what is to be examined. 1. What are the characteristics of native valve endocarditis? 1.1. What are causes and symptoms of endocarditis and who is susceptible to endocarditis? 1.2. What is the function and structure of the heart? 1.3. What is the shape and structure of a vegetation on the valvular leaflets? 1.4. How does a bacteria vegetation change and grow over time? 2. How do antibiotics act on a bacteria vegetation on a heart valve? 2.1. What is the current curing mechanism and treatment? 2.2. Does antibiotic diffuse through a bacteria vegetation? If so, to what extent? 2.3. Does antibiotic affect the boundary of a vegetation? 2.4. What factors could obstruct the eradication of bacteria? 3. What factors are relevant for modeling of the action of antibiotics on a vegetation and the treatment of endocarditis? 3.1. What kind of bacteria and antibiotics are to be studied?

16 6 2 Problem description 3.2. Is the situation (assumed to be) homogeneous and stationary? 3.3. What parameters or factors (derived from central questions 1 and 2) should be taken into account and are required to generate a realistic mathematical model? 4. What mathematical models are available and usable to construct an antibiotic model for endocarditis? 4.1. How could a model for growth and killing of bacteria be described if bacteria are not fixed in a vegetation, but able to move freely (in a petri-dish) and exposed to antibiotics? 4.2. Is a diffusion model a good representation of the functioning of antibiotics on a vegetation? What are the results of a diffusion model for the bacteria population? 4.3. How could the antibiotic attack on bacteria at the boundary of the vegetation be described? 4.4. What is the most realistic model? 4.5. How could the mode of administration, which is influenced by a doctor, be modeled? 5. How could we expand the models to a representative model for endocarditis? 5.1. What is a solution method for the chosen model? 5.2. What is the influence of the shape of the vegetation for our model? 5.3. How can the chosen model be improved or expanded? 2.3 Definitions In this report several definitions are used concerning endocarditis. In this paragraph the definitions are introduced which we define next. Endocarditis can occur in various forms: Subacute bacterial endocarditis (SBE) Acute bacterial endocarditis (ABE) Native valve endocarditis (NVE) Prosthetic valve endocarditis (PVE) Non-bacterial thrombotic endocarditis (NBTE) SBE usually has a more indolent course, evolving over several weeks or months. Alphahemolytic streptococci, enterococci, or haemophilus bacteria, usually in the setting of underlying structural valve disease, typically are the causative agents of this type of endocarditis. The progress of ABE is more hectic and evolves over days to weeks. ABE is most often caused by primary pathogens such as staphylococci, streptococci, pneumococci, or gonococci. NVE is an infection engrafted upon a heart valve without an underlying structural valve disease. PVE occurs in an early or late stadium. Early prosthetic valve endocarditis occurs within 60 days of valve implantation. Staphylococci, gram-negative bacilli, and Candida species are the

17 2 Problem description 7 common infecting organisms. Late prosthetic valve endocarditis is the infection of an artificial valve 60 days or more after valve implantation. Alpha-hemolytic streptococci, enterococci, and staphylococci are the common causative organisms. NBTE is non-infective, because the injuries, or lesions, inside the heart are thrombotic rather than inflammatory. NBTE usually results from trauma to the endothelial surface of the heart. However, transient bacteraemia may lead to seeding of lesions with adherent bacteria, and infective endocarditis may develop [16, 21]. Bacteria are microscopic, uni-cellar organisms. Bacteria are a natural component of the human body and important to human health. However, they are also the causative agent of many infectious diseases. Bacteraemia is a condition in which bacteria, that do not belong there, are present in the bloodstream. It may occur after minor surgery or infection and may be dangerous for people with a weakened immune system or abnormal heart valves. It is the principal means by which local infections spread to distant organs (referred to as hematogenous spread). Bacteraemia typically passes rather quickly [3], due to a vigorous immune system response when bacteria are detected in the blood. Under certain circumstances the bacteria are able to attach to a heart valve and form a vegetation. A bacterial vegetation is the crucial factor for endocarditis, because this may damage the heart valve. Antibiotics is a product derived from a living organism. Antibiotics can kill or prevent multiplication of bacteria in different ways. Many antibiotic species act by prevention of cell wall synthesis, a bactericidal action. Other antibiotics kill the bacteria by destruction of the ribosomes of a cell. The half-life of an antibiotic is the time it takes for one half of the original dose of the medication to leave the body. A model is a simplified description of reality. To develop a model the right assumptions should be considered. A model is useful if it explains or predicts the action and efficacy of antibiotics on a vegetation. For microbiologists it is common to use the term Colony Forming Unit (CFU) for the number of viable micro organisms (per gram or per ml). Say that 1 CFU is 1 bacteria cell. Effectiveness is defined by the time it takes to eradicate the entire bacteria population. If 10 7 CFU are present then the population is killed when all bacteria are eradicated, or the vegetation is sterile. 2.4 Assumptions Modeling means one has to define a framework for the research. Applying mathematics to model the treatment of endocarditis requires several assumptions because the model can only be a simplified description of a disease that is as complex as endocarditis. This paragraph will indicate boundaries for the research. Assumptions are a starting point to develop the mathematical model that describes the treatment of endocarditis with antibiotics. For this study we look at Subacute Bacterial Endocarditis (SBE) and mainly focus on a microbiological approach of endocarditis. Endocarditis will have a different clinical picture and different effect for every individual. A vegetation can grow in different ways. A general, or simplified picture is necessary to make a mathematical model. In this study individual aspects are not taken into account; a general model that is applicable to an ideal situation is described. As stated by a microbiologist (Dr. F.H. van Tiel, personal communication, , azm), we often think for bacteria, assumptions are made that might be correct or

18 8 2 Problem description incorrect. However, without assumptions it is not possible to make a model. If the model leads to a realistic outcome, the assumptions have no incorrect consequences. Here and in the following two sections, a picture is presented of endocarditis. The effects of only two bacteria are taken into account, those of Streptococcus and of Staphylococcus aureus. It is assumed for this study that the bacteria occur on a native heart valve. A distinction between a native heart valve and a prosthetic heart valve is that the last one could release antibiotics. The antibiotics are assumed to be homogeneously distributed through the blood. The blood flow is not taken into account, so a stationary situation is assumed. Antibiotics in the blood are either free molecules or bounded to proteins. Assumed is that the free fraction of antibiotics, the molecules that are unbounded, is able to penetrate into the bacteria and this is the fraction we are looking at and meant when we talk about the antibiotic concentration. The antibiotic molecule is small compared to bacteria cells. Blood itself can not enter the vegetation, while antibiotics can. The size of a drug molecule is assumed to be smaller than the spaces between bacteria, so that diffusion is not hindered [24]. The eradication of a bacteria population by antibiotics occurs at the boundary of the vegetation and from the inside when antibiotics penetrate into a vegetation. The effect of treatment is defined as the period it takes to sterilize a vegetation. We assume that sterilization requires an initial bioburden of 1 to decrease to 10 6 [23, p. 544]. Beyond this lower boundary of 10 6 bacteria are not measurable or able to multiply. For example microorganims are killed if 10 6 is reached, which is mathematically equal to a decay from 1 to More assumptions are formulated in Section 5 where a selection of relevant parameters is made.

19 3 Characteristics of endocarditis This section deals with the subquestions of Central question 1 posed in Paragraph 2.2. The characteristics of native valve endocarditis are described and a picture is given of the bacterial vegetation. The characteristics of the disease, its causes, characteristics and people that are susceptible to endocarditis are discussed in the first place. In this section the structure and functioning of the heart is described as the heart is the vital organ that is subject to endocarditis. The shape and structure of a vegetation is outlined. The dynamics of a growing vegetation is outlined in the last paragraph. 3.1 Causes, diagnosis, susceptibility Bacterial endocarditis occurs when bacteria in the bloodstream (bacteraemia) lodge on abnormal heart valves or other damaged heart tissue. Certain bacteria normally live on parts of the body, such as the mouth, the upper respiratory system, the intestinal and urinary tracts, and the skin. Normally the bacteria are not found in the blood but may enter the body by various means such as intravenous drug use, cardiac catheterization and other invasive procedures, cuts, bruises and minor surgical procedures, dental procedures etc. This may cause brief bacteraemia. Bacteraemia is common after many invasive procedures, but in susceptible individuals certainly may cause endocarditis. Although any organism may produce infective endocarditis, Streptococci and Staphylococci appear to be responsible for most of the cases. Gram-positive cocci and bacilli, gram-negative cocci and bacilli, yeasts and fungi and other etiologic organisms can cause endocarditis as well. See Weinstein [28, Table 4.1], Hurst [16] and Durak [12]. Endocarditis is not clearly identifiable. A number of symptoms is mentioned in the literature, but it is not clear in one glance whether someone has endocarditis. The diagnosis of infective endocarditis is straightforward in patients with classic Oslerian manifestations, such as bacteraemia or active valvulitis. In other patients, however, the classic signs of endocarditis may be few or absent. This may occur during acute courses of infective endocarditis which may evolve too quickly for the development of immunologic vascular phenomena, which are more characteristic for subacute endocarditis. Infective endocarditis can be proved at surgery or autopsy, i.e. open heart surgery or autopsy (pathologically definite) or well-defined microbiological criteria (high-grade bacteraemia or fungemia) plus echo-cardiographic data (clinically definite). Duke s criteria, which are not further investigated here, are defined to assess whether someone has endocarditis [8]. Symptoms are fatigue, weakness, fever, chills, night sweats, weight loss, muscle aches and pains, heart murmur, shortness of breath with activity, red painless skin spots located on the palms and soles (called Janeway lesions), red painful nodes in the pads of the fingers and toes (called Osler s nodes), joint pain or abnormal urine color. Some individuals are more at risk for endocarditis than others. Endocarditis rarely occurs in people with normal hearts. However, if one has certain preexisting heart conditions, the risk for endocarditis increases when a bacteraemia occurs. Some of these conditions include having an artificial heart valve, a history of previous endocarditis, damages of the heart valve (scarred) by conditions such as rheumatic fever, or congenital heart (valve) defects. Some

20 10 3 Characteristics of endocarditis congenital heart defects can be successfully repaired surgically. After this, one is no longer at increased risk for endocarditis. The procedures that carry greatest risk for endocarditis include most dental procedures likely to cause significant bleeding. Antibiotics may be recommended for other types of procedures if the tissue is infected. 3.2 Heart The heart is a vital organ for the human body. A patient with endocarditis has a damaged heart (valve) and a proper functioning is hindered. The heart is the organ that pumps blood, and antibiotics, through the body. Antibiotics are distributed to the location of bacteria by the blood. Blood passes the vegetation with bacteria that is located inside the heart. Figure 3.1 (left) shows a simplified image of a heart The blood flow The heart is divided into two sides. Each side is divided again into two chambers, the atrium (upper chamber) and ventricle (lower chamber). Blood vessels (veins) carry blood to the heart from the rest of the body. This blood carries carbon dioxide and cellular waste products. The blood goes into the right atrium and then to the right ventricle, where it is then pumped to the lungs to dispose of wastes and receive a fresh oxygen supply. From the lungs, the blood returns to the heart. It returns to the left atrium and then to the left ventricle. The blood is then pumped out of the heart by the left ventricle into the aorta. The left ventricle is the chamber of the heart that is responsible for pumping blood to all parts of the body. The aorta sends this blood to small arteries, which carry the oxygen-rich blood to the rest of the body [16]. See figure 3.1 (left) where the blood flow is indicated by arrows. Figure 3.1: Left: normal heart structure. Right: vegetation at heart valve, 1. bacteria at boundary of vegetation, 2. bacteria inside vegetation

21 3 Characteristics of endocarditis Heart valves There are four heart valves, the mitral valve, the aortic valve, the pulmonary valve and the tricuspid valve. The valves are all depicted in figure 3.1. They are all one-way valves to keep blood moving through the various chambers of the heart. The mitral valve guards the opening between the atrium and the ventricle in the left side of the heart. It allows blood to flow forward from the atrium to the ventricle, and prevents blood from flowing backwards. The mitral valve has tiny cords attached to the walls of the ventricles. This helps support the valve s small flaps or leaflets. The aortic valve, also called a semi-lunar valve, has three segments that prevent the return of the blood from the aorta to the left ventricle. Valves on the heart s left side need to withstand much pressure. Normally the passive soft tissue structure of the valves is elastic and strong enough to withstand this pressure. Sometimes they wear out and leak or become thick and stiff. The pulmonary valve is located at the junction of the pulmonary artery and the right ventricle. When the right ventricle contracts, the pulmonary valve opens, forcing the blood into the artery which leads to the lungs. It is also a semi-lunar valve. When the chamber relaxes, this valve closes and prevents a back-flow of the blood. The tricuspid valve is located between the upper and lower chamber in the right side of the heart. Its position corresponds to the mitral valve in the left side of the heart [16] Endothelium The inner lining of the heart consists of the endothelium of capillaries. This is an inert single layer of cells that is an important source of substances that cause contraction or relaxation of the vascular smooth muscle. The endothelium is a passive filter to permit passage of water and small molecules across the blood vessel wall, and to retain blood cells and large molecules (proteins) within the vascular compartment. One of the substances of the endothelium is prostacyclin. The primary function of prostacyclin is to inhibit platelet adherence to the endothelium and platelet aggregation, and thus prevent intravascular clot formation [4]. 3.3 Shape and structure of a vegetation Modeling requires a realistic representation of the vegetation. The vegetation is a biofilm of bacteria and hosts components on the valve. In this paragraph characteristics of a bacterial clot on the heart valve are described. If organs or tissues are infected with bacteria, antibiotics are transported to the organ or tissue and in most cases the colony is destroyed soon. The heart, a vital organ, requires an accurate treatment if endocarditis occurs. A vegetation on the heart valve is not eradicated easily. Cremieux [10] lists the following for vegetations: The difficulty encountered in sterilizing vegetations is usually explained by (i) the poor penetration of antibiotics into infected vegetations, (ii) the altered metabolic state of the organisms within the vegetation, and (iii) the absence of a host defence cellular response inside the vegetation which could cooperate with the antibiotic action. The organisms responsible for infective endocarditis are separated from the blood by layers of fibrin and platelets, the elements which compose vegetations. There is a potential barrier to antibiotics posed by fibrin. Antibiotics must be able to permeate into, or pass, the fibrin mass which surrounds the organisms, and must be able to remain in the blood long enough and at sufficiently high concentrations so that it can permeate the fibrin in a quantity sufficient for bactericidal purposes.

22 12 3 Characteristics of endocarditis The bacteria considered are Streptococcus and Staphylococcus aureus. The Streptococcus is a Gram positive bacterial organism with a diameter of µm. The spherical (or oval) shaped bacteria, with a volume of 0.5 µm 3, can form devastating colonies on the heart valve. Staph. aureus is also a Gram positive, spherical bacterium, about 1 µm in diameter. On microscopic examination Staph. aureus appears in pairs, short chains, or bunched, grape-like clusters. The size and the shape of a vegetation can be measured by a echo cardiography. Streptococcus is usually characterized by chains of bacteria cells while Staph. aureus form a triangle shaped colony of bacteria. Although the shape of a colony on microscale is known (partly), the shape of a vegetation can still vary widely. In a more developed stage the bacterial clot is fully covered by a fibrin layer, or as mentioned at the beginning of this paragraph, the vegetation is a mixed pile of bacteria clusters and protein clusters. Antibiotics can not get through the layer or clusters of fibrin, which makes it harder to reach bacteria inside the vegetation. Proteins, usually fibrin, that are also interwoven in the vegetation, and bacteria grow simultaneously. The growth and eradication of fibrin is a dynamic process; once new bacteria are created fibrin will grow to consolidate the vegetation. Similar to a scab on a wound the fibrin will be broken down once the wound is healed. As new bacteria grow new fibrin is created again. The bacteria cell maintains and preserves the fibrin. 3.4 Dynamics of vegetation The pathogenesis of bacterial endocarditis is thought to be a two-stage process. First, the endothelial surface of the valves is damaged or has become rough, which leads to activation of the coagulation system and the formation of a sterile fibrin-platelet thrombus. Second, the thrombus forming a suitable surface for adherence becomes infected by the bacteria present in the blood during a bacteraemic phase. The process of adherence is dependent on characteristics of both the bacteria and the vegetation. After adhering, the bacteria are covered by fibrin and proliferate to form colonies, thus completing the formation of an infected vegetation. The fibrin in which the bacteria are embedded makes the vegetation relatively inaccessible to host defence mechanisms and antibiotics. Bacteria are released from the infected vegetation into the blood circulation. They can affect the general condition of the host, but may also adhere again to the surface of the vegetation. This reseeding leads to a continuous growth of the vegetation [6]. Bacteria from the valve infection reseed into the blood stream, which causes continuous bacteraemia typical for infective endocarditis [18]. Before being covered by fibrin, microorganisms are present on the surface of the vegetations. For a short period these are readily accessible to host defence factors like phagocytosing cells [6]. The bacteria clot on the heart valve changes over time due to growth and crumbling of the vegetation. Growth is caused by multiplication of bacteria, growth of proteins, interference of white blood cells etc. The size of the vegetation may also decrease if pieces of the vegetation crumble into the blood stream or if the immune system is able to solve (some) bacteria. For the growth of bacteria nutrients are required. The nutrients come from the heart valve itself and from the blood stream through the heart supplying, for example, glucose. Nutrients like glucose will diffuse into the vegetation, like antibiotics. In a further stage a steady state of bacteria may occur at the core. In an earlier stage all the bacteria cells inside the vegetation are able to multiply. However, fewer nutrients may be available at the center, so the growth rate is smaller for bacteria closer to the core.

23 3 Characteristics of endocarditis 13 Normally the blood stream along the valves is in a certain direction. The vegetation often occurs at the downstream side of the valve, usually at the aortic or mitral valves. If the bacteria grow at the downstream side the vegetation grows in an area where a decreased flow (stasis) prevails. In the small area behind the valve the speed of the blood stream is rather low. A vegetation can also be caused by a high pressure jet, instead of an area where blood is more or less stagnant. An abscess then arises at a location where the pressure is high or where the blood flows at high speed. A high pressure jet can damage the endothelium. A lesion at the heart valve or endothelium results in the deposition of fibrin and platelets. The damaged endothelial cells serve as a nidus (a center in which an infection settles) for bacteria [15]. The bacteria multiply, proteins grow, white blood cells interfere and a clot or infection is born (stage 1). Then the endothelial cells of the heart valve become weak and bacteria damage the heart valve. Growth into the valve results in a bag with pus, or an abscess (stage 2). The abscess can break through, resulting in a bubble/bladder or in a kind of trouser-leg shape, such that blood flows through the hole and to some extent flushes the bacteria away, leaving an empty hole in the heart valve (stage 3). The antibiotic concentration of the blood at a certain location is nearly in a homogeneous and stationary state. The speed of the blood differs. By pressure on the heart valve, vortices may occur behind the valve. These factors may contribute to the growth of bacteria (and uptake and effect of antibiotics). Figure 3.2 shows an echo of a patient with mitral valve endocarditis. The two thin white ropes are the heart valves and the white thick clot is a vegetation. In every patient this looks different; the size, location, shape and state or condition may vary a lot! Figure 3.2: Echo mitral valve endocarditis 1 1 Picture from search terms: echo endocarditis,

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25 4 Antibiotics This section describes the functioning of antibiotics as it acts on a bacterial vegetation. In consequence of Central question 2 the characteristics of antibiotics are given. This section is about antibiotics, its functioning and the curing methods that are used nowadays. Diffusion of antibiotics into a vegetation is described in the second paragraph. Scraping or etching of the vegetation, affecting only the boundary of the vegetation, is described in Paragraph 4.3. The susceptibility of bacteria is treated in the final paragraph. Shortly the immune system of the body is mentioned and the influence of antibiotics. First note that antibiotics is administered if endocarditis has not reached too serious complications yet. Four reasons to call endocarditis serious, or serious enough for surgery and replacement of the valve, are 1. Emboli, which can firmly attach to a blood vessel and block the flow of blood in that vessel 2. Perforated heart valve, caused by an abces that has pierced the valve 3. Formation of abces, where an abces on the heart valve ring is worse than a small abces at the heart valve itself 4. Persistent fever, indicating an ongoing infection These four clinical pictures are too complicated to treat with antibiotics; antibiotics are not able to cure a patient whose symptoms are such an advanced stage. 4.1 Current curing methods and administration A general principle from Hurst [16] is as follows. The chief aims of management are to eradicate the infecting organism as soon as possible, to operate with correct timing if surgical intervention should be required, and to treat complications. To treat endocarditis, hospitalization is often required initially to administer intravenously antibiotics. Long-term antibiotic therapy is required to eradicate the bacteria from the heart chambers and vegetations on the valves. Therapy up to six or eight weeks is not uncommon. Antibiotics are administered intravenously in a few µg/ml. The chosen antibiotics must be specific for the organism causing the condition. This is determined by the blood culture and sensitivity tests. Activity is restricted to bed rest initially, then it is gradually increased as the condition improves. No special diet (such as a low-salt diet) is necessary, unless it is required because of an underlying heart disorder. After seven days of treatment with antibiotics the risk for emboli, the number one serious consequences of endocarditis, is suppressed, because a lot of bacteria killing has already been done. Antibiotics are a product from living organisms with bactericidal or bacteriostatic action on other micro-organisms, such as bacteria. Antibiotics kill or disrupt the growth of bacteria in different ways. The site and mode of action differs for every type of antibiotics: by destabilization of cell wall synthesis, disruption of cell division by targeting microtubules, cell membrane, or disrupting protein synthesis. Antibiotics that are often used in case of

26 16 4 Antibiotics endocarditis is penicillin. Penicillin prevents the cross-linking of small peptide chains in peptidoglycan, the main wall polymer of bacteria. Pre-existing cells are unaffected, but all newly produced cells grow abnormally, unable to maintain their wall rigidly, and they are susceptible to osmotic lysis. Factors such as solved fever, a decrease of CRP (C-reactive protein) level, white blood cells and the like tell a specialist whether a patient is free from infection. (CRP is a protein created when an infection arises). 4.2 Diffusion of antibiotics Diffusion is the spontaneous spreading of particles, the intermixing of molecules in solids and liquids, due to a difference in composition. The process tends to distribute the particles more uniformly. In the case of endocarditis diffusion could be looked at from the following point of view. A system of capillaries, the space between bacteria, is filled with proteins and tissue fluid. As soon as the antibiotic level of the blood increases, the antibiotics enter the capillairies and move from a concentrated area (the blood outside vegetation) to a less concentrated area (inside the vegetation). Bacteria inside the vegetation are able to grow and they need nutrients for that. As nutrients are able to diffuse into the vegetation, antibiotics are most likely too. Bacteria that are in steady state may not be provided by nutrients. Assume that the same holds for antibiotics; bacteria that are in steady state are not killed as antibiotics are not able to reach or affect those cells. Antibiotics whose function is to affect the cell-membrane during multiplication of a bacteria are not able to kill the bacteria if they do not multiply. Here we assume that antibiotics are able to penetrate through the vegetation. In literature [13, 17, 20] diffusion models are used to model the penetration of antibiotics and the eradication of bacteria. If it is assumed that antibiotics do indeed diffuse through a bacterial vegetation, but the question is to what extent. This depends on the bacteria density and fibrin density. Antibiotics can pass bacteria cells and affect bacteria cells. However, antibiotics cannot enter a fibrin cluster because fibrin is not porous. Antibiotics penetrates into the vegetation from the outside, from the bloodstream that is pumped through the heart. The penetration of antibiotics into a vegetation influences the time period that it takes to eradicate a bacteria clot. If a bacteria population is not clotted together the antibiotic can easily get through and kill bacterial cells relatively fast, but a high density of bacteria and fibrin often makes the sterilization of a vegetation difficult. 4.3 Etching Blood passes the vegetation, and antibiotics that are present in the blood enter the vegetation from the outside. Antibiotics are able to attack the bacteria at the boundary of the vegetation. The bacterial cells at the surface are easier to attack than bacteria cells at the core of the vegetation. If bacteria at the outer layer of the vegetation are attacked by antibiotics and if the cells are killed faster than their growth rate, then the outer layer is scraped off. This we will call etching, as antibiotics break down the boundary. If penetration of antibiotics into the vegetation is hard, the killing of bacteria at the boundary has the greatest effect.

27 4 Antibiotics Susceptibility of bacteria A patient with endocarditis has already reached a stage where the immune system of the body is not able to eradicate the bacteria that cause endocarditis. Then antibiotics are administered to eradicate the bacteria. The susceptibility of bacteria to antibiotics varies per patient. The immune system is a defence mechanism of the body against invading organisms such as bacteria. Phagocytic cells, such as macrophages, can swallow up and destroy bacteria. Note that the function of macrophages may be important as well. The macrophages contribute, to some extent, to the eradication of a bacteria vegetation. A vegetation as such is avascular. The cardiac valves of the heart do not actually receive any blood supply of their own, they are not vascularized, which may be surprising given their location. If blood vessels are present in cardiac valves, they are the result of an inflammatory process (such as the body s response to tissue injury or infection, mainly localized in the affected tissues and adjacent blood vessels) in the valvular leaflets [28]. Defence mechanisms (such as macrophages and white blood cells) cannot enter. So if an organism establishes hold on the valves, the body cannot get rid of it easily. Antibiotics will have a killing effect on bacteria. Several reasons make bacteria less susceptible to antibiotics. The combination of the kind of bacteria that causes endocarditis and the kind of antibiotics should be effective. Another upcoming issue is the resistance of bacteria to certain antibiotics. Besides, clotted bacteria make it hard for antibiotics to enter the vegetation and reach to all bacteria in the vegetation. If the bacteria would swim in a petri-dish antibiotics would have less difficulty in attacking all bacteria. Once a biofilm or bacterial vegetation is present, antibiotics may have difficulty penetrating to the site at which the organism resides and into the biofilm. However, recent evidence suggests that it may not be the antibiotic s inability to penetrate but rather inactivity of the antibiotic in the biofilm. In the article by Anderl et al. [1] results are presented of the Klebsiella Pneamoniae bacteria experiencing nutrient limitation locally within the biofilm, leading to zones in which the bacteria enter a stationary phase and are growing slowly or not at all. In these inactive regions bacteria are less susceptible to killing by antibiotics. Caiazza and O Toole [7] found that Staphylococcus aureus can persist in clinical settings and gain increased resistance to antimicrobial agents through biofilm formation. Wu et al. [29] also mention that sessile bacteria in biofilms are much more resistant to antibiotics than their planktonic (versus biofilms) counterparts. Bacteria deep inside a vegetation may have reached a steady state and are less susceptible to antibiotics. Note that several kinds of antibiotics act by affecting the wall synthesis; their function is to destroy the cell-wall during multiplication. If bacteria do not multiply then antibiotics have no chance to kill the bacteria cell. It can be concluded that the bacteria deep inside the vegetation are harder to kill than bacteria at the outer boundary. In Figure 3.1 the bacteria at the boundary are indicated by 1 and the bacteria at the core are located at 2.

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29 5 Formulation of parameters From the previous sections several factors appear to be important for the representation of a realistic model for endocarditis. As posed in Central question 3 the factors that influence the action of antibiotics on the vegetation are described. In the second paragraph of this section the most relevant factors are picked from the factors described in Paragraph Factors influencing treatment This paragraph addresses the factors that may play a role in the treatment of endocarditis. For example, the penetration of antibiotics into tissue fluids is limited by the rate of diffusion and depends on the amount of free serum antibiotics not bound to protein [13]. So the antibiotic concentration and the rate of diffusion are two of the parameters. Here a list of factors (as complete as possible) is given that could be taken into account for realistic modeling. The relevance of each of the factors is discussed. Kind of bacteria and antibiotics. Bacteria are Streptococcus and Staphylococcus aureus. Antibiotic penicillin or gentamicin. If the kind of bacteria and antibiotics is known, this should provide data or values to put in a model. Whether the values and outcomes of a model are representative for a data set of these bacteria and antibiotics should be a criterium. However, such a data set is not available and in this study values are predicted somehow, so the type of bacteria and antibiotics is not really taken into account. Structure of a bacterial vegetation. A possible imagination of the structure of a bacterial vegetation is represented in figure 5.1; bacteria cells with a fibrin framework in between. The vegetation is actually an unpredictable chaos of bacteria clusters and fibrin clusters. Antibiotic concentration in blood. The blood plasma concentration depends on the administration of antibiotics. Administrating intermittently will give an exponential decay of the concentration until the next administration. Continuous administration will lead to a steady state concentration of antibiotics in blood. Note that with antibiotic concentration we mean the free fraction of molecules that is not bound to proteins, as assumed earlier in Section 2. Antibiotic concentration in vegetation. The rate of change of the concentration of antibiotics in the vegetation depends on the diffusion coefficient which will depend on the bacterial density of the vegetation. If antibiotic is able to penetrate through the vegetation bacteria cells inside the vegetation can be killed. Then, either porosity increases, because an empty space occurs, or a piece of the vegetation breaks down, because piece becomes less attached, or new bacteria or fibrin grow in the empty space. Bacteria density in blood (plasma). Bacteria in the blood, indicating that the patient suffers from bacteraemia, may attach onto a heart valve if the valve is not smooth. Once bacteria establish hold on the valve the vegetation can grow by multiplication via doubling of bacteria. The concentration of bacteria in the blood is less important for our model, because bacteraemia does not indicate one has endocarditis.